Exact solutions corresponding to the viscous incompressible and conducting fluid flow due to a rotating disk

Abstract

AbstractA study is pursued in the present paper for the evaluation of the exact solution of the steady Navier‐Stokes equations governing the incompressible viscous Newtonian electrically conducting fluid flow motion over a disk rotating with a constant angular speed. Instead of the traditional von Kármán's axisymmetric evolution of the flow, the non‐axisymmetric conducting flow is taken into consideration here. The three‐dimensional equations of motion are treated analytically yielding derivation of exact solutions which differ from those of corresponding to the classical von Kármán's conducting flow. The effects of Alfven number on the flow field is better conceived from the exact velocity and induced magnetic field equations obtained. Making use of this solution, analytical formulas for the angular velocity and current density components as well as for the magnetic wall shear stresses are extracted. It is proved from the analytical results that the properly defined thicknesses of the flow decay as the magnetic field strength increases in magnitude, further enhanced by the increasing Reynolds number. Interaction of the resolved flow field with the surrounding temperature is then analyzed via the energy equation. The temperature field is shown to accord with the viscous dissipation and the Joule heating. As a result, exact formulas are obtained for the temperature field which takes different forms depending on whether isothermal wall or adiabatic wall conditions are considered. In accordance with the Fourier's heat law, a constant heat transfer from the disk to the fluid takes place, which is found to be effectively increased as the magnetic field becomes dominant on the fluid flow considered.