1. INTRODUCTION

Ferrofluids are unique types of fluids that exhibit both fluidic and magnetic properties. They are composed of stable colloidal suspensions of single-domain ferromagnetic or ferrimagnetic nanoparticles dispersed in a suitable liquid medium. The magnetic particles typically consist of ferromagnetic metals such as iron, nickel, and cobalt, or ferrimagnetic oxides like magnetite (Fe₃O₄) and spineltype ferrites. Common carrier liquids include water, ethylene glycol, and various oils. The specific combination of magnetic particles and carrier fluid is chosen based on the intended application of the ferrofluid. Ferrofluids are widely used in sealing, damping, heat transfer, bearings, and sensors. They act as coolants in systems like loudspeakers and transformers, and enhance heat transfer in devices such as heat exchangers. Their use in exclusion seals helps protect sensitive components, making them valuable in robotics, textiles, electronics, and machinery [1, 2].

Due to their vast practical applications, researchers have conducted numerous experimental and theoretical studies on ferrofluids. These studies cover areas such as synthesis and characterization, heat transfer and thermal properties, theoretical modelling and simulation, and industrial applications [3, 4, 5]. Discussing the thermal stability of ferrofluids is equally important as examining their other properties. Understanding thermal stability ensures that ferrofluids can be effectively and safely utilized in current and future technologies.

Thermal stability of ferrofluid in the existence of a vertical magnetic field has been investigated by Finlayson [6]. Schwab et al. examined experimentally the Finlayson’s problem [7]. Thermal stability of ferrofluid depends upon many factors such as density of ferrofluid, gravity acting on ferrofluid, medium of ferrofluid, Coriolis force on ferrofluid and hydrodynamic boundary conditions of ferrofluid. The influence of viscosity variation with magnetic field on thermal convection of ferrofluid has been explored by Sunil et al. [8]. whereas the viscosity variation with temperature field on thermal convection of ferrofluid for general boundary conditions has been investigated by Dhiman and Sharma [9, 10]. The porous medium plays a crucial role in directing geophysically detectable liquids into specific zones for imaging, controlled placement, or chemical treatment. Hence, analyzing the thermal stability of ferrofluids within a saturated porous layer is equally important. Vaidyanathan et al. [11] explored thermoconvective instability in a ferromagnetic fluid saturating a porous medium under the influence of a vertical magnetic field. Since rotation can significantly affect the thermal stability of a fluid layer, Venkatasubramanian and Kaloni [12] examined the impact of rotation on thermo-convective instability in a ferrofluid layer. Many authors have made remarkable contributions to exploring the impact of various parameters on ferrofluid convection (see references [13, 14, 15, 16, 17]).

There are many practical situations where ferrofluids may contain suspended particles. For instance, in industrial settings, ferrofluids are sometimes used as lubricants for machinery, particularly in high-precision equipment like bearings, seals, and pumps. However, in such environments, it is common for the ferrofluid to become contaminated with suspended particles. Therefore, studying ferrofluids with dust particles is equally important. Thermal stability of ferrofluid in the existence of dust particles is studied by Sunil et al. [18]. Sunil et al. [19] explored the impact of viscosity variation with magnetic field on the thermal convection of dusty ferrofluids for the case of free-free boundaries. Sunil et al. [20] conducted a theoretical study on how magnetic field-dependent (MFD) viscosity influences thermal convection in a ferromagnetic fluid saturated porous medium containing dust particles. Sharma and Kumar [21] carried out a theoretical analysis of the combined influence of MFD viscosity and rotation on ferroconvection in a dusty fluid exposed to a uniform transverse magnetic field, whereas Kumar et al. [22] explored the impact of viscosity variation with temperature on the thermal convection of dusty ferrofluids of permeable boundaries. For a detailed understanding of thermal convection in dusty ferrofluids under various effects, one may refer to references [23, 24, 25, 26]. For further insights into related studies involving fluid systems containing dust particles and their significant effects, readers are referred to references [27, 28, 29, 30, 31].

