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Abstract

The qualitative influence of internal heat and variable gravity field on the onset of convection in a sparsely packed porous layer with horizontal fluid-saturated are investigated. Linear stability analysis is performed using the normal mode technique. The dimensionless governing equations with four cases of gravity field variation: (1) $F (z) = - z$ , (2) $F (z) = - z^{2}$ , (3) $F (z) = - z^{3}$ , and (4) $F (z) = - (e^{z} - 1)$ are solved using bvp4c in MATLAB R2020a and the corresponding eigenvalue problem is calculated for three types of boundaries, namely free-free, rigid-free, and rigid-rigid, respectively. The effect of an internal heat parameter $(Q)$ , Darcy number $(D a)$ , and gravity variation parameter $(δ)$ for the stability of the system is investigated graphically. The critical value of the Rayleigh number and corresponding wavenumber are calculated for other fixed parameters. It is observed that internal heat parameters enhance the onset of convection, whereas the gravity variation parameter delays the same. It is also observed that the system becomes more stable for the exponential variable gravity function and less stable for the cubic variable gravity function.

Keywords

Linear stability analysis internal heat source eigenvalue problem variable gravity function

Introduction

The convective instabilities for horizontal fluid-saturated porous layer were first solved by Horton and Rogers.¹ An excellent survey on the topic of the porous medium has been given by Nield and Bejan.² The onset of convection for sparsely packed porous layer (SSPL) with modulation has been explained by Rudraiah and Malashetty.³ Bhadauria⁴ examined the Rayleigh-Bènard convection with thermal modulation in SPPL. Benerji et al.⁵ had investigated both linear as well as non-linear analyses of convection in rotating SSPL. Shekhar et al.⁶ studied the effect of rotation and cross-diffusion parameters with the linear analysis in a SSPL. Most of the previous studies are modeled using the Darcy model. Thus, they are well packed and low permeability porous medium. But, in a SPPL, Darcy’s law in its regular form cannot be applicable because SPPL involves big void spaces and gives rise to viscous shear except Darcy resistance.

Many studies are available in the literature on convection with heat generation. Tritton and Zarraga⁷ studied the thermal convection with internal heat experimentally and theoretically studied by Roberts⁸ and Thirlby.⁹ Convection with internal heat parameter in a biased fluid medium is observed by Takashima.^10,11 They deduced that onset of convection is extensively influenced by an inner heat source. Hill¹² studied the DDC in a porous medium with internal heat. Natural convection by the heat source outcome of heat source distribution tested by Tasaka and Takeda.¹³ Riahi¹⁴ studied the convection with the help of nonlinear analysis in a porous layer along with internal heat parameter and commented that non-uniform internal heat source might decrease or increase the size of the cell optimal Rayleigh value and cell size. Convection with internally heated parameter in the porous layer was explained by Gasser and Kazimi¹⁵ and Tveitereid.¹⁶ Choi et al.¹⁷ deduced the convection in a porous layer with consistent internal heat parameter in multiplicity aspects. The effect of internal heat-generating function in vertical cavities of the porous layer is explained by Du and Bilgen.¹⁸ The influence of an internal heat source has been observed by Char and Chiang.¹⁹ Cookey et al.²⁰ explored the onset of thermal convection in a porous medium with an internal heat for a low Prandtl number. The effect of chemical reaction on Maxwell fluid porous medium is studied by Reddy and Ragoju.²¹ The influence of internal heat parameter of porous media saturated by nanofluid for the onset of Darcy–Brinkman convection was deduced by Yadav et al.²² Bhadauria et al.²³ and Capone et al.²⁴ have added other effects, namely internal heat parameter, in their research investigation. Joseph and Shir²⁵ analyzed the impact of internal heat parameters on the onset of convection. Reddy and Ragoju²⁶ examined the instability of saturated porous layer in the presence of throughflow and internal heat. Joseph²⁷ has calculated critical Rayleigh number in a porous medium using non-linear energy method with an internal heat source. Rionero and Straughan²⁸ studied linear and non-linear stability analyses and calculated critical Rayleigh numbers.

