Question

A wide flat belt moves vertically upward at constant speed U, through a large bath containing a liquid of viscosity µ as shown in the figure. The belt (seen edge-on in the figure) carries with it a layer of liquid of constant thickness h. The motion is steady and fully-developed after a small distance above the liquid surface level. The external pressure is atmospheric (constant) everywhere.We are interested in finding the velocity profile inside the liquid layer. Denote u and v to be the x and y component of the fluid velocity. (a) Using the continuity equation, explain why v is a function of x alone. (b) Write down the simplified x- and y-momentum equations. State all your assumptions. (c) What are the boundary conditions at x = 0 and x = h? (d) Solve for the velocity profile v using the y-momentum equation and the boundary conditions. (e) Sketch the velocity profile. (f) Find the condition on U for the flow to have a volumetric flow rate Q > 0.

Fig: 1

Fig: 2

Fig: 3

Fig: 4

Fig: 5

Fig: 6

Fig: 7

Fig: 8