\batchmode
\documentclass[11pt]{ownart1}
\makeatletter
\usepackage{a4wide,strl,subeqn,epsfig,html}
\usepackage{latexsym}
\thispagestyle{empty}
\title{ISNaS - incompressible flow solver\\ [4ex]
Mathematical manual}
\author{Guus Segal\\ 
Hester Bijl\\ 
Kees Vuik\\ 
William Kuppen ?????\\ 
Marcel Zijlema\\ [8ex]
version 2.0 9-10-1996}
\date{ }



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{\newpage
\clearpage
\samepage \begin{figure}[htb]
\begin{minipage}[t]{7,3cm}
\hspace*{1cm}$\Omega$\\ 
\vspace*{-1,2cm}
\hspace*{1cm}
\centerline{\psfig{figure=fig21a.eps,height=6cm,angle=-90}}
\vspace*{-2cm}
$\begin{array}{lll}
y&\uparrow&\\ 
&\; \; \; \rightarrow&x
\end{array}$
\end{minipage}
\hfill
\begin{minipage}[t]{8cm}
$T$\\ 
$\rightarrow$\\ 
\vspace*{-1cm}
\centerline{\psfig{figure=fig21b.eps,height=5cm}}
\vspace{0,5cm}
\end{minipage}

\label{fig21}
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\begin{array}{ccl}
\bm{x}&=&(x^{1}, x^{2}, x^{3}) \mbox{\rm \ is the Cartesian co-ordinate
system,}\\ 
\bm{\xi}&=&(\xi^{1}, \xi^{2}, \xi^{3}) \mbox{\rm \ is the general
co-ordinate system,}\\ 
&&\mbox{\rm  \ i.e. the co-ordinate system corresponding to the computational 
grid.}
\end{array}\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{equation}T: x^{\alpha} = x^{\alpha} (\xi^{1}, \xi^{2}, \xi^{3})
\label{eq:2.1}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}J = |\frac{\partial x^{\alpha}}{\partial \xi^{\beta}}|
\label{eq:2.2}
\end{equation}
}

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{\newpage
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\end{equation}
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{\newpage
\clearpage
\samepage \begin{equation}\bm{a}^{(\alpha)} = \mbox{grad} \; \xi^{\alpha}
\label{eq:2.4}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\bm{a}^{(\alpha)} = \frac{1}{\sqrt{g}} \bm{a}_{(\beta)} \wedge
\bm{a}_{(\gamma)} \;\; \mbox{\rm \ for } \alpha, \beta, \gamma \mbox{\rm \
cyclic}
\label{eq:2.5}
\end{equation}
}

{\newpage
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{\newpage
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\samepage \begin{equation}\bm{u} = U^{\alpha} \bm{a}_{(\alpha)} = U_{\alpha} \bm{a}^{(\alpha)}
\label{eq:2.6}
\end{equation}
}

{\newpage
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\samepage \begin{equation}\bm{U}
= U^{\alpha}_{\beta} \bm{a}_{(\alpha)} \bm{a}^{(\beta)}
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\end{equation}
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{\newpage
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\samepage \begin{equation}U_{\alpha} = \bm{a}_{(\alpha)} \cdot \bm{u}, \; \; \; U^{\alpha} = 
\bm{a}^{(\alpha)}
\cdot \bm{u}
\label{eq:2.8}
\end{equation}
}

{\newpage
\clearpage
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\end{equation}
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{\newpage
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\label{eq:2.10}
\end{equation}
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{\newpage
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\label{eq:2.11}
\end{equation}
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{\newpage
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\end{equation}
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{\newpage
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\end{equation}
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{\newpage
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\samepage \begin{equation}\sqrt{g} = J
\label{eq:2.14}
\end{equation}
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\samepage \begin{equation}\mu_{, \alpha} = \frac{\partial \mu}{\partial \xi^{\alpha}}
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\end{equation}
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{\newpage
\clearpage
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\label{eq:2.16}
\end{equation}
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{\newpage
\clearpage
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\xi^{\alpha}}{\partial x^{\delta}} \frac{\partial^{2} x^{\delta}}{\partial
\xi^{\gamma}\partial \xi^{\beta}} = \chris{\alpha}{\beta}{\gamma}
\label{eq:2.17}
\end{equation}
}

{\newpage
\clearpage
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(\frac{\partial g_{\delta \gamma}}{\partial \xi^{\beta}} + \frac{\partial
g_{\delta \beta}}{\partial \xi^{\gamma}} - \frac{\partial g_{\gamma
\beta}}{\partial \xi^{\delta}})   
\label{eq:2.18}
\end{equation}
}

{\newpage
\clearpage
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\chris{\gamma}{\alpha}{\beta} U_{\gamma}
\label{eq:2.19}
\end{equation}
}

{\newpage
\clearpage
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\sqrt{g}U^{\alpha}}{\partial \xi^{\alpha}}
\label{eq:2.20}
\end{equation}
}

{\newpage
\clearpage
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\label{eq:2.21}
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{\newpage
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\label{eq:2.22}
\end{equation}
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{\newpage
\clearpage
\samepage \begin{equation}g^{\alpha \beta}_{, \beta} = 0
\label{eq:2.23}
\end{equation}
}

{\newpage
\clearpage
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\label{eq:2.24}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\rho_{, \alpha} = \sqrt{g} \frac{\partial \rho /
\sqrt{g}}{\partial \xi^{\alpha}}
\label{eq:2.25}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\sqrt{g}_{, \alpha} = 0
\label{eq:2.26}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\pderiv{\sqrt{g} \bm{a}^{(\alpha)}}{\xi^\alpha} = 0 \label{geomid}
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{\newpage
\clearpage
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  \bm{v}}{\xi^\alpha} = \frac{1}{\sqrt{g}} \bm{v} \cdot
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{\newpage
\clearpage
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\end{equation}
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{\newpage
\clearpage
\samepage \begin{equation}\int\limits_{\Omega} div \; \bm{u} d V = \oint\limits_{\Gamma} \bm{u}
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\label{eq:2.28}
\end{equation}
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{\newpage
\clearpage
\samepage \begin{equation}\int\limits_{\Omega} U^{\alpha}_{, \alpha} d\Omega = \oint\limits_{\Gamma}
U^{\alpha} d \Gamma_{\alpha}
\label{eq:2.29}
\end{equation}
}

\stepcounter{section}
\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{displaymath}\bm{a}_{(\alpha)} , g_{\alpha \beta}, \sqrt{g}, \chris{\alpha}{\beta}{\gamma}\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{figure}[htb]
\centerline{\psfig{figure=fig31.eps}}

\label{fig31}
\end{figure}
}

{\newpage
\clearpage
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{\newpage
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}

{\newpage
\clearpage
\samepage \begin{eqnarray}\bm{a}_{(1)} (0, \pm 1)&=&\bm{x} (1, \pm 1) - \bm{x} (-1, \pm 1)\label{drie1}\\ 
\bm{a}_{(2)} (\pm 1, 0) &=&\bm{x} (\pm 1, 1) - \bm{x}(\pm 1, -1)
\label{drie2}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}\bm{a}_{(1)} (0, 0)&=&\frac{1}{2}\{\bm{a}_{(1)}(0, 1) + \bm{a}_{(1)}
(0,-1)\}\label{drie3}\\ 
\bm{a}_{(2)} (0,0)&=&\frac{1}{2}\{\bm{a}_{(2)} (1,0) + \bm{a}_{(2)} (-1,
0)\}\label{drie4}\\ 
\bm{a}_{(1)} (-1,0)&=&\frac{1}{4} \{\bm{a}_{(1)} (0,1) + \bm{a}_{(1)}
(0,-1) + \bm{a}_{1}(-2,1) + \bm{a}_{1}(-2, -1)\} \label{drie5}\\ 
\bm{a}_{(2)}(0,-1)&=&\frac{1}{4}\{\bm{a}_{(2)} (1,0) + \bm{a}_{(2)} (-1, 0)
+ \bm{a}_{2}(-1,-2) + \bm{a}_{2}(1, -2)\}\label{drie6}\\ 
\bm{a}_{(1)} (-1,-1)&=&\frac{1}{2} \{\bm{a}_{(1)} (-1,0) + \bm{a}_{(1)}
(-1,-2)\}\label{drie7}\\ 
\bm{a}_{(2)}(-1,-1)&=&\frac{1}{2}\{\bm{a}_{(2)} (0,-1) + \bm{a}_{(2)}
(-2,-1)\}
\label{drie8}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{equation}g_{\alpha \beta} = \bm{a}_{(\alpha)} \cdot \bm{a}_{(\beta)}
\label{drie9}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}g_{\alpha \beta} (-1, 0)&=&\frac{1}{2} \{g_{\alpha \beta} (-2, 0) + g_{\alpha
\beta} (0,0)\}
\label{eq:3.10}\\ 
g_{\alpha \beta} (-1, -1)&=&\frac{1}{4} \{g_{\alpha \beta} (0, 0) + g_{\alpha
\beta} (-2,0) +
g_{\alpha \beta} (0, -2) + g_{\alpha \beta} (-2,-2)\} \label{eq:3.11}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sqrt{g}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline10841: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\sqrt{g} = \sqrt{ | det\; (g_{\alpha \beta}) |}
\label{eq:3.12}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\chris{\alpha}{\beta}{\gamma}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline10845: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray}\chris{1}{1}{1}&=&\frac{1}{2} g^{1\delta} (\frac{\partial g_{\delta
1}}{\partial \xi^{1}} + \frac{\partial g_{\delta 1}}{\partial \xi^{1}} -
\frac{\partial g_{11}}{\partial \xi^{\delta}})\nonumber\\ 
&=&\frac{1}{2} g^{11}\frac{\partial g_{11}}{\partial \xi^{1}} + g^{12}
(\frac{\partial g_{21}}{\partial \xi^{1}} - \frac{1}{2}\frac{\partial
g_{11}}{\partial \xi^{2}})\nonumber\\ 
&=&\frac{1}{2}\frac{g_{22}}{g} \frac{\partial g_{11}}{\partial \xi^{1}} - \frac{g_{12}}{g}
(\frac{\partial g_{21}}{\partial \xi^{1}} -
\frac{1}{2}\frac{g_{11}}{\partial \xi^{2}})
\label{eq:3.13}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}\chris{1}{1}{2}&=&\frac{1}{2} g^{1\delta}(\frac{\partial g_{\delta
1}}{\partial \xi^{2}} + \frac{\partial g_{\delta 2}}{\partial \xi^{1}} -
\frac{\partial g_{12}}{\partial \xi^{\delta}})\nonumber\\ 
&=&\frac{1}{2} g^{11} (\frac{\partial g_{11}}{\partial \xi^{2}} +
\frac{\partial g_{12}}{\partial \xi^{1}} - \frac{\partial g_{12}}{\partial
\xi^{1}}) + \frac{1}{2} g^{12} (\frac{\partial g_{21}}{\partial \xi^{2}} +
\frac{\partial g_{22}}{\partial \xi^{1}} - \frac{\partial g_{12}}{\partial
\xi^{2}})\nonumber\\ 
&=&\frac{1}{2} \frac{g_{22}}{g}\frac{\partial g_{11}}{\partial \xi^{2}} - \frac{1}{2}
\frac{g_{12}}{g} \frac{\partial g_{22}}{\partial \xi^{1}}\label{eq:3.14}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}\chris{1}{2}{2}&=&\frac{1}{2} g^{11} (\frac{\partial g_{12}}{\partial
\xi^{2}} + \frac{\partial g_{21}}{\partial \xi^{2}} - \frac{\partial
g_{22}}{\partial \xi^{1}} ) + \frac{1}{2}g^{12}(\frac{\partial g_{22}}{\partial \xi^{2}} +
\frac{\partial g_{22}}{\partial \xi^{2}} - \frac{\partial g_{22}}{\partial
\xi^{2}})\nonumber\\ 
&=&\frac{g_{22}}{g} (\frac{\partial g_{12}}{\partial \xi^{2}} - \frac{1}{2}
\frac{\partial g_{22}}{\partial \xi^{1}} ) - \frac{1}{2}\frac{g_{12}}{ g}
\frac{\partial g_{22}}{\partial \xi^{2}}\label{eq:3.15}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}\chris{2}{1}{2}&=&\frac{1}{2} g^{21} (\frac{\partial g_{11}}{\partial
\xi^{2}} + \frac{\partial g_{12}}{\partial \xi^{1}} - \frac{\partial
g_{12}}{\partial \xi^{1}}) + \frac{1}{2} g^{22} (\frac{\partial
g_{21}}{\partial \xi^{2}} + \frac{\partial g_{22}}{\partial \xi^{1}} -
\frac{\partial g_{12}}{\partial \xi^{2}})\nonumber\\ 
&=&-\frac{1}{2}\frac{g_{21}}{g} \frac{\partial g_{11}}{\partial \xi^{2}} +
\frac{1}{2} \frac{g_{11}}{g} \frac{\partial g_{22}}{\partial \xi^{1}}
\label{eq:3.16}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}\chris{2}{1}{1}&=&\frac{1}{2} g^{21}(\frac{\partial g_{11}}{\partial
\xi^{1}} + \frac{\partial g_{11}}{\partial \xi^{1}} - \frac{\partial
g_{11}}{\partial \xi^{1}}) + \frac{1}{2} g^{22}(\frac{\partial
g_{21}}{\partial \xi^{1}} + \frac{\partial g_{21}}{\partial \xi^{1}} -
\frac{\partial g_{11}}{\partial \xi^{2}})\nonumber\\ 
&=&- \frac{1}{2} \frac{g_{21}}{g} \frac{\partial g_{11}}{\partial \xi^{1}}
+ \frac{g_{11}}{g} (\frac{\partial g_{21}}{\partial \xi^{1}} - \frac{1}{2}
\frac{\partial g_{11}}{\partial \xi^{2}})\label{eq:3.17}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}\chris{2}{2}{2}&=&\frac{1}{2} g^{21} (\frac{\partial g_{12}}{\partial
\xi^{2}} + \frac{\partial g_{12}}{\partial \xi^{2}} - \frac{\partial
g_{22}}{\partial \xi^{1}} ) + \frac{1}{2} g^{22} (\frac{\partial
g_{22}}{\partial \xi^{2}} + \frac{\partial g_{22}}{\partial \xi^{2}} -
\frac{\partial g_{22}}{\partial \xi^{2}})\nonumber\\ 
&=&- \frac{g_{21}}{g} (\frac{\partial g_{12}}{\partial \xi^{2}} -
\frac{1}{2} \frac{\partial g_{22}}{\partial \xi^{1}} ) + \frac{1}{2}
\frac{g_{11}}{g} \frac{\partial g_{22}}{\partial \xi^{2}}
\label{eq:3.18}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{equation}\left[\begin{array}{cc}
g^{11} &g^{12}\\ g^{12} &g^{22}\end{array}\right] = \frac{1}{g}
\left[\begin{array}{rr}
g_{22} &-g_{12}\\ -g_{12} &g_{11}
\end{array} \right]
\label{eq:3.19}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\partial\cdot/\partial \xi^{\alpha}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline10851: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\sqrt{g}_{(\xi, \eta)} = | a^{1}_{(1)} a^{2}_{(2)} - a^{2}_{(1)}
a^{1}_{(2)} |_{(\xi, \eta)}
\label{eq:3.19a}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\bm{a}^{(1)} = \frac{1}{\sqrt{g}}(\bm{a}^{2}_{(2)}, \; -\bm{a}^{1}_{(2)}),
\; \;\; \bm{a}^{(2)} = \frac{1}{\sqrt{g}} (- \bm{a}^{2}_{(1)}, \;
a^{1}_{(1)})
\label{eq:3.20}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{figure}[htb]
\centerline{\psfig{figure=fig32.eps}}

\label{fig32}
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{equation}\bm{x}_{i,0} = 2 \bm{x}_{i,1} - \bm{x}_{i,2}
\label{eq:3.21}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\bm{x}_{0,0} = \frac{1}{2} [(2x_{1,0} - x_{2,0}) + (2 x_{0,1} - x_{0,2})]
\label{eq:3.22}
\end{equation}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\xi^{\alpha}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline10877: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{figure}[htb]
\centerline{\psfig{figure=fig33.eps,height=4cm}}

\label{fig33}
\end{figure}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\bm{a}_{(1)}, \bm{a}_{(2)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline10887: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\bm{a}_{(3)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline10889: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray}\bm{a}_{(1)}(0, i, j)&=&\bm{x}(1, i, j) - \bm{x} (-1, i, j)
\label{eq:3.24}\\ 
\bm{a}_{(2)}(i, 0, j)&=&\bm{x}(i, 1, j) - \bm{x}(i, -1,
j)\label{eq:3.25}\\ 
\bm{a}_{(3)} (i, j, 0)&=&\bm{x}(i, j, 1) - \bm{x}(i, j, -1)
\label{eq:3.26}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$i, j, \in \{-1, 1\}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline10891: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray}\bm{a}_{(1)} (-1, -1, -1)&=&\frac{1}{2}\{\bm{a}_{(1)} (-2,-1, -1) +
\bm{a}_{(1)} (0,-1,-1)\}
\label{eq:3.27}\\ 
\bm{a}_{(1)} (-1, 0, -1)&=&\frac{1}{4} \{\bm{a}_{(1)} (-2, -1, -1) +
\bm{a}_{(1)}(0, -1, -1) + \bm{a}_{(1)} (-2, 1, -1) + \nonumber\\ 
&&\bm{a}_{(1)}(0, 1,
-1)\}
\label{eq:3.28}\\ 
\bm{a}_{(1)} (-1, -1, 0)&=&\frac{1}{4} \{\bm{a}_{(1)} (-2, -1, -1) +
\bm{a}_{(1)} (0, -1, -1) +\bm{a}_{(1)} (-2, -1, 1) +\nonumber\\ 
&& \bm{a}_{(1)} (0, -1,
1)\}\label{eq:3.29}\\ 
\bm{a}_{(1)} (-1, 0, 0)&=&\frac{1}{8}\{\bm{a}_{(1)}(-2, -1, -1) +
\bm{a}_{(1)}(0, -1, -1) + \bm{a}_{(1)} (-2, 1, -1) + \bm{a}_{(1)}(0, 1, -1)
+ \nonumber\\ 
&&\bm{a}_{(1)} (-2, -1, 1) + \bm{a}_{(1)} (0, -1, 1) + \bm{a}_{(1)} (-2, 1,
1) + \bm{a}_{(1)} (0, 1, 1)\}
\label{eq:3.30}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}\sqrt{g}_{(i,j,k)}&=&(a^{1}_{(1)} a^{2}_{(2)} a^{3}_{(3)} - a^{1}_{(1)}
a^{3}_{(2)} a^{2}_{(3)} + \nonumber\\ 
&&a^{2}_{(1)} a^{3}_{(2)} a^{1}_{(3)} - a^{2}_{(1)} a^{1}_{(2)} a^{3}_{(3)}
+\nonumber\\ 
&&a^{3}_{(1)} a^{1}_{(2)} a^{2}_{(3)} - a^{3}_{(1)} a^{2}_{(2)} a^{1}_{(3)}
)_{(i,j,k)}.
\label{eq:3.31}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{equation}(g_{\alpha \beta})_{(i,j,k)} = (\bm{a}_{(\alpha)} \cdot \bm{a}_{(\beta)}
)_{(i,j,k)}
\label{eq:3.32}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}(g^{\alpha \beta})_{(i,j,k)} = (\frac{(-1)^{\alpha + \beta} det \; (G_{\alpha
\beta})}{g})_{(i,j,k)}
\label{eq:3.33}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}G_{\alpha \beta} = \begin{array}{c}\mbox{minor}\\  \alpha \beta\end{array}
\left[ \begin{array}{ccc}
g_{11} &g_{12} &g_{13}\\ 
g_{21} &g_{22} &g_{23}\\ 
g_{31} &g_{32} &g_{33}
\end{array}
\right]\; .
\label{eq:3.34}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\chris{1}{1}{1}, \chris{1}{1}{2}, ...,
\chris{1}{3}{3}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline10905: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\chris{2}{1}{1},
\chris{2}{1}{2}, ..., \chris{2}{3}{3}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline10907: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\chris{3}{1}{1}, ..., \chris{3}{3}{3}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline10909: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{figure}[htb]
\begin{minipage}[t]{9cm}
\centerline{\psfig{figure=fig34.eps,height=4cm}}
\end{minipage}
\hfill
\begin{minipage}[t]{6cm}
\vspace*{-3cm}
1: $\chris{1}{1}{1}, \; \chris{1}{1}{2}, ..., \chris{1}{3}{3}$\\ 
2: $\chris{2}{1}{1}, \; \chris{2}{1}{2}, ..., \chris{2}{3}{3}$\\ 
3: $\chris{3}{1}{1}, \; \chris{3}{1}{2}, ..., \chris{3}{3}{3}$
\end{minipage}