The thermal stability of the liquid is determined by the thermal and hydrodynamic conditions at the surfaces that border it. In past works, fluid boundaries have been mainly either free-free or rigidrigid. However, in many cases boundaries are neither purely free-free nor purely rigid-rigid. For instance, porous structures are used in cooling systems of electronic devices so as to facilitate ferrofluid movement and controlled heat transfer. Magnetic seals in rotating machinery consist of field-permeable membranes that maintain fluid position while allowing for fluid exchange and pressure fluctuations. For example, targeted drug delivery and hyperthermia treatment require ferrofluids to pass through semi-permeable biological membranes. Also heat exchangers, magnetic field-controlled filtration systems, ferrofluid-based microfluidic devices and aerospace thermal management all rely on boundary permeability for effective thermal convection. Siddheshwar [32] has documented the convective instability of ferromagnetic fluids confined by fluid-permeable and magnetically active boundaries. More recently, Nanjundappa et al. [26] explored penetrative ferro-thermal convection (FTC) driven by internal heating in a porous layer saturated with ferrofluid, considering various temperature, velocity, and magnetic potential boundary conditions. Additionally, Surya [33] examined convective instability of a liquid layer with permeable boundaries under the influence of variable gravitational force.

To the best of our knowledge, no prior studies have examined the combined effects of viscosity variation with magnetic fields and fluid-permeable magnetic boundary surfaces on the thermal convection of ferrofluids containing dust particles. This research is motivated by this gap in the literature. The study will provide different insights on how thermal convection can be optimized for various engineering applications such as electronics cooling, aerospace systems, biomedical technologies and environmental engineering through dust particles effects on it and variation of viscosity with magnetic field and fluid – permeable magnetic boundaries.

2. RESEARCH METHODOLOGY

The analytical investigation of convective instability in a dusty ferromagnetic fluid layer considering magnetic field-dependent viscosity and fluid-permeable, magnetically active boundaries under a uniform transverse magnetic field follows a structured approach aligned with the objectives of the study. A brief overview of the methodology is presented below as per the defined sections.

3. MATHEMATICAL MODELLING

Consider a static layer of ferromagnetic fluid with a finite vertical thickness d and an infinite horizontal extent, heated from below, and containing dust particles. This fluid layer is subjected to a uniform vertical magnetic field H and experiences a gravitational force represented by and this system is confined between two horizontal fluid-permeable magnetic boundaries (as depicted in Fig. 1)

Fig. 1.

Geometrical Configuration

We assume the fluid is incompressible, electrically non-conduc-tive, and has a viscosity depend upon the magnetic field, expressed as μ = μ₁(δ.B + 1) [35]. The fluid's viscosity when the magnetic field is zero is μ1, the isotropic coefficient of viscosity change is δ, and the magnetic induction is represented by B. For the model described, the governing equations under the Boussinesq approxi-mation are provided by [6, 19]:

The equation of state for density is

In the aforementioned equations: q represents the ferrofluid’s velocity, q indicates the dust particles’ velocity, pertains to pressure, ρ corresponds to density, N_d signifies the number density of the dust particles, M stands for magnetization, B represents magnetic induction, T is the temperature, C_H,V characterizes the specific heat at constant magnetic field and volume, C_pt indicates the specific heat of dust particles, K₁ designates the thermal conductivity, α represents the coefficient of thermal expansion, and K = 6πμr signifies the Stokes drag coefficient, where r represents the radius of the dust particle. Additionally, ρ₀ and μ₀ correspond to the density and magnetic permeability at the reference temperature.

The buoyant force acting on the dust particles is neglected [18, 23, 26]. It is further assumed that the spacing between the particles is much larger than their diameters, allowing inter-particle interactions to be disregarded. The influence of gravity, pressure, and viscous forces on the particles is assumed to be negligibly small and is thus omitted from consideration. An additional force term with this opposite direction must be included in the particles' equation of motion, because the force applied by the ferrofluid on the dust particles is equal in magnitude but opposite in direction to the force exerted by the dust particles on the ferrofluid. Let mN_d denote the mass of dust particles per unit volume; the motion and continuity equations for the dust particles, under these assumptions, are given as follows [19, 22]:

Due to assumption that fluid is electrically non-conductive, the equations of Maxwell in the case where displacement current is absent for a non-conductive fluid can be stated as follows [6]:

Thus, from equations (7) and (8), we get

The alignment of magnetization is governed by both the mag-netic field's strength and temperature, resulting in the following re-lationship

Where M₀ is the magnetization when temperature is T₀ and magnetic field is H₀, while

Equations (1) to (11) are solved for zero flow at base state and thus the basic state solutions are given by

Further following Finlayson [6] perturbations are added to initial basic state as:

Where the variables

On introducing equation (13) into equations (1) to (11) and applying the basic state solutions, we derive the linearized perturbation equations in the form.

