The effect of magnetic field dependent viscosity on ferroconvection in a rotating sparsely distributed porous medium
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 , where η1 is taken as viscosity of the fluids when the applied magnetic field is absent.
The variation coefficient of viscosity has been
Exact solution for free boundaries
The boundary conditions for stress-free non-conducting boundaries are
Following the analysis of Finlayson [8], the exact solution satisfying the boundary condition (23) is thenwhere A, B and C are constants and σ is the time factor. Substitution of Eq. (24) in , and dropping asterisks, for convenience leads to
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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