A particle of fluid is flowing in a wind tunnel where the velocity components are given by vx = U 1 + a 2 r 2 − 2a 2x 2 r 4 , vy = − 2U a2xy r 4 , vz = 0, where r 2 = x 2+y 2 and where U and a are given constants. A fixed electrical conductor on the z-axis creates the electrostatic potential field P given by P = A cos(qt) ln(x 2 + y 2 ), where A and q are constants. If the moving particle is at (a, a, 0) and t = t1, express the rate of change of P as seen by the particle at t1 in terms of the given quantities.

Along 8-cm diameter steam pipe whose external surface temperature is 900c connects two buildings. the pipe is exposed to ambient air at 70c with a wind speed of 50 km/hr blowing across the pipe. determine the heat loss from the pipe per unit length. (b) air at 500c enters a section of a rectangular duct (15 cm x 20 cm) at an average velocity of 7 m/s. if the walls of the duct are maintained at 100c. a) the length of the tube for an exit temperature of the air to be 40 0c. b)the rate of heat transfer from the air. c) the fan power needed to overcome the pressure drop in this section of the duct.