Convection in a Ferromagnetic Fluid Layer Influenced by Changeable Gravity and Viscosity

Abstract

The motive of this work is to numerically evaluate the effect of changeable gravitational fields and varying viscosity on the beginning of convection in ferromagnetic fluid layer. The fluid layer is constrained by two free boundaries and varying gravitational fields that vary with distance across the layer. The authors hypothesized two categories of gravitational field variation, which can be subdivided into six distinct cases: (i) $f\left(z\right)=z,$ (ii) $f\left(z\right)=e^z,$ (iii) $f\left(z\right)=log(1+z),$ (iv) $f\left(z\right)=-z,$ (v) $f\left(z\right)=-z^2,$ and (vi) $f\left(z\right)=z^2-2z.$ The normal mode method was applied, and the single term Galerkin approach was used to solve the ensuing eigenvalue problem. The results imply that, in the first three cases, the gravity variation parameter speeds up the commencement of convection, while, in the last three cases, the viscosity variation parameter and gravity variation parameter slow down the onset of convection. It was also observed that, in the absence of the viscosity variation parameter, the non-buoyancy magnetization parameter destabilizes the impact on the beginning of convection but, in the presence of the viscosity variation parameter, it destabilizes or stabilizes impact on the beginning of convection. In the case of oscillatory convection, the results illustrate that oscillatory modes are not permitted, suggesting the validity of the theory of exchange of stabilities. Additionally, it was also discovered that the system is more stable for case (vi) and more unstable for case (ii).

1. Introduction

Ferrohydrodynamics is the study of the mechanics of fluid motion when it is influenced by strong magnetic polarisation forces. The concept of ferrohydrodynamics was introduced in [1]. Ferromagnetic fluids, often known as “ferrofluids" are colloidal magnetic fluids that are electrically non-conducting. Ferromagnetic fluids act as a continuous homogeneous medium and exhibit a range of attractive characteristics. These fluids are not found in the natural world and must be created chemically. Numerous applications of ferromagnetic fluids are described in [2]. The exploration of ferromagnetic fluids has attracted the attention of researchers in the last millennium because of their applications in instrumentation, liquid-cooled loudspeakers [3], loudspeaker drivers, computer disk drives [4], vacuum technology, lubrication, recovery of metals, vibration dampening, acoustics, and biomedical applications, among others. These fluids are widely used in locations that require reliable sealing with low friction. Some of their most vital uses are in the medical domain, such as for drug delivery to an injured site and tumor excision.

The monograph [5] provides an authoritative introduction to research on ferromagnetic fluids, and the examination of magnetization effects, revealing interesting results. In general, a magnetic field, a fluid’s temperature and a fluid’s density all play a role in magnetization. Any change in these parameters can result in a change in the distribution of fluid body forces. Convection occurs in ferromagnetic fluids when there is a gradient in the magnetic fields. This process is referred to as ferro-convection, which is analogous to Bénard-convection. The acclaimed work by [6] provides a full analysis of the experimental investigations and theoretical explanations of the initiation of Bénard-convection in fluids under a variety of hydrodynamic and hydromagnetic assumptions.

Generalization of the classical Rayleigh–Bénard problem to ferromagnetic fluids using linear theory was first studied by [7]. An energy approach to explore the thermoconvective stability of ferromagnetic fluids was used in [8]. Later, a similar analysis was reported in [9], but with the fluid contained between ferromagnetic plates and using the Galerkin method to solve the disturbance equations. Thermal convection in a magnetic fluid was investigated in [10]. Finlayson’s problem was explored experimentally by the authors of [11] who found that their conclusions were consistent with [7]. The problem presented by [11] was later enlarged by [12] by assuming that, in the presence of a strong magnetic field, the effective shear viscosity is dependent on colloid concentration and temperature. The authors of [13] investigated the thermoconvective instability of ferromagnetic fluids using Brinkman model saturating in a porous medium heated from below. The effects of temperature, rotation and porous medium on ferromagnetic fluids were investigated in [14,15,16,17]. Additionally, refs. [18,19,20,21] conducted several significant investigations on the impact of magnetic field dependent viscosity on the beginning of convection. In [22], the single term Galerkin technique was used to conduct a linear stability study for the initiation of natural convection in a horizontal nanofluid layer. The start of convection in a nanofluid layer with a magnetic field using a linear stability analysis was investigated in [23]. The results demonstrated that magnetic fields stabilized the nanofluid layer for both stationary and oscillatory convection. Recently, several significant studies on nanofluid flow have been reported in [24,25,26,27,28,29,30,31].

In most laboratory studies, the Earth’s gravitational field is treated as a constant factor. Due to the fact that the Earth’s gravity varies with height above its surface, in many large-scale atmospheric convection phenomena, gravity must be modeled as a variable quantity. The instability of heated viscous fluid trapped by two horizontal planes subjected to changing gravitational field that varied spatially with height was investigated in [32]. Non-linear energy theory and linear instability theory were employed by [33] to analyse convection in a fluctuating gravitational environment. For porous media convection, ref. [34] studied the effects of altering gravity and internal heat sources. Later, ref. [35] examined the onset of convection in an anisotropic porous medium subjected to gravity field that varied in accordance with the layer depth. The energy technique was used to investigate the non-linear convection in a porous layer with an inclination temperature gradient and varying gravity field in [36]. The impact of changing gravity on the thermal instability in a layer of nanofluid in a porous medium were investigated in [37]. Fluid-saturated porous media with an interior heat source and variable gravitational fields were studied by [38] in order to understand convection. Several significant studies on the impact of variable gravity on convection initiation include the following [39,40,41,42,43]. The initiation of convection in thin magnetic nanofluid heated from the bottom, subject to a magnetic field and a variable gravity field, was investigated in [44]. The authors developed a model that integrates Brownian diffusion, magnetophoresis and thermophoresis, and then determined the numerical solutions using the Chebyshev pseudospectral approach. The impact of a changing gravity field and throughflow on the onset of convection in a porous medium was investigated in [45], with the finding that both parameters delayed convective motion. The single term Galerkin technique was employed in [46] to study the qualitative influence of variable internal heat source and changeable gravity field on the onset of convection in a horizontal fluid layer saturating in a porous medium. The higher-terms Galerkin weighted residual approach was used by [47] to investigate the impact of rotation and variable gravitational strength on the beginning of thermal convection in a porous medium layer. The Galerkin technique was used in [48] to investigate the effect of altering gravitational field and flow on the onset of nanofluid convective instability in a permeable medium layer. Very recently, ref. [49] investigated the influence of varying gravity on rotating convection in a sparsely packed porous layer, and used bvp4c in MATLAB R2020b to solve the eigenvalue problem numerically for free–free and rigid–rigid boundaries. A summary of the literature review is provided in

Table 1. Some Selected Literature Review Table Related to our Field of Study.

