Consider how pressure gradients and skin friction develop in an empty wind tunnel or water tunnel test section when the flow is incompressible. Here the fluid has viscosity $\mu$ and density $\rho ,$ and flows into a horizontal cylindrical pipe of length$L$ with radius $R$ at a uniform horizontal velocity $U_0.$ The inlet of the pipe lies at $x=0.$ Boundary layer growth on the pipe's walls induces the horizontal velocity on the pipe's centerline to be $U_L$ at $x=L;$ however, the pipe-wall boundary layer thickness remains much smaller than $R$. Here, $L/R$ is of order 10 , and $\rho U_{o}R/\mu \gg 1$ The radial coordinate from the pipe centerline is $r$a) Determine the displacement thickness, ${\delta }_L^{\ast }$, of the boundary layer at $x=L$ in terms of $U_o$ $U_L,$ and $R.$ Assume that the boundary layer displacement thickness is zero at $x=0$ [The boundary layer displacement thickness, ${\delta }^{\ast },$ is the thickness of the zero-flow-speed layer that displaces the outer flow by the same amount as the actual boundary layer. For a boundary layer velocity profile $u(y)$ with $y=$ wall-normal coordinate and $U=\text{ outer flow velocity, }{\delta }^{\ast }\text{ is defined by: }{\delta }^{\ast }={\int }_0^{\infty }(1-(u/U))dy.]$b) Determine the pressure difference, $\Delta P=P_L-P_o,$ between the ends of the pipe in terms of $\rho ,U_o,$ and $U_L$c) Assume the horizontal velocity profile at the outlet of the pipe can be approximated by: $u(r,x=L)=U_L\left(1-(r/R{)}^n\right)$ and estimate average skin friction, ${\bar{\tau }}_w,$ on the inside of the pipe between $x=0$ and $x=L$ in terms of $\rho$ $U_o,U_L,R,L,$ and $n$d) Calculate the skin friction coefficient, $c_f={\bar{\tau }}_w/(1/2)\rho U_o^2,$ when $U_o=20.0\mathrm{m}/\mathrm{s},U_L=$ $20.5\mathrm{m}/\mathrm{s},R=1.5\mathrm{m},L=12\mathrm{m},n=80,$ and the fluid is water, i.e., $\rho =10^3\mathrm{kg}/{\mathrm{m}}^3$