申論題

2. Two horizontal, infinite, parallel plates are spaced a distance b apart, please see the following figure. A viscous liquid is contained between the plates. The bottom plate is fixed (y = 0), and the upper plate (y = b) moves parallel to the bottom plate with a velocity U. Because of the no-slip boundary condition, the liquid motion is caused by the liquid being dragged along by the moving boundary. Assuming that there is no pressure gradient in the direction of flow (x-direction) and the flow is steady and incompressible without the y-component velocity. (a) Derive the governing equation of the above 2-D flow from the x-direction Navier-Stokes equation (see below). (b) Determine the velocity distribution between the plates. (c) Determine an expression for the flowrate per unit width passing between the plates. Express your answers of (b) and (c) in terms of b and U. (25%)