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. 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.
Flow description Sources: Consider an infinitely long plate which is oscillating with a velocity U cos ω t {\displaystyle U\cos \omega t} in the x {\displaystyle x} direction, which is located at y = 0 {\displaystyle y=0} in an infinite domain of fluid, where ω {\displaystyle \omega } is the frequency of the oscillations. The incompressible Navier–Stokes equations reduce to
∂ u ∂ t = ν ∂ 2 u ∂ y 2 {\displaystyle {\frac {\partial u}{\partial t}}=\nu {\frac {\partial ^{2}u}{\partial y^{2}}}}
where ν {\displaystyle \nu } is the kinematic viscosity. The pressure gradient does not enter into the problem. The initial, no-slip condition on the wall is
u ( 0 , t ) = U cos ω t , u ( ∞ , t ) = 0 , {\displaystyle u(0,t)=U\cos \omega t,\quad u(\infty ,t)=0,}
and the second boundary condition is due to the fact that the motion at y = 0 {\displaystyle 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 Sources: 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 ℜ [ e i ω t f ( y ) ] {\displaystyle u=U\Re \left[e^{i\omega t}f(y)\right]}
because cos ω t = ℜ [ e i ω t ] {\displaystyle \cos \omega t=\Re \left[e^{i\omega t}\right]} . Substituting this into the partial differential equation reduces it to ordinary differential equation
f ″ − i ω ν f = 0 {\displaystyle f''-{\frac {i\omega }{\nu }}f=0}
with boundary conditions
f ( 0 ) = 1 , f ( ∞ ) = 0 {\displaystyle f(0)=1,\quad f(\infty )=0}
The solution to the above problem is
f ( y ) = exp [ − 1 + i 2 ω ν y ] {\displaystyle f(y)=\exp \left[-{\frac {1+i}{\sqrt {2}}}{\sqrt {\frac {\omega }{\nu }}}y\right]}
u ( y , t ) = U e − ω 2 ν y cos ( ω t − ω 2 ν y ) {\displaystyle 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 travels as the transverse wave through the fluid, but it is highly damped by the exponential factor. The depth of penetration δ = 2 ν / ω {\displaystyle \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
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