Researcher(s)	Ferrofluid/ Nanofluid	Free-Free Boundary	Galerkin Single Term Approach	Magnetic Field Dependent Viscosity	Increasing Gravitational Field Variation	Decreasing Gravitational Field Variation
[7]	✓	✓
[18]	✓	✓		✓
[20]	✓	✓		✓
[22]	✓	✓	✓
[23]	✓	✓	✓
[40]	✓	✓	✓		✓	✓
[43,44]	✓	✓			✓	✓
[50]	✓					✓
[46]			✓			✓
[48]	✓					✓
[49]						✓
Present Paper	✓	✓	✓	✓	✓	✓

below:

The current investigation is significant because it has numerous engineering applications, including for biological and chemical engineering, and the investigation of a variety of geophysical and astronomical phenomena, pollutant passage, material processing, crystal growth, the Earth’s crust, and saturated soils. To demonstrate the importance of the topic, we examine the consequences of changeable gravitational fields and variable viscosity. In this paper, the effect of changeable gravity and variable viscosity on the beginning of convection for six distinct cases of gravitational field variation is analyzed, including: (i) $f\left(z\right)=z$ (linear increasing), (ii) $f\left(z\right)=e^z$ (convex increasing), (iii) $f\left(z\right)=log(1+z)$ (concave increasing), (iv) $f\left(z\right)=-z$ (linear decreasing), (v) $f\left(z\right)=-z^2$ (concave decreasing), and (vi) $f\left(z\right)=z^2-2z$ (convex decreasing). The set of eigenvalue problems is solved numerically using the single term Galerkin approach for free-free boundary conditions. Within the appropriate limits, our findings corroborate those of [7,20,37,43,44,46].

2. Mathematical Analysis

We analyse the scenario of an endless horizontal layer of incompressible ferromagnetic fluid with a thickness d that is electrically non-conducting and is bordered by the surfaces $z=0$ and $z=d$ having a variable viscosity which is defined as $\mu ={\mu }_1(1+\overrightarrow{\delta }·\overrightarrow{B})$ [18]. It is assumed that $\delta$ depends on the magnetic field and is isotropic in nature. The fluid layer is represented in a three-dimensional Cartesian coordinate system $(x,y,z)$ in which the z-axis is taken to be normal to the fluid layer. From below, the fluid layer is heated; the temperature T at the bottom surface $z=0$ is taken to be $T_0$ and on the top surface $z=d$ is taken to be $T_1$, respectively (See Figure 1). It is clear that $T_0>T_1.$ The fluid layer is heated in such a way that a steady adverse temperature gradient, which is defined as $\beta =\mid \frac{dT}{dz}\mid$, is retained within the fluid. A consistent vertical magnetic field $\overrightarrow{H}=(0,0,H_0)$ and changeable gravitational field $\overrightarrow{g}=(0,0,-g),g=(1+ϵf)g_0,$ (following [37,43,44]) where $f,$ a function of $z,$ is a changeable gravity function and $ϵ$ is a gravity variation parameter (assumed to be constant, in the range from 0 to 1), which acts on the entire system. For different changeable gravity function, the variation of $\frac{|\overrightarrow{g}|}{g_0}$ with z is shown in Figure 2.

The mathematical equations, following [7,17,20] that describe the movement of ferromagnetic fluid are as follows:

$$\begin{array}{cc}\hfill & ▿·\overrightarrow{q}=0\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill & {\rho }_0\frac{D\overrightarrow{q}}{Dt}+▿p-\rho \overrightarrow{g}-▿·\left(\overrightarrow{H}\overrightarrow{B}\right)-\mu {▿}^2\overrightarrow{q}=\overrightarrow{0}\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill & \rho ={\rho }_0\{1-\alpha (T-T_a)\}\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill & [{\rho }_0{C}_{V,H}-{\mu }_0\overrightarrow{H}·{\left(\frac{\partial \overrightarrow{M}}{\partial T}\right)}_{V,H}]\frac{DT}{Dt}+{\mu }_{0}T{\left(\frac{\partial \overrightarrow{M}}{\partial T}\right)}_{V,H}·\frac{D\overrightarrow{H}}{Dt}-k_1{▿}^{2}T=0\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill & ▿·\overrightarrow{B}=0\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill & ▿\times \overrightarrow{H}=\overrightarrow{0}\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill & \overrightarrow{B}={\mu }_0(\overrightarrow{H}+\overrightarrow{M})\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill & \overrightarrow{M}=\frac{\overrightarrow{H}}{H}M(H,T)\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill & M=M_0+\chi (H-H_0)-K(T-T_a)\hfill \end{array}$$

(9)

Equations (1) to (9) are the mathematical equations for this current problem. The nomenclature section explains the numerous notations used in these mathematical equations.

3. The Initial State

The fluid is believed to be at rest in the initial state, with a steady adverse temperature gradient $\beta$ maintained throughout the fluid. In mathematical terms, the initial state is $\overrightarrow{q}=\overrightarrow{q_b}=(0,0,0),\rho ={\rho }_b\left(z\right),p=p_b\left(z\right),$$T=T_b\left(z\right),$$\overrightarrow{H_b}=(0,0,H_b\left(z\right))$ and $\overrightarrow{M_b}=(0,0,M_b\left(z\right)).$

Putting these values in the governing Equations (1) to (9), we get $T=-\beta z+T_a,$ where $\beta =\frac{T_0-T_1}{d},H_b\left(z\right)=H_0-\frac{K\beta z}{(1+\chi )},M_b\left(z\right)=M_0+\frac{K\beta z}{(1+\chi )}$ and $H_b\left(z\right)+M_b\left(z\right)=C$.

