Consider the fluid flow between a fixed upper plate and an oscillating lower plate as shown in Fig.

$P1,$ where $h$ is the distance between two plates and

$u$ is the kinematic viscosity. Boundary conditions

are

$y=0$:$u=U{e}^{it}$

$y=h$:$u=0$

$($a$)$ Find the velocity profile of the flow between two flat plates.

Hint: Calculation will be easy if you use the following replacement: $=\sqrt[\phantom{2}]{\frac{i}{?}}$

$u.$ Here $$ is the

same definition in Section $6\mathrm{.}1$ of my handout.

$($b$)$ The velocity profile changes according to $h0(h$ approaches to $0,$ but $h0)$ and $h,$

respectively. Find the velocity profile for each case.