In above all investigations, the onset of convective with a porous medium is studied in the presence of a smooth gravity field, but there may be many situations that exist in nature where the gravitational field varies with distance. Some examples are crystal growth and larger scaled flow in outer and interior cores of the Earth, where not only gravity field increases, but the gravity field disparity is there with height also (see Alex and Patil²⁹ and Hirt et al.³⁰). The non-linear analysis of variable gravity for convection was investigated and employed the energy method by Straughan.³¹ Yadav³² explained throughflow at the onset of convection with a porous layer when variable gravity is present and found that the launch of convective activity holds up by them. The effect of gravity variation on convection with a porous layer had explained by Harfash³³ when the magnetic field is there. The influence of variable gravity for the onset of convection in nanofluid observed by Mahajan and Sharma³⁴ and Chand et al.³⁵ Suma et al.³⁶ and Gangadharaiah et al.³⁷ used one term Galerkin method to study the influence of variable gravity in the porous medium when throughflow is there. Chen and Chen³⁸ studied the influence of gradient of gravity in pure fluid on salt finger convection. The onset of convection with variable gravity and the magnetic nanofluid layer was studied by Amit and Mahesh.³⁹ Rionero and Straughan⁴⁰ deduced linear and non-linear theory in porous layer for convection under an internal heat parameter and variable gravity fields. Alex et al.⁴¹ investigated the varying gravity effects on convection with internal heat parameter in the anisotropic porous layer for a tiled temperature gradient. Herron⁴² investigated the onset of convection with variable gravity and internal heat for a porous layer. Mahabaleshwar et al.⁴³ explained the convection with variable gravity and variable internal heat parameter in a porous medium. Tzou⁴⁴ studied stress-free boundary conditions. Meanwhile, Tzou⁴⁵ investigated the combination of two different types of boundary surfaces and two rigid boundary surfaces were discussed.

This literature survey shows that there is no study has been investigated on the effects of internal heat and variable gravity field on the onset of convection in a sparsely packed porous medium. The study of variable gravity field and internal heat sources on the onset of convection in a sparsely packed porous medium is of very importance. However, the study of inconsistent gravity with an internal heat source in a porous layer is very limited. It has many applications in the transport of groundwater, chemical engineering, flows in the ocean, and Earth sciences. Therefore, in present problem three different types of boundary conditions with variable gravity are considered.

When the gravity field changes with height, the buoyancy force applied by the fluid also changes, as a result, some parts of the fluid layer tend to become unstable, while the rest will remain stable. Therefore, in large-scale convection problems, gravity variations with height must be taken into account. There are many examples of large-scale flows, some of them are: flows into the ocean, mantle, or atmosphere. So fluid experience distinct buoyancy forces at different points and as a result, it is very important to study the effect of variable gravity field on the onset of convection.

In this article, the impact of varying gravitational force with internal heat source are analysed for onset of convective in a sparsely packed porous layer for four types of gravity variations namely $F (z) = - z$ linear, $F (z) = - z^{2}$ parabolic, $F (z) = - z^{3}$ cubic, and $F (z) = - (e^{z} - 1)$ exponential. Three types of boundary surfaces are considered in this study: free-free, rigid-rigid, and rigid-free. Using bvp4c in MATLAB R2020a, the solution of the eigenvalue problems is solved numerically. The organisation of an article is as follows. Mathematical problem is described in section “Mathematical formulation.” In section “Basic state,” the basic state is discussed. The linear stability analysis is described in section “Linear stability analysis”. In section “Solution methodology,” the numerical method is described. The results and discussions are written in section “Results and discussions.” The article ends with the “Conclusion” section.

Mathematical formulation

Thermally conducting fluid is considered and placed between two infinitely parallel a horizontal layer at $z = 0$ and $z = d$ . Assuming that:

System holds Brinkman’s law;

Oberbeck-Boussinesq approximations are valid;

The viscous dissipation can be negligible.

