Ferromagnetic Convection in a Densely Packed Porous Medium with Magnetic-Field-Dependent Viscosity – Revisited

1 Introduction

A ferrofluid is a colloidal suspension containing magnetic nanoparticles, with typical dimensions of 10 nm, covered by a surfactant, for preventing their aggregation, and suspended in a non-conducting fluid, e.g. water, kerosene, or an ester. A ferrofluid does not occur in nature but is synthesised in the laboratory. In the recent past, studies on ferrofluids have attracted the attention of many researchers, which has resulted in several interesting applications such as magnetic drug targeting hyperthermia, pressure seals of compressors and blowers, contrast enhancement of magnetic resonance imaging (MRI), novel zero-leakage rotary shaft seals used in computer disk drives, etc. [1], [2].

The magnetic properties of ferrofluids have been studied for a considerable time since the 1930 [3]. Several researchers have contributed to the development of ferrohydrodynamics. Finlayson [4] studied the convective instability of a ferrofluid layer heated from below and derived an exact solution for the case of free boundaries and approximate solutions (for stationary convection) for rigid boundaries. Lalas and Carmi [5] investigated the thermoconvective stability of ferrofluids without considering buoyancy effects. Schwab [6] performed an analytical weakly nonlinear analysis of convective heat transfer in a ferrofluid of infinite magnetic susceptibility between free horizontal boundaries. The thermal convection in ferrofluids has been studied by several authors. For further details on the study of ferroconvection, one may refer to Shliomis [7], Schwab et al. [8], Stiles and Kagan [9], Gupta and Gupta [10], Rudraiah and Shekar [11], Qin and Kaloni [12], Souhar et al. [13], Auernhammer and Brand [14], Aniss et al. [15], Snyder et al. [16], Siddheshwar and Abraham [17], Sunil and Mahajan [18], and Prakash [19], [20], [21].

The study of ferroconvection in a porous medium has also been given considerable attention because of its importance in the controlled emplacement of liquids or treatment of chemicals and emplacement of geophysically imageable liquids into particular zones for subsequent imaging [22]. Rosensweig et al. [23] investigated experimentally the penetration of ferrofluids in a Hele-Shaw cell. Vaidyanathan et al. [24] studied the thermal convection of a ferrofluid layer saturating a porous medium in the presence of a vertical magnetic field using the Darcy–Brinkman model for the case of free boundaries. Sunil et al. [25] carried out the energy stability analysis of thermoconvective magnetised ferrofluids saturating a porous medium using an integral inequality technique. Sekar et al. [26] studied the Soret effect due to thermoconvective instability in a ferrofluid layer using Brinkman and Darcy models. Nanjudappa et al. [27] investigated the influence of Coriolis force on the onset of thermomagnetic convection in a ferromagnetic fluid saturating a porous layer in presence of a uniform vertical magnetic field. Shivakumara et al. [28] studied the effect of vertical heterogeneity on the onset of ferroconvection in a Brinkman porous medium. Mojumder et al. [29] numerically investigated the magneto-hydrodynamic convection in a half-moon-shaped cavity filled with a ferrofluid.

The convection in ferrofluids has gained much importance because of their astounding physical properties. One such property is the viscosity of ferrofluids. Several studies have been undertaken in the recent past considering the influence of magnetic-field-dependent (MFD) viscosity on ferrofluid flows. For details of such investigations, one may refer to Shliomis [7], [30], Vaidyanathan and Sekar [31], Sunil et al. [32], [33], Shivakumara et al. [22], Prakash and Gupta [34], and Prakash [35].

In the present communication, particular attention has been given to the above-cited paper by Sunil et al. [36] on the MFD viscosity, where the analysis has carried out by considering the MFD viscosity in the form μ=μ1(1+δ→⋅B→), where μ₁ is fluid viscosity in the absence of a magnetic field B→, and δ→ is the variation coefficient of viscosity. The authors split μ into the components μ_x, μ_y, and μ_z, which is technically incorrect since μ, being a scalar quantity, cannot be decomposed in such a manner. Though they have studied a very important problem of ferrohydrodynamics, their result cannot be relied upon because of the wrong assumption. Recently, Prakash and Bala [37] and Prakash et al. [38] pointed out this mistake and made corrections in the respective problems. Keeping in view this fact, we have reformulated the basic equations considered by Sunil et al. [36] and then carried out mathematical and numerical analysis to remedy the weaknesses in the existing results.

2 Mathematical Formulation

Consider a ferromagnetic Boussinesq fluid layer of infinite horizontal extension and finite vertical thickness d saturating a densely packed porous medium heated from below and is kept under the action of a uniform vertical magnetic field H→ (see Fig. 1). The flow in the porous medium is described by the Darcy law.

Figure 1:

Geometrical configuration.