4. The Perturbation Equation

To ascertain the stability of a given system, the original situation is disturbed infinitesimally and the behaviour of the related equations in the perturbed values is investigated. Let $\overrightarrow{q^{\prime }}=(u_1,u_2,u_3),{\rho }^{\prime },p^{\prime },T^{\prime },\overrightarrow{H^{\prime }}=(H_1^{\prime },H_2^{\prime },H_3^{\prime })$ and $\overrightarrow{M^{\prime }}=(M_1^{\prime },M_2^{\prime },M_3^{\prime })$ denote, respectively, the small perturbations in the fluid velocity, fluid density, fluid pressure, temperature, consistent vertical magnetic field and magnetization. Use of these in Equations (1) to (9) and linearizing, the linearized perturbation equations are

$$\begin{array}{cc}\hfill & \frac{\partial u_1}{\partial x}+\frac{\partial u_2}{\partial y}+\frac{\partial u_3}{\partial z}=0\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill & {\rho }_0\frac{\partial u_1}{\partial t}+\frac{\partial p^{\prime }}{\partial x}-{\mu }_0(M_0+H_0)\frac{\partial H_1^{\prime }}{\partial z}-{\mu }_1{▿}^2{u}_1=0\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill & {\rho }_0\frac{\partial u_2}{\partial t}+\frac{\partial p^{\prime }}{\partial y}-{\mu }_0(M_0+H_0)\frac{\partial H_2^{\prime }}{\partial z}-{\mu }_1{▿}^2{u}_2=0\hfill \\ \hfill & {\rho }_0\frac{\partial u_3}{\partial t}+\frac{\partial p^{\prime }}{\partial z}-{\mu }_0(M_0+H_0)\frac{\partial H_3^{\prime }}{\partial z}-{\mu }_1{▿}^2{u}_3+{\mu }_{0}K\beta H_3^{\prime }-\frac{{\mu }_0{K}^2\beta T^{\prime }}{(1+\chi )}\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill & -\alpha {\rho }_0{g}_0(1+ϵf)T^{\prime }-{\mu }_1{▿}^2\delta {\mu }_0(M_0+H_0)u_3=0\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill & {\rho }^{\prime }=-\alpha {\rho }_0{T}^{\prime }\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill & {\rho }_{0}C\frac{\partial T^{\prime }}{\partial t}-{\mu }_0{T}_{a}K\frac{\partial }{\partial t}\left(\frac{\partial {\varphi }^{\prime }}{\partial z}\right)=k_1{▿}^2{T}^{\prime }+\left({\rho }_{0}C\beta -\frac{{\mu }_0{T}_a{K}^2\beta }{(1+\chi )}\right)u_3\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill & H_1^{\prime }+M_1^{\prime }=H_1^{\prime }\left(1+\frac{M_0}{H_0}\right)\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill & H_2^{\prime }+M_2^{\prime }=H_2^{\prime }\left(1+\frac{M_0}{H_0}\right)\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill & H_3^{\prime }+M_3^{\prime }=H_3^{\prime }(1+\chi )-K{T}^{\prime }\hfill \end{array}$$

(18)

where ${\rho }_{0}C={\rho }_0{C}_{V,H}+{\mu }_{0}K{H}_0$ and $(1+\chi )H_0\gg K\beta d$ is assumed. With the help of Equation (6), $\overrightarrow{H^{\prime }}=▿{\varphi }^{\prime }$.

From Equations (16)–(18), we have

$$\left(1+\frac{M_0}{H_0}\right){{▿}_1}^2{\varphi }^{\prime }+(1+\chi )\frac{{\partial }^2{\varphi }^{\prime }}{\partial z^2}-K\frac{\partial T^{\prime }}{\partial z}=0$$

(19)

From Equations (10)–(13), we have

$$\begin{array}{cc}\hfill & \left[{\rho }_0\frac{\partial }{\partial t}-{\mu }_1{▿}^2\right]{▿}^2{u}_3=\left[-{\mu }_{0}K\beta \left({{▿}_1}^2\frac{\partial {\varphi }^{\prime }}{\partial z}\right)+\alpha {\rho }_0{g}_0{{▿}_1}^2(1+ϵf)T^{\prime }\right]\hfill \\ \hfill & +\left[\frac{{\mu }_0{K}^2\beta }{(1+\chi )}{{▿}_1}^2{T}^{\prime }+{\mu }_1{▿}^2\delta {\mu }_0(M_0+H_0){{▿}_1}^2{u}_3\right]\hfill \end{array}$$

(20)

Since the fluid layer is enclosed by the surfaces $z=0$ and $z=d,$ certain boundary requirements must be satisfied on these two surfaces. We suppose that both the boundaries are free-free and ideal heat conductors. For the aforementioned system, the required boundary conditions are [6]

$$\begin{array}{cc}\hfill T^{\prime }=0,u_3=0,\frac{{\partial }^2{u}_3}{\partial z^2}=0,H_3^{\prime }=0& \mathrm{at}\mathrm{the}\mathrm{bottom}\mathrm{surface}z=0\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill T^{\prime }=0,u_3=0,\frac{{\partial }^2{u}_3}{\partial z^2}=0,H_3^{\prime }=0& \mathrm{at}\mathrm{the}\mathrm{top}\mathrm{surface}z=d\hfill \end{array}$$

(22)

5. Normal Mode Analysis

Now, by adopting the following form, we may evaluate the perturbation quantities $T^{\prime },u_3$ and ${\varphi }^{\prime }$ in terms of two-dimensional periodic waves [6].

$$(T^{\prime },u_3,{\varphi }^{\prime })=[\theta \left(z\right),W\left(z\right),\Phi \left(z\right)]e^{[i(x{k}_x+y{k}_y)+nt]}$$

(23)

where $k=\sqrt{{k_x}^2+{k_y}^2}.$

Using Equations (15), (19), (20) and (23) becomes

$$\begin{array}{cc}\hfill & (1+\chi )\frac{{\partial }^2\Phi }{\partial z^2}-\left(1+\frac{M_0}{H_0}\right)k^2\Phi -K\frac{\partial \theta }{\partial z}=0\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill & {\rho }_{0}Cn\theta -{\mu }_0{T}_{a}Kn\left(\frac{\partial \Phi }{\partial z}\right)=k_1\left(\frac{{\partial }^2}{\partial z^2}-k^2\right)\theta +\left({\rho }_{0}C\beta -\frac{{\mu }_0{T}_a{K}^2\beta }{(1+\chi )}\right)W\hfill \\ \hfill & \left[{\rho }_{0}n-{\mu }_1\left(\frac{{\partial }^2}{\partial z^2}-k^2\right)\right]\left(\frac{{\partial }^2}{\partial z^2}-k^2\right)W={\mu }_{0}K\beta k^2\frac{\partial \Phi }{\partial z}-\alpha {\rho }_0{g}_0{k}^2(1+ϵf)\theta \hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill & -\frac{{\mu }_0{K}^2\beta k^2}{(1+\chi )}\theta -{\mu }_1{k}^2\left(\frac{{\partial }^2}{\partial z^2}-k^2\right)\delta {\mu }_0(M_0+H_0)W\hfill \end{array}$$

(26)

Now, simplifying the perturbation Equations (24)–(26) by introducing the following dimensionless quantities

