--- trunk/doc/user/stokesflow.tex 2010/10/21 00:37:42 3291 +++ trunk/doc/user/stokesflow.tex 2010/10/22 01:56:02 3295 @@ -28,35 +28,35 @@ tensor notation (compact notation): % $$--(\eta(v\hackscore{i,j} + v\hackscore{j,i})),\hackscore{j} - p,\hackscore{i} = f\hackscore{i}, +-(\eta(v_{i,j} + v_{j,i})),_{j} - p,_{i} = f_{i}, \label{GENERAL NAVIER STOKES COM}$$ % with the incompressibility condition % $$--v\hackscore{i,i} = 0. +-v_{i,i} = 0. \label{INCOMPRESSIBILITY COM}$$ % -The subscript comma $i$ denotes the derivative of the function with respect to $x\hackscore{i}$. -%A linear relationship between the deviatoric stress $\sigma^{'}\hackscore{ij}$ and the stretching $D\hackscore{ij} = \frac{1}{2}(v\hackscore{i,j} + v\hackscore{j,i})$ is defined as \cite{GROSS2006}: +The subscript comma $i$ denotes the derivative of the function with respect to $x_{i}$. +%A linear relationship between the deviatoric stress $\sigma^{'}_{ij}$ and the stretching $D_{ij} = \frac{1}{2}(v_{i,j} + v_{j,i})$ is defined as \cite{GROSS2006}: % %$$-%\sigma^{'}\hackscore{ij} = 2\eta D^{'}\hackscore{ij}, +%\sigma^{'}_{ij} = 2\eta D^{'}_{ij}, %\label{STRESS} %$$ % -%where the deviatoric stretching $D^{'}\hackscore{ij}$ is defined as +%where the deviatoric stretching $D^{'}_{ij}$ is defined as % %$$-%D^{'}\hackscore{ij} = D^{'}\hackscore{ij} - \frac{1}{3}D\hackscore{kk}\delta\hackscore{ij}. +%D^{'}_{ij} = D^{'}_{ij} - \frac{1}{3}D_{kk}\delta_{ij}. %\label{DEVIATORIC STRETCHING} %$$ % -%where $\delta\hackscore{ij}$ is the Kronecker $\delta$-symbol, which is a matrix with ones for its diagonal entries ($i = j$) and zeros for the remaining entries ($i \neq j$). +%where $\delta_{ij}$ is the Kronecker $\delta$-symbol, which is a matrix with ones for its diagonal entries ($i = j$) and zeros for the remaining entries ($i \neq j$). The body force $f$ in \eqn{GENERAL NAVIER STOKES COM} is the gravity acting in -the $x\hackscore{3}$ direction and is given as $f=-g\rho\delta\hackscore{i3}$. +the $x_{3}$ direction and is given as $f=-g\rho\delta_{i3}$. The Stokes equation is a saddle point problem, and can be solved using a Uzawa scheme. A class called \class{StokesProblemCartesian} in \escript can be used to solve for velocity and pressure. A more detailed discussion of the class can be @@ -101,10 +101,10 @@ The gravitational force is calculated based on the fluid density and the acceleration due to gravity. The boundary conditions are set for a slip condition at the base and the left -face of the domain. At the base fluid movement in the $x\hackscore{0}$-direction -is free, but fixed in the $x\hackscore{1}$-direction, and similarly at the left -face fluid movement in the $x\hackscore{1}$-direction is free but fixed in -the $x\hackscore{0}$-direction. +face of the domain. At the base fluid movement in the $x_{0}$-direction +is free, but fixed in the $x_{1}$-direction, and similarly at the left +face fluid movement in the $x_{1}$-direction is free but fixed in +the $x_{0}$-direction. An instance of the \class{StokesProblemCartesian} class is defined for the given computational mesh, and the solver tolerance set. Inside the \code{while} loop, the boundary conditions, viscosity and body