The Stokes equations for chyme flowing down the gut

Equip the gut with a coordinate system with coordinates $(x,y,z)$ and let $x$ be the direction pointing along the length of the gut. Write $U=(u,v,w)$ for the velocity of the chyme. Here $u$ is the component of the velocity in the direction of the $x$-axis and $u$ and $v$ are the velocity components in the other two directions. Write $\mu $ for the viscosity of the chyme (a measure of its stickiness) and $P$ for the pressure forcing the chyme to flow down the intestine. The Stokes equations relate velocity, pressure and viscosity as follows:

  \[ \mu \nabla ^2 U=\nabla P, \nabla .U=0. \]    

Here $\mu \nabla ^2 U$ is the vector $\mu (\nabla ^2 u, \nabla ^2 v, \nabla ^2w)$, where $\nabla ^2 u = \frac{\partial ^2 u}{\partial x^2}+\frac{\partial ^2 u}{\partial y^2}+\frac{\partial ^2 u}{\partial z^2}$ (similarly for $\nabla ^2 v$ and $\nabla ^2 w$).

The expression $\nabla P$ stands for the vector

  \[ \nabla P = \left( \frac{\partial P}{\partial x}, \frac{\partial P}{\partial y}, \frac{\partial P}{\partial z} \right) \]    

and $\nabla .U$ is defined as

  \[ \nabla .U = \left( \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+ \frac{\partial w}{\partial z}\right). \]    

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