Here ρC₁ = μ₀H₀K₂ + ρ₀C_H,V, L0=(mK∂∂t+1), also,

Here we have considered that βdK₂ < <(χ + 1)H₀. (Finlay-son [6]).

Now, eliminating p′, u′, v′ between equations (15)–(17) using equation (14), we get

Also, from equations (19) and (20), we have

Now, following normal mode analysis assuming that all quanti-ties characterizing the perturbation depend on t, x, y, and z in the form

In this context, we have wave numbers represented as k_x1and k_y1 for the x and y directions, respectively. Additionally, n denotes the growth rate, and k is defined as the magnitude of the resultant wave number, calculated as the square root of the sum of the squares of k_x1 and k_y1 and using non dimensional parameters:

A real independent variable in the range 0 ≤ ≤ 1 is used in the aforementioned equations. Here, the square of the wave number is represented by a², while differentiation with respect to z is shown by D=ddz. The variables p_r, R, M₁, M₂ and M₃ correspond to the Prandtl number, Rayleigh number, magnetic number, a non-dimensional parameter, and measure of nonlinearity of the magnetization parameter, respectively. The parameters h and f are related to dust particles. The complex constant ω = ω_r + lω_i signifies the complex growth rate, where ω_i and ω_r are real constants. The variables, w(z) = w_i(z)l + w_r(z), ϕ(z) = ϕ_i(z)l + ϕ_r(z), and θ(z) = θ_i(z)l + θ_r(z), are all complex functions of the real variable z, while w_r, w_i, ϕ_r, ϕ_i, θ_r, θ_i, are the real components of these functions.

From a physical standpoint, equations (24) to (26) define an eigenvalue problem for R, which governs ferromagnetic convection within a ferrofluid layer containing dust particles. In these equations, w represents the vertical velocity, θ indicates the temperature, and ϕ denotes the amplitude of the magnetic potential. Since M₂ is of a very small magnitude (on the order of 10⁻⁶) [6], its influence is negligible for further analysis, allowing equation (25) to be simplified as.

For the scenario of stationary convection, setting ω = 0 simplifies the eigenvalue problem into the following form:

Since the ferrofluid layer is confined between two thermally conducting and fluid-permeable magnetic surfaces. Hence, we use the following fluid-permeable magnetic boundaries condition as proposed by ([6, 32, 36, 37]):

4. MATHEMATICAL ANALYSIS

The non-dimensionalized linear ODEs (28) to (30) along with boundary conditions (31) form the eigen value problem with as Rayleigh number. To determine the Rayleigh number for the given boundary conditions (31), we will employ a single-term Galerkin method. Let’s consider a single term in the expansions of w, θ, and ϕ as follows:

Here, A₁₁, B₁₁, and C₁₁are constants, and w₁(z), θ₁(z), and ϕ₁(z) are appropriately selected trial functions that satisfy the necessary boundary conditions (31). By replacing w, θ, and ϕ with their respective expressions in equations (28) to (30), then multiplying each resulting equation by w₁(z), θ₁(z), and ϕ₁(z) accordingly, and integrating by parts the necessary number of times while applying the boundary conditions, we derive three homogeneous equations in A₁₁, B₁₁, and C₁₁. The coefficients of these equations are expressed as definite integrals. This system can be expressed in matrix form as follows:

For simplicity, the subscript 1 is omitted, and the functions w₁(z), θ₁(z), and ϕ₁(z) are henceforth denoted as w, θ, and ϕ respectively.

For a non-trivial solution to occur, the determinant of the coefficient matrix must be zero. This results in the following expression, which establishes a first-order relationship between the Rayleigh number R and the wave number a.

Let us now select the following trial functions that satisfy the specified boundary conditions.

The initially proposed trial function for the magnetic potential, denoted by ϕ, does not meet the boundary conditions given in equation (31). To resolve this inconsistency, the boundary residual method, as described by Finlayson [6], is applied to the function ϕ.

