An incompressible, Newtonian fluid flows steadily between two parallel plates. The bottom plate moves to the left at a constant speed U1 and the upper plate moves to the right at a constant speed U2. For a small gap width, h, determine the following using the :
a. The steady, fully-developed velocity distribution, u(y) as a function of the pressure gradient dP/dx (you may start from the simplified x-momentum equation (4) that we used for Couette flow in class). Sketch the shape of the velocity distribution for the cases dP/dx = 0, dP/dx >0 and dP/dx <0.
b. Develop a relation between dP/dx, U1 and U2 that ensures that the net volume flowrate between the two plates is zero.
c. Determine the magnitude and direction of the shear stress acting on the upper plate.