(Solved): 2. (30 points) Consider steady, incompressible flow in a circular pipe with a radius R. In a fully ...

2. (30 points) Consider steady, incompressible flow in a circular pipe with a radius $R$. In a fully developed, steady, incompressible pipe flow, the velocity profile can be approximated by a quadratic function $V=V_{\max }\left(1-\frac{r^2}{R^2}\right),$ where $V_{\max }$ is the velocity at the centerline of the pipe (cf. Figure 2). The velocity vector is defined such that $\begin{array}{rl}\vec{V}& =V_r\hat{r}+V_{\theta }\hat{\theta }+V_z\hat{z}\\ V_r& =0,V_{\theta }=0,V_z=V_{\max }\left(1-\frac{r^2}{R^2}\right).\end{array}$ (a) Find an expression for the strain rate ${\epsilon }_{rz}=\frac{1}{2}\left(\frac{\partial V_r}{\partial z}+\frac{\partial V_z}{\partial r}\right),$ in cylindrical coordinates. (b) Suppose that the stress is linearly proportional to the rate of strain and that ${\tau }_{rz}=2\mu {\epsilon }_{rz}$, where $\mu$ is the viscosity coefficient. Assume that all the other components of the stress tensor vanish (with the exception of ${\tau }_{zr}$ ). Find an expression for the frictional force per unit length exerted on the pipe by the fluid. Fully Figure 2: Fully developed pipe flow.