By applying these trial functions to the integrals A₁ through A₆, we derive the following formula for Rayleigh number as

Using MATLAB R2023a software, the square of the critical wave number x_c is determined by finding the positive roots of the equation dRdx=0. Additionally, the associated critical Rayleigh number R_c is numerically calculated.

Also, we have the following formula for the magnetic Rayleigh number N for significantly large values of M₁derived from expression (35) utilizing Finlayson [6] analysis.

5. RESULTS AND DISCUSSION

This study explores the convective instability of a dusty ferromagnetic fluid, incorporating the effects of viscosity that depend on the magnetic field. The system is analyzed within a Rayleigh-Bènard configuration, where the fluid is contained between permeable boundaries that are also magnetically active. Additionally,the setup is subject to a uniform transverse magnetic field. Given the complexity of the boundary conditions, the Galerkin method was employed to calculate the critical eigenvalue. These findings can contribute to better control of instability in engineering systems and enhance the efficiency of applications where stability and precise thermal regulation are essential.

In the present analysis, the nonlinearity of the magnetization parameter M₃ is considered to vary from 0 to 25, as suggested by Finlayson [6]. The MFD viscosity parameter δ ranges from 0 to 0.09, as per the work of Prakash et al. [14]. The dust particle parameter h₁ is taken to vary between 1 and 9, following the suggestion of Sunil et al. [18]. Furthermore, the magnetic susceptibility of boundaries and the permeability parameter of boundaries χ are assumed to range from 10⁰ to 10⁶ and from 0 to ∞, respectively, as suggested by Siddheshwar [32].

First, we discuss the accuracy of the results presented in this study. For the case of ordinary fluid, in the absence of a magnetic parameter (M₁ = 0 and M₃ = 0), χ → ∞ and without dust particles (h₁ = 1), the critical Rayleigh number (R_c) and wave number (ac2) are found to be R_c = 664.5251 at ac2=4.9594 for free boundaries. This closely matches to R_c = 657.551 at ac2=4.9328 as obtained by Surya [33]. Additionally, for rigid boundaries, we have R_c = 1750.0 at ac2=9.7127, which closely matches R_c = 1715.070 at ac2=9.6969, as reported by Surya [33].

Also, Tab. 1 provide a qualitative comparison of the numerical results computed at χ → ∞, without dust particle (h₁ = 1), M₃ = 10, M₁ = 1,5 for the case of free-free and rigid-rigid boundaries with Dhiman [10]. The table clearly shows that our results align exceptionally well with previously published data, confirming the accuracy of our numerical procedure.

Tab. 2 presented the numerical values of critical magnetic Rayleigh number (N_c) and square of critical wave number (ac2) for different combination of δ, M₃, χ at fixed h₁=3 and Da_s = 1. Tab. 4 presented the numerical values of critical magnetic Rayleigh number (N_c) and square of critical wave number (ac2) for different combination of h₁, Da_s and M₃ at fixed δ=0.03 and χ = 10.

Fig. 2 depict the variation of critical magnetic Rayleigh numbers N_c versus χ with different values of M₃ and fixed values of δ, h₁ and Da_s. From Fig. 2 and numerical values of critical magnetic Rayleigh numbers N_c presented in Tab. 3 we can observe that critical magnetic Rayleigh numbers increases for increasing values of χ, which yields that χ has stabilizing effect on the system. This stabilizing effect of χ directly depend upon M₃, χ has strong stabilizing effect for smaller values of M₃ as compare to larger values of M₃. Also from Tab.3 we can observe that the critical wave number increases as χ increases which indicate that χ reduces the size of convection cell.

Fig. 2.

