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^[1]^[2]

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^[3]^[4]

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 }{\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 decreases 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.

Stokes problem in cylindrical geometry

Torsional oscillation

Consider an infinitely long cylinder of radius $a$ exhibiting torsional oscillation with angular velocity $\Omega \cos \omega t$ where $\omega$ is the frequency. Then the velocity is given by^[5]

v_{\theta }=a\Omega \ \Re \left[{\frac {K_{1}(r{\sqrt {i\omega /\nu }})}{K_{1}(a{\sqrt {i\omega /\nu }})}}e^{i\omega t}\right]

where $K_{1}$ is the modified Bessel function of the second kind.

Axial oscillation

If the cylinder oscillates in the axial direction with velocity $U\cos \omega t$ , then the velocity field is

u=U\ \Re \left[{\frac {K_{0}(r{\sqrt {i\omega /\nu }})}{K_{0}(a{\sqrt {i\omega /\nu }})}}e^{i\omega t}\right]

where $K_{0}$ is the modified Bessel function of the second kind.

Stokes-Couette flow^[6]

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 $y=0$ and the upper wall at $y=h$ is executing an oscillatory motion with velocity $U\cos \omega t$ , then the velocity field is given by

u=U\ \Re \left\{{\frac {\sin ky}{\sin kh}}\right\},\quad {\text{where}}\quad k={\frac {1+i}{\sqrt {2}}}{\sqrt {\frac {\omega }{\nu }}}.

The frictional force per unit area on the moving plane is $-\mu U\Re \{k\cot kh\}$ and on the fixed plane is $\mu U\Re \{k\csc kh\}$ .

References

↑ Batchelor, George Keith. An introduction to fluid dynamics. Cambridge university press, 2000.
↑ Lagerstrom, Paco Axel. Laminar flow theory. Princeton University Press, 1996.
↑ Acheson, David J. Elementary fluid dynamics. Oxford University Press, 1990.
↑ Landau, Lev Davidovich, and Evgenii Mikhailovich Lifshitz. "Fluid mechanics." (1987).
↑ Drazin, Philip G., and Norman Riley. The Navier–Stokes equations: a classification of flows and exact solutions. No. 334. Cambridge University Press, 2006.
↑ Landau, L. D., & Sykes, J. B. (1987). Fluid Mechanics: Vol 6. pp. 88

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[1] Batchelor, George Keith. An introduction to fluid dynamics. Cambridge university press, 2000.

[2] Lagerstrom, Paco Axel. Laminar flow theory. Princeton University Press, 1996.

[3] Acheson, David J. Elementary fluid dynamics. Oxford University Press, 1990.

[4] Landau, Lev Davidovich, and Evgenii Mikhailovich Lifshitz. "Fluid mechanics." (1987).

[5] Drazin, Philip G., and Norman Riley. The Navier–Stokes equations: a classification of flows and exact solutions. No. 334. Cambridge University Press, 2006.

[6] Landau, L. D., & Sykes, J. B. (1987). Fluid Mechanics: Vol 6. pp. 88

Stokes problem

Flow description[1][2]

Solution[3][4]