(d) (45 points) Planar Couette (simple shear) flow
An incompressible Newtonian fluid is flowing through two parallel plates of length $L$, which are
separated by $H$ from each other. The plates are tilted up by an angle $\theta$ so that gravitational force
is involved in the fluid dynamics. Here we take the fluid flows in the $z$-direction and the plates are
separated in the $y$-direction. We make the top plate move with a constant velocity $V$ in the $z$
direction with fixing the bottom plate. There is a pressure difference ($P_1 - P_0$) between inlet ($P_0$) and
outlet ($P_1$). Also, assume a laminar flow and $v_y$ and $v_x$ are zero. While there is no slip of the fluid
motion at both plates. Set up the mass and momentum balance equations together with
appropriate boundary conditions, and then derive and depict the steady-state velocity profile by
solving them. Also, find the maximum velocity and its location ($y$-value).
