A viscous fluid is contained between two infinite, horizontal plates. The fluid moves between the plates under the action of a pressure gradient, and the upper plate moves with a velocity U while the bottom plate is fixed. A U-tube manometer connected between two points along the bottom indicates a differential reading h. a) Derive an expression for the velocity profile u (as a function of y, including the pressure gradient in the x-direction) assuming that the lubrication approximation is valid, and the Navier Stokes equations reduce to the form shown in the equation sheet. Note: it can be shown that u = u(y) only, and p = p(x) only. Assume this is true. b) Calculate the pressure gradient over the distance L using the manometer reading. In fact, this is the pressure gradient dp/dx factored into the velocity profile. h = 2.5 mm d = 25 mm U = 6 mm/s L= 15 cm (1) (2) viscous fluid manometer fluid viscous fluid: p = 1280 kg/m³ 1.43 kg/ms manometer fluid: P = 1600 kg/m³