A layer of viscous incompressible fluid of thickness H lies on top of a solid wall that oscillates simple harmonically w/ angular frequency Ω. u(wall)=Acos(Ωt). Ignore the motion of air above the fluid layer and find the shear stress at the wall. (Shear stress on free surface must be zero.)
Equation of motion and Navier-Stokes equations in cartesian coordinates.
Shear stress at wall = μ(∂u/∂y) for y=0
The Attempt at a Solution
Boundary conditions: For y=0, u(wall)=Acos(Ωt) and v=w=0. Solution is independent of x and z so ∂/∂x=∂/∂z=0. And where y=H you have μ(∂u/∂y).
From here, the governing equations simplify to just ∂u/∂t = μ(∂^2u/∂y^2)
Solving this PDE is where I'm running into trouble. I believe there is a way to simplify the problem by converting it to complex, but that's where I am stuck.