6. The flow pattern in bearing lubrication can be illustrated by the figure below, where a viscous oil (p,µ) is forced into the gap h(x) between a fixed slipper block

and a wall moving at velocity U. If the gap is thin, h « L, it can be shown that the pressure and velocity distributions are of the form p = p(x),u = u(y), v = w = 0. Neglecting gravity, reduce the Navier-Stokes equations to a single differential equation for u(y). What are the proper boundary conditions? Integrate and show that u=\frac{1}{2 \mu} \frac{d p}{d x}\left(y^{2}-y h\right)+U\left(1-\frac{y}{h}\right) where h = h(x) may be an arbitrary slowly varying gap width.