Dip coating is a process in which an immersed substrate is withdrawn through a gas$-$liquid interface. The problem here is not focused on what occurs near the surface of a pool of liquid $($Fig$.$ A$),$ as a solid sheet is pulled upward at a constant velocity V$.$ The concern in this question is what happens farther up $($Fig$.$ B$).$ Above a certain height, the liquid film on the sheet will achieve a constant thickness H $($Fig$.$ B$).$ At even larger x$,$ suppose that evaporation or reaction causes the film to solidify. It is desired to predict the thickness $H_{}$ of the solid film.

$($a$)$ Assuming that the air pressure is constant at $P_0$ and that viscous stresses in the gas are negligible, determine the local liquid velocity $v_x(y)$ and the mean liquid velocity $U.$

$($b$)$ If the liquid and solid densities are the same, how is $\frac{H_{}}{H}$ related to $H,V,$ and the liquid properties?