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.

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

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

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