To reduce pumping power requirements in oil transport, water is introduced into the pipe line yielding a concentric flow of oil and water. The less-viscous water forms a uniform annulus around the oil core as shown in the figure. Oil and water are assumed to be immiscible, and gravitational effects can be neglected. The viscosity of water is uw and that of oil is o. The radius of the pipe is R and that of the oil core is R, and the flow is steady state. The pressure drop over length L of the pipe is Ap, so the axial pressure gradient is ap/z = -Ap/L). Our goal is to find the velocity profile for oil flow. Water, w Oil. Ho P (a) (10 points) The Navier-Stokes equation for z-direction is given by du 1 a du u du du + r r de Fot =+U Side View at =+U z z Ro +pg+ R du . r dr dr + R Water oil End View 1 d'u du_ + r 20 Oz By deleting the zero terms, reduce the equation as much as you can. Briefly specify the reasons why the terms are zero. (b) (10 points) What are the boundary conditions defining the flow of both the water and oil? (c) (30 points) Obtain the velocity profile for oil flow.