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 three-dimensional (3D) incompressible Navier–Stokes equations are analytically solved by postulating the polynomial profiles for the axial and circumferential velocity components and by employing the open-end condition of zero pressure difference and 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 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). A simple theoretical relationship between the radial and axial pressure gradients is deduced to hinge the radial and circumferential velocity components together. The values of the compression and rotation Reynolds numbers suitable to this theory are also suggested for the problem of rotating disk flows at low Reynolds numbers. The validity of the theoretical predictions for the circumferential, radial, and axial velocity components is partially verified through comparison with previous steady experimental and numerical results. These analytical results have the immediate engineering applications of fluid flows with varying gap widths, including wet brakes, wet clutches, hydrostatic bearings, face seals, and rotating heat exchangers.