Rayleigh problem

Lore & Background

The problem describes an infinitely long plate at y=0 that is suddenly moved with constant velocity U in the x direction, in an infinite domain of fluid initially at rest. The incompressible Navier-Stokes equations reduce to a one-dimensional diffusion equation for the velocity u, with no imposed pressure gradient. The solution is self-similar and expressed in terms of the complementary error function, yielding u = U erfc(y / sqrt(4νt)).

Reader's Guide

The Rayleigh problem is significant because it provides an exact solution to the Navier-Stokes equations for an unsteady flow, serving as a benchmark for numerical methods and theoretical studies. The force per unit area on the plate is given by F = -ρ sqrt(νU²/(πt)). The problem can be extended to arbitrary wall motion, where the solution involves an integral of the wall velocity function. Additionally, a cylindrical version exists for a rotating cylinder, with the velocity expressed using modified Bessel functions, approaching a rigid vortex as time goes to infinity.

Did You Know?

• The problem is also known as Stokes first problem.
• The impulse movement of a semi-infinite plate was studied by Keith Stewartson.
• The self-similar variable introduced is η = y / sqrt(νt).
• The force per unit area on the plate is F = -ρ sqrt(νU²/(πt)).