Double-Sided 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(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:

 (lower plate velocity),

 (upper plate velocity).

This problem has an analytic solution:

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

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