Scaling for unsteady thermo-magnetic convection boundary layer of paramagnetic fluids of Pr > 1 in micro-gravity conditions

Abstract

This work incorporates scaling analysis to characterise unsteady boundary-layer development for thermo-magnetic convection of paramagnetic fluids with Prandtl numbers ( Pr) greater than one. Under consideration is a square cavity with a quiescent isothermal, Newtonian fluid placed in a micro-gravity condition ( g ≈ 0 ), and under a uniform vertical gradient magnetic field. A distinct magnetic convection boundary layer is produced by the sudden imposition of a higher temperature on the left-hand side vertical sidewall due to the effect of the magnetic body force generated on the paramagnetic fluid. This magnetic force is proportional to the magnetic susceptibility and the gradient of the square of the magnetic induction. According to Curie’s law, the magnetic susceptibility of a paramagnetic fluid is inversely proportional to the absolute temperature. Thermal convection of a paramagnetic fluid can therefore take place even in zero-gravity environments as a direct consequence of temperature differences occurring within the fluid placed within a magnetic field gradient. Scaling predictions presented here are verified by numerical simulations It is shown that the transient flow behaviour of the resulting boundary layer can be described by three stages: a start-up stage, a transitional stage and a steady state. Special attention in this work is paid to the dependency of the flow development upon the Prandtl number, varied over the range of 5–100, thus representing various fluids. Also, the effect of the magnetic momentum parameter, m, and the quantity γ Ra, upon the flow development obtained in numerical simulations confirms the accuracy of new scaling predictions for paramagnetic fluids.