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WPE 2014, Fluid Mechanics
A circular disk of mass M and radius R2 is placed a short distance h below a wall, just underneath a circular orifice of radius R1. An incompressible fluid at atmospheric pressure Pa is pumped with constant velocity U through the orifice so that it impinges on the disk as shown. Neglect the weight of the fluid.
R1
U, Pa
R2
g
h
R1
z
R2
r
Side view
Top view
a) Assuming an inviscid fluid, find an expression for the total force of the fluid on the disk. Show any control volumes useful to your analysis. Assume that uniform radial flow enters the gap at r = R 1, and state any other assumptions needed.
b) Is there a distance from the wall for which the disk is in equilibrium? If so, find an expression for this distance hc, and in a few sentences describe physically the balance of forces that create this equilibrium. Is this equilibrium stable? (In other words, if the disk is displaced vertically a small distance from hc, will it return to that location?)
c) Suppose now that the fluid has viscosity. From the Navier­Stokes equations and continuity, develop a differential equation for the pressure variation in the small gap between wall and disk. Assume u z = 0 in the gap. Describe briefly how viscosity modifies the inviscid solution. For example, how would the pressure field in part (a) be modified by inclusion of viscosity? (Note: You don't necessarily need to solve the equation you derived!) To aid in your description, consider cases of fluid with very high and very low viscosity.
d) For viscous flow, consider what happens to the total force upon the disk as the gap height h is slowly reduced from hc (in part (b)) to zero. In this case, are there any additional equilibrium configurations?
e) In parts (a) and (c) we assumed a purely radial flow in the gap (u z = 0). How would you expect real flow to differ from this assumption, especially in the vicinity of the corner at r = R 1? What pressure variations would be expected in this region?