On the Unsteady Rotationally-symmetric Flow Between a Stationary and a Finite Rotating Disk with a Given Change in the Axial Velocity


 We consider the scenario of an unsteady viscous flow between two coaxial finite disks, one stationary and the other rotating with an axial velocity changing impulsively from zero to a constant value. The incompressible Navier-Stokes equations are analytically solved by applying an integral approach. It is shown that the time-dependent squeezing of the fluid between the disks and the edge effects of the finite open-ended disks in the flow domain are uniquely determined by the compressing Reynolds number and the rotating Reynolds number for the case of laminar flow at low Reynolds numbers and small aspect ratios. The general explicit formulae are derived for the velocity and pressure distributions as a function of the compressing and rotating Reynolds numbers in this unsteady flow process (and the steady-state solutions are then obtainable as the compressing Reynolds number vanishes). These formulae fully characterize the nature of this unsteady, low Reynolds number, small aspect ratio, finite rotational disk flow with moving boundaries. A simple theoretical relationship between the radial and axial pressure gradients is also deduced to hinge the radial and circumferential velocity components together. The validity of the theoretical predictions for the circumferential and radial velocity components is partially verified through comparison with previous steady experimental measurements for low rotating Reynolds numbers. The analytical results have the potential to be applied to various applications of fluid flows with varying gap widths, including wet brakes, wet clutches, hydrostatic bearings, face seals and rotating heat exchangers.