The non-dimensional equations are (see Yadav et al.,²² Shekhar et al.⁴⁶) (Figure 1):

\nabla \cdot {V} = 0

(1)

\begin{aligned} (2) & \frac{1}{M^{2} ϕ P r} \frac{\partial {V}}{\partial t} + \frac{1}{M^{2} ϕ^{2} P r} ({V} \cdot \nabla) {V} + \frac{1}{M D a} {V} \\ + \frac{1}{M^{2} P r} \nabla P = \frac{Λ}{M} \nabla^{2} {V} + R a (1 + δ F (z)) T \hat{e_{z}} \end{aligned}

(2)

\frac{\partial T}{\partial t} + \frac{1}{M} ({V} \cdot \nabla) T = \nabla^{2} T + Q

(3)

and boundary conditions are given by

\begin{aligned} \begin{matrix} {V} = 0, T = 0 on z = 1 \\ {V} = 0, T = 1 on z = 0 \end{matrix}} \end{aligned}

(4)

All quantities which are used in the above equations have explained in Notation section. The re-scaling used to obtain equations (1) to (2) with equation (3) are

\begin{aligned} \begin{matrix} x = \frac{x^{*}}{d}, & y = \frac{y^{*}}{d}, & z = \frac{z^{*}}{d} \\ {V} = \frac{{V}^{*}}{\frac{k_{T}}{M d}}, & T = \frac{T^{*}}{Δ T}, & P = \frac{P^{*}}{ρ_{0} k_{T}^{2} M^{- 2} d^{- 2}} \\ P r = \frac{μ}{ρ_{0} k_{T}}, & R a = \frac{ρ_{0} α g_{0} Δ T d^{3}}{μ k_{T}}, & D a = \frac{K_{p}}{d^{2}} \\ t = \frac{t^{*}}{\frac{M d^{2}}{k_{T}}}, & Q = \frac{Q * d^{2}}{k_{T} Δ T}, & Λ = \frac{μ_{e}}{μ} \end{matrix} \end{aligned}

Figure 1.

Physical configuration of problem.

For simplicity the asterisks “*” are omitted. All quantities which are used in the above equations have explained in Notation section.

Basic state

The basic steady-state flow of equations (1) to (2) is

u_{b} = 0, v_{b} = 0, w_{b} = 0

(5)

where

b

indicates the basic state. On substituting equation (4) into equation (2), the solution of basic state temperature is

T_{b} = \frac{1}{2} (2 + Q z - 2 z - Q z^{2})

(6)

Linear stability analysis

The perturbation is introduced in basic state which is in the form of

{V} = {V}_{b} + ϵ {\vec{V}}^{'}, T = T_{b} + ϵ T^{'}, P = P_{b} + ϵ P^{'}

(7)

where

ϵ << 1

. By substituting equation (6) into equations (1) to (3) and neglect the terms of

O (ϵ^{2})

or higher, the linear governing equations are obtained as follows:

\begin{aligned} \frac{1}{M^{2} ϕ P r} \frac{\partial {V}^{'}}{\partial t} + \frac{1}{M D a} {V}^{'} + \frac{1}{M^{2} P r} \nabla P^{'} = \frac{Λ}{M} \nabla^{2} {V}^{'} \\ + R a (1 + δ F (z)) T^{'} \hat{e_{z}} \end{aligned}

(8)

\frac{\partial T^{'}}{\partial t} + \frac{1}{M} G (z) w^{'} = \nabla^{2} T^{'}

(9)

w^{'} = 0, T^{'} = 0 at z = 0, 1

(10)

Here

G (z) = \frac{1}{2} (- 2 + Q - 2 Q z)

. By taking the third components of curl of equation (7), the resultant equations are

\begin{aligned} (\frac{1}{M^{2} ϕ P r} \frac{\partial}{\partial t} + \frac{1}{M D a} - \frac{Λ}{M} \nabla^{2}) w_{z} = 0 \\ (\frac{1}{M^{2} ϕ P r} \frac{\partial}{\partial t} + \frac{1}{M D a} - \frac{Λ}{M} \nabla^{2}) \nabla^{2} w^{'} = \end{aligned}

(11)

(R a (1 + δ F (z)) \nabla_{h}^{2} T^{'}

(12)

where

w_{z} = (\nabla \times {V}^{'})

Now the normal modes are introduced by writing the perturbations in the form of the following equation:

