For a two dimensional incompressible flow in a square domain 0 <= x <= 2Ï€ and 0 <=y <= 2Ï€ of u(x, y)= sin (x cos(y) and y and component of velocity v (x, 0) at the lower boundary y =0. Required:
a. Find the expression of v(x, y)
b. Find the stream function for this flow.
c. Does the potential function φ exist for this flow, show your proof.
d. Show that the velocity field is NOT a valid solution of Navier-Stoke equation.

To save energy, the air supply to a 2000 ft office has been shut off overnight and the room temperature has dropped to 40°f. in the morning, the thermostat is reset to 70°f and warm air at 120 f begins to flov in at 200ft'/min. the air is well mixed within the room, and an equal mass flow of air at room temperature (changing with time) is withdrawn through a return duct. the air pressure is nearly 1 atm everywhere. ignoring heat transfer with the surroundings and kinetic and potential energy effects, estimate how long it takes for the room temperature to reach 70°f. plot the room temperature as a function of time.