Difference between revisions of "Governing equations"

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where <big>&sigma;</big> is the full stress tensor.
where <big>&sigma;</big> is the full stress tensor.
===Constitutive Relation===

Revision as of 12:54, 30 July 2009

Stokes Equations

In Cartesian coordinates, the Stokes equations are given by,

  & 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 \\ 

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.

Constitutive Relation