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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.