Understanding PDE: Parallel Plate Motion with Newtonian and Incompressible Fluid

[skhan84]

There are two parallel plates, upper plate is static and bottom plate is porous and in motion. At the same time there are types of motion in bottom plate. Plate is oscillating with velocity Uo*e^(iwt) and also moving forward along X-Axis with constant velocity Co. Due to motion in plate Newtonian and Imcompressible fluid is injected with velocity Wo
Velocity field is given
V=[u(y,t),Wo,0]
BC's: u(0,t)=Uo*e^(iwt)
u(d,t)=0
IC's: u(y,0)=0
In the start we used the "Galilean transform"
Let y=Y-c0t

 

[lippyka]

u(y,t)=U(Y,t)Then U(Y,t)=Uo*e^(iwt)-CoBC's: U(0,t)=Uo*e^(iwt)-CoU(d-c0t,t)=0IC's: U(Y,0)=Uo-CoThen we solved the equations to get U(Y,t)=(Uo-Co)*sin(K*Y)+Uo*e^(iwt)*cos(K*Y)-Co*e^(iwt)where K=(w/c0)^0.5Now I am stuck how to get velocity field from this? Please help me.