$\sigma =\frac{n{d}^2}{\nu }$, $W^{*}=\frac{d}{\nu }W$, ${\Phi }^{*}=\frac{(1+\chi )k_{1}a{R}^{\frac{1}{2}}}{K{\rho }_{0}C\beta \nu d^2}\Phi$, ${\delta }^{*}=\delta {\mu }_0{H}_0(1+\chi )$, ${\Theta }^{*}=\frac{k_{1}a{R}^{\frac{1}{2}}}{{\rho }_{0}C\beta \nu d}\theta$, $a=kd$, $z^{*}=\frac{z}{d}$, $D=\frac{\partial }{\partial z^{*}}$

Thus, we have

$$\begin{array}{cc}\hfill & D^2{\Phi }^{*}-a^2{M}_3{\Phi }^{*}-D{\Theta }^{*}=0\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill & P\sigma ({\Theta }^{*}-M_{2}D{\Phi }^{*})=(D^2-a^2){\Theta }^{*}+a{R}^{\frac{1}{2}}(1-M_2)W^{*}\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill & [\sigma -(D^2-a^2)](D^2-a^2)W^{*}=a{R}^{\frac{1}{2}}(1+ϵf)\left[M_{1}D{\varphi }^{*}-{\Theta }^{*}-M_1{\Theta }^{*}\right]\hfill \\ \hfill & -a^2(D^2-a^2){\delta }^{*}{M}_3{W}^{*}\hfill \end{array}$$

(29)

where $R=\frac{g_0\alpha \beta d^4{\rho }_{0}C}{\nu k_1}$, $M_1=\frac{{\mu }_0{K}^2\beta }{(1+\chi )\alpha {\rho }_{0}g}$, $M_2=\frac{{\mu }_0{T}_a{K}^2}{(1+\chi ){\rho }_{0}C}$, $M_3=\frac{\left(1+\frac{M_0}{H_0}\right)}{(1+\chi )}$ and $P=\frac{\nu }{k_1}\left({\rho }_{0}C\right)$. Since $M_2$ is of a very small order (~${10}^{-6}$), it is disregarded in the further analysis [7]. Each of the parameters defined above has an effect on the system’s stability in one way or another.

According to normal mode analysis, the system’s boundary conditions (21) and (22) are

$$\begin{array}{cc}\hfill {\Theta }^{*}=W^{*}=D^2{W}^{*}=D{\Phi }^{*}=0& \mathrm{at}z=0\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill {\Theta }^{*}=W^{*}=D^2{W}^{*}=D{\Phi }^{*}=0& \mathrm{at}z=1\hfill \end{array}$$

(31)

Here we analyze only the case in which $\chi \to \infty .$

For convenience, we shall disregard the asterisks in the Equations (27)–(29) and boundary conditions (30) and (31). We obtain

$$\begin{array}{cc}\hfill & (D^2-a^2{M}_3)\Phi -D\Theta =0\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill & (D^2-a^2-P\sigma )\Theta +a{R}^{\frac{1}{2}}W=0\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill & [(D^2-a^2-\sigma )(D^2-a^2)-a^2\delta M_3(D^2-a^2)]W+a{R}^{\frac{1}{2}}(1+ϵf)M_{1}D\Phi \hfill \\ \hfill & -a{R}^{\frac{1}{2}}(1+ϵf)(1+M_1)\Theta =0\hfill \end{array}$$

(34)

and

$$\begin{array}{cc}\hfill \Theta =W=D^{2}W=D\Phi =0& \mathrm{at}z=0\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill \Theta =W=D^{2}W=D\Phi =0& \mathrm{at}z=1\hfill \end{array}$$

(36)

6. Solution Methodology

Equations (32) to (34), in addition to the constraints (35), (36), form a double eigenvalue problem in terms of $\sigma$ and $R.$ The Galerkin approach [51] is used to find the critical stability parameters. The variables are expressed in this manner as a sequence of basis functions.

$$\begin{array}{cc}\hfill & \left[W\left(z\right),\Theta \left(z\right),\Phi \left(z\right)\right]=\sum _{i=1}^k\left[A_i{W}_i\left(z\right),B_i{\Theta }_i\left(z\right),C_i{\Phi }_i\left(z\right)\right]\hfill \end{array}$$

(37)

where the test functions $W_i\left(z\right),$${\Theta }_i\left(z\right),$${\Phi }_i\left(z\right)$ will be selected in such a manner that they satisfy the boundary conditions (35) and (36) and $A_i,B_i,C_i,i=1,2,3,\dots ,k$ are constants. Equation (37) is used in Equations (32) to (34), and Equation (32) is multiplied by ${\Phi }_j\left(z\right)$, Equation (33) is multiplied by ${\Theta }_j\left(z\right)$, and Equation (34) is multiplied by $W_j\left(z\right)$, and, integrating by parts with respect to variable z between the limits $z=0$ to $z=1,$ we obtain a set of $3k$ linear-homogeneous equations in which there are $3k$ unknown coefficients.

$$\begin{array}{cc}\hfill P_{ji}{B}_i+Q_{ji}{C}_i& =0\hfill \end{array}$$

(38)

$$\begin{array}{cc}\hfill R_{ji}{A}_i+S_{ji}{B}_i& =0\hfill \end{array}$$

(39)

$$\begin{array}{cc}\hfill T_{ji}{A}_i+U_{ji}{B}_i+V_{ji}{C}_i& =0\hfill \end{array}$$

(40)

The coefficients $P_{ji}-V_{ji}$ involve inner products of the basis functions and are expressed as

$$\begin{array}{cc}\hfill P_{ji}& =-<{\Phi }_{j}D{\Theta }_i>\hfill \\ \hfill Q_{ji}& =[<{\Phi }_j{D}^2{\Phi }_i>-a^2{M}_3<{\Phi }_j{\Phi }_i>]\hfill \\ \hfill R_{ji}& =a{R}^{\frac{1}{2}}<{\Theta }_j{W}_i>\hfill \\ \hfill S_{ji}& =-[<D{\Theta }_{j}D{\Theta }_i>+(a^2+P\sigma )<{\Theta }_j{\Theta }_i>]\hfill \\ \hfill T_{ji}& =<W_j{D}^4{W}_i>+(a^4+\sigma a^2+a^4\delta M_3)<W_j{W}_i>+(2a^2+a^2\delta M_3)<D{W}_{j}D{W}_i>\hfill \\ \hfill U_{ji}& =-a{R}^{\frac{1}{2}}(1+M_1)<W_j(1+ϵf){\Theta }_i>\hfill \\ \hfill V_{ji}& =a{R}^{\frac{1}{2}}{M}_1<W_j(1+ϵf)D{\Theta }_i>\hfill \end{array}$$

where

$$<\dots >={\int }_0^1(\dots )dz$$

For a non-trivial solution of homogeneous Equations (38)–(40), the determinant of coefficient matrix must be vanish, i.e.,

$$\left|\begin{array}{ccc}0& P_{ji}& Q_{ji}\\ R_{ji}& S_{ji}& 0\\ T_{ji}& U_{ji}& V_{ji}\end{array}\right|=0$$