The fluid is assumed to be incompressible with a variable viscosity given by μ=μ1(1+δ→⋅B→), where μ₁ is the viscosity of the fluid when there is no magnetic field applied, μ is the MFD viscosity, and B→ is the magnetic induction. The variation coefficient of viscosity δ→ has been taken to be isotropic, i.e. δ₁=δ₂=δ₃=δ. As a first approximation, for small field variation, linear variation of the magneto viscosity has been used. The basic equations that govern the motion of ferromagnetic fluid in a porous medium are given by [36]

where q→,ε, t, p, H→,B→,μ,g→=(0, 0, −​g),k₁, C_V,H, μ₀, T, M→,K₁, and Φ denote, respectively, the velocity, porosity of the medium, time, pressure, magnetic field, magnetic induction, variable viscosity, acceleration due to gravity, permeability of the porous medium, heat capacity at constant volume and magnetic field, magnetic permeability, temperature, magnetisation, thermal conductivity, and viscous dissipation function containing second order terms in velocity gradient.

The equation of state is given by

where ρ₀ is the density of fluid at T=T₀, and α is the coefficient of volume expansion.

For a non-conducting fluid with no displacement current, Maxwell’s equations are given by

where the magnetic induction B→ is given by

Combining (5a) and (6), we get

We assume that the magnetisation is aligned with the magnetic field, but allow a dependence on the magnitude of the magnetic field as well as the temperature as

The linearised magnetic equation of state is given by

where M₀ is the magnetisation when magnetic field is H₀ and temperature T₀,χ= (∂M→∂H→)H0,T0 is magnetic susceptibility, and  K2=−(∂M→∂T)H0,T0 is the pyromagnetic coefficient.

The basic state is assumed to be stationary. Thus the initial stationary state solution is given by

Now, following Finlayson [4], we analyse the stability of the basic state by introducing the following perturbations:

where q→′=(u′, v′, w′), ρ′, p′, θ′, H→′, and M→′ are infinitesimal perturbations in velocity, density, pressure, temperature, magnetic field intensity, and magnetisation. Substituting (11) into (1)–(9) and using the basic state solutions (10), we get the following linearised perturbation equations:

where

and

where ϕ′ is the perturbed magnetic potential, and

where we have assumed K₂βd=(1+χ)H₀.

Eliminating u′, v′, p′ between (13), (14), and (15) and using (12), we obtain

Now (19) and (20) can be combined to obtain

Now we analyse the perturbations w′, θ′, and ϕ′ into two-dimensional periodic waves by considering disturbances characterised by a particular wave number k̅. Thus we ascribe to all quantities describing the perturbation a dependence on x, y, and t of the form

where k_x and k_y are the wave numbers along the x and y directions, respectively, and k¯=kx2+ky2 is the resultant wave number.

Using (23) in (21), (16), and (22) and non-dimensionalising the variables by setting

we obtain (dropping the asterisks for simplicity)

In the above equations, z is the real independent variable such that 0≤z≤1; D is differentiation with respect to z; a² is square of the wave number; P_r>0 is the Prandtl number; ω is the complex growth rate; R>0 is the Rayleigh number; M₁>0 is the magnetic number that defines ratio of magnetic forces due to temperature fluctuation to buoyant forces; M₃>0 is the measure of the nonlinearity of magnetisation; M₂>0 is a non-dimensional parameter that defines the ratio of thermal flux due to magnetisation to magnetic flux; ω=ω_r+iω_i is a complex constant in general such that ω_r and ω_i are real constants and as a consequence the dependent variables w(z)=w_r(z)+iw_i(z), θ(z)=θ_r(z)+iθ_i(z); and ϕ(z)=ϕ_r(z)+iϕ_i(z) are complex valued functions of the real variable z such that w_r(z), w_i(z), θ_r(z), θ_i(z), ϕ_r(z) and ϕ_i(z) are real-valued functions of the real variable z.

Since, M₂ is of very small order [4], it is neglected in the subsequent analysis and thus (26) takes the form

The boundary conditions for stress-free non-conducting boundaries are

It may further be noted that (25) and (27)–(29) describe an eigenvalue problem for ω and govern ferromagnetic convection, with MFD viscosity, in a porous medium (Darcy model) heated from below.

3 Mathematical Analysis

The case of free-free bounding surfaces is discussed in the present analysis. Though the case of two free boundaries is of little physical interest, it is mathematically important because we can obtain an exact solution in this case whose properties guide our analysis. Following the analysis of Finlayson [4], the exact solution satisfying the boundary conditions (29) is given by

w=Asin(πz), θ=Bsin(πz), ϕ=−Cπcos(πz), Dϕ=Csin(πz),

where A, B, and C are constants. Using above solutions in (25), (27), and (28), we obtain a system of three linear homogeneous algebraic equations in the unknowns A, B, and C. For the existence of non-trivial solutions of this system, the determinant of the coefficients of A, B, and C must vanish. This determinant, on simplification, yields

where

Substituting ω=iω_i in (30), when ω_i=0, we obtain condition for stationary convection at the marginal state.

From (30), the Rayleigh number for stationary convection can be easily written as

where R1=Rπ4,x=a2π2,P_l=π²k₁.

