Problem 10.5. Consider an isotropic fluid contained in a rectangular box with sides of length Lx =L, Ly=2L, and L = 3L. Assume that the temperature of the fluid at time t=0 has a distribution T(r, 0), but the fluid is initially at rest. Assume that the thermal expansivity of the fluid is very small so that coupling to pressure variations can be neglected.

(a) Show that under these conditions the temperature variations satisfy the heat equation, 8T(r,t)/dt = -kV2T(r,t). What is k?

(b) If the walls of the box conduct heat and are maintained at temperature, To, approximatlely how long does it take for the system to reach equilibrium.

(c) If the walls of the box are insulators, approximately how long does it take for the system to reach equilibrium? (Hint: No heat currents flow through the walls of insulators.)