This gives an equation involving the physical parameters of the form $f(R,P,M_1,M_3,ϵ,\delta ,a^2)=0.$ Solving equation $f(R,P,M_1,M_3,ϵ,\delta ,a^2)=0$ numerically for various physical parameter values by using the Newton–Raphson technique to obtain the minimum value of R (=$R_c$) among all possible wave numbers, in the same fashion, we can find $N_c$ for $M_1\to \infty .$

7. Results and Explanation

First, we look in-depth at the case where the changeable gravity function $f\left(z\right)=z$.

For the first approximation, we take $k=1,$ to be the appropriate test functions which fulfill the boundary criteria defined in the following format:

$$(W,\Theta ,D\varphi )=(A_1,B_1,C_1)sin\pi z$$

(41)

Using Equation (41) in Equations (32) to (34), and using the boundary conditions (35) and (36), we obtain the set of equations that can be written in a matrix form as

$$\left[\begin{array}{ccc}0& -{\pi }^2& {\pi }^2+a^2{M}_3\\ a{R}^{\frac{1}{2}}& -({\pi }^2+a^2+P\sigma )& 0\\ ({\pi }^2+a^2+\sigma +a^2\delta M_3)({\pi }^2+a^2)& -(1+\frac{ϵ}{2})(1+M_1)a{R}^{\frac{1}{2}}& (1+\frac{ϵ}{2})M_{1}a{R}^{\frac{1}{2}}\end{array}\right]\left[\begin{array}{c}{A}_1\\ B_1\\ C_1\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$$

On simplifying this, the condition for a non-zero solution is

$$A{\sigma }^2+B\sigma +C=0$$

(42)

where

$$\begin{array}{cc}\hfill A& =P(a^2+{\pi }^2)({\pi }^2+a^2{M}_3)\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill B& =(a^2+{\pi }^2)\{P(a^2+{\pi }^2)({\pi }^2+a^2{M}_3)+(a^2+{\pi }^2)({\pi }^2+a^2{M}_3)+\hfill \\ \hfill & \delta a^2{M}_{3}P({\pi }^2+a^2{M}_3)\}\hfill \end{array}$$

(44)

$$\begin{array}{cc}\hfill C& ={(a^2+{\pi }^2)}^3({\pi }^2+a^2{M}_3)+{(a^2+{\pi }^2)}^2\delta a^2{M}_3({\pi }^2+a^2{M}_3)-\hfill \\ \hfill & a^{2}R\left(1+\frac{ϵ}{2}\right)({\pi }^2+a^2{M}_3+a^2{M}_1{M}_3)\hfill \end{array}$$

(45)

7.1. Stationary Convection

For stationary convection, inserting $\sigma =0$ into the Equation (42), the equation representing the marginal state is obtained.

$$R=\frac{{(a^2+{\pi }^2)}^2({\pi }^2+a^2{M}_3)({\pi }^2+a^2+\delta a^2{M}_3)}{\left(1+\frac{ϵ}{2}\right)a^2[{\pi }^2+a^2{M}_3+a^2{M}_1{M}_3]}$$

(46)

Equation (46) indicates that R is a function of a, $M_1$, $M_3$, $\delta$ and $ϵ$. The nomenclature section provides an explanation of the various notations used in this equation.

If we take gravity and viscosity as constant (i.e., $ϵ=\delta =0$), then from equation, we get

$$R=\frac{{(a^2+{\pi }^2)}^3({\pi }^2+a^2{M}_3)}{a^2[{\pi }^2+a^2{M}_3+a^2{M}_1{M}_3]}$$

(47)

This relation agrees with the relation derived by [7].

For constant gravity ($ϵ=0$) and the absence of a magnetic field parameter, the Rayleigh number R is given by

$$R=\frac{{(a^2+{\pi }^2)}^3}{a^2}$$

(48)

One re-establishes the well-known result that the critical Rayleigh number $R_c=\frac{27}{4}{\pi }^4=657.5$ for $a_c=\frac{1}{2}{\pi }^2$. This is in line with the results of the Rayleigh–Benard problem for Newtonian fluids. Normally, when a single term Galerkin approach is used in this context, the result is overestimated, but, in this situation, the approximation produces the exact value.

For $R=R_1{\pi }^4$ and $a^2=x{\pi }^2,$ it is also possible to write Equation (46) as

$$R_1=\frac{{(1+x)}^2(1+x+\delta x{M}_3)(1+x{M}_3)}{x\left(1+\frac{ϵ}{2}\right)(1+x{M}_3+x{M}_1{M}_3)}$$

(49)

To examine the impact of a viscosity variation parameter, gravity variation parameter, and a non-buoyancy magnetization parameter on stationary convection, analytically, we investigate the behavior of $\frac{d{R}_1}{d\delta },\frac{d{R}_1}{dϵ}$ and $\frac{d{R}_1}{d{M}_3}$.

$$\begin{array}{cc}\hfill & \frac{d{R}_1}{d\delta }=\frac{{(1+x)}^2(1+x{M}_3)M_3}{(1+\frac{ϵ}{2})(1+x{M}_3+x{M}_1{M}_3)}\hfill \end{array}$$

(50)

$$\begin{array}{cc}\hfill & \frac{d{R}_1}{dϵ}=-\frac{{(1+x)}^2(1+x{M}_3)(1+x+\delta x{M}_3)}{2{(1+\frac{ϵ}{2})}^{2}x(1+x{M}_3+x{M}_1{M}_3)}\hfill \end{array}$$

(51)

$$\begin{array}{cc}\hfill & \frac{d{R}_1}{d{M}_3}=-\frac{{(1+x)}^2}{(1+\frac{ϵ}{2})}\left[\frac{(1+x)M_1-\delta [1+x^2{M_3}^2+x^2{M_3}^2{M}_1+2x{M}_3]}{{(1+x{M}_3+x{M}_1{M}_3)}^2}\right]\hfill \end{array}$$

(52)

