Navier-Stokes Equations

Find the velocity distribution and shear stress for the following cases by the exact solution of Navier-Stokes equations: Plane Poiseuille flow and Plane Couette flow. The x-momentum equation is given as: $$\rho \left[ \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z} \right] = - \frac{\partial p}{\partial x} + \mu \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} \right) + \rho b_x$$

The viscous, incompressible flow between the parallel plates shown in Fig. 7(b) is caused by both the motion of the top plate and a pressure gradient $\partial p/\partial x$. Find the required pressure $\partial p/\partial x$ to achieve a zero net volume flow rate across a cross-section.