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\section{Stokes Flow} 
\section{Stokes Flow} 
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\label{STOKES FLOW CHAP} 
\label{STOKES FLOW CHAP} 
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In this section we will simulate the flow of fluid under the influence of gravity. The StokesProblemCartesian class will be used to calculate the velocity and pressure of the fluid. 
In this section we will look at Computational Fluid Dynamics (CFD) to simulate the flow of fluid under the influence of gravity. The StokesProblemCartesian class will be used to calculate the velocity and pressure of the fluid. 
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The fluid dynamics is governed by the Stokes equation. In geophysical problems the velocity of fluids are low; that is, the inertial forces are small compared with the viscous forces, therefore the inertial terms in the NavierStokes equations can be ignored. For a body force, $f$, the governing equations are given by: 
The fluid dynamics is governed by the Stokes equation. In geophysical problems the velocity of fluids are low; that is, the inertial forces are small compared with the viscous forces, therefore the inertial terms in the NavierStokes equations can be ignored. For a body force, $f$, the governing equations are given by: 
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\begin{equation} 
\begin{equation} 
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where $\delta t$, $v$, and $h$ are the timestep, velocity, and the width of an element in the mesh, respectively. The velocity $v$ may be chosen as the maximum velocity in the domain. In this problem the timestep size was calculated for a Courant number of 0.4. 
where $\delta t$, $v$, and $h$ are the timestep, velocity, and the width of an element in the mesh, respectively. The velocity $v$ may be chosen as the maximum velocity in the domain. In this problem the timestep size was calculated for a Courant number of 0.4. 
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The following PYTHON script is the setup for the simulation. It starts off by importing the classes, such as the StokesProblemCartesian class, for solving the Stokes equation and the incompressibility condition for velocity and pressure. Physical constants are define for the viscosity and density of the fluid, along with the acceleration due to gravity. Solver settings are set for the maximum iterations and tolerance for the solver; the default solver used is PCG. The mesh is defined as a rectangle, to represent the body of fluid. The gravitational force is calculated base on the fluid density, and the boundary conditions are set for a slip condition at the base of the mesh. An instance of the StokesProblemCartesian is defined for the given computational mesh, and the solver tolerance set. Inside the while loop, the boundary conditions, viscosity and body force are initialized. The Stokes equation is then solved for velocity and pressure. The timestep size is calculated base on the Courant condition, to ensure stable solutions. The nodes in the mesh are then displaced based on the current velocity and timestep size, to move the body of fluid. The output of the simulation of velocity and pressure is then save to file for visualization. 
The following PYTHON script is the setup for the Stokes flow simulation, and is available in the example directory as 'fluid.py'. It starts off by importing the classes, such as the StokesProblemCartesian class, for solving the Stokes equation and the incompressibility condition for velocity and pressure. Physical constants are defined for the viscosity and density of the fluid, along with the acceleration due to gravity. Solver settings are set for the maximum iterations and tolerance; the default solver used is PCG. The mesh is defined as a rectangle, to represent the body of fluid. The gravitational force is calculated base on the fluid density and the acceleration due to gravity. The boundary conditions are set for a slip condition at the base of the mesh; fluid movement in the xdirection is free, but fixed in the ydirection. An instance of the StokesProblemCartesian is defined for the given computational mesh, and the solver tolerance set. Inside the while loop, the boundary conditions, viscosity and body force are initialized. The Stokes equation is then solved for velocity and pressure. The timestep size is calculated base on the Courant condition, to ensure stable solutions. The nodes in the mesh are then displaced based on the current velocity and timestep size, to move the body of fluid. The output for the simulation of velocity and pressure is then save to file for visualization. 
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\begin{python} 
\begin{python} 
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from esys.escript import * 
from esys.escript import * 
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print "Time = %s seconds"%( time ) 
print "Time = %s seconds"%( time ) 
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solution.initialize(fixed_u_mask=boundary_cond,eta=eta,f=Y) 
solution.initialize(fixed_u_mask=boundary_cond,eta=eta,f=Y) 
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velocity,pressure=solution.solve(velocity,pressure,max_iter=max_iter,verbose=verbose,useUzawa=useUzawa) 
velocity,pressure=solution.solve(velocity,pressure,max_iter=max_iter, \ 
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verbose=verbose,useUzawa=useUzawa) 
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print "Max velocity =", Lsup(velocity), "m/s" 
print "Max velocity =", Lsup(velocity), "m/s" 
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t = t+1.0 
t = t+1.0 
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\end{python} 
\end{python} 
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The results from the simulation can be viewed with \mayavi, by executing the following command: 
The results from the simulation can be viewed with \mayavi, by executing the following command: 
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\begin{python} 
\begin{python} 
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mayavi d vel.00.vtu m SurfaceMap 
mayavi d vel.00.vtu m SurfaceMap 
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\end{python} 
\end{python} 
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Figures \ref{FLUID OUTPUT1} and \ref{FLUID OUTPUT2} shows the simulation output. Velocity vectors and a colour map for pressure are shown. As the time progresses the body of fluid falls under gravity. 
Colour coded scalar maps and velocity flow fields can be viewed by selecting them in the menu. The timesteps can be swept through to view a movie of the simulation. 
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Figures \ref{FLUID OUTPUT1} and \ref{FLUID OUTPUT2} shows the simulation output. Velocity vectors and a colour map for pressure are shown. As the time progresses the body of fluid falls under the influence of gravity. 
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\begin{figure} 
\begin{figure} 
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\center 
\center 
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\subfigure[t=1]{\label{FLOW OUTPUT 01}\includegraphics[scale=0.25]{figures/stokesfluidt01.eps}} 
\subfigure[t=1]{\label{FLOW OUTPUT 01}\includegraphics[scale=0.25]{figures/stokesfluidt01.eps}} 
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\caption{Simulation output for Stokes flow. Fluid body starts off as a rectangular shape, then progresses downwards under the influence of gravity. Color coded distribution represents the scalar values for pressure. Velocity vectors are displayed at each node in the mesh to show the flow field. Computational mesh used was 20$\times$20 elements.} 
\caption{Simulation output for Stokes flow. Fluid body starts off as a rectangular shape, then progresses downwards under the influence of gravity. Color coded distribution represents the scalar values for pressure. Velocity vectors are displayed at each node in the mesh to show the flow field. Computational mesh used was 20$\times$20 elements.} 
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\label{FLUID OUTPUT1} 
\label{FLUID OUTPUT1} 
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\end{figure} 
\end{figure} 
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\begin{figure} 
\begin{figure} 
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\center 
\center 
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\subfigure[t=40]{\label{FLOW OUTPUT 30}\includegraphics[scale=0.25]{figures/stokesfluidt30.eps}} 
\subfigure[t=40]{\label{FLOW OUTPUT 30}\includegraphics[scale=0.25]{figures/stokesfluidt30.eps}} 
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\caption{Simulation output for Stokes flow.} 
\caption{Simulation output for Stokes flow.} 
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\label{FLUID OUTPUT2} 
\label{FLUID OUTPUT2} 
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\end{figure} 
\end{figure} 
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The view used here to track the fluid is the Lagrangian view, since the mesh moves with the fluid. One of the disadvantages of using the Lagrangian view is that the elements in the mesh become severely distorted after a period of time and introduce solver errors. To get around this limitation the Level Set Method is used, with the Eulerian point of view for a fixed mesh. The Level Set Method is discussed in Section \ref{LEVELSET CHAP}. 