Effect of hall current on hydromagnetic instability of a couple-stress ferromagnetic fluid in the presence of varying gravitational field through a porous medium

Introduction

The growing importance of the usage of non-Newtonian fluids in modern technology and industries has led various researchers to strive numerous flow problems related with several non-Newtonian fluids. One such fluid that has attracted research workers during the last four decades is the couple-stress fluid. The theory of couple-stress fluids initiated by Stokes [1] is a generalization of the classical theory of viscous fluids, which allows for the presence of couple-stresses and body couples in the fluid medium. Sunil et al. [2] have mentioned the effect of Hall currents on thermosolutal instability of compressible Rivlin-Ericksen fluid. Rani and Tomar [3,4] have investigated thermal and thermosolutal convection problem of micropolar fluid subjected to Hall current. Kumar [5] have examined the effect of Hall currents on thermal instability of compressible dusty visco-elastic fluid saturated in a porous medium subjected to vertical magnetic field. There are various stability problems on ferromagnetic fluids. The convective instability, also known as Bénard convection (Chandrasekhar [6] ) is one of the instability of ferromagnetic fluid. Rosensweig [7] has given a definite introduction about magnetic liquids in his monograph. Finlayson [8] have studied the convective instability of ferromagnetic fluid for a fluid layer heated from below in the presence of uniform vertical magnetic field. The effect of Hall current on thermal instability has been mentioned by many authors (Raghavachar et al. [9] , Sharma et al. [10] , Gupta et al. [11,12] ). Aggarwal and Makhija [13] have studied the effect of Hall current on thermal stability of ferromagnetic fluids heated from below in porous medium in the presence of horizontal magnetic field.

In the present study, we have discussed the effect of Hall current on hydromagnetic instability of couple-stress ferromagnetic fluid in the presence of variable gravity field saturating in a porous medium. Recently, some stability problems about couple-stress ferromagnetic fluid have been discussed by Nadian et al. [14][15][16] . In the present problem, we have assumed that the gravity is varying as

where 0 g is the value of g at the surface of the Earth, which is always positive and λ can be positive or negative.

Mathematical Formulation of the Problem

We consider an infinite, incompressible, electrically non-conducting and thin layer of couple-stress ferromagnetic fluid which is bounded by the planes 0 z = and z d = as shown in Fig. 1. The fluid layer is heated from below so that a uniform temperature gradient dT dz β = is maintained within the fluid. The whole system is acted upon by a uniform vertical magnetic field ( ) 0, 0, H H and variable gravity field ( )

. Furthermore, the couple-stress ferromagnetic fluid layer is assumed to be flowing through an isotropic and homogeneous porous medium of porosity ε . ( ) ( )

, , u u u v is the fluid velocity, p is the fluid pressure, ρ is the fluid density, 0 ρ is the reference density, T is the temperature, 0

T is the reference temperature, α is the thermal coefficient of expansion, e µ is the magnetic permeability, ' µ is the couple-stress viscosity, ν is the kinematic viscosity, κ is the thermal diffusivity, ( )

ρ denote the density of the solid (porous) material, heat capacity of the solid (porous) material and heat capacity of the fluid at constant volume), M is the magnetization, ∇H is the magnetic field gradient.

In presence of Hall current, the Maxwell's equations are given by,

And . 0, ∇ = H (6) where, H is the magnetic field intensity. We consider that the magnetization is independent of magnetic field intensity but depend upon the temperature so that ( )

As a first approximation, we consider that ( ) 7) http://www.mathsjournal.com where,

 and 0 M is the magnetization at 0 T T = with 0 T being the reference temperature.

Basic State and Perturbation Equations

The basic state of which we wish to examine the stability is characterized by,

Here, β may be either positive or negative.

The perturbed flow may be represented as,

and M δ denote respectively the perturbations in fluid velocity, magnetic field, temperature, density, pressure and magnetization. Linearizing the equations in perturbation and reading the perturbation into normal modes, we anticipate that the perturbation quantities are of the form,

is the resultant wave number of the disturbance and n is the frequency of any arbitrary disturbance (that is generally a complex constant).

We eliminate the physical quantities using the non-dimensional parameters

Now, eliminating , , , X K Z Θ among Eqs. ( 11) -( 15), we obtain the stability governing equation,

is the Rayleigh number for ferromagnetic fluid,

is the Hall current parameter. Now, the appropriate boundary condition (when we take the case of two free boundaries) are,

Here, it is obvious that all even order derivatives of W at the boundaries vanish. Therefore, the solution of Eq. ( 16) characterizing the lowest mode is,

, where 0 W is a constant. (18)

Now, using Eq. ( 18) in Eq. ( 16), we get, ( ) ( ) ( ) ( )

Analytical Discussion

 Stationary Convection

When stability sets in at stationary convection, the marginal state will be characterized by 1 0 σ = . So, put 1 0 σ = in Eq. ( 19), we get, ( ) ( ) ( )

Now, to study the effect of medium permeability parameter, couple-stress parameter, magnetic field parameter and Hall current parameter, we examine the behavior of

which indicates that medium permeability has a stabilizing effect on the system under the condition,

. Also, medium permeability has a destabilizing effect on the system under the condition,

In the absence of magnetic field, Eq. ( 21) becomes,

which indicates that medium permeability has a stabilizing effect for 0 λ < and destabilizing effect for 0 λ > .

which indicates that couple-stress has a stabilizing effect on the system under the condition,

Also, couple-stress has a destabilizing effect on the system under the condition,

In the absence of magnetic field or hall current, Eq. ( 22) becomes, ( )

which clearly indicates that couple-stress has a stabilizing effect for 0 λ > and destabilizing effect for 0 λ < .

which clearly indicates that magnetic field has a stabilizing effect on the system for 0 λ > and destabilizing effect for 0 λ < .

which clearly indicates that Hall current has a stabilizing effect on the system for 0 λ < and destabilizing effect for 0 λ > .

