The permanent laminar flow of an incompressible viscous fluid in the space between two parallel plates can be described by a linear ODE for
is the dynamic viscosity of the fluid and
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 plate velocity.
Snapshot 1: pressure induced parabolic flow (Poiseuille solution)
Snapshot 2: linear flow induced by the translation of the upper plate (without pressure gradient)
Snapshot 3: general solution