\label{fig34}
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{figure}[htb]
\centerline{\psfig{figure=fig35.eps,height=15cm}}

\label{fig35}
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{equation}\bm{x}_{0,j,k} = 2 \bm{x}_{1,j,k} - \bm{x}_{2,j,k}\; ,
\label{eq:3.35}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\bm{x}_{0,0,k} = \frac{1}{2} [(2 \bm{x}_{1,0,k} - \bm{x}_{2,0,k}) + (2
\bm{x}_{0,1,k} - \bm{x}_{0,2,k})]\; ,
\label{eq:3.36}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$i = 0, \; j = 0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline10925: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\bm{x}_{0,0,0} = \frac{1}{3} [(2\bm{x}_{1,0,0} - \bm{x}_{2,0,0}) +
(2\bm{x}_{0,1,0} - \bm{x}_{0,2,0}) + (2 \bm{x}_{0,0,1} - \bm{x}_{0,0,2})
]\; .
\label{eq:3.37}
\end{equation}
}

\stepcounter{section}
{\newpage
\clearpage
\samepage \begin{equation}\frac{\partial}{\partial t} (\rho U^{\alpha}) + (\rho U^{\alpha} 
U^{\beta})_{,\beta} - R^{\alpha \beta}_{,\beta} + (g^{\alpha \beta}p)_{,\beta}
- \tau^{\alpha \beta}_{,\beta} = \rho f^{\alpha}
\label{eq:4.1}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tau^{\alpha \beta}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline10935: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\tau^{\alpha \beta} = \mu (g^{\alpha \gamma}U^{\beta}_{, \gamma} + g^{\gamma
\beta} U^{\alpha}_{,\gamma}).
\label{eq:4.2}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$U^{\alpha}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline10937: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\rho$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline10939: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$f^{\alpha}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline10943: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\mu$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline10945: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$R^{\alpha \beta} \equiv -\rho 
\overline{{\widehat U}^{\alpha}{\widehat U}^{\beta}}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline10947: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\widehat U}^\alpha$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline10949: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}U^{\alpha}_{, \alpha} = 0
\label{eq:4.3}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$V^{\alpha} = \sqrt{g} U^{\alpha}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline10953: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{figure}[htb]
\centerline{\psfig{figure=fig41.eps,height=5cm}}

\label{fig41}
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{equation}V^{1}|^{(1,0)}_{(-1,0)} + V^{2}|^{(0,1)}_{(0,-1)} = 0,
\label{eq:4.4}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\frac{\partial}{\partial t} (\rho V^{\alpha}|_{(0,0)}),
\label{eq:4.5}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$V^{\alpha}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline10957: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\rho f^{\alpha} \sqrt{g}|_{(0,0)}
\label{eq:4.6}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\rho \sqrt{g} (a_{1}^{(\alpha)} f_{1} + a_{2}^{(\alpha)} f_{2})|_{(0,0)}\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{equation}V^{\alpha} V^{\beta} \approx V^{\alpha} \bar{V}^{\beta} + \bar{V}^{\alpha}
V^{\beta} - \bar{V}^{\alpha}\bar{V}^{\beta}
\label{eq:4.7}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\bar{V}^{\alpha}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline10961: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$V^{1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline10963: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\frac{\rho}{\sqrt{g}}(V^{1})^{2}|^{(1,0)}_{(-1,0)} + \frac{\rho}{\sqrt{g}}
V^{1}V^{2}|^{(0,1)}_{(0,-1)}
+ \frac{\rho}{\sqrt{g}}\chris{1}{\gamma}{\beta} V^{\gamma}
V^{\beta}|_{(0,0)}
\label{eq:4.8}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$V^{2}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline10965: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\frac{\rho}{\sqrt{g}} V^{2} V^{1} |^{(1,0)}_{(-1,0)} + \frac{\rho}{\sqrt{g}}
(V^{2})^{2}|^{(0,1)}_{(0,-1)} +
\frac{\rho}{\sqrt{g}}\chris{2}{\gamma}{\beta} V^{\gamma}V^{\beta}|_{(0,0)}
\label{eq:4.9}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}- \sqrt{g} \mu (g^{11}U^{1}_{, 1} + g^{12} U^{1}_{, 2} )|^{(1,0)}_{(-1,0)}
- \sqrt{g} \mu (g^{11}U^{2}_{,1} + g^{22}U^{1}_{, 2}) |^{(0,1)}_{(0,-1)}
- \chris{1}{\gamma}{\beta} \tau^{\gamma \beta} \sqrt{g}|_{(0,0)}
\label{eq:4.10}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}- \sqrt{g} \mu (g^{11} U^{2}_{, 1} + g^{22}U^{1}_{, 2})|^{(1,0)}_{(-1,0)}
- \sqrt{g} 2\mu (g^{12}U^{2}_{, 1} +
g^{22}U^{2}_{,2})|^{(0,1)}_{(0,-1)}
- \chris{2}{\gamma}{\beta} \tau^{\gamma \beta} \sqrt{g}|_{(0,0)}
\label{eq:4.11}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$U^{\alpha}_{, \beta}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline10971: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\frac{\partial U^{\alpha}}{\partial \xi^{\beta}}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline10975: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray}\frac{\partial U^{\alpha}}{\partial \xi^{1}} |_{(\xi, \eta)}&=&U^{\alpha}|_{(\xi
+ 1, \eta)} - U^{\alpha}|_{(\xi -1, \eta)}\label{eq:4.12}\\ 
\frac{\partial U^{\alpha}}{\partial \xi^{2}}|_{(\xi,
\eta)}&=&U^{\alpha}|_{(\xi, \eta+1)} - U^{\alpha}|_{(\xi, \eta-1)}.
\label{eq:4.13}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$(\xi, \eta)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline10977: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$V^{\alpha}/\sqrt{g}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline10981: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sqrt{g}U^{\alpha}_{, \beta}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline10985: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$V^{\alpha}_{, \beta}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline10987: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sqrt{g}_{,\beta} = 0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline10989: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}(p_{(1,0)} - p_{(-1,0)})(g^{\alpha 1}\sqrt{g})_{(0,0)} + (p_{(0,1)} -
p_{(0,-1)}) (g^{\alpha 2}\sqrt{g})_{(0,0)}
\label{eq:4.14}
\end{equation}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{figure}[htb]
\centerline{\psfig{figure=fig42.eps,height=15cm}}

\label{fig42}
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{equation}V^{1}|^{(1,0,0)}_{(-1,0,0)} + V^{2}|^{(0,1,0)}_{(0,-1,0)} + 
V^{3}|^{(0,0,1)}_{(0,0,-1)} = 0\; ,
\label{eq:4.15}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\tau^{\alpha \beta} = \mu (g^{\alpha \gamma} \frac{\partial
U^{\beta}}{\partial \xi^{\gamma}} + g^{\gamma \beta} \frac{\partial
U^{\alpha}}{\partial \xi^{\gamma}} - \frac{\partial g^{\alpha
\beta}}{\partial \xi^{\gamma}}U^{\gamma})\; .
\label{eq:4.16}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$g^{\alpha \beta}_{, \gamma} = 0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline10995: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\frac{\partial}{\partial t}( (\rho V^{\alpha})|_{(0,0,0)} )\; ,
\label{eq:4.17}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}(\rho f^{\alpha} \sqrt{g}) |_{(0,0,0)}\; .
\label{eq:4.18}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}V^{\alpha}-\mbox{ cell }:&&\nonumber\\ 
&&\frac{\rho}{\sqrt{g}} V^{\alpha} V^{1}
|^{(1,0,0)}_{(-1,0,0)} + \frac{\rho}{\sqrt{g}} V^{\alpha} V^{2}
|^{(0,1,0)}_{(0,-1,0)} + \frac{\rho}{\sqrt{g}}V^{\alpha}
V^{3}|^{(0,0,1)}_{(0,0,-1)} +\nonumber\\ 
&&\frac{\rho}{\sqrt{g}} \chris{\alpha}{\gamma}{\beta} V^{\gamma} V^{\beta}
|_{(0,0,0)} \; \;\; \mbox{ for } \; \; \; \alpha \in \{1,2,3\}\; .
\label{eq:4.19}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{equation}V^{\alpha} V^{\beta} \approx V^{\alpha} \bar{V}^{\beta} + \bar{V}^{\alpha}
V^{\beta} - \bar{V}^{\alpha} \bar{V}^{\beta}
\label{eq:4.20}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}V^{\alpha}-\mbox{ cell }:&&\nonumber\\ 
&&-\sqrt{g} \tau^{\alpha 1}|^{(1,0,0)}_{(-1,0,0)} - \sqrt{g} \tau^{\alpha
2}|^{(0,1,0)}_{(0,-1,0)} - \sqrt{g}\tau^{\alpha 3} |^{(0,0,1)}_{(0,0,-1)}\nonumber\\ 
&& -
(\chris{\alpha}{\gamma}{\beta} \tau^{\gamma \beta}\sqrt{g})|_{(0,0,0)}\; ,
\label{eq:4.21}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{equation}\frac{\partial U^{\alpha}}{\partial \xi^{\beta}}|_{(i,j,k)} = U^{\alpha}|_{(i,j,k)+\Delta_{\beta}}
- U^{\alpha}|_{(i,j,k) - \Delta_{\beta}}
\label{eq:4.22}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\Delta_{\beta} = (\frac{1}{2}|\beta - 2|\; |\beta - 3|,\; |\beta - 1|\;
|\beta - 3|, \; \frac{1}{2} |\beta - 1|\; |\beta - 2|)\; .
\label{eq:4.23}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\sqrt{g} \tau^{\alpha \beta} = \mu(g^{\alpha \gamma} \frac{\partial
V^{\alpha}}{\partial \xi^{\gamma}} + g^{\gamma \beta} \frac{\partial
V^{\alpha}}{\partial \xi^{\gamma}} - \frac{\partial g^{\alpha
\beta}}{\partial \xi^{\gamma}} V^{\gamma})
\label{eq:4.24}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}V^{\alpha}-\mbox{ cell }:&& \nonumber\\ 
&&(g^{\alpha 1} \sqrt{g})|_{(0,0,0)} (p_{(1,0,0)} - p_{(-1,0,0)}) +
(\sqrt{g}^{\alpha 2} \sqrt{g})|_{(0,0,0)} (p_{(0,1,0)} - p_{(0,-1,0)}) +
\nonumber\\ 
&&(g^{\alpha 3} \sqrt{g})|_{(0,0,0)} (p_{(0,0,1)} - p_{(0,0,-1)})\; .
\label{eq:4.25}
\end{eqnarray}
}

\stepcounter{section}
\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{equation}\frac{\partial c^* \phi}{\partial t} + \bm{u}\cdot \nabla(c^* \phi) -
 \mbox{div} \; (k \nabla \phi) + D\phi = f^*,
\label{eq:4.26}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$c^*$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11021: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$d \times d$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11027: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$f^*$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11031: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\frac{\partial c^* \phi}{\partial t} + (c^* U^\alpha \phi)_{, \alpha} - 
(K^{\alpha \beta} \phi_{,\beta})_{, \alpha} + D \phi = f^*,
\label{eq:4.27}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$K^{\alpha \beta} = a^{(\alpha)}_{\gamma} a^{(\beta)}_{\delta}
k^{\gamma \delta}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11037: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\frac{1}{\sqrt{g}}
  \frac{\partial \sqrt{g} Q^\alpha}{\partial \xi^\alpha} = f^* - D\phi -
  \frac{\partial c^* \phi}{\partial t}
  \label{equaphi}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}Q^\alpha = c^* U^\alpha \phi - K^{\alpha \beta} \phi_{,\beta}
  \label{trnsptensor}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}\int_{\Omega_{ijk}} \frac{1}{\sqrt{g}} \pderiv{\sqrt{g} Q^\alpha}{\xi^\alpha}
   d\Omega &=& \int_{G_{ijk}} \pderiv{\sqrt{g} Q^\alpha}{\xi^\alpha} d\xi^1
   d\xi^2 d\xi^3 \nonumber \\ 
   &\approx& \sqrt{g} Q^1 |_{(i-1/2,j,k)}^{(i+1/2,j,k)} +
             \sqrt{g} Q^2 |_{(i,j-1/2,k)}^{(i,j+1/2,k)} +
             \sqrt{g} Q^3 |_{(i,j,k-1/2)}^{(i,j,k+1/2)}
   \label{discrphi}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{equation}\int_{\Omega_{ijk}} (f^* - D\phi - \pderiv{c^* \phi}{t}) d\Omega \approx
   \sqrt{g} (f^* - D\phi - \pderiv{c^* \phi}{t})|_{(i,j,k)}
\end{equation}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$k^{\alpha \beta}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11043: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\Phi$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11047: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\bm{x}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11049: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\pderiv{\Phi}{x^\beta} = \frac{1}{\sqrt{g}} \pderiv{}{\xi^\gamma}(\sqrt{g} 
                        a^{(\gamma)}_{\beta} \Phi) \label{exprderiv}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\frac{\partial c^* \phi}{\partial t} +
  \frac{\partial c^* u^\beta \phi}{\partial x^\beta} -
  \frac{\partial}{\partial x^\beta} (k^{\beta \gamma} 
  \frac{\partial \phi}{\partial x^\gamma}) + D\phi = f^*,
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\pderiv{c^* \phi}{t} + \frac{1}{\sqrt{g}}
                          \pderiv{c^* V^\alpha \phi}{\xi^\alpha} - 
  \frac{1}{\sqrt{g}} \pderiv{}{\xi^\alpha} (\sqrt{g} a^{(\alpha)}_\beta
   k^{\beta \gamma} \pderiv{\phi}{x^\gamma}) + D\phi = f^* 
  \label{equaphi2}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\Omega_{ijk}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11051: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray}\int_{\Omega_{ijk}} \frac{1}{\sqrt{g}} \pderiv{c^* V^\alpha \phi}{\xi^\alpha} 
  d \Omega &&= 
  \int_{G_{ijk}} \pderiv{c^* V^\alpha \phi}{\xi^\alpha} d\xi^1 d\xi^2 d\xi^3 \nonumber \\ 
  &&\approx
   c^* V^1 \phi |^{(i+1/2,j,k)}_{(i-1/2,j,k)} + 
   c^* V^2 \phi |^{(i,j+1/2,k)}_{(i,j-1/2,k)} +
   c^* V^3 \phi |^{(i,j,k+1/2)}_{(i,j,k-1/2)}
   \label{convterm}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}&&\int_{\Omega_{ijk}} \pderiv{c^* \phi}{t} d\Omega \approx \sqrt{g}_{(i,j,k)} 
  \pderiv{c^* \phi}{t}|_{(i,j,k)}, \quad
  \int_{\Omega_{ijk}} D\phi d\Omega \approx \sqrt{g}_{(i,j,k)} D\phi|_{(i,j,k)},\nonumber \\ 
  &&\int_{\Omega_{ijk}} f^* d\Omega \approx \sqrt{g}_{(i,j,k)} f^*|_{(i,j,k)}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{equation}\frac{1}{\sqrt{g}} \pderiv{}{\xi^\alpha} (\sqrt{g} a^{(\alpha)}_\beta
  k^{\beta \gamma} \pderiv{\phi}{x^\gamma}) = 
  \frac{1}{\sqrt{g}} \pderiv{}{\xi^\alpha} (\sqrt{g} 
  {\bm{a}}^{(\alpha)} \cdot k \nabla \phi)
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}- \int_{G_{ijk}} \pderiv{}{\xi^\alpha} (\sqrt{g} 
    {\bm{a}}^{(\alpha)} \cdot k \nabla \phi) d \xi^1 d \xi^2 d \xi^3
    &\approx&
  - \sqrt{g} {\bm{a}}^{(1)} \cdot k \nabla \phi
    |^{(i+1/2,j,k)}_{(i-1/2,j,k)} 
    - \sqrt{g} {\bm{a}}^{(2)} \cdot k \nabla \phi
    |^{(i,j+1/2,k)}_{(i,j-1/2,k)} \nonumber \\ 
    &&- \sqrt{g} {\bm{a}}^{(3)} \cdot k \nabla \phi
    |^{(i,j,k+1/2)}_{(i,j,k-1/2)}
    \label{diffterm}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$k \nabla \phi$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11061: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\nabla \phi$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11065: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}k \nabla \phi|_{(i+1/2,j,k)} = 
      \phi|^{(i+1,j,k)}_{(i,j,k)} {\bm{c}}^{(1)} + \frac{1}{4}
      (\phi|^{(i,j+1,k)}_{(i,j-1,k)} + \phi|^{(i+1,j+1,k)}_{(i+1,j-1,k)})
      {\bm{c}}^{(2)} +
      \frac{1}{4}
      (\phi|^{(i,j,k+1)}_{(i,j,k-1)} + \phi|^{(i+1,j,k+1)}_{(i+1,j,k-1)})
      {\bm{c}}^{(3)}
      \label{wesbk}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\bm{c}^{(\alpha)} = \frac{1}{C} (\bm{c}_{(\beta)} \wedge \bm{c}_{(\gamma)}),
  \quad \alpha, \beta, \gamma \quad \mbox{cyclic}
  \label{dried1}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}C = \bm{c}_{(1)} \cdot (\bm{c}_{(2)} \wedge \bm{c}_{(3)})
  \label{dried2}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}{\bm{c}}_{(1)} = 
  \frac{1}{2} (\frac{{\bm{a}}_{(1)}}{k}|_{(i,j,k)} +
               \frac{{\bm{a}}_{(1)}}{k}|_{(i+1,j,k)})
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}{\bm{c}}_{(2)} &=& \frac{1}{8} \frac{{\bm{a}}_{(2)}}
                            {k}|_{(i,j-1,k)} +
                            \frac{1}{4} \frac{{\bm{a}}_{(2)}}
                            {k}|_{(i,j,k)} +
                            \frac{1}{8} \frac{{\bm{a}}_{(2)}}
                            {k}|_{(i,j+1,k)} + \nonumber \\ 
                            &&\frac{1}{8} \frac{{\bm{a}}_{(2)}}
                            {k}|_{(i+1,j-1,k)} +
                            \frac{1}{4} \frac{{\bm{a}}_{(2)}}
                            {k}|_{(i+1,j,k)} +
                            \frac{1}{8} \frac{{\bm{a}}_{(2)}}
                            {k}|_{(i+1,j+1,k)}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}{\bm{c}}_{(3)} &=& \frac{1}{8} \frac{{\bm{a}}_{(3)}}
                            {k}|_{(i,j,k-1)} +
                            \frac{1}{4} \frac{{\bm{a}}_{(2)}}
                            {k}|_{(i,j,k)} +
                            \frac{1}{8} \frac{{\bm{a}}_{(2)}}
                            {k}|_{(i,j,k+1)} + \nonumber \\ 
                            &&\frac{1}{8} \frac{{\bm{a}}_{(2)}}
                            {k}|_{(i+1,j,k-1)} +
                            \frac{1}{4} \frac{{\bm{a}}_{(2)}}
                            {k}|_{(i+1,j,k)} +
                            \frac{1}{8} \frac{{\bm{a}}_{(2)}}
                            {k}|_{(i+1,j,k+1)}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$k^{\alpha \beta} = k \delta^{\alpha \beta}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11075: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}{\bm{c}}^{(1)} = \frac{1}{C} (c^{2}_{(2)}, -c^{1}_{(2)})^\top, \quad
  {\bm{c}}^{(2)} = \frac{1}{C} (-c^{2}_{(1)}, c^{1}_{(1)})^\top
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}C = c^{2}_{(2)} c^{1}_{(1)} - c^{2}_{(1)} c^{1}_{(2)}
\end{equation}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\xi^1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11081: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\phi_{i+1/2}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11083: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$i+\frac12$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11087: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{figure}[htb]
\centerline{\psfig{figure=onedimfv.eps,height=35mm}}

