Shear stress on oscillating wall in viscous incompressible fluid

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Homework Statement


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.)


Homework Equations


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.
 

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