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: $$\mu {\mathrm{\nabla }}^{2}U=\mathrm{\nabla }P,\mathrm{\nabla }.U=0.$$ Here $\mu {\mathrm{\nabla }}^{2}U$ is the vector $\mu ({\mathrm{\nabla }}^{2}u,{\mathrm{\nabla }}^{2}v,{\mathrm{\nabla }}^{2}w)$, where ${\mathrm{\nabla }}^{2}u=\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}+\frac{{\partial }^{2}u}{\partial z^2}$ (similarly for ${\mathrm{\nabla }}^{2}v$ and ${\mathrm{\nabla }}^{2}w$). The expression $\mathrm{\nabla }P$ stands for the vector $$\mathrm{\nabla }P=(\frac{\partial P}{\partial x},\frac{\partial P}{\partial y},\frac{\partial P}{\partial z})$$ and $\mathrm{\nabla }.U$ is defined as $$\mathrm{\nabla }.U=(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}).$$