# Diff of /trunk/doc/user/stokesflow.tex

revision 3291 by caltinay, Thu Oct 21 00:37:42 2010 UTC revision 3295 by jfenwick, Fri Oct 22 01:56:02 2010 UTC
# Line 28  Alternatively, the Stokes equations can Line 28  Alternatively, the Stokes equations can
28  tensor notation (compact notation):  tensor notation (compact notation):
29  %  %
30
31  -(\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},
32  \label{GENERAL NAVIER STOKES COM}  \label{GENERAL NAVIER STOKES COM}
33
34  %  %
35  with the incompressibility condition  with the incompressibility condition
36  %  %
37
38  -v\hackscore{i,i} = 0.  -v_{i,i} = 0.
39  \label{INCOMPRESSIBILITY COM}  \label{INCOMPRESSIBILITY COM}
40
41  %  %
42  The subscript comma $i$ denotes the derivative of the function with respect to $x\hackscore{i}$.  The subscript comma $i$ denotes the derivative of the function with respect to $x_{i}$.
43  %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}:  %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}:
44  %  %
45  %  %
46  %\sigma^{'}\hackscore{ij} = 2\eta D^{'}\hackscore{ij},  %\sigma^{'}_{ij} = 2\eta D^{'}_{ij},
47  %\label{STRESS}  %\label{STRESS}
48  %  %
49  %  %
50  %where the deviatoric stretching $D^{'}\hackscore{ij}$ is defined as  %where the deviatoric stretching $D^{'}_{ij}$ is defined as
51  %  %
52  %  %
53  %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}.
54  %\label{DEVIATORIC STRETCHING}  %\label{DEVIATORIC STRETCHING}
55  %  %
56  %  %
57  %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 ($i \neq j$).
58  The body force $f$ in \eqn{GENERAL NAVIER STOKES COM} is the gravity acting in  The body force $f$ in \eqn{GENERAL NAVIER STOKES COM} is the gravity acting in
59  the $x\hackscore{3}$ direction and is given as $f=-g\rho\delta\hackscore{i3}$.  the $x_{3}$ direction and is given as $f=-g\rho\delta_{i3}$.
60  The Stokes equation is a saddle point problem, and can be solved using a Uzawa scheme.  The Stokes equation is a saddle point problem, and can be solved using a Uzawa scheme.
61  A class called \class{StokesProblemCartesian} in \escript can be used to solve  A class called \class{StokesProblemCartesian} in \escript can be used to solve
62  for velocity and pressure. A more detailed discussion of the class can be  for velocity and pressure. A more detailed discussion of the class can be
# Line 101  expressed by Line 101  expressed by
101  The gravitational force is calculated based on the fluid density and the  The gravitational force is calculated based on the fluid density and the
102  acceleration due to gravity.  acceleration due to gravity.
103  The boundary conditions are set for a slip condition at the base and the left  The boundary conditions are set for a slip condition at the base and the left
104  face of the domain. At the base fluid movement in the $x\hackscore{0}$-direction  face of the domain. At the base fluid movement in the $x_{0}$-direction
105  is free, but fixed in the $x\hackscore{1}$-direction, and similarly at the left  is free, but fixed in the $x_{1}$-direction, and similarly at the left
106  face fluid movement in the $x\hackscore{1}$-direction is free  but fixed in  face fluid movement in the $x_{1}$-direction is free  but fixed in
107  the $x\hackscore{0}$-direction.  the $x_{0}$-direction.
108  An instance of the \class{StokesProblemCartesian} class is defined for the  An instance of the \class{StokesProblemCartesian} class is defined for the
109  given computational mesh, and the solver tolerance set.  given computational mesh, and the solver tolerance set.
110  Inside the \code{while} loop, the boundary conditions, viscosity and body  Inside the \code{while} loop, the boundary conditions, viscosity and body

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