$(50$ points$)$ 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 ${}_w$ and that of oil is ${}_0.$ The radius of

the pipe is $R$ and that of the oil core is $R_0$ and the flow is steady state. The pressure drop over length $L$

of the pipe is $p,$ so the axial pressure gradient is del$\frac{p}{d}$elz$=-\frac{p}{L}.$ Our goal is to find the velocity

profile for oil flow.

$($a$)(10$ points$)$ The Navier$-$Stokes equation for $z-$direction is given by

$(\frac{del{u}_z}{\text{d}\text{e}\text{l}\text{t}}+u_r\frac{del{u}_z}{\text{d}\text{e}\text{l}\text{r}}+\frac{u_{}}{r}\frac{del{u}_z}{\text{d}\text{e}\text{l}}+u_z\frac{del{u}_z}{\text{d}\text{e}\text{l}\text{z}})=-\frac{\text{d}\text{e}\text{l}\text{p}}{\text{d}\text{e}\text{l}\text{z}}+g_z+[\frac{1}{r}\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{r}}(r\frac{del{u}_z}{\text{d}\text{e}\text{l}\text{r}})+\frac{1}{r^2}\frac{de{l}^2{u}_z}{\text{d}\text{e}\text{l}{}^2}+\frac{de{l}^2{u}_z}{del{z}^2}].$

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.