The effect of magnetic field dependent viscosity on ferroconvection in a rotating sparsely distributed porous medium

doi:10.1016/S0304-8853(02)00355-4

Journal of Magnetism and Magnetic Materials

Volume 250, September 2002, Pages 65-76

https://doi.org/10.1016/S0304-8853(02)00355-4 Get rights and content

Abstract

The effect of magnetic field dependent viscosity on ferroconvection in a rotating sparsely distributed porous medium has been studied. The fluid layer is heated below and a linear stability analysis is being carried out for both stationary and oscillatory modes for various parameters. It is found that the increase in magnetisation dependent viscosity stabilises the system, which means the convection is inhibited. Numerical computations are made and illustrated graphically.

Introduction

In the last millennium, the investigation on the interaction of electromagnetic fields with fluids attracted researchers because of the increase of applications in areas such as chemical reactor, engineering, medicine and high-speed silent printers, etc. Ferrohydrodynamics deals with the interaction of magnetic fields on non-conducting ferromagnetic fluids, which has aroused a lot interest [1]. Much work has been carried out on the flow in the devices in engineering and industrial applications such as magneto fluid seals [2], [3], magnetic levitation [4] and energy conversion [5].

Shliomis [6] obtained linearized relation for normalized perturbation on velocity, temperature and pressure valid at the limit of instability. Lalas and Carmi [7] studied thermoconvective instability without considering buoyancy effects. Finlayson [8] investigated the convective instability of ferromagnetic fluids heated from below in the presence of vertical uniform magnetic field. Vaidyanathan et al. [9] studied ferroconvective instability of fluids saturating a porous medium of very large permeability. Sekar et al. [10] analysed the ferroconvection in fluids saturating a rotating densely packed porous medium.

Thermal convection of ferrofluids saturating a porous medium has vast applications in ferromagnetic fluids entrapped in earth crust. Many investigators analysed the effect of rotation in ordinary fluids saturating a rotating porous medium bounded by rigid and free boundaries. Shliomis [11] and Shliomis et al. [12] analysed the variation of magnetic viscosity.

All the above investigators did not consider the effect of magnetic field dependent viscosity on ferroconvection in a rotating sparsely distributed porous medium. In the present investigation, the authors have studied the effect of magnetic field dependent viscosity on ferroconvection of ferrofluid in a rotating sparsely distributed porous medium. The fluid layer is heated from below. A linear stability analysis is used. Investigations are made for both stationary and oscillatory modes for various values of coefficient of field dependent viscosity, Taylor number, and magnetization. Numerical computations are made and are illustrated graphically also.

Section snippets

Mathematical formulation

Consider an infinitely spread horizontal layer of thickness d of ferromagnetic Boussinesq fluid saturating a rotating sparsely distributed porous medium heated from below. A uniform magnetic field H₀ acts along the vertical direction which is taken as z-axis.

The fluid is assumed to be incompressible fluid having a variable viscosity, given by $η=η_{1} (1+ δ·B)$ , where η₁ is taken as viscosity of the fluids when the applied magnetic field is absent.

The variation coefficient of viscosity $δ$ has been

Exact solution for free boundaries

The boundary conditions for stress-free non-conducting boundaries are $w*=D^{2} w*=T*=Dw*=Dφ*=0 at z=0 and z=1.$

Following the analysis of Finlayson [8], the exact solution satisfying the boundary condition (23) is then $w*=A e^{σt*} cos πz*, T*=B e^{σt*} cos πx*,Dφ*=C e^{σt*} cos πz*, φ*= C π e^{σt*} sin πz*,$ where A, B and C are constants and σ is the time factor. Substitution of Eq. (24) in , and dropping asterisks, for convenience leads to $aR^{1/2} (1−M_{2})A−[π^{2} +a^{2} +Pr σ]B+M_{2} Pr σC=0,$ $−π^{2} B+(π^{2} +a^{2} M_{3})C=0.$

On elimination of ζ* from , and using

Discussion

The effect of magnetic field dependent viscosity on ferroconvection in a rotating sparsely distributed porous medium is studied by using Brinkman model. The permeability values proposed by Walker and Homsy [13] have been taken for the analysis. The Prandtl number Pr is varied from 0.01 to 0.09. The Taylor number Ta which decides the rotation, is varied from 1 to 10¹⁰. The magnetization parameter M₁ is assumed to be 1000 [14]. For these fluids M₂ is assumed to have negligible value and hence

Acknowledgements

The authors are highly thankful to the refree's comments which enabled them to bring the paper in the present form.

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The effect of magnetic field dependent viscosity on ferroconvection in a rotating sparsely distributed porous medium

Abstract

Introduction

Section snippets

Mathematical formulation

Exact solution for free boundaries

Discussion

Acknowledgements

Int. J. Eng. Sci.

Int. J. Eng. Sci.

J. Magn. Magn. Mater.

Ferrohydrodynamics

Mach. Des.

ASLE Trans.

Nature (London)