\label{onedfv}
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{equation}\phi_{i+1/2} = \frac12 (\phi_{i} + \phi_{i+1})
  \label{cds}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}Pe^1_{i+1/2} = \frac{\rho_{i+1/2} V^1_{i+1/2}}
                      {2 \sqrt{g}_{i+1/2} (K^{11})_{i+1/2}}
                      \label{mshPe}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\phi_{i+1/2} = \frac12 [1 + \mbox{sign}(V^1_{i+1/2})] \phi_{i} 
               + \frac12 [1 - \mbox{sign}(V^1_{i+1/2})] \phi_{i+1}
  \label{uds}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\phi_{i+1/2} = [1 - s(Pe^1_{i+1/2})] \phi_C + s(Pe^1_{i+1/2}) \phi_U
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\phi_C$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11095: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\phi_U$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11097: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$0 \leq s(Pe) \leq 1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11103: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}s(Pe) = 
          \left\{ 
             \begin{array}{ll}
                0, & |Pe| \leq 1 \\ 
                1, & |Pe| > 1
             \end{array}
          \right.
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}s(Pe) = 1 - \mbox{min} (1, \frac{1}{|Pe|}) \label{swfunc}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\phi_{i+1/2} =
            \left\{ 
             \begin{array}{ll}
                \phi_{i} + \frac{1}{2} \Psi (r_{i+1/2}^{\mbox{\tiny +}})
                (\phi_{i} - \phi_{i-1}), & V^1_{i+1/2} > 0 \\ 
                \\ 
                \phi_{i+1} - \frac{1}{2} \Psi (r_{i+1/2}^{\mbox{\small -}})
                (\phi_{i+2} - \phi_{i+1}), & V^1_{i+1/2} < 0 
             \end{array}
           \right.
  \label{fluxlim1}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}r_{i+1/2}^{\mbox{\tiny +}} = \frac{\phi_{i+1} - \phi_{i}}{\phi_{i} - \phi_{i-1}}
  \quad
  \mbox{ and } \quad
  r_{i+1/2}^{\mbox{\small -}} = 
  \frac{\phi_{i+1} - \phi_{i}}{\phi_{i+2} - \phi_{i+1}}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\begin{array}{ll}
\vspace*{0.3cm}
    \Psi_{\kappa} (r) = \frac12 (1 + \kappa) r + \frac12 (1 - \kappa) & 
       \mbox{Linear } \kappa\mbox{-scheme} \\ 
\vspace*{0.3cm}
    \Psi_{\mbox{\tiny $\Phi$}} (r) = \max[\, 0, \, \min(\mbox{\small $\Phi$} r, 
        \, 1), \, \min(r, \, \mbox{\small $\Phi$}) \, ] & \mbox{Sweby } 
       \mbox{{\small $\Phi$}-limiter} \\ 
\vspace*{0.3cm}
  \Psi_\kappa (r) = 
        \left\{ 
          \begin{array}{ll}
             (r + |r|)\, \frac{-r^2 + (3+\kappa)r - \kappa}{(1 + r)^2}, & 
              r \leq 1,\; \; -1 \leq \kappa < 0 \\ 
             \\ 
             \frac{(2+\kappa)r - \kappa}{1+r}, & r \geq 1,\; \; -1 \leq \kappa < 0 \\ 
             \\ 
             (r + |r|)\, \frac{(1 + \kappa) r + 1 - \kappa}{(1 + r)^2}, &
             0 \leq \kappa \leq 1
          \end{array}
        \right.
    &  
    \mbox{R-}\kappa \mbox{ limiter} \\ 
\vspace*{0.3cm}
  \Psi_\kappa (r) = 
        \left\{ 
          \begin{array}{ll}
             r + |r|, & r \leq \frac{1}{3}\\ 
             \\ 
             \frac{(8 + 9\kappa)r^2 + (2 - 12\kappa)r + 2 + 3\kappa}{3(1 + r)^2},
             & r \geq \frac{1}{3}\\ 
          \end{array}
        \right.
    &  
    \mbox{USR-}\kappa \mbox{ limiter} \\ 
\vspace*{0.3cm}
    \Psi_{\kappa,\mbox{\tiny M}} (r) = \max [0,\, \min (M,\, \frac12 (1 + \kappa) 
                      r + \frac12 (1 - \kappa),\, 2r )] &  
    \mbox{PL-}\kappa \mbox{ limiter} \\ 
\vspace*{0.3cm}
    \Psi_{\kappa,\mbox{\tiny M}} (r) = \max [0,\, \min (2r, \frac23),\,
                     \min (M,\, \frac12 (1 + \kappa) 
                      r + \frac12 (1 - \kappa),\, 2r )] &  
    \mbox{USPL-}\kappa \mbox{ limiter} \\ 
\vspace*{0.3cm}
    \Psi_{\kappa} (r) = \max [0,\, \min (2,\, \frac12 (1 + 
                      \kappa) r + \frac12 (1 - \kappa),\\ 
        \qquad \qquad \; \; \frac12 (1 - \kappa) r + \frac12 (1 + 
                \kappa),\, 2r )] &  
    \mbox{Symmetric PL-}\kappa \mbox{ limiter}
  \end{array}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$-1 \leq \kappa \leq 1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11119: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$1 \leq \mbox{\small $\Phi$} \leq 2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11121: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$1 \leq M \leq 4$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11123: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\mbox{\small $\Phi$} = 1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11125: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\mbox{\small $\Phi$} = 2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11127: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\frac12$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11129: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\kappa = \frac13$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11131: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\frac13$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11133: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\kappa = \frac12$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11141: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\kappa = 0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11145: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\kappa$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11149: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\kappa \neq \frac13$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11153: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\Psi(r)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11155: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\phi_{i+1/2} = \phi_{i+1/2}^{\mbox{\tiny {LOS}}} +
                 ( \phi_{i+1/2}^{\mbox{\tiny {HOS}}} - 
                   \phi_{i+1/2}^{\mbox{\tiny {LOS}}} )^o
  \label{defcor}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\phi_{i+1/2}^{\mbox{\tiny {LOS}}}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11163: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\phi_{i+1/2}^{\mbox{\tiny {HOS}}}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11165: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{equation}- \sqrt{g} K^{11} \pderiv{\phi}{\xi^1}|_{(i+1/2,j,k)} -
    \sqrt{g} K^{12} \pderiv{\phi}{\xi^2}|_{(i+1/2,j,k)} -
    \sqrt{g} K^{13} \pderiv{\phi}{\xi^3}|_{(i+1/2,j,k)}
  \label{exprdiff}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$(i+\frac12,j,k)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11173: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\bm{x} (\bm{\xi})$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11175: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$K^{\alpha \beta}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11179: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\pderiv{\phi}{\xi^1}|_{(i+1/2,j,k)} = \phi_{(i+1,j,k)} - \phi_{(i,j,k)}
  \label{cendif1}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\partial \phi /\partial \xi^2 |_{(i+1/2,j,k)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11181: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray}\pderiv{\phi}{\xi^2} |_{(i+1/2,j,k)} &=& 
  \phi_{(i+1/2,j+1/2,k)} - \phi_{(i+1/2,j-1/2,k)} \nonumber \\ 
  &=& \frac14
  (\phi_{(i+1,j+1,k)} + \phi_{(i,j+1,k)} - \phi_{(i+1,j-1,k)} - \phi_{(i,j-1,k)})
  \label{cendif2}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\phi_{(i,j-1,k)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11185: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\phi_{(i+1,j-1,k)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11187: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$K^{12}_{(i+1/2,j,k)} > 0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11189: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$K^{12}_{(i+1/2,j,k)} < 0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11191: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\phi_{(i,j+1,k)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11193: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\phi_{(i+1,j+1,k)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11195: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\partial \phi / \partial \xi^1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11197: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray}\pderiv{\phi}{\xi^2} |_{(i+1/2,j,k)} &=& 
  \phi_{(i+1/2,j+1/2,k)} - \phi_{(i+1/2,j-1/2,k)} \nonumber \\ 
  &=& \frac12 (\phi_{(i,j,k)} + \phi_{(i+1,j+1,k)}) - 
        \frac12 (\phi_{(i,j-1,k)} + \phi_{(i+1,j,k)})
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{equation}\pderiv{\phi}{\xi^2} |_{(i+1/2,j,k)} =
  \frac12 (\phi_{(i,j+1,k)} + \phi_{(i+1,j,k)} - \phi_{(i,j,k)} - 
           \phi_{(i+1,j-1,k)})
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\phi_{(i+1,j,k-1)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11209: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\phi_{(i+1,j,k+1)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11211: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$K^{11}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11223: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$K^{12}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11225: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\xi^2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11227: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\pderiv{\phi}{\xi^2}|_{(i+1/2,j,k)} =
    \left\{ 
           \begin{array}{ll}
              \frac12 \phi_{(i+1,j+1,k)} + \frac12 \phi_{(i,j+1,k)} - 
              \phi_{(i,j,k)}, & K^{12}_{(i+1/2,j,k)} > 0 \\ 
              \\ 
              \phi_{(i,j,k)} - \frac12 \phi_{(i+1,j-1,k)} - 
              \frac12 \phi_{(i,j-1,k)}, & K^{12}_{(i+1/2,j,k)} < 0
           \end{array}
    \right.
   \label{methiii}
\end{equation}
}

\stepcounter{section}
\stepcounter{subsection}
\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$R^{\alpha \beta}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11235: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\mu_t$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11237: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}R^{\alpha \beta} = \mu_t(g^{\beta \gamma} U^\alpha_{,\gamma}
  + g^{\alpha \gamma} U^\beta_{,\gamma}) - \frac{2}{3} \rho k g^{\alpha \beta}
  \label{bouss}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}k = \frac{1}{2} g_{\alpha \beta} R^{\alpha \beta} \label{defk}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\frac{2}{3} \rho k g^{\alpha \beta}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11243: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$p + \frac{2}{3} \rho k$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11245: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray}R^{\alpha \beta} &=& -\frac23 \rho k g^{\alpha \beta} + 2 \mu_t S^{\alpha \beta}
  -\frac{4 \mu_t^2}{\rho c_\mu k} [c_{\tau 1} (g^{\delta \beta} S^{\alpha \gamma}
  S_{\gamma \delta} 
  - \frac13 S^{\delta \varepsilon} S_{\delta \varepsilon}
  g^{\alpha \beta}) \nonumber \\ 
  &&+c_{\tau 2} (g^{\delta \beta} \Omega^{\alpha \gamma} S_{\gamma \delta} + 
    g^{\delta \alpha} \Omega^{\beta \gamma}
    S_{\gamma \delta}) + c_{\tau 3}
  (g^{\delta \beta} \Omega^{\alpha \gamma} \Omega_{\beta \delta} - 
   \frac13 \Omega^{\delta \varepsilon}
  \Omega_{\delta \varepsilon} g^{\alpha \beta})]
  \label{aniso}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$S^{\alpha \beta}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11251: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$S_{\alpha \beta}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11253: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\Omega^{\alpha \beta}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11255: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\Omega_{\alpha \beta}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11257: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}S^{\alpha \beta} = \frac{1}{2}(g^{\beta \gamma} U^\alpha_{,\gamma} +
                     g^{\alpha \gamma} U^\beta_{,\gamma}), \quad 
  S_{\alpha \beta} = \frac{1}{2}(U_{\alpha, \beta} +
                      U_{\beta, \alpha})
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\Omega^{\alpha \beta} = \frac{1}{2}(g^{\beta \gamma} U^\alpha_{,\gamma} -
                          g^{\alpha \gamma} U^\beta_{,\gamma}), \quad 
  \Omega_{\alpha \beta} = \frac{1}{2}(U_{\alpha, \beta} -
                          U_{\beta, \alpha})
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\beta$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11261: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$c_\mu$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11263: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$c_{\tau 1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11265: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$c_{\tau 2}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11267: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$c_{\tau 3}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11269: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{table}[htb]
\begin{center}
\begin{tabular}{|l|c|c|c|c|}
\hline
AEVM               &$c_\mu$ &$c_{\tau 1}$ &$c_{\tau 2}$ &$c_{\tau 3}$\\ 
\hline
Speziale           &0.09    &0.1512       &0.1512       &0           \\ 
\hline
Rubinstein-Barton  &0.085   &0.68         &0.14         &-0.56       \\  
\hline
Nisizima-Yoshizawa &0.09    &-0.7881      &0.1769       &1.0675      \\ 
\hline
Myong-Kasagi       &0.09    &0.275        &0.2375       &0.05        \\ 
\hline
\end{tabular}
\end{center}

\label{tabaniso}
\end{table}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\omega$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11301: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\mu_t = \rho c_\mu \frac{k^2}{\varepsilon}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\pderiv{\rho k}{t} + (\rho U^\alpha k)_{,\alpha} -
       ((\mu+\frac{\mu_t}{\sigma_k}) g^{\alpha \beta} k_{,\beta} )_{,\alpha} =
       P_k - \rho \varepsilon
   \label{invaequak}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\pderiv{\rho \varepsilon}{t} + (\rho U^\alpha \varepsilon)_{,\alpha} -
       ((\mu+\frac{\mu_t}{\sigma_\varepsilon}) g^{\alpha \beta} \varepsilon_{,\beta} )_{,\alpha} =
       \frac{\varepsilon}{k} (c^*_{\varepsilon 1} P_k - c_{\varepsilon 2} \rho \varepsilon)
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$P_k$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11317: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}P_k = g_{\alpha \gamma} R^{\alpha \beta} U^\gamma_{,\beta} \label{prod1}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma_k$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11321: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma_\varepsilon$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11323: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$c^*_{\varepsilon 1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11325: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$c_{\varepsilon 2}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11327: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\eta = S k/\varepsilon$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11331: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$S = (2 S^{\alpha\beta} S_{\alpha \beta})^\frac{1}{2}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11333: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\eta$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11335: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\eta = \sqrt{P_k/\rho c_\mu \varepsilon}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11337: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}c^*_{\varepsilon 1} = 
          \left\{ 
             \begin{array}{ll}
                c_{\varepsilon 1}, & \mbox{standard model} \\ 
                c_{\varepsilon 1} - \frac{\eta (1 - \eta/\eta_0)}{1 + \gamma \eta^3}, & \mbox{RNG model} \\ 
                c_{\varepsilon 1} + c_{\varepsilon 3} c_\mu \eta^2,  & \mbox{extended model}
             \end{array}
          \right.
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\gamma = 0.012$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11345: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\eta_0 = 4.38$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11347: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{table}[htb]
\begin{center}
\begin{tabular}{|l|c|c|c|c|c|c|}
\hline
$k$-$\varepsilon$ variant  &$c_\mu$ &$\sigma_k$ &$\sigma_\varepsilon$ &$c_{\varepsilon 1}$ &$c_{\varepsilon 2}$ &$c_{\varepsilon 3}$\\ 
\hline
Standard                   &0.09    &1.0        &1.3                  &1.44                &1.92                &-                  \\ 
\hline
RNG                        &0.085   &0.7179     &0.7179               &1.42                &1.68                &-                  \\  
\hline
Extended                   &0.09    &0.75       &1.15                 &1.35                &1.9                 &0.05               \\ 
\hline
\end{tabular}
\end{center}

