Determine the velocity at the interface between the two fluids and the force per unit area needed to maintain this motion. Express your answers in terms of U, h, pi and p2.

Engineering by Geoffrey

5. Two immiscible, incompressible, viscous fluids, having the same densities but different viscosities, are contained between two infinite, horizontal, parallel plates. The bottom plate is fixed, and the upper plate moves with a constant velocity U. Determine the velocity at the interface between the two fluids and the force per unit area needed to maintain this motion. Express your answers in terms of U, h, pi and p2. (13 points)
U ,_ jp■- i■

I Y P, P2 h x
P, Pi
a a a
i
( Fixed plate
Hints: The motion of the fluid is caused entirely by the movement of the upper plate; that is, there is no pressure gradient or gravitational field in the x direction. The fluid velocity and shearing stress are continuous across the interface between the two fluids. Assume laminar flow.