Problem $3$

A fluid flows in the positive $x-$direction through a long flat duct of length $L,$ width $W,$ and thickness $B,$ where $LW>.$ The duct has porous walls at $y=0$ and $y=B,$ so that a constant cross flow can be maintained, with $v_y=v_0,$ a constant, everywhere. Flows of this type are important in connection with separation processes using the sweep diffusion effect. By carefully controlling the cross flow, one can concentrate the larger constituents $($molecules$,$ dust

particles, etc.$)$ near the upper wall.

a$.$ Assuming that the lubrication approximation can be applied in the gap, determine the radial

velocity in terms of $d\frac{p}{d}x$ and B$.$