ALLING FILM PROBLEM. An incompressible liquid ows upward and out the end of a vertical tube with outer radius R at a rate Q , and then downward under gravity in laminar ow on the outside of the tube, to form a uniform liquid falling lm. Presume the liquid lm has constant thickness, occupying R < r < aR; a > 1 in a cylindrical coordinate system whose z axis is along the tube centerline. Assume also v = (0; 0; vz (r)) in the lm and that p is constant (= po) along the lms outer surface, at r = aR: Model the steady ow using the NS equations to show vz (r) = gR2 4 1 r R 2 2a2 ln r R Find a relation between Q and a . 1