Triple- Diffusive Convection in a Magnetized Ferrofluid with MFD Viscosity Saturating a Porous Medium: A Nonlinear Stability Analysis

Suresh Chand¹, S.K. Kango² and Vikram Singh³

¹Department of Mathematics, SCVB Govt. College, Palampur, HP-176061 India

²Department of Mathematics, Govt. College, Haripur (Manali), HP-175136 India

³Department of Mathematics, Jwalaiji Degree College , Jwalamukhi, HP-176072 India

*Corresponding Author:

ABSTRACT:

A nonlinear stability analysis is performed for a triple- diffusive convection in a magnetized ferrofluid with magnetic field –dependent viscosity (MFD)saturating a porous for stress- free boundaries. The major mathematical emphasis is on how to control the non-linear terms caused by magnetic body force and inertia forces. A suitable generalized energy functional is introduced to perform the nonlinear energy stability analysis. It is found that nonlinear critical stability magnetic thermal Rayleigh number does not coincide with that of linear instability, and thus indicate that the subcritical instabilities are possible. However, it is noted that in case of non-ferrofluid global nonlinear stability Rayleigh number is exactly same as that of linear instability. For lower values of magnetic parameters, this coincidence is immediately lost. The effects of magnetic parameter , solute gradients & , Darcy number D_a and MFD viscosity parameter _, on the subcritical instability region have also been analyzed. The solutes gradients & have stabilizing effect, _, increases as solute gradients increases and Darcy number has a destabilizing effect, _, decreases as D_a increases. It has also been observed that in the presence of MFD viscosity ( , , decrease for lower values of and increase for higher values of .

KEY WORDS: Nonlinear stability, magnetized ferrofluid, triple- diffusive convection, MFD viscosity , , porous medium, magnetization.

CONCLUSIONS:

In this paper a nonlinear stability analysis of triple- diffusive convection in a magnetized ferrofluid with magnetic field –dependent viscosity has been investigated. It has been observed that the boundaries of nonlinear stability and linear instability analyses do not intersect. The MFD viscosity and solute gradients always delay the onset of convection, whereas medium permeability hastens the onset of convection. We have derived a nonlinear stability threshold very close to the linear instability one. It has been seen that the magnetic mechanism alone can induce subcritical region of instability. The comparison between the linear and energy stability reveals that for convection problem in ferrofluids, the linear critical magnetic thermal Rayleigh number is higher in values than the nonlinear (energy) critical magnetic thermal Rayleigh number, which shows the possibility of the existence of subcritical instability. It is important to realize that the subcritical instability region decreases as magnetization and medium permeability increase, whereas with the increase of MFD viscosity and the solute gradients, the gap between the linear and energy stability widens. We also observe that solute gradients cannot induce subcritical region of instability, but in magnetic mechanism, this region expands with the increase of solute gradients. In non-ferrofluids, it is verified that the global stability Rayleigh number is exactly the same as that of linear instability.

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Received on 20.01.2013 Accepted on 05.02.2013

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