Thus, when the changeable gravity function $f\left(z\right)=z$, the gravity variation parameter has a destabilizing impact on stationary convection, while the viscosity variation parameter always has a stabilizing impact. Additionally, if viscosity is assumed to be constant (i.e., $\delta =0$), the non-buoyancy magnetization parameter has a destabilizing impact; however, if viscosity is dependent on the magnetic field, the non-buoyancy magnetization parameter has a stabilizing (or destabilizing) impact, on condition that

$$\delta [1+x^2{M_3}^2+x^2{M_3}^2{M}_1+2x{M}_3]>(\mathrm{or}<)(1+x)M_1$$

(53)

For finding $R_c$, Equation (49) is differentiated with regard to x and equal to 0. In variable $x,$ a fourth degree polynomial equation is obtained whose coefficients are derivatives of parameters affecting the instability of the system.

$$\begin{array}{cc}\hfill & 2M_3(M_1{M}_3+\delta M_1{M_3}^2+M_3+\delta {M_3}^2)x^4+M_3(4+4\delta M_3+M_3+M_1+M_1{M}_3\hfill \\ \hfill & \delta M_1{M}_3)x^3+(2M_3-{M_3}^2-M_1{M}_3-M_1{M_3}^2-\delta M_1{M_3}^2+2\delta M_3+2)x^2\hfill \\ \hfill & +(1-2M_3-2M_1{M}_3)x-1=0\hfill \end{array}$$

(54)

We obtain the findings for the magnetic mechanism using $M_1\to \infty$. The magnetic Rayleigh number corresponding to this is defined as

$$N=R{M}_1=\frac{{\pi }^4{(1+x)}^2(1+x{M}_3)(1+x+\delta x{M}_3)}{(1+\frac{ϵ}{2})x^2{M}_3}$$

(55)

Obtaining $x_c$ and its related $N_c,$ we repeat the procedure above and obtain a third degree polynomial equation in variable $x.$

$$2M_3(\delta M_3+1)x^3+(M_3+\delta M_3+1)x^2-(M_3+\delta M_3+1)x-2=0$$

(56)

7.2. Oscillatory Convection

In this section, we look into the possibilities of oscillatory modes arising from the existence of magnetization parameter, viscosity variation parameter, and gravity variation parameter on the stability problem.

For oscillatory instability to occur, $\sigma =i\omega$, where $\omega$ represents the real dimensionless frequency. Put $\sigma =i\omega$ in Equation (42), and, equating the imaginary parts, we obtain

$$\left[(a^2+{\pi }^2)\{P(a^2+{\pi }^2)({\pi }^2+a^2{M}_3)+(a^2+{\pi }^2)({\pi }^2+a^2{M}_3)+\delta a^2{M}_{3}P({\pi }^2+a^2{M}_3)\}\right]\omega =0$$

(57)

Here, the value inside the brackets in (57) is positive definite. As a result, Equation (57) is only satisfied if $\omega =0$. This demonstrates that oscillatory modes are not permitted, and so the theory of exchange of stability is valid. This agreement is also supported by various researchers (see [7,20]).

8. Numerical Results and Discussion

The influence of changeable gravity and varying viscosity on the beginning of convection is investigated in ferromagnetic fluid layer heated from below. The present problem is discussed in terms of free-free boundary conditions, and the associated eigenvalue problem is solved using the single term Galerkin approach. The analysis focused on two categories of gravitational field variation:

•  Increasing gravitational field variation
•  Decreasing gravitational field variation

Both of which are further classified into three distinct cases, described in

Table 2. Different cases for gravitational field variation.

Increasing Gravitational Field Variation
Case (i)	$f\left(z\right)=z$	Linear Increasing
Case (ii)	$f\left(z\right)=e^z$	Convex Increasing
Case (iii)	$f\left(z\right)=log(1+z)$	Concave Increasing
Decreasing Gravitational Field Variation
Case (iv)	$f\left(z\right)=-z$	Linear Decreasing
Case (v)	$f\left(z\right)=-z^2$	Concave Decreasing
Case (vi)	$f\left(z\right)=z^2-2z$	Convex Decreasing

.

Now, using the approach mentioned in (7), we construct a table corresponding to different cases for changeable gravity function $f\left(z\right)$. It is clear from

Table 3. The magnetic Rayleigh number and corresponding non-linear equations in variable x for different changeable gravity function $f\left(z\right)$.

Case	f(z)	N	Non-Linear Equation
(i)	z	${\displaystyle \frac{{\pi }^4{(1+x)}^2(1+x{M}_3)(1+x+\delta x{M}_3)}{(1+\frac{ϵ}{2})x^2{M}_3}}$	$2M_3(\delta M_3+1)x^3+(M_3+\delta M_3+1)x^2-(M_3+\delta M_3+1)x-2=0$
(ii)	$e^z$	${\displaystyle \frac{{\pi }^4{(1+x)}^2(1+x{M}_3)(1+x+\delta x{M}_3)}{(1+ϵ(e-1))x^2{M}_3}}$
(iii)	$log(1+z)$	${\displaystyle \frac{{\pi }^4{(1+x)}^2(1+x{M}_3)(1+x+\delta x{M}_3)}{(1+ϵ(2log2-1))x^2{M}_3}}$
(iv)	$-z$	${\displaystyle \frac{{\pi }^4{(1+x)}^2(1+x{M}_3)(1+x+\delta x{M}_3)}{(1-\frac{ϵ}{2})x^2{M}_3}}$
(v)	$-z^2$	${\displaystyle \frac{{\pi }^4{(1+x)}^2(1+x{M}_3)(1+x+\delta x{M}_3)}{(1-\frac{ϵ}{3})x^2{M}_3}}$
(vi)	$z^2-2z$	${\displaystyle \frac{{\pi }^4{(1+x)}^2(1+x{M}_3)(1+x+\delta x{M}_3)}{(1-\frac{2ϵ}{3})x^2{M}_3}}$

that the non-linear equation in each case is the same. The following equation is numerically solved using the Newton–Raphson technique for different values of $M_3$ and $\delta ,$ the critical wave number $x_c$ is obtained (See

Table 4. $x_c$ for the case of stationary convection for different values of $\delta$ and $M_3$.