(w^{'}, T^{'}) = (W (z), T (z)) e^{i (l x + m y) + ω t}

(13)

where

ω

is the growth rate and

q = (l, m, 0)

is the wave vector with

q = \sqrt{l^{2} + m^{2}}

expressing the wavenumber. In general,

ω

is a complex quantity in such a way that

ω = ω_{r} + i ω_{i}

. If

ω_{r} < 0

system is always stable, and the system will become unstable for

ω_{r} > 0

. For neutral stability of system

ω_{r} = 0

Substitute the above expression into equations (11) and (8), we obtain

\begin{aligned} (\frac{ω}{M^{2} ϕ P r} + \frac{1}{M D a} - \frac{Λ}{M} (D^{2} - q^{2})) (D^{2} - q^{2}) W \\ + q^{2} R a (1 + δ F (z)) T = 0 \end{aligned}

(14)

(ω - (D^{2} - q^{2})) T + \frac{1}{M} G (z) W = 0

(15)

It is assumed that principle of exchange of instabilities holds. In other words, the marginal stability condition happens to stationary modes. Hence the value of

ω = 0

is substituted into the above equations and the resultant equations are:

\begin{aligned} (\frac{1}{M D a} - \frac{Λ}{M} (D^{2} - q^{2})) (D^{2} - q^{2}) W \\ + q^{2} R a (1 + δ F (z)) T = 0 \end{aligned}

(16)

((D^{2} - q^{2})) T - \frac{1}{M} G (z) W = 0

(17)

Now the above eigenvalue problems are solved for following boundary conditions.

Free-free boundary conditions

w = D^{2} w = T = 0 a t z = 0, 1

(18)

Rigid-rigid boundary conditions

w = D w = T = 0 a t z = 0, 1

(19)

Rigid-free boundary conditions

\begin{aligned} w = D w = T = 0 a t z = 0 \\ w = D^{2} w = T = 0 a t z = 1 \end{aligned}

(20)

Solution methodology

The eigenvalue problem obtained by equations (15) and (16) is solved with the help of bvp4c routine in MATLAB R2020a. It is considered that $w^{'} (0) = 1$ as a normalization condition for obtaining a non-trivial solution of a given eigenvalue problem. To get eigenvalue $R a$ , the above normalization condition is used. The critical value of the Rayleigh number and the critical value of wavenumber are acquired using the index-linked instruction in MATLAB R2020a. The comparative and conclusive patience have been assumed as $10^{- 6}$ and $10^{- 10}$ , respectively, to obtain higher-order exactness.

Results and discussions

The influence of internal heat parameter and variable gravity field on the convective instability in a sparsely packed porous media is examined with the linear stability theory. The four cases of variable gravity function: (1) $F (z) = - z$ , (2) $F (z) = - z^{2}$ , (3) $F (z) = - z^{3}$ , and (4) $F (z) = - (e^{z} - 1)$ are considered for investigated. Non-dimensional parameters existing in the governing equations for the onset of convection are internal heat parameter $(Q)$ , Darcy number $(D a)$ , and Rayleigh number $(R a)$ . The eigenvalue problem of three types of boundary conditions is solved in MATLAB R2020a by employing the bvp4c routine. In all the figures, to show the results correspond for free-free boundaries blue lines are used and red lines are used to show the results correspond to rigid-rigid boundaries. Black lines are used in all the figures for rigid-free boundaries.

The results that exist in the literature and results coming from the current study are compared for validation of the current method of solution. The governing equation of this article may be reduced by Chandrasekhar⁴⁷ in the absence of SPPL and internal heat with constant gravity. Table 1 shows the comparison between critical Rayleigh number, $R a_{c}$ , and corresponding critical wavenumber, $a_{c}$ , given by Chandrasekhar⁴⁷ and our numerical value of $R a_{c}$ and wavenumber, $a_{c}$ . See page no. 43 of Chandrasekhar.⁴⁷

Table 1.

Comparison table.

Nature of boundaries	Chandrasekhar⁴⁷		Present study
Nature of boundaries	$R a_{c}$	$q_{c}$	$R a_{c}$	$q_{c}$
Free-free	657.511	2.221	657.511	2.221
Rigid-free	1100.650	2.682	1100.649	2.681
Rigid-rigid	1707.762	3.117	1707.762	3.117

Before writing a discussion of results, the authors discuss on the relation between the present problem and real application. The external regulation of the onset of convection is essential in studying convection in a porous medium. Controlling the onset of convection is a significant issue in systems where fluids work as a medium, especially when a fluid system has internal heat. So in the present article, four parameters, such as $(1)$ internal heat source ( $Q$ ), $(2)$ Darcy number, $(3)$ amplitude of modulation, and $(4)$ different types of the gravity fields are considered for advancing or delaying for the onset of convective as observed in a real-life application. The parameters $M$ , $Λ$ , and $ϕ$ belong to the sparsely packed porous medium. Internal heat source ( $Q$ ) and modulation amplitude ( $δ$ ) are external factors for controlling the onset of convection.

