Rotating Rayleigh–Be ´nard convection under the influence of transverse magnetic field Hirdesh Varshney, Mirza Faisal Baig * Department of Mechanical Engineering, A.M.U., Aligarh 202 002, India Received 19 January 2007; received in revised form 13 August 2007 Available online 7 March 2008 Abstract In the present numerical study the effect of constant transverse magnetic field on convection of low Prandtl number liquid metal rotat- ing in a cubical cavity with an aspect-ratio of 8:8:1 has been investigated. The bottom wall is heated while the top-wall is cooled and all the other walls are kept thermally insulated. The governing equations of mass, momentum, energy and magneto-hydrodynamic for a frame rotating with the enclosure, subject to Boussinesq approximation applied to gravity and centrifugal force terms, have been solved on a collocated grid using a semi-implicit finite difference method. The simulations have been carried out for liquid metal flows having a fixed Prandtl number Pr ¼ 0:01, Rayleigh number Ra ¼ 10 7 , and magnetic Prandtl number Pm ¼ 4:0  10 4 while Chandrasekhar num- ber Q varies from 5:0625  10 4 to 1:21  10 6 and non-dimensional rotation rate X is varied from zero to 10 5 . The increase in strength of transverse magnetic field (from Q l ¼ 5:0625  10 4 to Q h ¼ 1:21  10 6 ) till Q ’ Ta leads to slight increase in convective heat transfer as well as formation of two-dimensional coherent structures aligned along the direction of magnetic field. For cases pertaining to Q < Ta the two-dimensionality of the flow breaks down and the rolls distort in their alignment which leads to decrease in magnitude of vertical heat transfer. For cases where Q  Ta, the increased Coriolis forces lead to generation of large-scale circulation which forms a large cylindrical rotating column of fluid in consonance with Taylor–Proudman theorem. On increasing the strength of magnetic field the component of rms velocity in the direction of magnetic field gets suppressed while there is increase in other two components. Ó 2008 Elsevier Ltd. All rights reserved. Keywords: Rotating Rayleigh–Be ´nard convection; Transverse magnetic field; Two-dimensional coherent structures; Coriolis forces 1. Introduction The turbulent Rayleigh–Be ´nard convection is primarily due to the instability of Boussinesq fluid. The turbulent convection with rotation is an important phenomena in many industrial applications as well as in astrophysical and geophysical flows. Further introduction of magnetic field makes the flow more complex and can have profound effect on the convection. In the present paper we focus on the behavior of flow of an electrically conducting Bous- sinesq liquid metal in a rotating rectangular enclosure heated from bottom while being subjected to an applied magnetic field transverse to the temperature gradient. The controlling non-dimensional parameters in rotating magneto-convection are  Rayleigh number Ra ¼ agH 3 MT mj is the ratio of buoyancy forces to viscous forces and represents the driving force of convection where a is the thermal expansion coeffi- cient, g is the magnitude of the acceleration due to grav- ity, m is the kinematic viscosity and j ¼ k=qc p is the thermal diffusivity with the thermal conductivity k, den- sity q and the specific heat capacity c p .  Chandrasekhar number Q ¼ Ha 2 ¼ rH 2 B 2  q  m is the square of the Hartmann number Ha and represents the ratio of Lorentz forces F L ¼ j  B, that are produced by the 0017-9310/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2007.11.046 * Corresponding author. Tel.: +91 9897517071. E-mail address: drmfbaig@yahoo.co.uk (M.F. Baig). www.elsevier.com/locate/ijhmt Available online at www.sciencedirect.com International Journal of Heat and Mass Transfer 51 (2008) 4095–4108