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.
Consider an infinitely long plate which is oscillating with a velocity formula_1 in the formula_2 direction, which is located at formula_3 in an infinite domain of fluid, where formula_4 is the frequency of the oscillations. The incompressible Navier–Stokes equations reduce to
formula_5
where formula_6 is the kinematic viscosity. The pressure gradient does not enter into the problem. The initial, no-slip condition on the wall is
formula_7
and the second boundary condition is due to the fact that the motion at formula_3 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
formula_9
because formula_10.
Substituting this into the partial differential equation reduces it to ordinary differential equation
formula_11
with boundary conditions
formula_12
The solution to the above problem is
formula_13
formula_14
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 formula_15 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
formula_16
There is a phase shift between the oscillation of the plate and the force created.
Vorticity oscillations near the boundary.
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. 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.
Fluid bounded by an upper wall.
If the fluid domain is bounded by an upper, stationary wall, located at a height formula_17, the flow velocity is given by
formula_18
where formula_19.
Fluid bounded by a free surface.
Suppose the extent of the fluid domain be formula_20 with formula_17 representing a free surface. Then the solution as shown by Chia-Shun Yih in 1968 is given by
formula_22
where formula_23
Flow due to an oscillating pressure gradient near a plane rigid plate.
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 formula_24 far away from the plate and a vanishing velocity at the plate formula_25. Unlike the stationary fluid in the original problem, the pressure gradient here at infinity must be a harmonic function of time. The solution is then given by
formula_26
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.
Stokes problem in cylindrical geometry.
Torsional oscillation.
Consider an infinitely long cylinder of radius formula_27 exhibiting torsional oscillation with angular velocity formula_28 where formula_4 is the frequency. Then the velocity approaches after the initial transient phase to
formula_30
where formula_31 is the modified Bessel function of the second kind. This solution can be expressed with real argument as:
formula_32
where
formula_33
formula_34 and formula_35 are Kelvin functions and formula_36 is to the dimensionless oscillatory Reynolds number defined as formula_37, being formula_38 the kinematic viscosity.
Axial oscillation.
If the cylinder oscillates in the axial direction with velocity formula_39, then the velocity field is
formula_40
where formula_41 is the modified Bessel function of the second kind.
Stokes–Couette flow.
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 formula_3 and the upper wall at formula_17 is executing an oscillatory motion with velocity formula_39, then the velocity field is given by
formula_45
The frictional force per unit area on the moving plane is formula_46 and on the fixed plane is formula_47.