1) Problem 1 (50 pts): Consider a long cylindrical pipe with a radius of Ro. The pipe is initially fully filled with a fluid having the viscosity p and the density p. Then, high- pressure air is continuously applied in the center of pipe so that air gap with a radius of Ri(constant) forms in the center. Assume that gravitational acceleration is acting downwards. Also, assume that no external pressure gradient is applied to the system. In addition, assume that viscosity of trapped air is much smaller than that of the fluid. Obtain an expression for the velocity profile of the fluid inside the pipe under steady-state conditions. Ro Ri Water Trapped Air gravity, g