(Solved): Consider flow between parallel plates, as shown in FIGURE Q2 where the top plate moves at a velocit ...

Consider flow between parallel plates, as shown in FIGURE Q2 where the top plate moves at a velocity $U=XX\mathrm{m}/\mathrm{s}(\mathrm{XX}$ is the last 2 digits of your Student ID). The height of the channel, $h=1\mathrm{m}$. The flow is governed by the simplified Navier-Stokes equation $\frac{\partial p}{\partial x}=\mu \frac{{\partial }^2{u}_x}{\partial y^2}$ The general velocity profile can be obtained by integrating the above equation, resulting in the following expression $u_x=\frac{1}{\mu }\left(\frac{dp}{dx}\right)\frac{y^2}{2}+C_{1}y+C_2$ Upper plate. Velocity, U $⟶$ Shear stress, $\tau \to$ FIGURE Q2 (a) Obtain the expression of $u_x(y)$ for the following boundary conditions: $\begin{array}{l}\text{ at }y=0,u_x(0)=0;\\ \text{ at }y=\mathrm{h},u_x(\mathrm{h})=U\mathrm{m}/\mathrm{s}.\end{array}$ (b) Obtain the dimensionless expression, $u_x/U$ from (a) by substituting into the equation the dimensionless pressure gradient $\mathrm{P}=-\frac{h^2}{2\mu U}\left(\frac{dp}{dx}\right)$ (c) Plot the graph of $u_x/U$ vs $y/h$, for $\frac{dp}{dx}=0\frac{dp}{dx}=2\mu U$ าd $\frac{dp}{dx}=-2\mu U$