Write down the Navier-Stokes equation for the velocity $\mathbf{u}(\mathbf{x}, t)$ of an incompressible viscous fluid of density $\rho$ and kinematic viscosity $\nu$. Cast the equation into dimensionless form. Define rectilinear flow, and explain why the spatial form of any steady rectilinear flow is independent of the Reynolds number.

(i) Such a fluid is contained between two infinitely long plates at $y=0, y=a$. The lower plate is at rest while the upper plate moves at constant speed $U$ in the $x$ direction. There is an applied pressure gradient $d p / d x=-G \rho \nu$ in the $x$ direction. Determine the flow field.

(ii) Now there is no applied pressure gradient, but baffles are attached to the lower plate at a distance $L$ from each other $(L \gg a)$, lying between the plates so as to prevent any net volume flux in the $x$ direction. Assuming that far from the baffles the flow is essentially rectilinear, determine the flow field and the pressure gradient in the fluid.