CFD solutions for magnetohydrodynamic natural convection over horizontal and vertical surfaces Kaustav Pradhan ⁎, Abhijit Guha Mechanical Engineering Department, Indian Institute of Technology Kharagpur, Kharagpur 721302, India abstract article info Article history: Received 18 March 2016 Received in revised form 7 March 2017 Accepted 29 March 2017 Available online 02 April 2017 This paper investigates the effects of a magnetic field on the natural convective boundary layer flow of an electri- cally conducting fluid adjacent to horizontal as well as vertical surfaces. This has allowed us to establish overall similarities and several subtle differences between the two cases. Previously published studies concentrated on obtaining self-similar solutions at the cost of assuming very restrictive variation of the magnetic field along the surface. In the present work, a numerical model and an in-house computer program have been developed to solve directly the non-linear boundary layer equations which can accommodate any arbitrary variation of the magnetic field. Special emphasis is given to the case of uniform magnetic field which perhaps represents the most practical case and which cannot be solved by the similarity theory. Computations show that the Nusselt number and the skin-friction coefficient decrease as the magnetic field increases. It is shown that the detailed characteristics of the velocity profiles and the values of Nusselt number and skin-friction coefficient for the case of a magnetic field which admits similarity are significantly different from those when a uniform magnetic field is applied, thus showing the importance of the present model. © 2017 Elsevier B.V. All rights reserved. Keywords: MHD Natural convection Non-similarity CFD Time-marching 1. Introduction Magnetohydrodynamics (MHD) is the study of the interaction be- tween a moving fluid and a magnetic field. When a magnetic field is ap- plied perpendicular to the main flow direction, the magnetic lines offer a resistance to the flow and cause a retardation [1]. The study of magne- tohydrodynamic natural convection has gained much importance due to its application in the field of geophysical engineering, enhanced oil recovery and nuclear sciences [2]. The flow control achieved by the ap- plication of a magnetic field is of particular use in metallurgical and polymer processing industries [3]. Relevant examples of Newtonian fluids, for which a magnetic field may have an effect, include liquid metals, ionized gases, electrolytic solutions and certain water-based nanofluids. The laminar natural convection of electrically conducting fluids past a heated vertical surface in the presence of a magnetic field has been studied by many researchers [4–7]. Riley [4] used a method of “matching ‘outer’ and ‘inner’ solutions in the moving layer of fluid” in his studies for strong magnetic fields. Lykoudis [6] obtained similarity solutions for a specific variation of the magnetic field. Sparrow and Cess [7] found that the application of a magnetic field significantly af- fects the free convection heat transfer to liquid metals. Self-similar solutions for magnetohydrodynamic natural convection past a vertical plate exist only when the strength of the magnetic field varies as the in- verse of the fourth root of the distance from the leading edge [5,6]. In spite of the existence of numerous studies on the magnetohydrodynam- ic natural convection over a vertical plate, the effect of a uniform mag- netic field (which is of greater physical significance but does not admit self-similar solutions) has not been investigated thoroughly. Natural convective boundary layer flow over a horizontal surface is quite different from its counterpart on a vertical surface and the flow is set up indirectly by the buoyancy force acting normal to the surface. This is why Schlichting and Gersten referred to this as “indirect natural convection” [8]. Theoretical and numerical studies of this type of flow for various types of fluids and boundary conditions may be found in [9–15]. Natural convection over a heated horizontal surface under the influence of a vertical magnetic field has been analysed by Gupta [16] using the momentum-integral method. Gupta [16] considered self- similar solutions for two cases: (i) the magnetic field varying as the in- verse of the two-fifth power of the distance along the plate (from the leading edge) when the surface temperature is constant, and, (ii) the temperature difference varying as the square of the distance and the boundary layer thickness being held constant for a uniform magnetic field. Similar studies using the integral technique have been performed by Singh [17,18] and Singh and Cremers [19]. Samanta and Guha [20] performed a similarity analysis for the magnetohydrodynamic natural convection over an isothermal horizontal plate, assuming the magnetic field to vary as the inverse of the two-fifth power of the distance along Journal of Molecular Liquids 236 (2017) 465–476 ⁎ Corresponding author. E-mail addresses: kaustav.pradhan@mech.iitkgp.ernet.in (K. Pradhan), a.guha@mech.iitkgp.ernet.in (A. Guha). http://dx.doi.org/10.1016/j.molliq.2017.03.110 0167-7322/© 2017 Elsevier B.V. All rights reserved. Contents lists available at ScienceDirect Journal of Molecular Liquids journal homepage: www.elsevier.com/locate/molliq