Figure 2 gives a visual representation between the critical value of Rayleigh number $(R a_{c})$ and $(Q)$ for different values of $D a = 0.01, 0.1, 1$ and other parameters are fixed. From Figure 2, it is clear that as $Q$ and $D a$ increase, $R a_{c}$ decreases for three types of boundary conditions. So Darcy number, $D a$ and internal heat parameter, $Q$ have destabilized the system, therefore, the onset of convection is enhanced. It is also clear from Figure 2 and Tables 2 to 4 that for rigid-rigid boundary system is more stable and less stable in the case of free-free boundaries. It is also concluded from Figure 2 that system is more stable for the exponential case of gravity function and least stable for the cubic variation of gravity function.

Figure 2.

Variation of $R a_{c}$ with $Q$ for distinct values of $D a$ for four instances 1, 2, 3, and 4 when $Λ = 8, M = 0.9$ , and $δ = 0.1$ .

Table 2.

Evaluation of $R a_{c}$ and $q_{c}$ for different values of $D a$ and $δ$ when $Λ = 8, M = 0.9, Q = 2$ for free-free boundaries.

		Case 1		Case 2		Case 3		Case 4
Da	$δ$	$R a_{c}$	$q_{c}$	$R a_{c}$	$q_{c}$	$R a_{c}$	$q_{c}$	$R a_{c}$	$q_{c}$
	0.00	9438.817	2.522	9438.817	2.522	9438.817	2.522	9438.820	2.521
	0.25	10897.566	2.518	10255.415	2.518	9945.864	2.518	11562.042	2.512
0.01	0.50	12883.072	2.510	11222.106	2.510	10507.748	2.514	14886.405	2.497
	0.75	15737.222	2.498	12382.821	2.502	11133.330	2.506	20764.365	2.479
	1.00	20168.214	2.485	13799.540	2.493	11833.380	2.502	33399.478	2.476
	0.00	5670.705	2.269	5670.705	2.269	5670.699	2.271	5670.700	2.271
	0.25	6537.212	2.265	6152.303	2.265	5968.138	2.268	6928.088	2.263
0.1	0.50	7713.793	2.261	6721.563	2.261	6297.480	2.265	8890.144	2.253
	0.75	9400.788	2.253	7404.205	2.257	6663.956	2.262	12354.255	2.239
	1.00	12014.281	2.245	8236.764	2.253	7073.948	2.256	19897.592	2.229
	0.00	5277.986	2.233	5277.986	2.233	5277.984	2.231	5277.985	2.233
	0.25	6083.252	2.227	5725.114	2.229	5553.933	2.229	6446.022	2.225
1	0.50	7176.322	2.221	6253.520	2.225	5859.441	2.225	8267.776	2.215
	0.75	8742.994	2.215	6887.041	2.217	6199.363	2.221	11483.241	2.201
	1.00	11169.290	2.206	7659.576	2.213	6579.629	2.217	18494.986	2.190

Table 3.

Evaluation of $R a_{c}$ and $q_{c}$ for different values of $D a$ and $δ$ when $Λ = 8, M = 0.9, Q = 2$ for rigid-free boundaries.

