A wide moving belt passes through a container of a viscous liquid. The belt moves vertically upward with a constant velocity of ????0, as shown below. Because of viscous forces, the belt picks up a film of liquid with thickness ℎ. Gravity tends to make the fluid drain down the belt. Assume that the flow is laminar, steady, and fully developed. Using the Navier-Stokes Equations (Cartesian), develop an expression for the profile of the velocity in the vertical y-direction.(A) Express your final equation in dimensionless form.(B) Determine the expression for the volumetric flow rate of the fluid (per unit width).(C) In your answer for part A, you should notice a dimensionless constant, C, that is multiplied by the dimensionless distance (i.e., x/ℎ). Generate a plot of the dimensionless velocity (i.e., ????/????0) vs. the dimensionless distance for the following values of C: 0.01, 0.5, 1, 1.5, and 2. (Hint: assume there is no pressure gradient in this flow. Pressure changes due to the change in elevation of the fluid as it rises are already captured by the gravity term in the Navier-Stokes equations.) HINT: (B: ???? = ???????????? − (????????^????)/(????????))
