Problem 4. (25 points) Consider in compressible flow in the entrance of a circular tube having radius R. The inlet flow is uniform, u, =U,. The flow at section 2

is turbulent, fully-developed 1/7rpipe flow having a velocity profile of u,(r) =Ucı|R. In your solution, you need to work out the entire analysis starting with the conservation of mass and momentum equations from Lesson F.3 Slide 2. 0=\frac{d}{d t}\left(\iiint_{C F} \rho d \nu\right)+\iint_{C s} \rho(\vec{V} \bullet \vec{n}) d \mathcal{A} \sum \vec{F}=\frac{d}{d t}\left(\iiint_{C T} \rho \vec{V} d \nu\right)+\iint_{C s} \rho \vec{V}(\vec{V} \bullet \vec{n}) d \mathscr{A} a. Use conservation of mass to find Ug as a function (multiple) of U,. (Ans: 60/49 U, b. Find the wall drag force F between the inlet and location 2 as a function of P, P,, p,U,, R. Hint: Use the substitution u=1-r/R to integrate. Show all work in the integration. Do not use an integration table or online solver. c. Find the momentum-flux correction factor for the velocity profile at location 2. Give the correction factor as a fraction and decimal. Hint: You can find this value from your calculations in Part b.