Graph between N_c vs χ at δ = 0.05, h₁ = 3 and Da_s = 1

Fig. 3 epict the variation of critical magnetic Rayleigh numbers N_c versus Da_s with different values of and fixed values of M₃, h₁ and χ. From Fig. 3 and numerical values of critical magnetic Rayleigh numbers N_c presented in Tab. 4 we can observe that critical magnetic Rayleigh numbers increases for increasing values of Da_s, which yields that Da_s has stabilizing effect on the system. Also we can conclude that (N_c)_Free ≤ (N_c)_Permeable ≤ (N_c)_Rigid. Because (N_c)_Free and (N_c)_Rigid can be acquired from (N_c)_Permeble in the limit Da_s → 0 and ∞ respectively. Therefore the equality sign in (N_c)_Free ≤ (N_c)_Permeable ≤ (N_c)_Rigid is understandable. This analysis effectively demonstrates the connection between the results for free-free and rigidrigid boundary conditions. This behavior arises because free boundaries impose minimal resistance to fluid motion near the surfaces, allowing the temperature gradient to drive convection more easily. As a result, the fluid becomes more responsive to instabilities, leading to a lower critical magnetic Rayleigh number and an earlier onset of convection compared to the case with rigid boundaries. Further, it is evident from Tab. 4 that the critical wave number increases as Da_s increases which indicate that Da_s reduces the size of convection cell.

Fig. 3.

Graph between N_c vs Da_s at h₁ = 3, χ = 10 and M₃ = 5

Fig. 4 depict the variation of critical magnetic Rayleigh numbers N_c versus δ with different values of M₃ and fixed values of χ, h₁ and Da_s. From Fig. 3, Fig. 4, Fig. 5, Fig. 7 and numerical values of critical magnetic Rayleigh numbers N_c presented in Tab. 3 we can observe that critical magnetic Rayleigh numbers increases for increasing values of δ, which yields that δ has stabilizing effect on the system. This is consistent with the physical understanding that higher viscosity suppresses disturbances and restricts fluid movement, resulting in a delayed onset of convection. Also, it is noted from table Tab. 3 that the critical wave number has no influence of MFD viscosity δ, i.e. size of convection cell is independent of MFD viscosity δ.

Fig. 4.

Graph between N_c vs δ at χ = 10, Da_s = 1 and h₁ = 3

Fig. 5.

Graph between N_c vs h₁ at Da_s = 10, χ = 10 and M₃ = 5

Fig. 7.

Graph between N_c vs M₃ at χ = 1, Da_s = 1 and h₁ = 3

The behavior of the critical magnetic Rayleigh number N_c as a function of M₃ is illustrated in Fig. 6 for several values of χ, while keeping M₃, h₁ and δ constant. Similarly, Fig. 7 depicts the relationship between N_c and M₃ for different values of δ, with h₁, M₃, and χ held fixed. The critical magnetic Rayleigh number N_c initially decreases for increasing values of M₃, but after a certain value of M₃, it increases for increasing values of M₃, as shown by Fig. 2, Fig. 4, Fig. 6, Fig. 7 and Tab. 2 as well as Tab. 3. This destabilizing or stabilizing effect of M₃ varies upon the value of. The destabilizing effect range of M₃is larger for small values of than it is at large values of. From Tab. 2 and Tab. 3 it is observed that critical wave number reduces with raise in M₃ which means that M₃ increases the size of convection cell.

Fig. 6.

Graph between N_c vs M₃ at δ = 0.03, Da_s = 1 and h₁ = 3

The behavior of the critical magnetic Rayleigh number N_c as a function of dust particles parameter h₁ is illustrated in Fig. 5 for several values of, while keeping h₁, M₃, and χ constant. From Fig. 2 and numerical values of critical magnetic Rayleigh numbers N_c presented in table Tab. 2 we can observe that critical magnetic Rayleigh numbers decreases for increasing values of h₁, which shows that h₁ has destabilizing effect on the system as the overall heat capacity of the fluid increases due to the additional contribution from the dust particles. Moreover Tab. 2 demonstrates that the critical wave number does not depend on the parameter h₁ of dust particles, meaning that the size of the convection cell is not affected by dust particles parameter h₁.

6. CONCLUSION

For a Rayleigh-Bénard situation between fluid-permeable, magnetic boundaries, we investigate the impact of dust particles and viscosity variation with magnetic field on the convective instability of a ferro magnetic fluid layer using the single term Galerkin technique for stationary mode of convection. The main conclusion from our research can be summarized as follows:

These discoveries advance our knowledge of the variables affecting convection of ferrofluid and offer useful data for a number of disciplines, including fluid dynamics and geophysics. Additional investigation in this field may clarify our understanding of convection phenomena and examine the practical consequences of these results.

Nomenclature