When M₁ is very large, the critical magnetic Rayleigh number N_c=R₁M₁ for stationary mode can be obtained as

To find the minimum value of N_c (the critical magnetic Rayleigh number) with respect to wave number, we differentiate (35) with respect to x and equate it to zero, and the following polynomial in x is obtained:

The above equation is solved numerically by using the software Scientific Work Place, for various values of δ and M₃ (see Tab. 1) and the minimum value of x is obtained each time; hence the critical wave number is obtained. Using this in (35), we obtain the critical magnetic Rayleigh number above which the instability sets in as stationary motion.

4 Discussion and Conclusion

In this paper, the effect of MFD viscosity on the thermal convection in a ferrofluid layer heated from below saturating a densely packed porous medium in the presence of uniform vertical magnetic field was studied using the Darcy model. The permeability values as proposed by Walker and Homsy [39] were used. The magnetisation parameter M₁ was considered to be 1000 [36]. The value of M₂, being negligible [4], was taken to be zero. The values of the coefficient of MFD viscosity δ were varied from 0.01 to 0.09, and the values of the parameter M₃ were varied from 1 to 7.

Emphasis was given to a paper published recently by Sunil et al. [36]. These authors had performed their analysis by considering MFD viscosity as μ= μ1(1+δ→⋅B→) and further decomposed μ into components μ_x, μ_y, and μ_z, which is technically wrong because μ being a scalar quantity cannot be decomposed into components. Because of the above fact, the basic equations were reformulated accordingly and then mathematical and numerical analysis was carried out. The results derived here are considerably different from the existing results which were obtained by using wrong assumption.

From Table 1 and Figure 2, it is evident that as the permeability P_l of the medium increases, the critical value of the Rayleigh number N_c=(R₁M₁)_c decreases. Hence the permeability of the medium has a destabilizing effect on the system, a result also obtained by Sunil et al. [36].

Figure 2:

Variation of the critical magnetic Rayleigh number (N_c) with MFD viscosity parameter (δ) for M₃=1 for P_l=0.001 (curve I), P_l=0.002 (curve II), P_l=0.003 (curve III), and P_l=0.004 (curve IV).

It is also found from Table 1 and Figure 3 that, as the magnetisation parameter M₃ increases, the critical value of the Rayleigh number, N_c=(R₁M₁)_c, decreases for lower values of δ and increases for higher values of δ. Therefore, lower values of N_c are needed for the onset of convection with an increase in M₃ for lower values of δ, whereas higher values of N_c are needed for the onset of convection with an increase in M₃ for higher values of δ, hence predicting correctly the destabilizing behaviour of the magnetisation M₃ and the stabilizing behaviour of the MFD viscosity δ.

Figure 3:

Variation of critical magnetic Rayleigh number (N_c) with MFD viscosity parameter (δ) for P_l=0.001, M₃=3 (curve I), M₃=5 (curve II), and M₃=7 (curve III).

The role of the permeability of the medium, the MFD viscosity, and the magnetic parameters can also be illustrated with the help of Figures 4–8. From Figures 4 and 5, it is clear that the permeability of the medium prepones the onset of convection, whereas the MFD viscosity postpones the onset of convection as the Rayleigh number decreases and increases with the increase in the permeability parameter and the MFD viscosity parameter, respectively.

Figure 4:

Variation of Rayleigh number (R₁) with wave number (x) for M₁=1000, M₃=1; δ=0.05; P_l=0.001 (curve I), P_l=0.002 (curve II), P_l=0.003 (curve III), and P_l=0.004 (curve IV).

Figure 5:

Variation of Rayleigh number (R₁) with wave number (x) for M₁=1000, P_l=0.001, M₃=1; δ=0.01 (curve I), δ=0.03 (curve II), δ=0.05 (curve III) and δ=0.07 (curve IV).

Figure 6:

Variation of Rayleigh number (R₁) with wave number (x) for M₁=1000, P_l=0.001, δ=0; M₃=1 (curve I), M₃=3 (curve II), M₃=5 (curve III), and M₃=7 (curve IV).

Figure 7:

Variation of Rayleigh number (R₁) with wave number (x) for M₁=1000, P_l=0.001, δ=0.05; M₃=1 (curve I), M₃=3 (curve II), M₃=5 (curve III), and M₃=7 (curve IV).

Figure 8:

Variation of Rayleigh number (R₁) with wave number (x) for M₃=1, P_l=0.001, δ=0.01; M₁=0 (curve I), M₁=1 (curve II), M₁=5 (curve III), and M₁=10 (curve IV).

From Figures 6 and 7, it is clear that the magnetisation prepones the onset of convection in the absence of MFD viscosity as the Rayleigh number decreases with the increase in the magnetisation parameter, whereas in the presence of MFD viscosity, the magnetisation prepones the onset of convection for smaller values of wave numbers and postpones the onset of convection for higher values of wave numbers.

From Figure 8, it is clear that the Rayleigh number decreases with increase in the parameter M₁, thereby showing its destabilizing effect on the system. Thus, M₁ prepones the onset of convection, and in its absence, i.e. M₁=0, higher values of R₁ are needed for the onset of convection.