\end{table}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$c_{\varepsilon 1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11373: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$c_{\varepsilon 3}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11375: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$c_{\varepsilon 1} = 1.15$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11377: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$c_{\varepsilon 3} = 0.25$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11379: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$c_{\varepsilon 1} = 1.35$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11381: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$c_{\varepsilon 3} = 0.05$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11383: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}c_\mu \leftarrow f_\mu c_\mu, \quad c_{\varepsilon 1} \leftarrow 
  f_1 c_{\varepsilon 1}, \quad c_{\varepsilon 2} \leftarrow f_2
  c_{\varepsilon 2}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}&&f_\mu = (1 - e^{-0.0165 R_y})^2\, (1 + \frac{20.5}{Re_{\mbox{\tiny T}}})\\ 
  &&f_1 = 1 + (\frac{0.05}{f_\mu})^3\\ 
  &&f_2 = 1 - e^{-Re_T^2}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{equation}R_y = \frac{\rho \sqrt{k} Y}{\mu}, \quad Re_{\mbox{\tiny T}} = 
  \frac{\rho k^2}{\varepsilon \mu}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\bm{u} = 0, \quad k = 0, \quad \varepsilon = \nu \pderiv{^2 k}{Y^2} |_{Y=0}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\pderiv{\varepsilon}{Y} = 0 \quad \mbox{at} \quad Y = 0 \label{bndeps}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\omega\,(\equiv \varepsilon/k)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11413: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\mu_t = \rho \frac{k}{\omega}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\pderiv{\rho k}{t} + (\rho U^\alpha k)_{,\alpha} -
       ((\mu + \sigma^*\mu_t) g^{\alpha \beta} k_{,\beta} )_{,\alpha} =
       P_k - \beta^* \rho k \omega
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\pderiv{\rho \omega}{t} + (\rho U^\alpha \omega)_{,\alpha} -
       ((\mu + \sigma \mu_t) g^{\alpha \beta} \omega_{,\beta} )_{,\alpha} =
       \alpha \frac{\omega}{k} P_k - \beta \rho \omega^2
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\alpha = \frac{5}{9}, \quad \beta = \frac{3}{40}, \quad \beta^* = \frac{9}{100},
  \quad \sigma = \frac{1}{2}, \quad \sigma^* = \frac{1}{2} \label{conskw}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\lim_{Y \rightarrow 0} \omega = \frac{N_\omega \nu}{Y^2}
  \label{bndomega}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}N_\omega =
             \left\{
               \begin{array}{ll}
                  6/\beta, \quad \mbox{without viscous corrections} \\ 
                  \\ 
                  2/\beta^*, \quad \mbox{with viscous corrections}.
               \end{array}
             \right.
  \label{nomega}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}P_k = \mu_t S^2 \label{equaprod}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\Omega = (2 \Omega^{\alpha \beta} \Omega_{\alpha \beta})^{\frac12}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11431: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}P_k = \mu_t S \Omega \label{katolaunder}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}P_k = \mu_t S ( (1-\alpha) S + \alpha \Omega )
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$0 \leq \alpha \leq 1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11433: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\alpha = 0.85$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11435: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
\stepcounter{subsubsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$Q^{\alpha \beta}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11439: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}Q^{\alpha \beta} = R^{\alpha \beta} + \frac23 \rho k g^{\alpha \beta} - 
                     2 \mu_t S^{\alpha \beta} \label{hterm}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\rho f^\alpha \leftarrow \rho f^\alpha + Q^{\alpha \beta}_{,\beta}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\alpha = 1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11447: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray}&&\int_{\Omega_{i+1/2,j,k}} (\rho f^1 + \frac{1}{\sqrt{g}} 
   \pderiv{\sqrt{g} Q^{1 \beta}}{\xi^\beta} + \chris{1}{\beta}{\gamma}
   Q^{\beta \gamma}) d\Omega =
   \int_{\Omega_{i+1/2,j,k}} (\rho f^1 +
   \chris{1}{\beta}{\gamma} Q^{\beta \gamma}) d\Omega +\nonumber \\ 
   &&\int_{G_{i+1/2,j,k}} \pderiv{\sqrt{g} Q^{1 \beta}}{\xi^\beta} d\xi^1 
    d\xi^2 d\xi^3
   \approx \sqrt{g} (\rho f^1 + \chris{1}{\beta}{\gamma} Q^{\beta \gamma})
     |_{(i+1/2,j,k)} + \nonumber \\ 
   &&\sqrt{g} Q^{11} |^{(i+1,j,k)}_{(i,j,k)} +
     \sqrt{g} Q^{12} |^{(i+1/2,j+1/2,k)}_{(i+1/2,j-1/2,k)} +
     \sqrt{g} Q^{13} |^{(i+1/2,j,k+1/2)}_{(i+1/2,j,k-1/2)}
     \label{upmomrhs}
\end{eqnarray}
}

\stepcounter{subsubsection}
{\newpage
\clearpage
\samepage \begin{equation}U^{\alpha}_{,\beta} = \frac{1}{\sqrt{g}}(\frac{\partial V^\alpha}{\partial 
   \xi^\beta} - 
   \chris{\gamma}{\gamma}{\beta} V^\alpha + \chris{\alpha}{\beta}{\gamma}
   V^\gamma)
   \label{covdu}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\partial V^\alpha / \partial \xi^\beta$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11453: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}V^1_{i,-2} = 2V^1_{i,0} - V^1_{i,2}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}P_k = \mu_t (\pderiv{u_\alpha}{x_\beta} + \pderiv{u_\beta}{x_\alpha}) 
        \pderiv{u_\alpha}{x_\beta}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$x^\beta$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11465: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$u^\alpha$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11467: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\bm{\xi}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11469: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\pderiv{u^\alpha}{x^\beta} = \pderiv{\xi^\gamma}{x^\beta} 
  \pderiv{u^\alpha}{\xi^\gamma} = a^{(\gamma)}_{\beta} 
  \pderiv{u^\alpha}{\xi^\gamma} \label{chainr}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\pderiv{u^\alpha}{x^\beta}|_{(i,j,k)} \approx a^{(\gamma)}_{\beta}|_{(i,j,k)}
  \Delta_\gamma u^\alpha|_{(i,j,k)} 
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\Delta_\gamma u^\alpha|_{(i,j,k)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11473: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\xi^\gamma$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11479: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray}&&\Delta_1 u^\alpha|_{(i,j,k)} = u^\alpha_{(i+1/2,j,k)} - u^\alpha_{(i-1/2,j,k)}, \nonumber\\ 
  &&\Delta_2 u^\alpha|_{(i,j,k)} = u^\alpha_{(i,j+1/2,k)} - u^\alpha_{(i,j-1/2,k)}, \nonumber\\ 
  &&\Delta_3 u^\alpha|_{(i,j,k)} = u^\alpha_{(i,j,k+1/2)} - u^\alpha_{(i,j,k-1/2)}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{equation}\nabla u^\alpha = (\,\pderiv{u^\alpha}{x^1}, \,\pderiv{u^\alpha}{x^2}, 
                     \,\pderiv{u^\alpha}{x^3} \,)^\top
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\nabla u^\alpha|_{(i,j,k)} \approx \bm{a}^{(1)}|_{(i,j,k)} 
                                   u^\alpha|^{(i+1/2,j,k)}_{(i-1/2,j,k)} +
                                   \bm{a}^{(2)}|_{(i,j,k)} 
                                   u^\alpha|^{(i,j+1/2,k)}_{(i,j-1/2,k)} +
                                   \bm{a}^{(3)}|_{(i,j,k)} 
                                   u^\alpha|^{(i,j,k+1/2)}_{(i,j,k-1/2)}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\bm{u} = \frac{\bm{a}_{(\alpha)} V^\alpha}{\sqrt{g}}
  \label{ident5}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\bm{u}_{(i,j+1/2,k)} = \frac{\bm{a}_{(1)} V^1 +
                                      \bm{a}_{(2)} V^2+
                                      \bm{a}_{(3)} V^3}
                                     {\sqrt{g}}|_{(i,j+1/2,k)}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\bm{a}_{(2)}|_{(i,j+1/2,k)} \equiv \frac14 
                                       (\bm{a}_{(2)}|_{(i-1/2,j,k)} +
                                        \bm{a}_{(2)}|_{(i-1/2,j+1,k)} +
                                        \bm{a}_{(2)}|_{(i+1/2,j,k)} +
                                        \bm{a}_{(2)}|_{(i+1/2,j+1,k)})
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}V^1_{(i,j+1/2,k)} \equiv \frac14 (V^1_{(i-1/2,j,k)} + V^1_{(i-1/2,j+1,k)} +
                                  V^1_{(i+1/2,j,k)} + V^1_{(i+1/2,j+1,k)})
\end{equation}
}

\stepcounter{subsubsection}
{\newpage
\clearpage
\samepage \begin{equation}f_\mu = (1 - e^{-0.0165 R_y})^2\, (1 + \frac{20.5}{Re_{\mbox{\tiny T}}})
  \label{eqfmu}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}f_\mu =
    \left\{
      \begin{array}{ll}
         (1 - e^{-0.0165 R_y})^2\, (1 + \frac{20.5}{Re_{\mbox{\tiny T}}}), \quad  R_y > 1.0\\ 
         \\ 
         (0.0165 R_y)^2\, (1 + \frac{20.5}{Re_{\mbox{\tiny T}}}), \quad R_y \leq 1.0
      \end{array}
    \right.
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$f_\mu$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11485: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}k \sim \frac12 Y^2, \quad \varepsilon \sim \nu \quad \mbox{as} \quad Y
  \rightarrow 0
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}R_y \sim \frac{\sqrt{2} Y^2}{2\nu}, \quad Re_{\mbox{\tiny T}} \sim 
 \frac{Y^4}{4 \nu^2} \quad \mbox{as} \quad Y \rightarrow 0
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\lim_{Y \rightarrow 0} f_\mu = 0.01116225
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$Re_{\mbox{\tiny T}}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11487: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$R_y$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11489: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$Y \rightarrow 0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11493: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$Re_{\mbox{\tiny T}} < 1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11495: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$f_\mu > 1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11497: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$f_2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11505: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{figure}[htb]
\centerline{\psfig{figure=ndist.eps,height=8cm}}

\label{ndist}
\end{figure}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\bm{n}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11511: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}Y_{\mbox{\tiny P}} = \bm{n} \cdot \bm{BP} = \bm{n} \cdot
   (\bm{x}_{\mbox{\tiny P}} - \bm{x}_{\mbox{\tiny B}})
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\bm{x}_{\mbox{\tiny B}} = \frac12 (\bm{x}_1 + \bm{x}_2), \quad 
 \bm{x}_{\mbox{\tiny P}} = \frac14 (\bm{x}_3 + \bm{x}_4 + \bm{x}_5 + \bm{x}_6) 
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\bm{n} = (-t_2, t_1)^\top, \quad \bm{t} = 
  \frac{\bm{x}_2-\bm{x}_1}{||\bm{x}_2-\bm{x}_1||}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$Y_{\mbox{\tiny P}}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11513: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$k = 10^{-6}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11529: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\omega = \frac{N_\omega \nu}{Y_{\mbox{\tiny P}}^2}, \qquad
                   Y_{\mbox{\tiny P}}^+ < 5.0
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$N_\omega$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11533: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$Y^+ = \rho u_\tau Y/\mu$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11535: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$u_\tau = \sqrt{\tau_w/\rho}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11537: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tau_w$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11539: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsubsection}
{\newpage
\clearpage
\samepage \begin{equation}P_k - \rho \varepsilon = 2 \rho c_\mu \frac{k^2}{\varepsilon} G - \rho \varepsilon
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$G = S^{\alpha \beta} S_{\alpha \beta}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11555: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}-\rho \varepsilon^{n+1} \approx - \rho^2 c_\mu \frac{(k^{n+1})^2}{\mu^n_t} \approx
  -\frac{2 \rho^2 c_\mu k^n}{\mu^n_t} k^{n+1} + \rho^2 c_\mu 
  \frac{(k^n)^2}{\mu^n_t} = \rho \varepsilon^n - 2 \rho \frac{\varepsilon^n}{k^n} k^{n+1} 
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}D = 2 \rho \frac{\varepsilon^n}{k^n}, \quad f^* = P^n_k + \rho \varepsilon^n
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}c^* = \rho, \quad K^{\alpha \beta} = g^{\alpha \beta} (\mu+\frac{\mu^n_t}{\sigma_k})
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}c^* = \rho, \quad K^{\alpha \beta} = g^{\alpha \beta}
  (\mu+\frac{\mu^n_t}{\sigma_\varepsilon}), \quad 
  D = 2 \rho c_{\varepsilon 2} \frac{\varepsilon^n}{k^n}, \quad 
  f^* = c_{\varepsilon 1} \frac{\varepsilon^n}{k^n} P_k^n +
        \rho c_{\varepsilon 2} \frac{(\varepsilon^n)^2}{k^n}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}-\frac{\eta(1 - \eta/\eta_0)}{1+\gamma \eta^3}\, \frac{\varepsilon}{k}\, P_k
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}D + \frac{\eta P^n_k}{k^n (1 + \gamma \eta^3)}, \quad
  f^* + \frac{\eta^2 \varepsilon^n P^n_k}{\eta_0 k^n (1 + \gamma \eta^3)}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\eta = \sqrt{P^n_k/\rho c_\mu \varepsilon^n}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11579: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\varepsilon^{n+1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11583: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}P_k = 2 \mu_t S^{\alpha \beta} S_{\alpha \beta} + g_{\alpha \gamma}
        Q^{\alpha \beta} U^\gamma_{,\beta} \label{prod5}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}&&f^* := f^* + \max(0,g_{\alpha \gamma} Q^{\alpha \beta} U^\gamma_{,\beta}) \nonumber \\ 
  && \nonumber \\ 
  &&D := D - \frac{\min(0,g_{\alpha \gamma} Q^{\alpha \beta} U^\gamma_{,\beta})}{k^n}
\end{eqnarray}
}

\stepcounter{section}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma \phi + (k \nabla \phi) \cdot \bm{n}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11599: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\xi^{1} = nx$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11607: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
\stepcounter{subsubsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\bm{u} \cdot
\bm{n}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11613: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\bm{u} \cdot \bm{t}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11615: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray}U^{n}& =& \sqrt{g^{nn}} \bm{u} \cdot \bm{n}
\label{eq:5.1}\\ 
U^{t}&=&\frac{1}{g_{tt}}(\sqrt{g_{tt}} \bm{u} \cdot \bm{t} - g_{nt} U^{n})
\label{eq:5.2}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\bm{t}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11619: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$U^{t}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11621: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{figure}[htb]
\centerline{\psfig{figure=fig51.eps,height=5cm}}

\label{fig51}
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}p_{i, -2}&=&2p_{i,0} - p_{i,2}\label{eq:5.3}\\ 
V^{1}_{i,-2}&=&2V^{1}_{i,-1} - V^{1}_{i,0},
\label{eq:5.4}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$V^{1}_{i, -1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11623: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sqrt{g} U^{t}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11625: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsubsection}
{\newpage
\clearpage
\samepage \begin{equation}U^{\alpha} = \bm{a}^{(\alpha)} \cdot \bm{u}
\label{eq:5.5}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\xi^{n}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11629: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\bm{u} \cdot \bm{n}, \;
\bm{u} \cdot \bm{\tau}_{1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11631: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\bm{u} \cdot \bm{\tau}_{2}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11633: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\bm{\tau}_{1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11635: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\bm{\tau}_{2}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11637: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\underline{u} \cdot \underline{n}, \; \underline{u} \cdot
\underline{\tau}_{1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11639: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\underline{u} \cdot \underline{\tau}_{2}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11641: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\underline{\tau}_{1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11643: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\underline{\tau}_{2}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11645: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\|\underline{n}\| = \|\underline{\tau}_{1}\| = \|\underline{\tau}_{2}\| = 1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11647: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{figure}[htb]
\centerline{\psfig{figure=fig52.eps}}

\label{fig52}
\end{figure}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\bm{u} \cdot \bm{n}, \; \bm{u} \cdot \bm{\tau}_{1}, \; \bm{u} \cdot
\bm{\tau}_{2}, \; \bm{\tau}_{1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11649: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$U^{n}, \; U^{t_{1}}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11653: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$U^{t_{2}}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11655: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray}U^{n}&=&sign \; (\bm{a}^{(n)} \cdot \bm{n})\sqrt{g^{nn}} \bm{u} \cdot \bm{n}\label{eq:5.6}\\ 
&&\nonumber\\ 
\left[\begin{array}{c}
U^{t_{1}}\\ 
U^{t_{2}}
\end{array}
\right]&=&\left[\begin{array}{cc}
g_{t_{1}t_{1}} & g_{t_{1}t_{2}}\\ 
g_{t_{2}t_{1}} & g_{t_{2} t_{2}}
\end{array}
\right]^{-1} \left\{\left[\begin{array}{cc}
\alpha_{11} & \alpha_{12}\\ 
\alpha_{21} & \alpha_{22}
\end{array}
\right]^{-1}\left[\begin{array}{c}
\bm{u} \cdot \bm{\tau}_{1}\\ 
\bm{u} \cdot \bm{\tau}_{2}
\end{array}
\right] - U^{n} \left[\begin{array}{c}
g_{nt_{1}}\\ 
g_{nt_{2}}
\end{array}\right]\right\}
\label{eq:5.7}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{subequations}% latex2html id marker 8276
\begin{equation}
\alpha_{i1} = \frac{\left|\begin{array}{cc}\bm{\tau}_{i} \cdot \bm{a}_{(t_{1})}
&g_{t_{1}t_{2}}\\ 
\bm{\tau}_{i} \cdot \bm{a}_{(t_{2})}
&g_{t_{2}t_{2}}\end{array}\right|}{\left|\begin{array}{cc}g_{t_{1}t_{1}}
&g_{t_{1}t_{2}}\\  g_{t_{2}t_{1}} &g_{t_{2}t_{2}}\end{array}\right|}\; ,
\label{eq:5.8a}
\end{equation}
\begin{equation}
\alpha_{i2} = \frac{\left|\begin{array}{cc} g_{t_{1}t_{1}}&\bm{\tau}_{i} \cdot 
\bm{a}_{(t_{1})}\\ g_{t_{2}t_{1}} &\bm{\tau}_{i} \cdot
\bm{a}_{(t_{2})}\end{array}\right|}{\left|\begin{array}{cc}g_{t_{1}t_{1}}&g_{t_{1}t_{2}}\\ 
g_{t_{2}t_{1}}&g_{t_{2}t_{2}}\end{array}\right|}\; .
\label{eq:5.8b}
\end{equation}
\end{subequations}
}

\stepcounter{subsection}
\stepcounter{subsubsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$S^{nn}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11667: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$S^{nt}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11669: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma^{nn}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11675: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma^{nt}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11677: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray}\sigma^{nn}&=&g^{nn} S^{nn},\label{eq:5.9}\\ 
\sigma^{nt}&=&(\sqrt{g^{nn} g_{tt}} S^{nt} - g_{nt}
\sigma^{nn})/g_{tt},\label{eq:5.10}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma^{\alpha \beta}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11679: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\sigma^{\alpha \beta} = - g^{\alpha \beta} p + \tau^{\alpha \beta}.
\label{eq:5.11}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\sqrt{g} \sigma^{11} |^{(1,0)}_{(-1,0)} + \sqrt{g}
\sigma^{12}|^{(0,1)}_{(0,-1)} + \chris{1}{\gamma}{\beta} \sigma^{\gamma
\beta}\sqrt{g}|_{(0,0)}
\label{eq:5.12}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma^{12}|_{(0,-1)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11681: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{figure}[htb]
\centerline{\psfig{figure=fig53.eps,height=6cm}}

\label{fig53}
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{equation}\frac{1}{2} \sqrt{g} T^{21}|^{(1,1)}_{(-1,1)} + \sqrt{g} T^{22}
|^{(0,1)}_{(0,0)} + \frac{1}{2} \sqrt{g} \chris{2}{\gamma}{\beta} T^{\gamma
\beta} |_{(0,0)},
\label{eq:5.13}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}T^{\alpha \beta} = \rho U^{\alpha} U^{\beta} - \sigma^{\alpha \beta}
\label{eq:5.14}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\frac{1}{2} \rho f^{\alpha}\sqrt{g}|_{(0,0)}
\label{eq:5.15}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\frac{1}{2} \frac{\partial}{\partial t}(\rho V^{\alpha})|_{(0,0)}
\label{eq:5.16}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}T^{\alpha \beta} = \rho U^{\alpha} U^{\beta}
\label{eq:5.17}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}V^{1}_{0,0} = \frac{1}{2} (V^{1}_{1,1} + V^{1}_{-1,1})
\label{eq:5.18}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}RHS - \sqrt{g} \sigma^{22}|_{(0,1)} - \frac{1}{2} \sqrt{g} \chris{2}{1}{1}
\sigma^{11}|_{(0,0)},
\label{eq:5.19}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}RHS&=&-\frac{1}{2} \sqrt{g} \sigma^{21}|_{(1,0)} + \frac{1}{2} \sqrt{g}
\sigma^{21}|_{(-1,0)} + \sqrt{g}\sigma^{22}|_{(0,0)}\nonumber\\ 
&&-\frac{1}{2} \sqrt{g} \chris{2}{2}{2} \sigma^{22} |_{(0,0)} - \sqrt{g}
\chris{2}{1}{2} \sigma^{12}|_{(0,0)}
\label{eq:5.20}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma^{11}|_{(0,0)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11685: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$p_{0,1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11687: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$p_{0,0}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11689: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\frac{\partial U^{1}}{\partial \xi^{2}}|_{(0,0)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11691: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\frac{\partial U^{1}}{\partial \xi^{1}}|_{(0,0)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11693: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\frac{\partial U^{1}}{\partial \xi^{1}}|_{(0,1)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11695: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsubsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$S^{n\tau}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11705: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\bm{\tau}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11707: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\bm{S}_{_{\xi^{n}=const}} = S^{nn} \bm{n} + S^{n\tau}\bm{\tau}
\label{eq:5.21}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$S^{nn} \bm{n}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11713: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$S^{n \tau} \bm{\tau}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11715: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma^{nn}, \; \sigma^{nt_{1}}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11717: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma^{nt_{2}}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11719: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{figure}[htb]
\centerline{\psfig{figure=fig54.eps,height=6cm}}

