A fluid flows between two parallel plates held at temperature zero. At the inlet, fluidtemperature is To and initially the fluid is at temperature T1. If V is the speed of the fluid in the x direction, a problem describing the temperature u(x, y, t) is: + V k at ax 272 , x > 0, 0 0, u(x, 0, t) = 0, u(x, b, t) = 0, u(0, y, t) = To, u(x, y, 0) = T1. Make a separation of variables as above. State and solve the eigenvalue problem for Y . Show that Unk (*+ Vt ) Un(x, y, t) = $n(x - Vt)e 2V sin Any where In = un, for n = 1, 2, 3, ... are the eigen values for the Y problem, satisfies the PDE and boundary conditions at y = 0 and y = b, without restrictions of on (except differentiability)