$\mathit{\delta }\downarrow /{\mathit{M}}_3\to$	1	5	10	15	20
	${\mathit{x}}_{\mathit{c}}$	${\mathit{x}}_{\mathit{c}}$	${\mathit{x}}_{\mathit{c}}$	${\mathit{x}}_{\mathit{c}}$	${\mathit{x}}_{\mathit{c}}$
$0.00$	$1.000$	$0.690$	$0.613$	$0.581$	$0.564$
$0.02$	$0.995$	$0.672$	$0.582$	$0.537$	$0.508$
$0.04$	$0.990$	$0.657$	$0.556$	$0.504$	$0.469$
$0.06$	$0.986$	$0.642$	$0.534$	$0.478$	$0.439$
$0.08$	$0.981$	$0.630$	$0.516$	$0.456$	$0.416$

) and for different values of $ϵ$ the critical magnetic Rayleigh numbers are obtained. The values of $M_3$ and $\delta$ are nearly identical to the values provided by [20]. In

Table 5. $N_c$ for the cases (i), (ii), and (iii), for different values of $x_c$ and $ϵ$.

${\mathit{x}}_{\mathit{c}}\downarrow /\mathit{ϵ}\to$	0			$\mathbf{0.5}$			1
	${\mathit{N}}_{\mathit{c}}$			${\mathit{N}}_{\mathit{c}}$			${\mathit{N}}_{\mathit{c}}$
	Case (i)	Case (ii)	Case (iii)	Case (i)	Case (ii)	Case (iii)	Case (i)	Case (ii)	Case (iii)
$1.000$	$1558.560$	$1558.560$	$1558.560$	$1246.848$	$838.323$	$1306.260$	$1039.040$	$573.362$	$1124.263$
$0.690$	$878.932$	$878.932$	$878.932$	$703.146$	$472.763$	$736.651$	$585.955$	$323.341$	$634.016$
$0.613$	$775.667$	$775.667$	$775.667$	$620.534$	$417.218$	$650.102$	$517.111$	$285.352$	$559.525$
$0.581$	$738.581$	$738.581$	$738.581$	$590.865$	$397.270$	$397.270$	$492.388$	$271.709$	$532.774$
$0.564$	$719.323$	$719.323$	$719.323$	$575.458$	$386.912$	$602.879$	$479.549$	$264.624$	$518.882$
$0.995$	$1574.126$	$1574.126$	$1574.126$	$1259.301$	$846.696$	$1319.306$	$1049.418$	$579.089$	$1135.492$
$0.672$	$914.542$	$914.542$	$914.542$	$731.633$	$491.916$	$766.495$	$609.695$	$336.441$	$659.702$
$0.582$	$833.671$	$833.671$	$833.671$	$666.937$	$448.418$	$698.716$	$555.781$	$306.691$	$601.367$
$0.537$	$818.018$	$818.018$	$818.018$	$654.415$	$439.998$	$685.597$	$545.346$	$300.932$	$590.076$
$0.508$	$819.622$	$819.622$	$819.622$	$655.698$	$440.861$	$686.942$	$546.415$	$301.522$	$591.233$
$0.990$	$1589.655$	$1589.655$	$1589.655$	$1271.724$	$855.048$	$1332.321$	$1059.770$	$584.801$	$1146.694$
$0.657$	$949.649$	$949.649$	$949.649$	$759.720$	$510.800$	$795.920$	$633.100$	$349.357$	$685.027$
$0.556$	$890.034$	$890.034$	$890.034$	$712.027$	$478.734$	$745.955$	$593.356$	$327.425$	$642.024$
$0.504$	$894.200$	$894.200$	$894.200$	$715.360$	$480.975$	$749.446$	$596.133$	$328.958$	$645.029$
$0.469$	$914.696$	$914.696$	$914.696$	$731.757$	$491.999$	$766.625$	$609.797$	$336.498$	$659.814$
$0.986$	$1605.147$	$1605.147$	$1605.147$	$1284.117$	$863.381$	$1345.305$	$1070.098$	$590.500$	$1157.869$
$0.642$	$984.313$	$984.313$	$984.313$	$787.450$	$529.445$	$824.972$	$656.209$	$362.109$	$710.032$
$0.534$	$945.071$	$945.071$	$945.071$	$756.057$	$508.337$	$792.082$	$630.047$	$347.672$	$681.724$
$0.478$	$967.946$	$967.946$	$967.946$	$774.357$	$520.642$	$811.255$	$645.298$	$356.088$	$698.226$
$0.439$	$1006.107$	$1006.107$	$1006.107$	$804.886$	$541.168$	$843.238$	$670.738$	$370.126$	$725.753$
$0.981$	$1620.603$	$1620.603$	$1620.603$	$1296.482$	$871.694$	$1358.259$	$1080.402$	$596.186$	$1169.018$
$0.630$	$1018.580$	$1018.580$	$1018.580$	$814.864$	$547.877$	$853.692$	$679.054$	$374.715$	$734.750$
$0.516$	$999.011$	$999.011$	$999.011$	$799.209$	$537.351$	$837.291$	$666.008$	$367.516$	$720.634$
$0.456$	$1039.790$	$1039.790$	$1039.790$	$831.832$	$559.285$	$871.469$	$693.193$	$382.517$	$750.050$
$0.416$	$1094.779$	$1094.779$	$1094.779$	$875.824$	$588.863$	$917.556$	$729.853$	$402.747$	$789.716$

, we discuss $N_c$ for the first three cases and in

Table 6. $N_c$ for the cases (iv), (v), and (vi), for different values of $x_c$ and $ϵ$.