\label{fig54}
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{equation}\sigma^{nn} = g^{nn} S^{nn}\; ,
\label{eq:5.22}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma^{nt_{1}}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11723: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\left[\begin{array}{c}
\sigma^{nt_{1}}\\ 
\sigma^{nt_{2}}
\end{array}
\right] = sign\; (\bm{a}^{(n)} \cdot \bm{n})\sqrt{g^{nn}} S^{n\tau}\left[
\begin{array}{c}
\alpha_{1}\\ 
\alpha_{2}
\end{array}\right] - \sigma^{nn}\left[\begin{array}{cc}
g_{t_{1}t_{1}} &g_{t_{1}t_{2}}\\ 
g_{t_{2}t_{1}} &g_{t_{2}t_{2}}
\end{array} \right]^{-1} \left[\begin{array}{c}
g_{nt_{1}}\\ 
g_{nt_{2}}
\end{array}
\right]
\label{eq:5.23}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{subequations}% latex2html id marker 8278
\begin{equation}
\alpha_{1} = \frac{\left|\begin{array}{cc}\bm{\tau} \cdot \bm{a}_{(t_{1})}
&g_{t_{1}t_{2}}\\ 
\bm{\tau} \cdot \bm{a}_{(t_{2})}&g_{t_{2}t_{2}}\end{array}\right|}{\left|
\begin{array}{cc}
g_{t_{1}t_{1}} &g_{t_{1}t_{2}}\\  
g_{t_{2}t_{1}} 
&g_{t_{2}t_{2}}\end{array}\right|}\; .
\label{eq:5.24a};
\end{equation}
\begin{equation}
\alpha_{2} = \frac{\left|\begin{array}{cc}g_{t_{1}t_{1}}&\bm{\tau} \cdot
\bm{a}_{(t_{1})}\\ g_{t_{2}t_{1}}&\bm{\tau}\cdot
\bm{a}_{(t_{2})}\end{array}\right|}{\left|\begin{array}{cc}
g_{t_{1}t_{1}}
&g_{t_{1}t_{2}}\\ g_{t_{2}t_{1}}&g_{t_{2}t_{2}}\end{array}\right|}
\label{eq:5.24b}
\end{equation}
\end{subequations}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma^{\alpha \beta} = -
g^{\alpha \beta} p + \tau^{\alpha \beta}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11729: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{figure}[htb]
\centerline{\psfig{figure=fig55.eps}}

\label{fig55}
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}V^{1}_{i,j,-2}&=&2 V^{1}_{i,j,0} - V^{1}_{i,j,2}\label{eq:5.25}\\ 
V^{2}_{i,j,-2}&=&2 V^{2}_{i,j,0} - V^{2}_{i,j,2}\label{eq:5.26}\\ 
V^{3}_{i,j,-3}&=&2 V^{3}_{i,j,-1} - V^{3}_{i,j,1}
\label{eq:5.27}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{equation}\begin{array}{l}
- \sqrt{g}\sigma^{\alpha 1}|^{(1,0,0)}_{(-1,0,0)} - \sqrt{g} \sigma^{\alpha
2}|^{(0,1,0)}_{(0,-1,0)} - \sqrt{g} \sigma^{\alpha 3}
|^{(0,0,1)}_{(0,0,-1)}\\ 
- \sqrt{g}\chris{\alpha}{\gamma}{\beta} \sigma^{\gamma\beta}|_{(0,0,0)} \;
\; \; \mbox{\rm \ for }\; \; \; \alpha = 1, 2.
\end{array}
\label{eq:5.28}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma^{\alpha 3}|_{(0,0,-1)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11741: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{figure}[htb]
\centerline{\psfig{figure=fig56.eps}}

\label{fig56}
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{equation}\frac{1}{2} \frac{\partial}{\partial t} (\rho V^{3})|_{(0,0,0)}
\label{eq:5.29}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\frac{1}{2} (\rho f^{3} \sqrt{g})|_{(0,0,0)}\; .
\label{eq:5.30}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}&&\frac{1}{2} \frac{\rho}{\sqrt{g}} V^{3} V^{1} |_{(-1,0,0)}^{(1,0,0)} +
\frac{1}{2} \frac{\rho}{\sqrt{g}} V^{3} V^{2} |_{(0,-1,0)}^{(0,1,0)} +
\frac{\rho}{\sqrt{g}}V^{3}V^{3}|^{(0,0,1)}_{(0,0,0)}\nonumber\\ 
&&+\frac{1}{2} \frac{\rho}{\sqrt{g}} \chris{3}{\gamma}{\beta}
V^{\gamma}V^{\beta}|_{(0,0,0)}
\label{eq:5.31}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}V^{1}_{i,j,0}&=&V^{1}_{i,j,1}\label{eq:5.32}\\ 
V^{2}_{i,j,0}&=&V^{2}_{i,j,1}\;.\label{eq:5.33}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}V^{1}_{i,j,1}&=&\frac{1}{2}(V^{1}_{i-1,j,1} + V^{1}_{i+1,j,1})\label{eq:5.34}
\\ 
V^{2}_{i,j,1}&=&\frac{1}{2}(V^{2}_{i,j-1,1} +
V^{2}_{i,j+1,1})\; .\label{eq:5.35}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}&&- \frac{1}{2}\sqrt{g} \sigma^{31}|^{(1,0,0)}_{(-1,0,0)} - \frac{1}{2}
\sqrt{g} \sigma^{32}|^{(0,1,0)}_{(0,-1,0)} -
\sqrt{g}\sigma^{33}|^{(0,0,1)}_{(0,0,0)}\nonumber\\ 
&&- \frac{1}{2} \sqrt{g} \chris{3}{\gamma}{\beta}\sigma^{\gamma \beta}
|_{(0,0,0)}
\label{eq:5.36}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}RHS&-&\sqrt{g} \sigma^{33} |_{(0,0,1)} - \frac{1}{2} \sqrt{g}
\chris{3}{1}{1} \sigma^{11}|_{(0,0,0)}\nonumber\\ 
&-&\sqrt{g} \chris{3}{1}{2} \sigma^{12}|_{(0,0,0)} - \frac{1}{2} \sqrt{g}
\chris{3}{2}{2} \sigma^{22}|_{(0,0,0)}
\label{eq:5.37}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}RHS&=&- \frac{1}{2} \sqrt{g} \sigma^{31} |_{(-1,0,0)}^{(1,0,0)} -
\frac{1}{2} \sqrt{g} \sigma^{32}|^{(0,1,0)}_{(0,-1,0)} + \sqrt{g}
\sigma^{33}|_{(0,0,0)}\nonumber\\ 
&&- \sqrt{g} \chris{3}{1}{3} \sigma^{13}|_{(0,0,0)} - \sqrt{g}
\chris{3}{2}{3} \sigma^{23}|_{(0,0,0)} - \frac{1}{2} \sqrt{g}
\chris{3}{3}{3} \sigma^{33}|_{(0,0,0)}\; .
\label{eq:5.38}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma^{11}|_{(0,0,0)}, \; \sigma^{12}_{(0,0,0)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11755: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma^{22}_{(0,0,0)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11757: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma^{11}, \;
\sigma^{12}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11761: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma^{22}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11763: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$p_{0,0,0}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11765: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$p_{0,0,1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11767: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\frac{\partial U^{1}}{\partial \xi^{\alpha}}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11769: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\frac{\partial U^{2}}{\partial \xi^{\alpha}}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11771: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\alpha =
1, 2, 3$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11773: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\frac{\partial U^{1}}{\partial \xi^{1}}|_{(0,0,0)},
\; \frac{\partial U^{1}}{\partial \xi^{2}}|_{(0,0,0)}, $}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11775: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\frac{\partial U^{2}}{\partial \xi^{1}}|_{(0,0,0)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11777: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\frac{\partial U^{2}}{\partial \xi^{2}}|_{(0,0,0)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11779: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\frac{\partial U^{1}}{\partial \xi^{1}}|_{(0,0,1)}, \; 
\frac{\partial U^{1}}{\partial \xi^{2}} |_{(0,0,1)}, \; 
\frac{\partial U^{2}}{\partial
\xi^{1}}|_{(0,0,1)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11781: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\frac{\partial U^{2}}{\partial
\xi^{2}}|_{(0,0,1)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11783: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\frac{\partial U^{1}}{\partial
\xi^{3}}|_{(0,0,0)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11785: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\frac{\partial U^{2}}{\partial
\xi^{3}}|_{(0,0,0)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11787: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\frac{\partial U^{\alpha}}{\partial \xi^{3}}|_{(0,0,0)} =
U^{\alpha}|_{(0,0,3)} - U^{\alpha}|_{(0,0,1)}
\label{eq:5.39}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\alpha = 1, 2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11797: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
\stepcounter{subsubsection}
{\newpage
\clearpage
\samepage \begin{equation}\bm{u} \cdot \bm{t} \mbox{ and } S^{nn} \mbox{ given. }
\label{eq:5.40}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}U^{t} \mbox{ and } S^{nn} \mbox{ given}
\label{eq:5.41}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}&&- \frac{1}{2} \sqrt{g} \sigma^{21}|^{(1,0)}_{(-1,0)} - \sqrt{g}
\sigma^{22}|_{(0,1)} - \frac{1}{2}\sqrt{g} (\chris{2}{1}{1} \sigma^{11} + 2
\chris{2}{1}{2}\sigma^{12})|_{(0,0)}\nonumber\\ 
&&+ \sqrt{g} \sigma^{22}|_{(0,0)} - \frac{1}{2} \sqrt{g} \chris{2}{2}{2}
\sigma^{22}|_{(0,0)},
\label{eq:5.42}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma^{22}|_{(0,0)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11813: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\frac{1}{2} \rho \sqrt{g} U^{2} U^{\beta}|^{(1,0)}_{(-1,0)} + \rho \sqrt{g}
U^{2} U^{\beta} |^{(0,1)}_{(0,0)} + \frac{1}{2} \rho \sqrt{g}
\chris{2}{\gamma}{\beta} U^{\gamma}U^{\beta}|_{(0,0)}
\label{eq:5.43}
\end{equation}
}

\stepcounter{subsubsection}
{\newpage
\clearpage
\samepage \begin{equation}\bm{u} \cdot \bm{\tau}_{1}, \; \bm{u} \cdot \bm{\tau}_{2}, \;
\bm{\tau}_{1}, \; \bm{\tau}_{2} \; \;\; \mbox{ and } \; \;\; S^{nn} \;
\;\; \mbox{ are given .}
\label{eq:5.44}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$U^{t_{1}}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11815: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$g_{nt_{1}}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11819: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$g_{nt_{2}}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11821: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}U^{t_{1}}, \; U^{t_{2}} \; \; \; \mbox{ and } \; \;\; S^{nn} \; \; \; \mbox{
are
given ,}
\label{eq:5.45}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}V^{\alpha}_{i,j,-2}&=&2 V^{\alpha}_{i,j,-1} - V^{\alpha}_{i,j,0} \; \;\;
\mbox{ for }\; \;\; \alpha = 1, 2
\label{eq:5.46}\\ 
V^{3}_{i,j,-3}&=&2 V^{3}_{i,j,-1} - V^{3}_{i,j,1}\label{eq:5.47}\\ 
p_{i,j,-2}&=&2 p_{i,j,0} - p_{i,j,2}\; .
\label{eq:5.48}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}&&\frac{1}{2} \frac{\rho}{\sqrt{g}}V^{3}V^{1}|^{(1,0,0)}_{(-1,0,0)} +
\frac{1}{2} \frac{\rho}{\sqrt{g}} V^{3} V^{2}|^{(0,1,0)}_{(0,-1,0)} +
\frac{\rho}{\sqrt{g}} V^{3} V^{3}|_{(0,0,0)}^{(0,0,1)}\nonumber\\ 
&&+ \frac{1}{2} \frac{\rho}{\sqrt{g}}\chris{3}{\gamma}{\beta}
V^{\gamma}V^{\beta}|_{(0,0,0)}\; .
\label{eq:5.49}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$V^{3} V^{3}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11835: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray}&-& \frac{1}{2} \sqrt{g} \sigma^{31}|^{(1,0,0)}_{(-1,0,0)} - \frac{1}{2}
\sqrt{g} \sigma^{32}|^{(0,1,0)}_{(0,-1,0)} - 
\sqrt{g}\sigma^{33}|_{(0,0,0)}^{(0,0,1)} \nonumber\\ 
&-& \frac{1}{2} \sqrt{g} \chris{3}{\gamma}{\beta} \sigma^{\gamma \beta}
|_{(0,0,0)}\; ,
\label{eq:5.50}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma^{33}|_{(0,0,0)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11841: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$(g^{33} S^{33})|_{(0,0,0)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11843: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}(-1 + \frac{1}{2}\chris{3}{3}{3}) \sqrt{g} \sigma^{33}|_{(0,0,0)}\; .
\label{eq:5.51}
\end{equation}
}

\stepcounter{subsection}
\stepcounter{subsubsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$U^{n}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11845: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\sqrt{g^{nn}g_{tt}} S^{nt} = g_{nt} \sigma^{nn} + g_{tt}\sigma^{nt}
\label{eq:5.52}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}- \sqrt{g}\sigma^{11}|^{(1,0)}_{(-1,0)} - \sqrt{g}
\sigma^{12}|^{(0,1)}_{(0,-1)} - \chris{1}{\gamma}{\beta} \sigma^{\gamma
\beta}\sqrt{g}|_{(0,0)}
\label{eq:5.53}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}V^{1}_{i, -2} = 2V^{1}_{i,0} - V^{1}_{i,2}
\label{eq:5.54}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$(\sqrt{g} \sigma^{12})_{(0,-1)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11847: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\sqrt{g} \sigma^{12}|_{(0,-1)} = S^{nt}|_{(0,-1)} - \sqrt{g} \frac{g_{21}}{g_{11}} \sigma^{22}|_{(0,-1)},
\label{eq:5.55}
\end{equation}
}

\stepcounter{subsubsection}
{\newpage
\clearpage
\samepage \begin{eqnarray}&&\frac{\rho}{\sqrt{g}} V^{\alpha} V^{1} |_{(-1,0,0)}^{(1,0,0)} +
\frac{\rho}{\sqrt{g}} V^{\alpha} V^{2}|^{(0,1,0)}_{(0,-1,0)} +
\frac{\rho}{\sqrt{g}} V^{\alpha} V^{3}|^{(0,0,1)}_{(0,0,-1)}\nonumber\\ 
&&+ \frac{\rho}{\sqrt{g}}\chris{\alpha}{\gamma}{\beta}
V^{\gamma}V^{\beta}|_{(0,0,0)} \; \;\; \mbox{ for } \;\;\; \alpha = 1, 2\;
.
\label{eq:5.56}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$V^{3} = \sqrt{g} U^{3}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11855: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$V^{\alpha} V^{3}|_{(0,0,-1)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11857: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}V^{\alpha} V^{3}|_{(0,0,-1)} = \frac{1}{2} (3 V^{\alpha}|_{(0,0,0)} -
V^{\alpha}|_{(0,0,2)} ) V^{3}|_{(0,0,-1)}\; \;\; \mbox{ for } \;\;\; \alpha
= 1,2\; .
\label{eq:5.57}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}&&-\sqrt{g} \sigma^{\alpha 1}|_{(-1,0,0)}^{(1,0,0)} - \sqrt{g}\sigma^{\alpha
2}|^{(0,1,0)}_{(0,-1,0)} - \sqrt{g} \sigma^{\alpha 3}
|^{(0,0,1)}_{(0,0,-1)}\nonumber\\ 
&&- \sqrt{g} \chris{\alpha}{\gamma}{\beta} \sigma^{\gamma \beta} |_{(0,0,0)}\;\;\;
\mbox{ for } \;\;\; \alpha = 1, 2\; .
\label{eq:5.58}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\alpha
= 1 \; (U^{1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11861: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\sigma^{13}|_{(0,0,-1)} = \left(sign\; (\bm{a}^{(3)}\cdot \bm{n})\sqrt{g^{33}} S^{n\tau} \alpha_{1} - \sigma^{33}
\bm{e}^{T}_{1} \left[\begin{array}{cc}
g_{11} &g_{12}\\ 
g_{21} &g_{22}
\end{array}
\right]^{-1} \left[\begin{array}{c}
g_{31}\\ g_{32}\end{array}\right]\right) |_{(0,0,-1)}
\label{eq:5.59}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\bm{e}^{T}_{1} = [1 \;\; 0]$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11863: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\sigma^{13}|_{(0,0,-1)} = \left(sign\; (\bm{a}^{(3)}\cdot \bm{n})\sqrt{g^{33}}S^{n\tau}\alpha_{1} -
\sigma^{33}\frac{g_{22}g_{31} - g_{12} g_{32}}{g_{11}g_{22} - g_{12}
g_{21}}\right)|_{(0,0,-1)}\; .
\label{eq:5.60}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\alpha = 2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11867: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\sigma^{23}|_{(0,0,-1)} = \left(sign \; (\bm{a}^{(3)}\cdot \bm{n})\sqrt{g^{33}} S^{n\tau} \alpha_{2} - \sigma^{33}
\bm{e}^{T}_{2} \left[\begin{array}{cc}
g_{11} &g_{12}\\ 
g_{21} &g_{22}
\end{array}\right]^{-1} \left[\begin{array}{c}
g_{31}\\ g_{32}
\end{array}
\right]\right)|_{(0,0,-1)}
\label{eq:5.61}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\bm{e}^{T}_{2} = [0 \;\; 1]$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11869: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\sigma^{23}|_{(0,0,-1)} = \left(sign\; (\bm{a}^{(3)}\cdot \bm{n})\sqrt{g^{33}} S^{n\tau} \alpha_{2} - \sigma^{33}
\frac{g_{11} g_{32} - g_{21} g_{31}}{g_{11} g_{22} - g_{21}
g_{12}}\right)|_{(0,0,-1)}\; .
\label{eq:5.62}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\alpha_{1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11871: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\alpha_{2}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11873: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma^{33}|_{(0,0,-1)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11875: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
\stepcounter{subsubsection}
{\newpage
\clearpage
\samepage \begin{figure}[htb]
\centerline{\psfig{figure=fig57.eps,height=5cm}}

