On The Exact Solution Of IncompressibleViscous Flows With Variable Viscosity

Flows of variable viscosity fluids have many industrial applications in fluid mechanics and in engineering such as pump flows for highly viscous fluids. Carrying out a literature survey, it was found that in most cases the fluid viscosity is mainly temperature dependent. Numerical investigation of such flows involves the solution of the Navier-Stokes equations with an extra difficulty arising from the fact that the viscosity is not constant over the flow field. This article presents an analytical solution to the Navier-Stokes equations for the case of laminar flows in rotating systems with variable viscosity fluids. The equations are written in a cylindrical relative frame of reference rotating with a constant angular velocity. In the following, appropriate reference quantities are chosen to provide the non-dimensional form of the partial differential equations. The proposed solution that satisfies the continuity, the momentum and the energy equation, is expressed in terms of the Bessel function of the first kind and of exponential functions. Carrying out algebraic manipulations, it is proven that the proposed solution satisfies all the governing equations. Plots of the velocity, pressure and temperature distributions show the influence of the radius and of the axial coordinate to the flow field.