Couette Flow
Prepared by W. C. Guttner
The permanent laminar flow of an incompressible viscous fluid in the space between two parallel plates can be described by a linear ODE for u = u(y):  (d2u)/(dy2) = 1/μ (dp)/(dx),
where μ is the dynamic viscosity of the fluid and dp/dx  is the pressure gradient. The boundary conditions are:

u(0) = 0 (lower plate velocity),

u(h) = U0 (upper plate velocity).

This problem has an analytic solution:

u(y) = (U0/2)(1 + y/h) - (dp/dx)(h2/2μ)(1 - (y/h)2)

that varies as a function of the pressure gradient and the upper plate velocity.

Download the CDF file to view the simulation using the free Wolfram CDF player.