Problem 11: Couette flow Consider a viscous fluid between two horizontal rigid plates having the distance h. If the upper plate moves with a constant velocity Up to the right side, the flow between the two plates has the form of a stationary Couette flow: J = Uo ex Uo x (1) Figure 1: A fluid between to infinitely extended horizontal plates, where the upper plate moves with a constant velocity in x-direction. a) Show that (1) describes an incompressible fluid and that it is a solution of the Navier-Stokes eqs. P + ( V) ] = n - Vp +. t (2) Show also that (1) fulfills the no-slip boundary conditions. What do you find for the pressure p if f = -pg? b) Now let the plates be in contact with a heat bath with constant temperature To. Compute the stationary temperature distribution T(z) inside the fluid. Where, and how large is the maximal temperature? Compute the difference between the plates and the hottest point for water (v = 10-6 m/s, p = 1000 kg/m, A = 0.5 J/Ksm) and Ug = 1 m/s. How fast must the plate move to obtain the difference of 1 K? c) Compute the entropy production rate (z) for the case of an isothermal fluid (T = To) (neglect heat produced by friction). Compute the total production rate per area = A dz (z). (3) Show that is equal to the power needed to move the upper plate divided by To. d) Compute the entropy production rate (2) with T(2) from b). Show that the total rate is the same than that found in c). Why?