Stokes problem

In fluid dynamics, Stokes problem also known as Stokes second problem is a problem of determining the flow created by an oscillating plate, named after 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 is oscillating with a velocity $U\cos \omega t$ in the $x$ direction, which is located at $y=0$ in an infinite domain of fluid, where $\omega$ is the frequency of the oscillations. 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

$u(0,t)=U\cos \omega t,\quad u(\infty ,t)=0,$

the second condition is due to the fact that the motion at $y=0$ is not felt at infinity. The flow is only due to the motion of the plate, there is no imposed pressure gradient.

Solution

The initial condition is not required because of periodicity. Since both the equation and the boundary conditions are linear, the velocity can be written as the real part of some complex function

$u=U\Re \left[e^{i\omega t}f(y)\right]$

because $\cos \omega t=\Re e^{i\omega t}$.

Substituting this the partial differential equation, reduces it to ordinary differential equation

$f''-{\frac {i\omega }{\nu }}f=0$

with boundary conditions

$f(0)=1,\quad f(\infty )=0$

The solution to the above problem is

$f(y)=\exp \left[-{\frac {1+i}{\sqrt {2}}}{\sqrt {\frac {\omega }{2\nu }}}y\right]$

$u(y,t)=Ue^{-{\sqrt {\frac {\omega }{2\nu }}}y}\cos \left(\omega t-{\sqrt {\frac {\omega }{2\nu }}}y\right)$

The disturbance created by the oscillating plate is traveled as the transverse wave through the fluid, but it is highly damped by the exponential factor. The depth of penetration $\delta ={\sqrt {2\nu /\omega }}$ of this wave increases with the frequency of the oscillation, but increases with the kinematic viscosity of the fluid.

The force per unit area exerted on the plate by the fluid is

$F=\mu \left({\frac {\partial u}{\partial y}}\right)_{y=0}={\sqrt {\rho \omega \mu }}U\cos \left(\omega t-{\frac {\pi }{4}}\right)$

There is a phase shift between the oscillation of the plate and the force created.

See also

• Rayleigh problem