Stokes Boundary Layer For Laminar Flow Near A Wall
The oscillating flow is assumed to be uni-directional and parallel to the plane wall. The only non-zero velocity component is called u (SI measure in meter/second, or m/s) and is in the x-direction parallel to the oscillation direction. Moreover, since the flow is taken to be incompressible, the velocity component u is only a function of time t (in seconds) and distance from the wall z (in meter). The Reynolds number is taken small enough for the flow to be laminar. Then the Navier–Stokes equations, without additional forcing, reduce to:
with:
- ρ the mass density of the fluid (kg/m3), taken to be a constant,
- p the fluid pressure (SI: Pa),
- ν the kinematic viscosity of the fluid (m2/s), also taken constant.
and
- u the velocity of the fluid along the plate (m/s)
- x the position along the plate (m)
- z the distance from the plate (m)
- t the time (s)
Because the velocity u is not a function of position x along the plate, the pressure gradient ∂p/∂x is also independent of x (but the pressure p varies linearly with x). Moreover, the Navier–Stokes equation for the velocity component perpendicular to the wall reduces to ∂p/∂z = 0, so the pressure p and pressure gradient ∂p/∂x are also independent of the distance to the wall z. In conclusion, the pressure forcing ∂p/∂x can only be a function of time t.
The only non-zero component of the vorticity vector is the one in the direction perpendicular to x and z, called ω (in s-1) and equal to:
Taking the z-derivative of the above equation, ω has to satisfy
As usual for the vorticity dynamics, the pressure drops out of the vorticity equation.
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