In fluid dynamics, Stokes problem also known as Stokes second problem or sometimes referred to as Stokes boundary layer or Oscillating boundary layer is a problem of determining the flow created by an oscillating solid surface, named after Sir George Stokes. This is considered one of the simplest unsteady problems that has an exact solution for the Navier–Stokes equations.[1] [2] In turbulent flow, this is still named a Stokes boundary layer, but now one has to rely on experiments, numerical simulations or approximate methods in order to obtain useful information on the flow.
Consider an infinitely long plate which is oscillating with a velocity
U\cos\omegat
x
y=0
\omega
| \partialu | |
| \partialt |
=\nu
| \partial2u | |
| \partialy2 |
where
\nu
u(0,t)=U\cos\omegat, u(infty,t)=0,
and the second boundary condition is due to the fact that the motion at
y=0
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[ei\omegaf(y)\right]
because
\cos\omegat=\Reei\omega
Substituting this into the partial differential equation reduces it to ordinary differential equation
f''-
| i\omega | |
| \nu |
f=0
with boundary conditions
f(0)=1, f(infty)=0
The solution to the above problem is
f(y)=\exp\left[-
| 1+i | |
| \sqrt{2 |
u(y,t)=U
| ||||||
| e |
y}\cos\left(\omegat-\sqrt{
| \omega | |
| 2\nu |
The disturbance created by the oscillating plate travels 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}
The force per unit area exerted on the plate by the fluid is
F=\mu\left(
| \partialu | |
| \partialy |
\right)y=0=\sqrt{\rho\omega\mu}U\cos\left(\omegat-
| \pi | |
| 4 |
\right)
There is a phase shift between the oscillation of the plate and the force created.
An important observation from Stokes' solution for the oscillating Stokes flow is that vorticity oscillations are confined to a thin boundary layer and damp exponentially when moving away from the wall.[7] This observation is also valid for the case of a turbulent boundary layer. Outside the Stokes boundary layer – which is often the bulk of the fluid volume – the vorticity oscillations may be neglected. To good approximation, the flow velocity oscillations are irrotational outside the boundary layer, and potential flow theory can be applied to the oscillatory part of the motion. This significantly simplifies the solution of these flow problems, and is often applied in the irrotational flow regions of sound waves and water waves.
If the fluid domain is bounded by an upper, stationary wall, located at a height
y=h
u(y,t)=
| U | |
| 2(\cosh2λh-\cos2λh) |
[e-λ(y-2h)\cos(\omegat-λy)+eλ(y-2h)\cos(\omegat+λy)-e-λ\cos(\omegat-λy+2λh)-eλ\cos(\omegat+λy-2λh)]
where
λ=\sqrt{\omega/(2\nu)}
Suppose the extent of the fluid domain be
0<y<h
y=h
u(y,t)=
| U\cosh/\deltacoshh/\delta | |
| 2(\cos2h/\delta+sinh2h/\delta) |
\Re\left\{W+W*-itanhh/\delta\tanh/\delta(W-W*)\right\}, W=cosh[(1+i)(h-y)/\delta]ei\omega
where
\delta=\sqrt{2\nu/\omega}.
The case for an oscillating far-field flow, with the plate held at rest, can easily be constructed from the previous solution for an oscillating plate by using linear superposition of solutions. Consider a uniform velocity oscillation
u(infty,t)=Uinfty\cos\omegat
u(0,t)=0
u(y,t)=Uinfty\left[\cos\omegat-
| ||||
| e |
y}\cos\left(\omegat-\sqrt{
| \omega | |
| 2\nu |
which is zero at the wall y = 0, corresponding with the no-slip condition for a wall at rest. This situation is often encountered in sound waves near a solid wall, or for the fluid motion near the sea bed in water waves. The vorticity, for the oscillating flow near a wall at rest, is equal to the vorticity in case of an oscillating plate but of opposite sign.
Consider an infinitely long cylinder of radius
a
\Omega\cos\omegat
\omega
v\theta=a\Omega \real\left[
| K1(r\sqrt{i\omega/\nu | |
| )}{K |
| i\omegat | |
| 1(a\sqrt{i\omega/\nu})}e |
\right]
where
K1
\begin{align}v\theta\left(r,t\right)&=\Psi\left\lbrace\left[rm{kei}1\left(\sqrt{R\omega
\Psi=\left[
| 2 | |
| rm{kei} | |
| 1 |
\left(\sqrt{R\omega
kei
ker
R\omega
R\omega=\omegaa2/\nu
\nu
If the cylinder oscillates in the axial direction with velocity
U\cos\omegat
u=U \real\left[
| K0(r\sqrt{i\omega/\nu | |
| )}{K |
| i\omegat | |
| 0(a\sqrt{i\omega/\nu})}e |
\right]
where
K0
In the Couette flow, instead of the translational motion of one of the plate, an oscillation of one plane will be executed. If we have a bottom wall at rest at
y=0
y=h
U\cos\omegat
u=U \real\left\{
| \sinky | |
| \sinkh |
\right\}, where k=
| 1+i | |
| \sqrt{2 |
The frictional force per unit area on the moving plane is
-\muU\real\{k\cotkh\}
\muU\real\{k\csckh\}