Exact Solutions for the Incompressible Viscous Fluid of a Rotating Disk Flow

摘要

The present paper is devoted to a class of exact solutions to the steady Navier-Stokes equations for the incompressible Newtonian viscous fluid flow motion due to a disk rotating with a constant angular speed. In place of the traditional von Karman's axisymmetric evolution of the flow, the rotational non-axisymmetric stationary flow is taken into account here. As a consequence, the governing equations allow an exact solution to develop over the disk, which is influenced by a fixed point on the disk and also bounded everywhere in the normal direction to the wall. The three-dimensional equations of motion are treated analytically yielding derivation of exact solutions, which differ from those of corresponding to the classical von Karman's flow. Making use of this solution, analytical formulas for the wall shear stresses are extracted. The critical peripheral locations at which extrema of the local skin friction occur are also determined. Analytic calculations show that for the specific flow the thicknesses take the same value. Interaction of the resolved flow field with the surrounding temperature is further analyzed via the energy equation. According to the Fourier's heat law, a constant heat transfer from the disk to the fluid occurs, which is proportional to product of the thermal conduction, Reynolds number and temperature difference.