${\mathit{x}}_{\mathit{c}}\downarrow /\mathit{ϵ}\to$	0			$0.5$			1
	${\mathit{N}}_{\mathit{c}}$			${\mathit{N}}_{\mathit{c}}$			${{N}}_{\mathit{c}}$
	Case (iv)	Case (v)	Case (vi)	Case (iv)	Case (v)	Case (vi)	Case (iv)	Case (v)	Case (vi)
$1.000$	$1558.560$	$1558.560$	$1558.560$	$2078.080$	$1870.272$	$2337.840$	$3117.120$	$2337.840$	$4675.680$
$0.690$	$878.932$	$878.932$	$878.932$	$1171.910$	$1054.719$	$1318.399$	$1757.865$	$1318.399$	$2636.797$
$0.613$	$775.667$	$775.667$	$775.667$	$1034.222$	$930.800$	$1163.500$	$1551.334$	$1163.500$	$2327.001$
$0.581$	$738.581$	$738.581$	$738.581$	$984.775$	$886.298$	$1107.872$	$1477.163$	$1107.872$	$2215.744$
$0.564$	$719.323$	$719.323$	$719.323$	$959.097$	$863.188$	$1078.984$	$1438.646$	$1078.984$	$2157.969$
$0.995$	$1574.126$	$1574.126$	$1574.126$	$2098.835$	$1888.952$	$2361.190$	$3148.253$	$2361.190$	$4722.379$
$0.672$	$914.542$	$914.542$	$914.542$	$1219.389$	$1097.450$	$1371.813$	$1829.083$	$1371.813$	$2743.625$
$0.582$	$833.671$	$833.671$	$833.671$	$1111.562$	$1000.406$	$1250.507$	$1667.343$	$1250.507$	$2501.014$
$0.537$	$818.018$	$818.018$	$818.018$	$1090.691$	$981.622$	$1227.028$	$1636.037$	$1227.028$	$2454.055$
$0.508$	$819.622$	$819.622$	$819.622$	$1092.830$	$983.547$	$1229.433$	$1639.245$	$1229.433$	$2458.867$
$0.990$	$1589.655$	$1589.655$	$1589.655$	$2119.540$	$1907.586$	$2384.482$	$3179.310$	$2384.482$	$4768.964$
$0.657$	$949.649$	$949.649$	$949.649$	$1266.199$	$1139.579$	$1424.474$	$1899.299$	$1424.474$	$2848.948$
$0.556$	$890.034$	$890.034$	$890.034$	$1186.712$	$1068.040$	$1335.051$	$1780.067$	$1335.051$	$2670.101$
$0.504$	$894.200$	$894.200$	$894.200$	$1192.266$	$1073.040$	$1341.300$	$1788.400$	$1341.300$	$2682.599$
$0.469$	$914.696$	$914.696$	$914.696$	$1219.595$	$1097.635$	$1372.044$	$1829.392$	$1372.044$	$2744.088$
$0.986$	$1605.147$	$1605.147$	$1605.147$	$2140.195$	$1926.176$	$2407.720$	$3210.293$	$2407.720$	$4815.440$
$0.642$	$984.313$	$984.313$	$984.313$	$1312.417$	$1181.176$	$1476.469$	$1968.626$	$1476.469$	$2952.939$
$0.534$	$945.071$	$945.071$	$945.071$	$1260.094$	$1134.085$	$1417.606$	$1890.141$	$1417.606$	$2835.212$
$0.478$	$967.946$	$967.946$	$967.946$	$1290.595$	$1161.536$	$1451.920$	$1935.893$	$1451.920$	$2903.839$
$0.439$	$1006.107$	$1006.107$	$1006.107$	$1341.476$	$1207.328$	$1509.160$	$2012.214$	$1509.160$	$3018.321$
$0.981$	$1620.603$	$1620.603$	$1620.603$	$2160.804$	$1944.723$	$2430.904$	$3241.205$	$2430.904$	$4861.808$
$0.630$	$1018.580$	$1018.580$	$1018.580$	$1358.107$	$1222.296$	$1527.870$	$2037.161$	$1527.870$	$3055.741$
$0.516$	$999.011$	$999.011$	$999.011$	$1332.015$	$1198.814$	$1498.517$	$1998.023$	$1498.517$	$2997.034$
$0.456$	$1039.790$	$1039.790$	$1039.790$	$1386.387$	$1247.748$	$1559.685$	$2079.580$	$1559.685$	$3119.370$
$0.416$	$1094.779$	$1094.779$	$1094.779$	$1459.706$	$1313.735$	$1642.169$	$2189.559$	$1642.169$	$3284.338$

, we deal with $N_c$ for cases (iv), (v), and (vi).

Table 5 and Figure 3, Figure 4 and Figure 5 indicate that, as the values of $ϵ$ increase, $N_c$ drops, and hence lower values of $N_c$ are required. This validates the destabilizing impact of an increasing gravitational field on stationary convection. This happened because, for the case of increasing gravitational field, an increase in the estimate of $ϵ$ increases the gravitational force. Since the system’s disturbance persists as gravitational force increases, it accelerates the commencement of convection. Furthermore, when the gravitational field was convex rather than linear or concave, the system was more disturbed.

In Table 6 and Figure 3, Figure 4 and Figure 5, it is shown that, as the value of $ϵ$ grows, $N_c$ increases for the decreasing gravitational field, and therefore greater values of $N_c$ are necessary for the beginning of convection with an increase in $ϵ$. It demonstrates the stabilizing impact of decreasing gravitational field on stationary convection. This occurred because, for the case of decreasing gravitational field, an increase in the estimate of $ϵ$ decreases the gravitational force. Since the disruption in the system reverses as the gravitational force decreases, the onset of convection is delayed. It was also observed that the system was more stable when the gravitational field was convex rather than linear or concave.

The relationships between $N_c$ and $\delta$ for various gravitational fields are shown in Table 5 and Table 6 and Figure 6. It was observed that, as $\delta$ increases, $N_c$ increases as well, delaying the beginning of convection. This happened because the magnetic field made the system more viscous, which made it more stable. It was also found that, the impact of variable viscosity is more stable in the case of decreasing gravitational field $f\left(z\right)=z^2-2z$ compared to other scenarios.

The relationship between $N_c$ and $M_3$ for various gravitational fields is shown in Table 5 and Table 6 and the plots in Figure 7. As seen plots (a1), (a2), (a3) in Figure 7 with lower values of $\delta ,$$N_c$ drops as $M_3$ grows, and hence smaller values of $N_c$ are necessary for the beginning of convection. Additionally, at larger values of $\delta ,$ $Nc$ drops for smaller values of $M_3$ and subsequently rises for higher values of $M_3$, implying that higher values of $N_c$ are necessary with an increase in $M_3$ for the beginning of convection. It demonstrates that non-buoyancy magnetization can have a destabilizing or stabilizing impact. This occurred because when variable viscosity grows, the domain of non-buoyancy magnetization’s destabilizing influence decreases. This is an odd occurrence. This is due to the increased viscosity caused by the magnetic field, which in turn impacts the $M_3$ value, resulting in the influence of the magnetization parameter being counteracted by the variable viscosity.

9. Conclusions

The effect of changeable gravity and variable viscosity on the beginning of convection in ferromagnetic fluid layer heated from below in the presence of a uniform vertical magnetic field was investigated. The problem was described in terms of free-free boundary conditions, and the corresponding eigenvalue problem was addressed using the single term Galerkin approach. The analysis was carried out for six different scenarios of gravitational field variation: (i) $f\left(z\right)=z$, (ii) $f\left(z\right)=e^z$, (iii) $f\left(z\right)=log(1+z),$ (iv) $f\left(z\right)=-z$, (v) $f\left(z\right)=-z^2$, and (vi) $f\left(z\right)=z^2-2z$. In the first three cases, the size of the convection cell increased on raising the gravity variation parameter, while, in the last three cases, the size of the convection cell decreased on raising the viscosity and gravity variation parameter. Additionally, it was noted that the system was more disturbed for case (ii), whereas it was more stable for case (vi). The present work can be extended for free-rigid and rigid-rigid boundary conditions.