Difference between revisions of "Governing equations"

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(New page: In Cartesian coordinates, the Stokes equations are given by, <math>\begin{align} & x:\quad \frac{\partial \tau _{xx}}{\partial x}-\frac{\partial P}{\partial x}+\frac{\partial \tau _{xy...)
 
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<math>\tau _{ij}=\sigma _{ij}+P\delta _{ij}</math>,
+
<big><math>\tau _{ij}=\sigma _{ij}+P\delta _{ij}</math></big>,
  
  
 
where <big>&sigma;</big> is the full stress tensor.
 
where <big>&sigma;</big> is the full stress tensor.

Revision as of 11:53, 30 July 2009

In Cartesian coordinates, the Stokes equations are given by,


\begin{align}
  & x:\quad \frac{\partial \tau _{xx}}{\partial x}-\frac{\partial P}{\partial x}+\frac{\partial \tau _{xy}}{\partial y}+\frac{\partial \tau _{xz}}{\partial z}=0 \\ 
 & y:\quad \frac{\partial \tau _{yy}}{\partial y}-\frac{\partial P}{\partial y}+\frac{\partial \tau _{xy}}{\partial x}+\frac{\partial \tau _{yz}}{\partial z}=0 \\ 
 & z:\quad \frac{\partial \tau _{zz}}{\partial z}-\frac{\partial P}{\partial z}+\frac{\partial \tau _{zy}}{\partial y}+\frac{\partial \tau _{xz}}{\partial x}=\rho g \\ 
\end{align},


where P is the pressure and τ is the deviatoric stress tensor. The latter is given by


\tau _{ij}=\sigma _{ij}+P\delta _{ij},


where σ is the full stress tensor.