To see the effect of magnetization, we examine the effect of 0 dR dM analytically.

which clearly indicates that magnetization has a stabilizing effect on the system for both 0 λ > and 0 λ < .

 Oscillatory convection

Multiplying Eq. ( 11) by the conjugate of W i.e. * W and integrate over the range of z and making use of Eqs. ( 12) -( 15) together with boundary conditions (17), we get, ( ) ( )

then the terms in bracket are positive definite, which implies that 0 i σ = . Therefore, oscillatory modes are not allowed and the principle of exchange of stabilities is satisfied if

Numerical Discussion

The dispersion relation ( 20) is analyzed numerically also. The numerical value of thermal Rayleigh number 1 R is decided for numerous values of medium permeability parameter P , couple-stress parameter 1 F , magnetic field parameter 1 Q , Hall current parameter h M and magnetization parameter 0 M . Also, graphs have been plotted between 1 R and these parameters as shown in Figures ( 2) -( 15),

Medium Permeability (P) -3000 In fig. 2, critical Rayleigh number 1 R increases with increase in medium permeability parameter P for 50 λ = , which indicates that medium permeability has a stabilizing effect on the system. In fig. 3, critical Rayleigh number 1 R decreases with increase in medium permeability parameter P for 2 λ = , which indicates that medium permeability has a destabilizing effect on the system. In fig. 4, critical Rayleigh number 1 R increases with increase in medium permeability parameter P for 5 λ = -, which indicates that medium permeability has a stabilizing effect on the system. In fig. 5, critical Rayleigh number 1 R decreases with increase in medium permeability parameter P for 0.00001 λ = -, which indicates that medium permeability has a destabilizing effect on the system. In fig. 6, critical Rayleigh number 1 R increases with increase in couple-stress parameter 1 F for 5 λ = , which indicates that couplestress has a stabilizing effect on the system. In fig. 7, critical Rayleigh number 1 R decreases with increase in couple-stress parameter 1 F for 50 λ = , which indicates that couple-stress has a destabilizing effect on the system. In fig. , which indicates that couple-stress has a destabilizing effect on the system. In fig. 10, critical Rayleigh number 1 R increases with increase in magnetic field parameter 1 Q for 3 λ = , which indicates that magnetic field has a stabilizing effect on the system. In fig. 11, critical Rayleigh number 1 R decreases with increase in magnetic field parameter 1 Q for 15 λ = -, which indicates that magnetic field has a destabilizing effect on the system. In fig. 12, critical Rayleigh number 1 R decreases with increase in hall current parameter h M for 4 λ = , which indicates that magnetic field has a destabilizing effect on the system. In fig. 13, critical Rayleigh number 1 R increases with increase in hall current parameter h M for 0.5 λ = -, which indicates that magnetic field has a stabilizing effect on the system. In fig. 14, critical Rayleigh number 1 R increases with increase in magnetization parameter 0 M for 0.2 λ = , which indicates that magnetic field has a stabilizing effect on the system. In fig. 15, critical Rayleigh number 1 R increases with increase in magnetization parameter 0 M for 0.25 λ = -, which indicates that magnetic field has a stabilizing effect on the system.

Conclusions

The main results from the evaluation of the present paper are as below:  For stationary convection, 1. Medium permeability has both stabilizing and destabilizing effect on the system for 0 λ > and 0 λ < under certain conditions. Furthermore, in the absence of magnetic field, medium permeability has a stabilizing effect on the system for 0 λ < and destabilizing effect for 0 λ > . 2. Couple-stress has both stabilizing and destabilizing effect on the system for 0 λ > and 0 λ < under certain conditions. Furthermore, in the absence of magnetic field, couple-stress has a stabilizing effect on the system for 0 λ > and destabilizing effect for 0 λ < . 3. Magnetic field has a stabilizing effect on the system for 0 λ > and destabilizing effect for 0 λ < . 4. Hall current has a stabilizing effect on the system for 0 λ < and destabilizing effect for 0 λ > . 5. Magnetization has a stabilizing effect on the system for both 0 λ > and 0 λ < .  The principle of exchange of stabilities is not valid for the present problem under consideration, whereas in the absence of magnetic field (hence hall current), it is valid for the present problem if 0 0 0 . M H g γ λ ρ α ∇ ≥

Acknowledgment

The corresponding author acknowledges CSIR, New Delhi for providing financial assistance through JRF Vide letter No. 08/688(0001)/2018-EMR-I.

References

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