\label{fig57}
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{figure}[htb]
\centerline{\psfig{figure=fig58.eps,height=5cm}}
 
\label{fig58}
\end{figure}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$V^{1}_{(0,0)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11895: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$V^{1}_{(2,-2)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11897: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}V^{1}_{(2,-2)} = 2 V^{1}_{(2,0)} - V^{1}_{(2,2)}
\label{eq:5.63}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{figure}[htb]
\centerline{\psfig{figure=fig59.eps,height=5cm}}

\label{fig59}
\end{figure}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$u_{t}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11921: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsubsection}
{\newpage
\clearpage
\samepage \begin{figure}[htb]
\centerline{\psfig{figure=fig510a.eps,height=5cm}}

\label{fig510a}
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{figure}[htb]
\centerline{\psfig{figure=fig511a.eps,height=8.3cm}}

\label{fig511a}
\end{figure}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$V^{2}_{(1,1,-2)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11941: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}V^{2}_{(1,1,-2)} = 2 V^{2}_{(1,1,-1)} - V^{2}_{(1,1,0)}\; ,
\label{eq:5.64a}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}V^{2}_{(1,1,-2)} = 2V^{2}_{(1,1,0)} - V^{2}_{(1,1,2)}\; .
\label{eq:5.65a}
\end{equation}
}

\stepcounter{subsection}
\stepcounter{subsubsection}
{\newpage
\clearpage
\samepage \begin{figure}[htb]
\centerline{\psfig{figure=fig510.eps,height=5cm}}

\label{fig510}
\end{figure}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$V^{2}_{(0,i)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11951: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$V^{2}_{(-2,i)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11953: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsubsection}
\stepcounter{paragraph}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$4^{2} = 16$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11963: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\left(\begin{array}{c}4+2-1\\ 2\end{array}\right) =
10$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11965: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{table}[htb]
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
&\multicolumn{2}{|c|}{boundary condition type}\\  \cline{2-3}
combination &left boundary &bottom boundary\\ 
            &$(\xi^{2} = 0)$ &$(\xi^{3} = 0)$\\ 
\hline
(i) &1 &1\\ 
(ii) &1 &2\\ 
(iii) &1 &3\\ 
(iv) &1 &4\\ 
(v)  &2 &2\\ 
(vi) &2 &3\\ 
(vii) &2 &4\\ 
(viii) &3 &3\\ 
(ix)   &3 &4\\ 
(x)    &4 &4\\ 
\hline
\end{tabular}
\end{center}

\label{tab51}
\end{table}
}

{\newpage
\clearpage
\samepage \begin{figure}[htb]
\centerline{\psfig{figure=fig513a.eps}}

\label{fig513a}
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}V^{\alpha}_{(i,j,-2)}&=&2V^{\alpha}_{(i,j,-1)} - V^{\alpha}_{(i,j,0)}
\;\;\;\;\; \mbox{for} \;\;\; \alpha = 1, 2 \;\;\;\;\; \mbox{and} \;\;\; j
\neq -2\\ 
V^{\alpha}_{(i,-2,k)}&=&2V^{\alpha}_{(i,-1,k)} -
V^{\alpha}_{(i,0,k)} \;\;\;\;\; \mbox{for} \;\;\; \alpha = 1, 3 \;\;\;\;\;
\mbox{and} \;\;\; k \neq -2\\ 
V^{2}_{(i,-3,0)}&=&2V^{2}_{(i,-1,0)} - V^{2}_{(i,1,0)}\\ 
V^{3}_{(i,0,-3)}&=&2V^{3}_{(i,0,-1)} - V^{3}_{(i,0,1)}\;\;.
\end{eqnarray*}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$V^{1}_{(0,-2,-2)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11977: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}V^{1}_{(0,-2,-2)} = 4V^{1}_{(0,-1,-1)} - 2V^{1}_{(0,-1,0)} -
2V^{1}_{(0,0,-1)} + V^{1}_{(0,0,0)}
\label{eq:5.66a}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}V^{1}_{(0,-2,-2)} = 2V^{1}_{(0,-1,-1)} - V^{1}_{(0,0,0)}\; .
\label{eq:5.67a}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{figure}[htb]
\centerline{\psfig{figure=fig514.eps}}

\label{fig514}
\end{figure}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma^{31}, \sigma^{32}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11991: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma^{33}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11993: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$U^{1}, U^{2}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11995: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\frac{1}{2}
\frac{\rho}{\sqrt{g}} V^{3}V^{2}|_{(0,-1,0)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline11999: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\frac{\partial
U^{1}}{\partial \xi^{2}}|_{(0,0,-1)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12001: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$V^{3}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12005: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\frac{\partial U^{1}}{\partial \xi^{2}}|_{(0,0,1)} =
\frac{1}{2}(U^{1}_{(0,2,1)} - U^{1}_{(0,-2,1)})$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12007: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\frac{\partial U^{1}}{\partial \xi^{2}}|_{(0,0,1)} =
\frac{1}{2}(U^{1}_{(0,2,1)} - (2U^{1}_{(0,-1,1)} - U^{1}_{(0,0,1)})
\label{eq:5.68a}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}U^{1}_{(0,i,j)} = \frac{1}{2}(U^{1}_{(-1,i,j)} + U^{1}_{(1,i,j)})\; .
\label{eq:5.69a}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$V^{1}_{(0,-2,-1)}, V^{2}_{(0,-3,-1)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12021: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$V^{3}_{(0,-2,-2)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12023: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$V^{1}_{(0,-2,-1)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12025: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$V^{1}_{(0,-3,-1)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12027: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}V^{1}_{(i,-2,-1)} = 4V^{1}_{(i,-1,0)} - 2V^{1}_{(i,0,0)} - 2V^{1}_{(i,-1,1)} +
V^{1}_{(i,0,1)}
\label{eq:5.70a}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}V^{2}_{(0,-3,-1)} = 4V^{2}_{(0,-1,0)} - 2V^{2}_{(0,1,0)} -
2V^{2}_{(0,-1,1)} + V^{2}_{(0,1,1)}\; ,
\label{eq:5.71a}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}V^{3}_{(0,-2,-2)} = 4V^{3}_{(0,0,0)} - 2V^{3}_{(0,2,0)} - 2V^{3}_{(0,0,2)}
+ V^{3}_{(0,2,2)}\; .
\label{eq:5.72a}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma^{21}, \sigma^{22}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12041: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma^{23}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12043: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma^{13}|_{(0,0,-1)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12057: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma^{12}_{(0,0,-1)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12059: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{figure}[htb]
\centerline{\psfig{figure=fig515.eps}}

\label{fig515}
\end{figure}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{equation}\phi_{.,0} = 2 \phi_{.,1/2} - \phi_{.,1}
\label{eq:5.64}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\phi_{0,0} = \frac{1}{2} (2 \phi_{1,0} - \phi_{2,0} ) + \frac{1}{2} (2 \phi_{0,1} -
\phi_{0,2})
\label{eq:5.65}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{figure}[htb]
\centerline{\psfig{figure=fig511.eps,height=5cm}}

\label{fig5.11}
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{equation}k^{\alpha \beta} \phi_{, \beta} n_{\alpha} = b - \sigma \phi
\label{eq:5.66}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}\lefteqn{
\int\limits_{\Omega} - (k^{\alpha \beta}\phi_{, \beta})_{, \alpha} d \Omega =
- \int\limits_{\Gamma} k^{\alpha \beta} \phi_{, \beta} n_{\alpha} d\Gamma }
\nonumber\\  
&&= - \int\limits_{\Gamma \setminus \Gamma_{b}} k^{\alpha \beta} \phi_{, \beta}
n_{\alpha} d\Gamma - \int\limits_{\Gamma_{b}} (b - \sigma \phi) d \Gamma \; ,
\label{eq:5.67}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\Gamma_{b}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12069: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\int\limits_{\Gamma_{b}} (b - \sigma \phi) d \Gamma \approx
  \sqrt{g} \sqrt{g^{11}} (b - \sigma \phi)
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}\int\limits_{\Gamma_{b}} (b - \sigma \phi) d \Gamma &&\approx
  \sqrt{(\sqrt{g} a^{(1)}_1)^2 + (\sqrt{g} a^{(1)}_2)^2}\; (b - \sigma \phi) \nonumber \\ 
  &&= \sqrt{(a^{2}_{(2)})^2 + (a^{1}_{(2)})^2}\; (b - \sigma \phi)
\end{eqnarray}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\bm{\tau}_w$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12083: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\bm{\tau}_w = \frac{\rho c^{1/4}_\mu \kappa \sqrt{k_{\mbox{\tiny P}}}}
           {\ln (E Y^+_{\mbox{\tiny P}})}
           {\bm{u}}^{t}_{\mbox{\tiny P}}
  \label{walllaw}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$Y^+$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12085: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}Y^+ = \frac{\rho c_\mu^{1/4} Y \sqrt{k}}{\mu}
   \label{yplus}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\bm{u}}^{t}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12087: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\approx$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12091: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$Y^+_{\mbox{\tiny P}} > 11.3$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12097: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\bm{\tau}_w = 
                \frac{\mu}{Y_{\mbox{\tiny P}}} {\bm{u}}^{t}_{\mbox{\tiny P}}
  \label{linlaw}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}S^{nt} = -|\bm{\tau}_w|,
  \quad
  \bm{t} = \frac{\bm{\tau}_w}{|\bm{\tau}_w|}
  \label{rvwspan}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$S^{nt} \bm{t}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12101: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\bm{u} \cdot \bm{n} = 0
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\bm{u}}^{n}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12105: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}{\bm{u}}^{t} = \bm{u} - {\bm{u}}^{n}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\bm{u} = U^\alpha \bm{a}_{(\alpha)} \quad \mbox{and} \quad
  {\bm{u}}^{n} = (\bm{u} \cdot \bm{n}) 
  \bm{n} = 
  \frac{U^n}{g^{nn}}\bm{a}^{(n)}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{figure}[htb]
\centerline{\psfig{figure=nwallcell.eps,height=10cm}}

\label{nearwll}
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{equation}Y_{\mbox{\tiny P}} = \bm{BP} \cdot \bm{n} = 
   \frac{\bm{BP} \cdot \bm{a}_{\mbox{\tiny B}}^{(n)}}
   {{\sqrt{g^{nn}_{\mbox{\tiny B}}}}}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}P_k = \bm{\tau}_w \cdot \pderiv{\bm{u}^{t}}{Y}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}{{\overline P}_k} = \frac{1}{Y_{\mbox{\tiny P}}} \int^{Y_{\mbox{\tiny P}}}_{0} \,
      \bm{\tau}_w \cdot \pderiv{\bm{u}^{t}}{Y}\, dY = 
      \bm{\tau}_w \cdot \frac{\bm{u}^{t}_{\mbox{\tiny P}}}{Y_{\mbox{\tiny P}}}
   \label{aveprod}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\varepsilon = \frac{c_\mu^{3/4} k^{3/2}}{\kappa Y} \label{expeps}  
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\varepsilon = \rho c_\mu \frac{k^2}{\mu} \label{expeps1}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\overline {\varepsilon} = \frac{1}{Y_{\mbox{\tiny P}}} \int^{Y_{\mbox{\tiny P}}}_{0} \varepsilon dY = 
     \left\{ 
       \begin{array}{ll}
          c_\mu^{3/4} k^{3/2}_{\mbox{\tiny P}} \frac{Y^+_{\mbox{\tiny P}}}{Y_{\mbox{\tiny P}}}, 
          & Y^+_{\mbox{\tiny P}} < 11.3 \\ 
          \\ 
          c_\mu^{3/4} k^{3/2}_{\mbox{\tiny P}} \frac{ \ln(E Y^+_{\mbox{\tiny P}})}{\kappa Y_{\mbox{\tiny P}}}, 
          & Y^+_{\mbox{\tiny P}} > 11.3 
       \end{array}
     \right.
     \label{aveeps}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$h_{\mbox{\tiny R}}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12131: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$h_{\mbox{\tiny R}}/L \ll 1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12133: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\frac{|\bm{u}^{t}|}{|{\bm{u}}_\tau|} = 
  \frac{1}{\kappa} \ln (\frac{Y}{h_{\mbox{\tiny R}}}) + 
  f(h_{\mbox{\tiny R}}^+) \label{eqlibwalllaw}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\bm{\tau}}_w = \rho |{\bm{u}}_{\tau}| {\bm{u}}_{\tau}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12137: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$h_{\mbox{\tiny R}}^+$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12139: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}h_{\mbox{\tiny R}}^+ =  \frac{\rho |{\bm{u}}_\tau| 
                          h_{\mbox{\tiny R}}}{\mu}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\frac{|\bm{u}^{t}| \, c_\mu^{1/4} \sqrt{k}}
  {|{\bm{\tau}}_w|/\rho} = 
  \frac{1}{\kappa} \ln (\frac{Y}{h_{\mbox{\tiny R}}}) + 
  f(h_{\mbox{\tiny R}}^+) \quad \mbox{ and } \quad h_{\mbox{\tiny R}}^+ = 
  \frac{\rho c_\mu^{1/4} \sqrt{k} h_{\mbox{\tiny R}}}{\mu}
  \label{walllaw2}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$h_{\mbox{\tiny R}}^+ \rightarrow 0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12145: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}f(h_{\mbox{\tiny R}}^+) \rightarrow \frac{1}{\kappa} \ln 
  (h_{\mbox{\tiny R}}^+) + 5.5 \quad \mbox{ as } h_{\mbox{\tiny R}}^+
  \rightarrow 0
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$f(h_{\mbox{\tiny R}}^+)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12149: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}f(h_{\mbox{\tiny R}}^+) \rightarrow 8.5 \quad \mbox{ if } 
   h_{\mbox{\tiny R}}^+ > 30
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$5 < h_{\mbox{\tiny R}}^+ < 30$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12153: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$Y^+ = 11.3$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12157: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$h_{\mbox{\tiny R}}^+ < 11.3$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12159: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\rho \bm{u}^{t} \, c_\mu^{1/4} 
  \sqrt{k} = \frac{{\bm{\tau}}_w}{\kappa} 
  \ln (E_r \frac{Y}{h_{\mbox{\tiny R}}}), \quad 
  h_{\mbox{\tiny R}}^+ > 11.3 \label{wfunc1}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$E_r \approx 30$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12161: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}{\overline P_k} = {\bm{\tau}}_w \cdot \frac{\bm{u}^{t}}{Y} 
   \quad \mbox{and} \quad
   {\overline \varepsilon}  = c_\mu^{3/4} k^{3/2} \frac{ \ln(E_r Y/h_{\mbox{\tiny R}})}
                     {\kappa Y}
\end{equation}
}

\stepcounter{section}
\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{equation}M \frac{d \bm{V}}{dt} + S(\bm{V}, \bm{\phi}_1,..,
  \bm{\phi}_{\mbox{\tiny {N}}}) + G \bm{p} = \bm{F}
  \label{eq:6.1}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}D \bm{V} = \mbox{\bf 0} \label{eq:6.2}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}M \frac{d \bm{\phi}_i}{dt} + T_i(\bm{V},
  \bm{\phi}_1,..,\bm{\phi}_{\mbox{\tiny {N}}}) = 
  \bm{S}_{i} + \bm{S}_{dc,i},
  \quad \forall i \in \{1,2,..,\mbox{N} \}
  \label{eq:6.3}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\bm{V}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12165: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\bm{p}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12167: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\bm{\phi}_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12169: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$i^{\mbox{\scriptsize {th}}}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12171: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$T_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12183: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\bm{F}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12187: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\bm{S}_{i}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12189: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\bm{\phi}_j$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12195: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\forall j \in \{1,..,\mbox{N}\}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12197: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\bm{S}_{dc,i}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12199: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\bm{\phi}_1,..,\bm{\phi}_{\mbox{\tiny {N}}}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12207: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray}M \frac{\bm{V}^{n+1} - \bm{V}^{n}}{\Delta t} + 
  \theta S(\bm{V}^{n+1}) &+& (1- \theta) S(\bm{V}^{n}) + 
  \theta G \bm{p}^{n+1}  + (1- \theta) G \bm{p}^{n} \nonumber \\ 
  &=& \theta \bm{F}^{n+1} + (1- \theta) \bm{F}^{n}
  \label{eq:6.4} \\ 
  D \bm{V}^{n+1} &=& \mbox{\bf 0} \label{eq:6.5}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\Delta t$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12219: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\theta = 0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12221: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\theta = 1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12223: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\theta = \frac12$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12225: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$0.5 \leq \theta \leq 1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12229: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$0 \leq \theta < 0.5$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12231: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\theta < 0.5$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12233: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$S(\bm{V}^{n+1})$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12237: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$n + \theta$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12239: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray}\bm{V}^{n+\theta}&=&\theta \bm{V}^{n+1} + (1 - \theta) \bm{V}^{n}
\nonumber\\ 
\bm{p}^{n+\theta}&=&\theta \bm{p}^{n+1} + (1 -\theta) \bm{p}^{n}
\label{eq:6.6}\\ 
\bm{F}^{n+\theta}&=&\theta \bm{F}^{n+1} + (1 - \theta) \bm{F}^{n}
\nonumber
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$S(\bm{V})$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12241: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$S(\bm{V}^{n+1}) \simeq A(\bm{V}^{n}) + B(\bm{V}^{n}) \bm{V}^{n+1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12243: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray}M \frac{\bm{V}^{n+1} - \bm{V}^{n}}{\Delta t}& +& \theta B
(\bm{V}^{n}) \bm{V}^{n+1} + (1 - \theta) B (\bm{V}^{n}) \bm{V}^{n} +
\theta G \bm{p}^{n+1} + (1 - \theta) G \bm{p}^{n} \nonumber \\ 
&+&A (\bm{V}^{n}) = \theta \bm{F}^{n+1} + (1- \theta) \bm{F}^{n}
\label{eq:6.7}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{equation}M \frac{\bm{V}^{n+\theta} - \bm{V}^{n}}{\theta \Delta t} + B
(\bm{V}^{n}) \bm{V}^{n+\theta} + G \bm{p}^{n+\theta} =
\bm{F}^{n+\theta} - A(\bm{V}^{n})
\label{eq:6.8}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}D \bm{V}^{n+\theta} = \mbox{\bf 0}
\label{eq:6.9}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}\bm{V}^{n+1}&=&\frac{1}{\theta}(\bm{V}^{n+\theta} - (1 - \theta)
\bm{V}^{n})\label{eq:6.10}\\ 
\bm{p}^{n+1}&=&\frac{1}{\theta}(\bm{p}^{n+\theta} - (1 - \theta)
\bm{p}^{n})
\nonumber
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$t^{n+1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12245: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\phi_i, \forall i \in \{1,..,N\}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12255: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$V^\alpha \rightarrow p$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12257: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$k \rightarrow \varepsilon \mbox{ or } \omega$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12265: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$V^\alpha \rightarrow p \rightarrow \phi_1 \rightarrow \,..\, \rightarrow \phi_N \rightarrow k \rightarrow \varepsilon \mbox{ or } \omega$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12267: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\phi_1 \rightarrow \,..\, \rightarrow \phi_N$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12271: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{section}
\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{equation}\left[\begin{array}{cc}
S &G\\ 
D &0
\end{array}\right] \left[\begin{array}{c}
\bm{V}^{n+\theta}\\  \bm{p}^{n+\theta} \end{array}
\right] = \left[ \begin{array}{c}
\bm{F}^{n+\theta}_{1}\\  \bm{F}^{n+\theta}_{2}
\end{array}\right]\; ,
\label{eq:7.1}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\bm{F}^{n+\theta}_{2}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12277: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\bm{V}^{*}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12281: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}M \frac{\bm{V}^{*} - \bm{V}^{n}}{\theta \Delta t} + B
(\bm{V}^{n}) \bm{V}^{*} + G \bm{p}^{n} = \bm{F}^{n+\theta} -
A(\bm{V}^{n})
\label{eq:7.2}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$t = t^{n} +
\theta \Delta t$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12285: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}M \frac{\bm{V}^{n+\theta} - \bm{V}^{*}}{\theta \Delta t} = - G
(\bm{p}^{n+\theta} - \bm{p}^{n})\; ,
\label{eq:7.3}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$B(\bm{V}^{n}) (\bm{V}^{n+\theta} - \bm{V}^{*})$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12289: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}D \bm{V}^{*} = \theta \Delta t D M^{-1} G
(\bm{p}^{n+\theta} - \bm{p}^{n})\; ,
\label{eq:7.4}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\bm{p}^{n+\theta}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12291: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\bm{V}^{n+\theta}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12293: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\bm{V}^{n+\theta} = \bm{V}^{*} - \theta \Delta t M^{-1} G
(\bm{p}^{n+\theta} - \bm{p}^{n})
\label{eq:7.5}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$D M^{-1} G$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12295: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{section}
\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{displaymath}M^{n+1} u^{n+1} = f^{n+1} \; \; , \;  \; u^{n+1} = \left(\begin{array}{c}
u^{n+1}_{1}\\ u^{n+1}_{2}\\ u^{n+1}_{3}
\end{array}\right) \; ,
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}P\Delta p^{n+1} = g^{n+1} \; \; , \; \; \Delta p^{n+1} = p^{n+1} - p^{n}\; ,
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\begin{array}{cccc}
C^{n+1}_{1} &c^{n+1}_{1} &= &d^{n+1}_{1} \; ,\\ 
            &\vdots      &  &     \\ 
C^{n+1}_{k} &c^{n+1}_{k} &= &d^{n+1}_{k} \; . 
\end{array}\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$n_{i}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12305: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$x_{i}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12307: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$n_{3} = 1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12309: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$n_{1} \cdot n_{2} \cdot n_{3}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12311: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$2 \cdot n_{1} \cdot n_{2}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12313: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$3 \cdot n_{1} \cdot n_{2} \cdot n_{3}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12315: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}\left(\begin{array}{ccc}
M_{11} &M_{12} &M_{13}\\ 
M_{21} &M_{22} &M_{23}\\ 
M_{31} &M_{32} &M_{33}
\end{array}\right) \left(\begin{array}{c}
u_{1}\\ u_{2}\\ u_{3}
\end{array}\right) = \left(\begin{array}{c}
f_{1}\\ f_{2}\\ f_{3}
\end{array}
\right) \; .
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$M_{ii},\; i = 1, 2, 3$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12317: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$M_{21}^{T}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12325: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{table}[htb]
\begin{center}
\begin{tabular}{|l|c|c|}
\hline
 &2D &3D\\ 
\hline
pressure matrix &$9 \cdot n_{1} \cdot n_{2}$ &$19 \cdot n_{1} \cdot n_{2}
\cdot n_{3}$\\ 
momentum matrix  &$13\cdot2 \cdot n_{1} \cdot n_{2}$ &$51\cdot3 \cdot n_{1} \cdot
n_{2} \cdot n_{3}$\\ 
\hline
\end{tabular}
\end{center}

