Rotating magnetic field effect on convection and its stability in a horizontal cylinder subjected to a longitudinal temperature gradient

摘要

A rotating magnetic field (RMF) is used in crystal growth applications during the solidification process in order to improve the crystal quality. Its influence on the convective flows in molten metals and on their stability is studied here in the case of a horizontal infinite cylindrical channel subjected to a longitudinal temperature gradient. The steady convective flows, which correspond to the usual longitudinal counterflow structure, with four vortices in the cross-section for non-zero Prandtl number, Pr, are modified by the RMF (parametrized by the magnetic Taylor number Tam). For zero Prandtl number, the flow in the cross-section corresponds to circular streamlines and the longitudinal flow structure is moved in the direction of the magnetic field rotation, with a decrease in its intensity and an asymptotic variation as 1/Tam. For non-zero Prandtl numbers, depending on the respective values of Tam on one side and Prandtl and Grashof numbers on the other side, different structures ranging from the circular streamlines with transport by rotation of the longitudinal velocity and the temperature field, to the more usual counterflow structure almost insensitive to the RMF with four cross-section vortices, can be obtained. The decrease in the flow intensity with increasing Tam is also delayed for non-zero Pr, but the same asymptotic limit is eventually reached. The stability analysis of these convective flows for Tam = 0 shows a steep increase of the thresholds around Pr = Prt,0≈ 3 10~(-4), corresponding to the transition between the usual counterflow shear mode and a new sidewall shear mode. This transition is still present with an RMF, but it occurs for smaller Pr values as Tam is increased. Strong stabilizing effects of the rotating magnetic field are found for Pr < Prt,0, particularly for Pr = 0 where an exponential increase of the threshold with Tam is found. For Pr > Prt,0 (i.e. in the domain where the sidewall instability is dominant), in contrast, the stabilization by the RMF is weak.