Numerical modelling of axisymmetric electromagnetically driven flows in thin layers

Abstract

We present the results of the numerical modelling of deceptively simple steady axisymmetric electromagnetically driven flows in thin disk-like layers of a weakly conducting electrolyte. The fluid motion is caused by an azimuthally acting Lorentz force appearing when a radial current flows in the electrolyte layer placed on top of a magnet with a vertical polarisation. The small layer thickness and the circumferential direction of the driving force suggest that the flow in such a system should be essentially uni-directional. However, it was found that not only is the flow fully three-dimensional, but multiple solutions can exist for the same set of governing parameters. References R. M. Digilov, Making a fluid rotate: circular flow of a weakly conducting fluid induced by Lorentz force. Am. J. Phys. , 75(4) :361–367, 2007. doi:10.1119/1.2372472 F.V. Dolzhanskii, V. A. Krymov and D. Y. Manin, Stability and vortex structures of quasi-two-dimensional shear flows. Sov. Phys. Usp. , 33(7) :495–520, 1990. doi:10.3367/UFNr.0160.199007a.0001 V. A. Dovzhenko, V. A. Krymov and V. M. Ponomarev, Experimental and theoretical study of a shear flow driven by an axisymmetric force. Izv. Akad. Nauk SSSR Fiz. Atm. Okeana (in Russian) , 20(8) :693–704, 1984. D. Hatziavramidis and H. C. Ku, An integral Chebyshev expansion method for boundary-value problems of O.D.E. type. Comput. Math. Appl. , 11(6) :581–586, 1985. doi:10.1016/0898-1221(85)90040-9 H. C. Ku and D. Hatziavramidis, Chebyshev expansion methods for the solution of the extended Graetz problem. J. Comput. Phys. , 56 :495–512, 1984. doi:10.1016/0021-9991(84)90109-8 J. Perez-Barrera, J. E. Ortiz, E. Ramos and S. Cuevas, Instability of electrolyte flow driven by an azimuthal Lorentz force. Magnetohydrodynamics , 51(2) :203–213, 2015. http://mhd.sal.lv/contents/2015/2/MG.51.2.4.R.html S. A. Suslov and S. Paolucci, Stability of mixed-convection flow in a tall vertical channel under non-Boussinesq conditions. J. Fluid. Mech. 302 :91–115, 1995. doi:10.1017/S0022112095004022 S. A. Suslov and S. Paolucci, Stability of natural convection flow in a tall vertical enclosure under non-Boussinesq conditions. Int. J. Heat Mass Transfer , 38 :2143–2157, 1995. doi:10.1016/0017-9310(94)00348-Y S. A. Suslov, J. Perez-Barrera and S. Cuevas, Electromagnetically driven flow of electrolyte in a thin annular layer: axisymmetric solutions. J. Fluid Mech. , to appear, 2017.