\label{tab81}
\end{table}
}

{\newpage
\clearpage
\samepage \begin{figure}[htb]
\vspace*{-1,5cm}
\centerline{\psfig{figure=fig81.eps,height=10cm}}

\label{fig81}
\end{figure}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\frac{13\cdot 2}{9} = 3$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12337: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\frac{51\cdot 3}{19} = 8$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12339: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{figure}[htb]
\centerline{\psfig{figure=fig82.eps,height=10cm}}
\vspace*{-1,5cm}

\label{fig82}
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{figure}[htb]
\centerline{\psfig{figure=fig83.eps,height=10cm}}
\vspace*{-1,5cm}

\label{fig83}
\end{figure}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{table}[htb]
\begin{center}
\begin{tabular}{|l|l|l|l|l|}
\hline
properties &\multicolumn{2}{|c}{bad} &\multicolumn{2}{c|}{good}\\ 
           &\multicolumn{2}{|c}{$\longleftarrow$}
&\multicolumn{2}{c|}{$\longrightarrow$}\\ 
\hline
memory     &GMRES     &GMRESR  &CGS    &LSQR\\ 
robustness &CGS       &GMRES   &GMRESR &LSQR\\ 
CPU-time   &LSQR      &GMRES   &CGS    &GMRESR\\ 
\hline
\end{tabular}
\end{center}

\label{tab82}
\end{table}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\|r_{k}\|_{2} = \|b - Ax_{k}\|_{2}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12357: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\|r_{k}\|_{2}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12359: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{eqnarray}U^{-1}L^{-1}A x&=&U^{-1}L^{-1}b\; ,\label{eq:8.1}\\ 
&&\nonumber\\ 
AU^{-1}L^{-1} y&=&b\; ,\label{eq:8.2}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{equation}L^{-1}AU^{-1} y = L^{-1}b\; .
\label{eq:8.3}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$U = diag \;
(A)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12383: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$LD^{-1}U$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12385: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$diag \; (L) = diag \; (U) = D $}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12393: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$diag \; (LD^{-1}U) = diag \; (A)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12401: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}\mbox{\rm \ rowsum } (LD^{-1}U) = \mbox{\rm \ rowsum } (A)\; \; \;
\mbox{\rm \ for every row}\; ,
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$D^{-1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12405: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$L^{-1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12407: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$U^{-1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12409: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$diag \; (L) = I$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12419: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$a_{ij}\neq 0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12427: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$(LU)_{ij} = a_{ij}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12429: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
\stepcounter{section}
\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{figure}[htb]
\centerline{\psfig{figure=fig91.eps,height=5cm}}

\label{fig91}
\end{figure}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\times$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12455: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\left[\begin{array}{c}
x \\  y
\end{array} \right] = \sum\limits_{i,j=1}^{2} \; \; \; \lambda_{i}(\xi)
\lambda_{j} (\eta) \left[ \begin{array}{c}
x \\  y \end{array} \right]_{ij} \; , 
\label{eq:9.1}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\lambda_{1} (\xi) = 1 - \xi, \; \; \lambda_{2}(\xi) = \xi$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12457: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\mbox{Scalar} \; \; (x,y) = \sum\limits_{i,j=1}^{2} \; \; \; \lambda_{i}
(\xi) \lambda_{j} (\eta) \; \mbox{scalar}\; \; (x_{ij}, y_{ij})
\label{eq:9.2}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\left[\begin{array}{c}
F_{1} (\xi, \eta)\\ 
F_{2} (\xi, \eta)
\end{array}
\right] = \sum\limits_{i,j=1}^{2}\; \; \;  \lambda_{i}(\xi) \lambda_{j}
(\eta) \left[\begin{array}{c}
x\\ y
\end{array} \right]_{ij} - \left[\begin{array}{c}
x\\ y
\end{array} \right]
\label{eq:9.3}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{displaymath}(\xi, \eta)^{1} = (1/2, 1/2)\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{equation}\left[\begin{array}{cc}
\partial F_{1}/\partial \xi &\partial F_{1}/\partial \eta\\ 
\partial F_{2}/\partial \xi &\partial F_{2}/\partial \eta
\end{array} \right]^{n} (\left[\begin{array}{c}
\xi\\  \eta \end{array}\right]^{n+1} - \left[\begin{array}{c} \xi \\  \eta
\end{array}\right]^{n} = - \left[\begin{array}{c}
F_{1}\\ F_{2}
\end{array}\right]^{n}
\; \;\; n = 1, 2, ...
\label{eq:9.4}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\|\bm{\xi}^{n+1} - \bm{\xi}^{n}\| < 0.001$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12465: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray}\bm{F} (\xi, \eta)&=&(1 - \xi)(1 - \eta) \bm{x}_{11} + \xi \eta \bm{x}_{21}
+ (1 - \xi) \eta \bm{x}_{12} - \bm{x}\nonumber\\ 
&=&\bm{x}_{11} + (\bm{x}_{11} + \bm{x}_{22} - \bm{x}_{21} - \bm{x}_{12})
\xi \eta + (\bm{x}_{21} - \bm{x}_{11})\xi + (\bm{x}_{12} - \bm{x}_{11})
\eta - \bm{x}
\label{eq:9.5}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}\frac{\partial \bm{F}}{\partial \xi}&=&(\bm{x}_{11} + \bm{x}_{22} -
\bm{x}_{21} - \bm{x}_{12}) \eta + \bm{x}_{21} - \bm{x}_{11}\label{eq:9.6}\\ 
\frac{\partial \bm{F}}{\partial \eta}&=&(\bm{x}_{11} + \bm{x}_{22} -
\bm{x}_{21} - \bm{x}_{12}) \xi + \bm{x}_{12} - \bm{x}_{11}
\label{eq:9.7}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{figure}[htb]
\centerline{\psfig{figure=fig92.eps,height=5cm}}

\label{fig92}
\end{figure}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{eqnarray}V^{1}_{(0,0)}&=&(V^{1}_{(1,0)} + V^{1}_{(-1,0)})/2\; , \label{eq:9.8}\\ 
V^{2}_{(0,0)}&=&(V^{2}_{(0,1)} + V^{2}_{(0,-1)})/2\;,
\label{eq:9.9}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{figure}[htb]
\centerline{\psfig{figure=fig93.eps}}

\label{fig93}
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\begin{array}{lll}
V^{\alpha}&=&\sqrt{g} U^{\alpha}\; ,\\ 
\bm{u}&=&U^{\alpha} \bm{a}_{\alpha}\; ,\end{array}\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{equation}\bm{u} = (V^{1} \bm{a}_{(1)} + V^{2} \bm{a}_{(2)}) /\sqrt{g}
\label{eq:9.10}
\end{equation}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\psi$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12481: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\psi = 0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12483: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\psi_{1,j+1} := \psi_{1,j} + V^{1}_{1, j}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12487: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\psi_{i+1,j+1} := \psi_{i, j+1} + V^{2}_{i,j+1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12491: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\appendix
\stepcounter{section}
{\newpage
\clearpage
\samepage \begin{equation}U^{n} = sign (\bm{a}^{(n)} \cdot \bm{n}) \sqrt{g^{nn}}\;  \bm{u} \cdot \bm{n}
\label{eq:A.1}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\left[\begin{array}{c}
U^{t_{1}}\\  U^{t_{2}}\end{array}\right] =
\left[\begin{array}{cc}g_{t_{1}t_{1}} &g_{t_{1}t_{2}}\\ 
g_{t_{2}t_{1}}
&g_{t_{2}t_{2}}\end{array}\right]^{-1}\left\{\left[\begin{array}{cc}
\alpha_{11} &\alpha_{12}\\  \alpha_{21} &\alpha_{22}\end{array} \right]^{-1}\left[\begin{array}{c}\bm{u} \cdot \bm{\tau}_{1} \\  \bm{u} \cdot
\bm{\tau}_{2}\end{array}\right] -
U^{n}\left[\begin{array}{c}g_{nt_{1}}\\ g_{nt_{2}}\end{array}\right]\right\}
\label{eq:A.2}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\alpha_{i1} = \frac{\left|\begin{array}{cc}
\bm{\tau}_{i} \cdot \bm{a}_{(t_{1})} &g_{t_{1}t_{2}}\\ 
\bm{\tau}_{i} \cdot \bm{a}_{(t_{2})}
&g_{t_{2}t_{2}}\end{array}\right|}{
\left|\begin{array}{cc}
g_{t_{1}t_{1}} &g_{t_{1}t_{2}}\\ 
g_{t_{2}t_{1}} &g_{t_{2}t_{2}}
\end{array}\right|}
\label{eq:A.3}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\alpha_{i2} = \frac{\left|\begin{array}{cc}
g_{t_{1}t_{1}} &\bm{\tau}_{i} \cdot \bm{a}_{(t_{1})}\\ 
g_{t_{2}t_{1}} &\bm{\tau}_{i} \cdot \bm{a}_{(t_{2})}\end{array}\right|}{
\left|\begin{array}{cc}
g_{t_{1}t_{1}} &g_{t_{1}t_{2}}\\ 
g_{t_{2}t_{1}} &g_{t_{2}t_{2}}
\end{array}\right|}
\label{eq:A.4}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{figure}[htb]
\centerline{\psfig{figure=figA1.eps}}

\label{figA1}
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\bm{u} \cdot \bm{n} = \bm{u} \cdot \frac{\bm{a}^{(n)}}{\|\bm{a}^{(n)}\|} =
\frac{U^{n}}{\sqrt{g^{nn}}}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}U^{n} = \sqrt{g^{nn}} \; \bm{u} \cdot \bm{n}\; .
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\bm{a}^{(n)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12647: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}U^{n} = - \sqrt{g^{nn}} \; \bm{u} \cdot \bm{n}\; .
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\Box$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12651: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\bm{\tau}_{i}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12653: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\bm{\tau}_{i} = \alpha_{i1} \bm{a}_{(t_{1})} + \alpha_{i2}\bm{a}_{(t_{2})}\;
,
\label{eq:A.5}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\underline{\tau}_{i}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12655: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{figure}[htb]
\centerline{\psfig{figure=figA2.eps}}

\label{figA2}
\end{figure}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\alpha_{ij}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12657: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\begin{array}{l}
\bm{a}_{(t_{1})} \cdot \alpha_{i2}\bm{a}_{(t_{2})} = \bm{a}_{(t_{1})} \cdot
(\bm{\tau}_{i} - \alpha_{i1} \bm{a}_{(t_{1})})\\ 
\bm{a}_{(t_{1})} \cdot \bm{a}_{(t_{1})} \alpha_{i1} + \bm{a}_{(t_{1})}
\cdot \bm{a}_{(t_{2})} \alpha_{i2} = \bm{a}_{(t_{1})} \cdot \bm{\tau}_{i}
\end{array}
\label{eq:A.6}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\begin{array}{l}
\bm{a}_{(t_{2})} \cdot \alpha_{i1} \bm{a}_{(t_{1})} = \bm{a}_{(t_{2})}
\cdot (\bm{\tau}_{i} - \alpha_{i2} \bm{a}_{(t_{2})})\\ 
\bm{a}_{(t_{2})} \cdot \bm{a}_{(t_{1})} \alpha_{i1} + \bm{a}_{(t_{2})}
\cdot \bm{a}_{(t_{2})} \alpha_{i2} = \bm{a}_{(t_{2})} \cdot \bm{\tau}_{i}\;
.
\end{array}
\label{eq:A.7}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\alpha_{i1} = \frac{\left|\begin{array}{cc}
\bm{a}_{(t_{1})} \cdot \bm{\tau}_{i} &\bm{a}_{(t_{1})} \cdot
\bm{a}_{(t_{2})}\\ 
\bm{a}_{(t_{2})} \cdot \bm{\tau}_{i} &\bm{a}_{(t_{2})} \cdot
\bm{a}_{(t_{2})}
\end{array}\right|}{\left|\begin{array}{cc}
\bm{a}_{(t_{1})} \cdot \bm{a}_{(t_{1})} &\bm{a}_{(t_{1})} \cdot
\bm{a}_{(t_{2})}\\ 
\bm{a}_{(t_{2})} \cdot \bm{a}_{(t_{1})} &\bm{a}_{(t_{2})} \cdot
\bm{a}_{(t_{2})}\end{array}\right|} = 
\frac{\left|\begin{array}{cc}
\bm{a}_{(t_{1})} \cdot \bm{\tau}_{i} &g_{t_{1}t_{2}}\\ 
\bm{a}_{(t_{2})} \cdot \bm{\tau}_{i} &g_{t_{2}t_{2}}
\end{array}\right|}{\left|\begin{array}{cc}
g_{t_{1}t_{1}} &g_{t_{1}t_{2}}\\ 
g_{t_{2}t_{1}} &g_{t_{2}t_{2}}
\end{array}\right|}
\label{eq:A.8}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\alpha_{i2} = \frac{\left|\begin{array}{cc}
\bm{a}_{(t_{1})} \cdot \bm{a}_{(t_{1})} &\bm{a}_{(t_{1})} \cdot
\bm{\tau}_{i}\\ 
\bm{a}_{(t_{2})} \cdot \bm{a}_{(t_{1})} &\bm{a}_{(t_{2})} \cdot
\bm{\tau}_{i}
\end{array}\right|}{\left|\begin{array}{cc}
\bm{a}_{(t_{1})} \cdot \bm{a}_{(t_{1})} &\bm{a}_{(t_{1})} \cdot
\bm{a}_{(t_{2})}\\ 
\bm{a}_{(t_{2})} \cdot \bm{a}_{(t_{1})} &\bm{a}_{(t_{2})} \cdot
\bm{a}_{(t_{2})}
\end{array}\right|} = \frac{\left|\begin{array}{cc}
g_{t_{1}t_{1}} &\bm{a}_{(t_{1})} \cdot \bm{\tau}_{i}\\ 
g_{t_{2}t_{1}} &\bm{a}_{(t_{2})} \cdot \bm{\tau}_{i}
\end{array}\right|}{\left|\begin{array}{cc}
g_{t_{1}t_{1}} &g_{t_{1}t_{2}}\\ 
g_{t_{2}t_{1}} &g_{t_{2}t_{2}}
\end{array}\right|}
\label{eq:A.9}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}\bm{u} \cdot \bm{\tau}_{i}&=&\bm{u} \cdot (\alpha_{i1} \bm{a}_{(t_{1})} +
\alpha_{i2} \bm{a}_{(t_{2})})\nonumber\\ 
&=&\alpha_{i1} \bm{u} \cdot g_{pt_{1}}\bm{a}^{(p)} + \alpha_{i2} \bm{u} \cdot
g_{pt_{2}}\bm{a}^{(p)}\nonumber\\ 
&=&(\alpha_{i1}g_{pt_{1}} + \alpha_{i2}g_{pt_{2}})U^{p}\; ,
\label{eq:A.10}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}\bm{u} \cdot \bm{\tau}_{1}&=&(\alpha_{11}g_{t_{1}t_{1}} +
\alpha_{12}g_{t_{1}t_{2}})U^{t_{1}} + (\alpha_{11}g_{t_{2}t_{1}} +
\alpha_{12}g_{t_{2}t_{1}})U^{t_{2}} +\nonumber\\ 
&&(\alpha_{11}g_{nt_{1}} + \alpha_{12}g_{nt_{2}})U^{n}\; ,\label{eq:A.11}\\ 
\bm{u} \cdot \bm{\tau}_{2}&=&(\alpha_{21}g_{t_{1}t_{1}} +
\alpha_{22}g_{t_{1}t_{2}})U^{t_{1}} + (\alpha_{21}g_{t_{2}t_{1}} +
\alpha_{22}g_{t_{2}t_{2}})U^{t_{2}} + \nonumber\\ 
&&(\alpha_{21}g_{nt_{1}} + \alpha_{22}g_{nt_{2}})U^{n}\; .
\label{eq:A.12}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\left[\begin{array}{cc}
\alpha_{11}g_{t_{1}t_{1}} + \alpha_{12}g_{t_{1}t_{2}}
&\alpha_{11}g_{t_{2}t_{1}} + \alpha_{12}g_{t_{2}t_{2}}\\ 
\alpha_{21}g_{t_{1}t_{1}} + \alpha_{22}g_{t_{1}t_{2}}
&\alpha_{21}g_{t_{2}t_{1}} + \alpha_{22}g_{t_{2}t_{2}}
\end{array}\right]\left[\begin{array}{c}
U^{t_{1}}\\ 
U^{t_{2}}
\end{array}\right] = \left[\begin{array}{c}
\bm{u}\cdot \bm{\tau}_{1} - (\alpha_{11}g_{nt_{1}} +
\alpha_{12}g_{nt_{2}})U^{n}\\ 
\bm{u}\cdot \bm{\tau}_{2} - (\alpha_{21}g_{nt_{1}} +
\alpha_{22}g_{nt_{2}})U^{n}\end{array}\right]
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\left[\begin{array}{cc}
\alpha_{11} &\alpha_{12}\\ 
\alpha_{21} &\alpha_{22}
\end{array}
\right]\left[\begin{array}{cc}
g_{t_{1}t_{1}} &g_{t_{1}t_{2}}\\ 
g_{t_{2}t_{1}} &g_{t_{2}t_{2}}
\end{array}\right]\left[
\begin{array}{c}
U^{t_{1}}\\ U^{t_{2}}
\end{array}\right] = \left[\begin{array}{c}
\bm{u} \cdot \bm{\tau}_{1}\\ 
\bm{u} \cdot \bm{\tau}_{2}
\end{array}\right] - U^{n}\left[\begin{array}{cc}
\alpha_{11} &\alpha_{12}\\ 
\alpha_{21} &\alpha_{22}
\end{array}
\right]\left[\begin{array}{c}
g_{nt_{1}}\\ 
g_{nt_{2}}
\end{array}\right]
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\left[\begin{array}{c}
U^{t_{1}}\\ 
U^{t_{2}}
\end{array}\right] = \left[\begin{array}{cc}
g_{t_{1}t_{1}} &g_{t_{1}t_{2}}\\ 
g_{t_{2}t_{1}} &g_{t_{2}t_{2}}
\end{array}\right]^{-1}\left\{
\left[\begin{array}{cc}
\alpha_{11} &\alpha_{12}\\ 
\alpha_{21} &\alpha_{22}
\end{array}\right]^{-1}\left[\begin{array}{c}
\bm{u} \cdot \bm{\tau}_{1}\\ 
\bm{u} \cdot \bm{\tau}_{2}
\end{array}\right] - U^{n}\left[\begin{array}{c}
g_{nt_{1}}\\ 
g_{nt_{1}}
\end{array}\right]\right\}
\footnote[1]{
This formula can be modified in the following way:
\begin{displaymath}\left[\begin{array}{c}
U^{t_{1}}\\ 
U^{t_{2}}
\end{array}\right] = \left[\begin{array}{cc}
g_{t_{2}t_{2}} &g_{t_{1}t_{2}}\\ 
-g_{t_{2}t_{1}} &g_{t_{1}t_{1}}
\end{array}\right] \left\{\left[\begin{array}{cc}
\beta_{11} &\beta_{12}\\ 
\beta_{21} &\beta_{22}
\end{array}\right]^{-1}\left[\begin{array}{c}
\underline{u} \cdot \underline{\tau_{1}}\\ 
\underline{u} \cdot \underline{\tau_{2}}
\end{array} \right] - \frac{U^{n}}{g_{t_{1}t_{1}} g_{t_{2}t_{2}} -
g_{t_{1}t_{2}}g_{t_{2}t_{1}}}\left[\begin{array}{c}
g_{nt_{1}}\\ 
g_{nt_{2}}
\end{array}
\right]\right\}
\end{displaymath}
where
\begin{displaymath}
\beta_{i1} = \left|\begin{array}{cc}
\underline{\tau_{i}} \cdot \underline{a}_{(t_{1})} &g_{t_{1}t_{2}}\\ 
\underline{\tau_{i}} \cdot \underline{a}_{(t_{2})} &g_{t_{2}t_{2}}
\end{array}\right|
\end{displaymath}
and
\begin{displaymath}
\beta_{12} = \left|\begin{array}{cc}
g_{t_{1}t_{1}} &\underline{\tau_{i}} \cdot \underline{a}_{(t_{1})}\\ 
g_{t_{2}t_{1}} &\underline{\tau_{i}} \cdot \underline{a}_{(t_{2})}
\end{array}\right| \end{displaymath}  
}
\end{displaymath}
}

