Details

As explained in the caption, this specific situation of laminar flow of a viscous fluid near a plate that oscillates with velocity provides one of the few pure analytical solutions of the Navier–Stokes equations:

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where . If we assume the flow to be unidimensional, with the only nonzero component of the velocity being , parallel to the plate, only one component of the three-dimensional Navier–Stokes equation remains:

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Since we assume that the flow is incompressible and that other forces are not present, the above equation reduces to

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Now, as the pressure gradient does not change with but only with time, its value must be the same for all . Since as goes to infinity, the velocity of the fluid, , goes to zero, the pressure gradient there must be zero; therefore it must be zero for all , and we can write

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We can solve this second-order partial differential equation, assuming a solution for of the form , so the equation can be written as

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With the boundary conditions of no slip at the plate, , and zero velocity far away from it, , we can obtain the analytical solution of the velocity profile:

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The parameter can be seen as a kind of "wavenumber", associated with a dimension or "wavelength":

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This is known as the Stokes boundary layer thickness. From the velocity equation, we can see that the oscillation of the plate propagates as a damped wave along the direction, and that the amplitude of this wave reduces to at a distance from the plate.

This Demonstration shows how the thickness changes with the viscosity of the fluid, the oscillation frequency of the plate, and the evolution in time of the velocity profile . Set the "time" slider motion to automatic and change the conditions of the problem to see how each one affects the velocity profile, which can be easily seen after a few moments.

Snapshot 1: viscosity greatly affects the layer thickness

Snapshot 2: frequency also affects the layer thickness

Snapshot 3: oscillation amplitude is best shown at