$15\mathrm{.}9.$ A constant$-$density Newtonian fluid is flowing as a thin film down a vertical wall in laminar flow; see Fig. $15\mathrm{.}9.$Find the velocity distribution and the volumetric flow rate per unit width of wall by using the Navier$-$Stokes equations $(z$ component$)$ on the assumptions that there is no flow in thex or $y$ direction, that the $z$ component of the velocity is zero at the solid wall, and that there is no shear stress at the liquid$-$air surface, and the flow is steady$-$state. $($Waves may appear on the fluid surface in this situation; ignore that possibility for this problem$).$