\stepcounter{section}
{\newpage
\clearpage
\samepage \begin{figure}[htb]
\centerline{\psfig{figure=figA3.eps}}

\label{figA3}
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{equation}\sigma^{nn} = g^{nn} S^{nn}
\label{eq:A.13}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\left[\begin{array}{c}
\sigma_{nt_{1}}\\  \sigma_{nt_{2}} \end{array}\right] = sign (\bm{a}^{(n)}
\cdot \bm{n}) \cdot \sqrt{g^{nn}}S^{n\tau} \left[\begin{array}{c}
\alpha_{1}\\  \alpha_{2}\end{array}\right] -
\sigma^{nn}\left[\begin{array}{cc} 
g_{t_{1}t_{1}} &g_{t_{1}t_{2}}\\ 
g_{t_{2}t_{1}} &g_{t_{2}t_{2}}
\end{array}\right]^{-1}\left[\begin{array}{c}g_{nt_{1}}\\ 
g_{nt_{2}}\end{array}\right]
\label{eq:A.14}
\footnote[2]{
This formula can also be modified in the following way:
\begin{displaymath}
\left[\begin{array}{c}
\sigma^{nt_{1}}\\ 
\sigma^{nt_{2}}
\end{array}\right] = \frac{1}{g_{t_{1}t_{1}}\cdot g_{t_{2}t_{2}} - 
g_{t_{1}t_{2}}\cdot g_{t_{2}t_{1}}} 
\left\{sign (a^{(n)} \cdot \underline{n}) \sqrt{g^{nn}}
S^{n\tau}\left[\begin{array}{c}\beta_{1}\\  \beta_{2}\end{array}\right] -
\sigma^{nn} \left[\begin{array}{rr}
g_{t_{2}t_{2}} &-g_{t_{1}t_{2}}\\ 
-g_{t_{2}t_{1}} &g_{t_{1}t_{1}}
\end{array}\right]\left[\begin{array}{c}
g_{nt_{1}}\\  g_{nt_{2}}\end{array}\right]\right\}
\end{displaymath}
where
\begin{displaymath}
\beta_{1} = \left|\begin{array}{cc}
\underline{\tau}\cdot \underline{a}_{(t_{1})} &g_{t_{1}t_{2}}\\ 
\underline{\tau}\cdot \underline{a}_{(t_{2})} &g_{t_{2}t_{2}}
\end{array}\right|\end{displaymath}
and
\begin{displaymath}
\beta_{2} = \left|\begin{array}{cc}
g_{t_{1}t_{1}} &\underline{\tau} \cdot \underline{a}_{(t_{1})}\\ 
g_{t_{2}t_{1}} &\underline{\tau} \cdot \underline{a}_{(t_{2})}
\end{array}\right|\; .
\end{displaymath}
}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\alpha_{1} = \frac{\left|\begin{array}{cc}
\bm{\tau} \cdot \bm{a}_{(t_{1})} &g_{t_{1}t_{2}}\\ 
\bm{\tau}\cdot \bm{a}_{(t_{2})}  &g_{t_{2}t_{3}}
\end{array}\right|}{\left|\begin{array}{cc}
g_{t_{1}t_{1}} &g_{t_{1}t_{2}}\\ 
g_{t_{2}t_{1}} &g_{t_{2}t_{2}}
\end{array}\right|}\; ,
\label{eq:A.15}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\alpha_{2} = \frac{\left|\begin{array}{cc}
g_{t_{1}t_{1}} &\bm{\tau}\cdot \bm{a}_{(t_{1})}\\ 
g_{t_{2}t_{1}} &\bm{\tau}\cdot \bm{a}_{(t_{2})}
\end{array}\right|}{\left|\begin{array}{cc}
g_{t_{1}t_{1}} &\bm{\tau}\cdot \bm{a}_{(t_{1})}\\ 
g_{t_{1}t_{1}} &g_{t_{1}t_{2}}\\ 
g_{t_{2}t_{1}} &g_{t_{2}t_{2}}
\end{array}\right|}\; .
\label{eq:A.16}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\bm{S}_{\xi^{n} = constant} = \bar{\sigma}^{ij} (\bm{n} \cdot \bm{e}_{i})
\bm{e}_{j}
\label{eq:A.17}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\bar{\sigma}^{ij}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12689: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\underline{S}_{\xi^{n} = constant} = S^{nn} \bm{n} +
S^{n\tau}\bm{\tau}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12691: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\bar{\sigma}^{ij}(\bm{n} \cdot \bm{e}_{i}) \bm{e}_{j} = S^{nn} \bm{n} +
S^{n\tau} \bm{\tau}
\label{eq:A.18}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}(\bar{\sigma}^{ij}(\bm{n} \cdot \bm{e}_{j}) \bm{e}_{j}) \cdot g^{(n)} =
(S^{nn}\bm{n} + S^{n\tau}\bm{\tau}) \cdot \bm{a}^{(n)} = S^{nn} \bm{n}
\cdot \bm{a}^{(n)}\; .
\label{eq:A.19}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\frac{1}{\|\bm{a}^{(n)}\|} \bm{a}^{(n)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12699: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$- \frac{1}{\|a^{(n)}\|}\bm{a}^{(n)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline12703: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\bm{n} = \frac{sign (\bm{a}^{(n)} \cdot \bm{n})}{\|\bm{a}^{(n)}\|}
\bm{a}^{(n)}\; .
\label{eq:A.20}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}&&(\bar{\sigma}^{ij}(\bm{a}^{(n)} \cdot \bm{e}_{i})\bm{e}_{j}) \cdot
\bm{a}^{(n)} = S^{nn} \bm{a}^{(n)} \cdot \bm{a}^{(n)}\\ 
&&\bar{\sigma}^{ij}(\bm{a}^{(n)})_{i} (\bm{a}^{(n)})_{j} = S^{nn} g^{nn}\\ 
&&\bar{\sigma}^{ij} \frac{\partial \xi^{n}}{\partial x^{i}} \frac{\partial
\xi^{n}}{\partial x^{j}} = g^{nn} S^{nn}
\end{eqnarray*}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\sigma^{nn} = g^{nn} S^{nn}\; .
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{equation}\bm{\tau} = \alpha_{1}\bm{a}_{(t_{1})} + \alpha_{2} \bm{a}_{(t_{2})}
\label{eq:A.21}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\alpha_{1} = \frac{\left|\begin{array}{cc}
\bm{\tau} \cdot \bm{a}_{(t_{1})} &g_{t_{1}t_{2}}\\ 
\bm{\tau} \cdot \bm{a}_{(t_{2})} &g_{t_{2}t_{1}}
\end{array}\right|}{\left|\begin{array}{cc}
g_{t_{1}t_{1}} &g_{t_{1}t_{2}}\\ 
g_{t_{2}t_{1}} &g_{t_{2}t_{2}}
\end{array}\right|}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\alpha_{2} = \frac{\left|\begin{array}{cc}
g_{t_{1}t_{1}} &\bm{\tau} \cdot \bm{a}_{(t_{1})}\\ 
g_{t_{2}t_{1}} &\bm{\tau} \cdot \bm{a}_{(t_{2})}
\end{array}\right|}{\left|\begin{array}{cc}
g_{t_{1}t_{1}} &g_{t_{1}t_{2}}\\ 
g_{t_{2}t_{1}} &g_{t_{2}t_{2}}
\end{array}\right|}\; .
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}&&(\bar{\sigma}^{ij}(\frac{sign(\bm{a}^{(n)} \cdot
\bm{n})}{\|\bm{a}^{(n)}\|} \bm{a}^{(n)} \cdot \bm{e}_{i})\bm{e}_{j}) \cdot
\bm{a}_{t_{l})} = S^{n\tau}(\alpha_{1}\bm{a}_{(t_{1})} +
\alpha_{2}\bm{a}_{(t_{2})}) \cdot \bm{a}_{(t_{l})}\\ 
&&sign(\bm{a}^{(n)} \cdot \bm{n}) \bar{\sigma}^{ij}
(\bm{a}^{(n)})_{i}(\bm{a}_{(t_{l})})_{j} = \| \bm{a}^{(n)}\|
S^{n\tau}(\alpha_{1} \bm{a}_{(t_{1})} \cdot \bm{a}_{(t_{l})} +
\alpha_{2}\bm{a}_{(t_{2})} \cdot \bm{a}_{t_{l})})\\ 
&&sign(\bm{a}^{(n)} \cdot \bm{n}) \bar{\sigma}^{ij}(\bm{a}^{(n)})_{i}
g_{pt_{l}}(\bm{a}^{(p)})_{j} = \sqrt{g^{nn}}S^{n\tau}
(\alpha_{1}g_{t_{1}t_{l}} + \alpha_{2}g_{t_{2}t_{l}})\\ 
&&sign(\bm{a}^{(n)}\cdot \bm{n}) g_{pt_{l}}
\bar{\sigma}^{ij}\frac{\partial \xi^{n}}{\partial x^{i}} \frac{\partial
\xi^{p}}{\partial x^{j}} = \sqrt{g^{nn}}
S^{n\tau}(\alpha_{1}g_{t_{1}t_{1}} + \alpha_{2}g_{t_{2}t_{l}})
\end{eqnarray*}
}

{\newpage
\clearpage
\samepage \begin{equation}sign(\bm{a}^{(n)}\cdot \bm{n}) g_{pt_{l}}\sigma^{np} = \sqrt{g^{nn}}
S^{n\tau} (\alpha_{1}g_{t_{1}t_{l}} + \alpha_{2}g_{t_{2}t_{l}})\; ,
\label{eq:A.22}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{displaymath}sign(\bm{a}^{(n)} \cdot \bm{n}) \{ g_{t_{1}t_{1}}\sigma^{nt_{1}} +
g_{t_{2}t_{1}} \sigma^{nt_{2}} + g_{nt_{1}} \sigma^{nn}\} =
\sqrt{g^{nn}}S^{n\tau} (\alpha_{1}g_{t_{1}t_{1}} + \alpha_{2}g_{t_{2}t_{1}})
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}sign(\bm{a}^{(n)} \cdot \bm{n}) \{g_{t_{1}t_{2}} \sigma^{nt_{1}} +
g_{t_{2}t_{2}} \sigma^{nt_{2}} + g_{nt_{2}}\sigma^{nn}\} =
\sqrt{g^{nn}}S^{n\tau}(\alpha_{1}g_{t_{1}t_{2}} + \alpha_{2}g{t_{2}t_{2}})
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\left[
\begin{array}{cc}
g_{t_{1}t_{1}} &g_{t_{2}t_{1}}\\ 
g_{t_{1}t_{2}} &g_{t_{2}t_{2}}
\end{array}\right]\left[\begin{array}{c}
\sigma^{nt_{1}}\\ 
\sigma^{nt_{2}}
\end{array}\right] + \sigma^{nn}\left[\begin{array}{c}
g_{nt_{1}}\\ 
g_{nt_{2}}
\end{array}\right] = sign(\bm{a}^{(n)} \cdot \bm{n})
\sqrt{g^{nn}}S^{n\tau}\left[\begin{array}{cc}
g_{t_{1}t_{1}} &g_{t_{2}t_{1}}\\ 
g_{t_{1}t_{2}} &g_{t_{2}t_{2}}
\end{array}\right]\left[\begin{array}{c}
\alpha_{1}\\ 
\alpha_{2}
\end{array}\right]
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\left[\begin{array}{c}
\sigma^{nt_{1}}\\ 
\sigma^{nt_{2}}
\end{array}\right] = sign (\bm{a}^{(n)} \cdot \bm{n})
\sqrt{g^{nn}}S^{n\tau} \left[\begin{array}{c}\alpha_{1} \\  \alpha_{2}
\end{array}\right] - \sigma^{nn} \left[\begin{array}{cc} g_{t_{1}t_{1}}
&g_{t_{2}t_{1}} \\ 
g_{t_{1}t_{2}} &g_{t_{2}t_{2}}\end{array}\right]^{-1}\left[\begin{array}{c}
g_{nt_{1}}\\ 
g_{nt_{2}}\end{array}\right]
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{equation}\left[\begin{array}{c}
\sigma^{nt_{1}}\\ 
\sigma^{nt_{2}}
\end{array}\right] = sign(\bm{a}^{(n)} \cdot \bm{n})
\sqrt{g^{nn}}S^{n\tau}\left[\begin{array}{c}
\alpha_{1}\\ 
\alpha_{2}
\end{array}\right] - \sigma^{nn}\left[\begin{array}{cc}
g_{t_{1}t_{1}} &g_{t_{1}t_{2}}\\ 
g_{t_{2}t_{1}} &g_{t_{2}t_{2}}
\end{array}\right]^{-1}\left[\begin{array}{c}
g_{nt_{1}}\\ 
g_{nt_{2}}
\end{array}\right]\; .
\label{eq:A.23}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\left[\begin{array}{c}
\sigma^{nt_{1}}\\ 
\sigma^{nt_{2}}
\end{array}\right] = \frac{1}{g_{t_{1}t_{1}}g_{t_{2}t_{1}} -
g_{t_{1}t_{2}}g_{t_{2}t_{1}}} \left\{ sign(\bm{a}^{(n)} \cdot \bm{n})
\sqrt{g^{nn}}S^{n\tau}\left[\begin{array}{c}
\beta_{1}\\ 
\beta_{2}
\end{array}\right] - \sigma^{nn}\left[\begin{array}{rr}
g_{t_{2}t_{2}}&-g_{t_{1}t_{2}}\\ 
-g_{t_{2}t_{1}} &g_{t_{1}t_{1}}
\end{array}\right]\left[\begin{array}{c}
g_{nt_{1}}\\ 
g_{nt_{2}}
\end{array}\right]\right\}
\label{eq:A.24}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\beta_{1} = \left|\begin{array}{cc}
\bm{\tau} \cdot \bm{a}_{(t_{1})} &g_{t_{1}t_{2}}\\ 
\bm{\tau} \cdot \bm{a}_{(t_{2})} &g_{t_{2}t_{2}}
\end{array}\right|
\label{eq:A.25}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\beta_{2} = \left|\begin{array}{cc}
g_{t_{1}t_{1}} &\bm{\tau} \cdot \bm{a}_{(t_{1})}\\ 
g_{t_{2}t_{1}} &\bm{\tau} \cdot \bm{a}_{(t_{2})}
\end{array}\right|\; .
\label{eq:A.26}
\end{equation}
}


\end{document}
