2. The objective of this problem is to apply what you learned to a realistic problem, forming connections between mathematical concepts and a physical process. You are using the coating process shown in figure below to coat a foil with a polymer, which may be considered as a Newtonian fluid with viscosity 980 cP and density 1 g/cm'. The foil is spooled and pulled through a tub filled with the polymer at upwards (z-directed) velocity v. A thin layer of the film of thickness d is dragged onto the foil and dries after being turned horizontal. The problem resembles the falling film problem except that the film in this case is rising. The gravity points down in negative z direction. Assume that during the rise the flow is laminar and v, is a function of x only and all velocities in all other directions are zero. (a) Derive an expression for the velocity profile vz(x). (b) Taking 8 = 1 mm, plot the velocity profile using MATLAB and the quiver function for the following cases: (1) v= 5 cm/s, (ii) v= 1 cm/s, (iii) v=0.5 cm/s, (iv) v=0.4 cm/s (v) v= 0.3 cm/s Explain what you see. Describe physically the reason behind what you are seeing. Hint: this is shear driven (by a moving boundary) and gravity driven flows battling. (c) Find an expression for the fluid velocity at the surface. Is there a situation where the surface of the film is stagnant (i.e., vz(=0))? Is there a situation where the surface of the film is flowing downwards? (d) What is the average velocity? Derive an expression for the volumetric flow rate. At a critical v there is no net flow. Why is this? What is that critical value? How does it depend on the viscosity? Does it make sense to you? (e) If the desired film thickness was 10 um (1 um = 104 cm), how fast does the foil have to travel upwards in order not to have flow reversal at the surface? (f) How would your answer change if the fluid was water? Does the difference make sense to you? film O Z kd coated spool V Liquid Uncoated spool