Rayleigh–Marangoni–Bénard instability of a ferrofluid layer in a vertical magnetic field

Abstract

A horizontal ferrofluid layer is submitted to a lateral heating and to a strong oblique magnetic field. The problem, combining the momentum and heat balance equations with the Maxwell equations, introduces two Rayleigh numbers, Ra the gravitational one and Ra m the magnetic one, to represent the buoyancy and the Kelvin forces, which induce motion, versus the momentum viscous diffusion and heat diffusion. Whatever the inclination of the magnetic field, the steady solution of the problem is presented as a power series of a small parameter ⑀ H measuring the ratio of variation of the magnetization across the layer divided by the magnitude of the external imposed field. For cases of physical relevance, comparisons between analytical and numerical studies have lead to a major statement: in the strong field region ͑⑀ H 1͒ the zero order solution is the product of the Birikh solution that corresponds to the usual Newtonian fluid submitted to a lateral gradient, multiplied by a modulating factor accounting for inclination and both Rayleigh numbers. Physically, this simplified solution is valid for microgravity conditions where the magnetic field competes enough with microgravity effects to invert the laminar flow and thus suppress the motion for two specific values of the inclination angle.

Magnetic fluid rheology and flows

“…aspects of volume and surface forces due to heating, flow rheology, and magnetism [146]. The Faraday instability is the parametric generation of standing waves on the free surface of a fluid subjected to vertical vibrations.…”

Onset of Benard-Marangoni Ferroconvection with Internal Heat Generation

“…The coupling between Marangoni and Rosensweig instabilities by considering two semi-infinite incompressible and immiscible viscous fluids of infinite lateral extent in which one of them is ferromagnetic and the other is a usual Newtonian liquid is addressed by Weilepp and Brand (2001). The effect of different forms of basic temperature gradients on the onset of ferroconvection driven by combined surface tension and buoyancy forces has been discussed by Shivakumara et al (2002), while Hennenberg et al (2005) have discussed RayleighBenard-Marangoni instability in a ferrofluid layer in the presence of a weak vertical magnetic field normal to the boundaries. Shivakumara and Nanjundappa (2006) have analyzed the onset of Marangoni ferroconvection with different initial temperature gradients with the object of understanding control of convection.…”

Thermocapillary Convection in a Deformable Ferrofluid Layer

“…The mixed of Rayleigh-Bénard with Marangoni-Bénard in the presence of temperature-dependent viscosity has been done by Skarda and Mccaughan [3]. Marangoni-Bénard and Rayleigh-Bénard convection in a ferrofluid layer system with magnetic field had been examined by Hennenberg et al, [4].…”