A circular disk of radius a is parallel to and at a distance h from a rigid plane, and the intervening, space is filled with fluid. The pressure at the edge of the disk is atmospheric. The disk is pulled slowly away from the rigid plane, in the y direction. There is no fluid beyond r = a. SEE the attached figure. You may assume Write down the full equation of motion in cylindrical polar coordinates (assuming axial symmetry) and by scale analysis reduce the equation to These equations imply that u(r, y) may be solved for assuming separation of variables in r and y- Why? Solve for the y-variation of u. Using the conservation of volume in the integral sense, find the average (depth integrated) inward radial velocity. Use the result a in (a) and (b) and the pressure at the edge of the disk, find the pressure distribution as a function of (mu, r, dh/dt, pram). Hence show that the force on the disk (due to this pressure distribution) Comment on the dependence of this force on mu, h. This explains why machine shafts use thin film of high viscosity oil rather than low viscosity water as a lubricant. Extend the above result to show that a constant force F applied to the disk will pull it well away from the plane in a time This implies that if (a/h)4 >> (4/3pi)rhoF/mu2 then the time-dependence can be neglected as assumed in deriving the reduced equations (Why?) Is u time-dependent?
