Effect of surface-tension gradients on a failing film 

(a) Repeat the determination of the shear-stress and velocity distributions of Example 2.1-1 in the presence of a small temperature gradient dT/dz in the direction of flow. Assume that this temperature gradient produces a constant surface-tension gradient dσ/dz = A but has no other effect on system physical properties. Note that this surface-tension gradient will produce a shear stress at the free surface of the film (see Problem 11C.6) and, hence, will require a nonzero velocity gradient there. Once again, postulate a stable, non-rippling, laminar film. 

(b) Calculate the film thickness as a function of the net downward flow rate and discuss the physical significance of the result.