Rayleigh problem

In fluid dynamics, Rayleigh problem also known as Stokes first problem is a problem of determining the flow created by a sudden movement of a plane from rest, named after Lord Rayleigh and Sir George Stokes. This is considered as one of the simplest unsteady problem that have exact solution for the Navier-Stokes equations.

Flow description

Consider an infinitely long plate which suddenly made to move with constant velocity $U$ in the $x$ direction, which is located at $y=0$ in an infinite domain of fluid, which is at rest initially everywhere. The incompressible Navier-Stokes equations reduce to

${\frac {\partial u}{\partial t}}=\nu {\frac {\partial ^{2}u}{\partial y^{2}}}$

where $\nu$ is the kinematic viscosity. The initial and the no-slip condition on the wall are

$u(y,0)=0,\quad u(0,t>0)=U,\quad u(\infty ,t>0)=0,$

the last condition is due to the fact that the motion at $y=0$ is not felt at infinity.