		Case 1		Case 2		Case 3		Case 4
Da	$δ$	$R a_{c}$	$q_{c}$	$R a_{c}$	$q_{c}$	$R a_{c}$	$q_{c}$	$R a_{c}$	$q_{c}$
	0.00	12186.767	2.826	12186.768	2.826	12186.771	2.827	12186.767	2.826
	0.25	14202.151	2.821	13356.627	2.821	12930.500	2.821	15176.425	2.817
0.01	0.50	17007.169	2.815	14768.049	2.816	13766.562	2.818	20060.848	2.806
	0.75	21168.829	2.807	16501.501	2.809	14712.275	2.812	29335.766	2.794
	1.00	27945.743	2.798	18675.699	2.803	15789.276	2.809	52006.953	2.827
	0.00	8505.628	2.699	8505.628	2.699	8505.628	2.699	8505.628	2.699
	0.25	9628.458	2.692	9202.587	2.692	8871.524	2.692	11159.458	2.687
0.1	0.50	11875.469	2.691	10310.956	2.691	9610.434	2.693	14020.363	2.685
	0.75	14792.585	2.685	11526.738	2.688	10273.482	2.689	20584.442	2.677
	1.00	19565.871	2.679	13056.693	2.683	11030.410	2.687	37214.723	2.703
	0.00	8132.989	2.682	8132.989	2.682	8132.990	2.682	8132.989	2.682
	0.25	9479.458	2.678	8914.845	2.679	8630.028	2.678	10131.510	2.676
1	0.50	11355.824	2.674	9859.662	2.675	9189.657	2.676	13408.298	2.668
	0.75	14146.554	2.669	11022.857	2.671	9824.001	2.672	19695.126	2.660
	1.00	18715.629	2.663	12487.203	2.667	10548.369	2.670	35692.766	2.686

Table 4.

Evaluation of $R a_{c}$ and $q_{c}$ for different values of $D a$ and $δ$ when $Λ = 8, M = 0.9, Q = 2$ for rigid-rigid boundaries.

		Case 1		Case 2		Case 3		Case 4
Da	$δ$	$R a_{c}$	$q_{c}$	$R a_{c}$	$q_{c}$	$R a_{c}$	$q_{c}$	$R a_{c}$	$q_{c}$
	0.00	17923.256	3.168	17923.256	3.168	17923.256	3.168	17923.253	3.170
	0.25	20673.310	3.163	19422.823	3.163	18823.379	3.163	21881.082	3.158
0.01	0.50	24406.636	3.153	21186.889	3.153	19813.280	3.158	28018.933	3.139
	0.75	29752.456	3.142	23288.958	3.148	20906.142	3.153	38690.948	3.121
	1.00	37999.788	3.127	25830.858	3.137	22117.605	3.142	60861.117	3.115
	0.00	13998.344	3.132	13998.344	3.132	13998.344	3.132	13998.348	3.130
	0.25	16135.394	3.127	15158.572	3.127	14691.943	3.127	17068.466	3.122
0.1	0.50	19033.502	3.117	16522.349	3.117	15454.308	3.122	21821.980	3.106
	0.75	23178.818	3.107	18146.280	3.112	16295.582	3.117	30080.300	3.090
	1.00	29568.704	3.097	20109.029	3.102	17227.847	3.112	47322.054	3.082
	0.00	13603.502	3.127	13603.502	3.127	13603.502	3.127	13603.502	3.127
	0.25	15679.092	3.122	14729.810	3.122	14276.518	3.122	16584.737	3.117
1	0.50	18493.530	3.112	16053.610	3.112	15016.215	3.117	21199.794	3.102
	0.75	22518.665	3.102	17629.796	3.107	15832.432	3.112	29216.826	3.087
	1.00	28722.675	3.092	19534.749	3.097	16736.897	3.107	45963.971	3.077

The influence of gravity variation parameter $(δ)$ to critical Rayleigh number ( $R a_{c}$ ) against internal heat source $Q$ are shown in Figure 3. In this figure, it is observed that $R a_{c}$ decreases as the value of $Q$ increases for different values of $δ = 0.1, 0.2, 0.3$ for all three types of boundary conditions. Figure 3 indicates that for increasing the value of the gravity variation parameter $(δ)$ system is going to stabilize, so the onset of convection for the system is delayed. It is also concluded from Figure 3 that the system is least unstable for the exponential case of gravity function and least stable for the cubic variation of gravity function.

Figure 3.

Variation of $R a_{c}$ with $Q$ for distinct values of $δ$ for four instances 1, 2, 3, and 4 when $Λ = 8, M = 0.9$ , and $D a = 0.01$ .

Figure 4 gives the visualization of critical value of Rayleigh number $R a_{c}$ versus amplitude of modulation $δ$ for different values of $D a = 0.01, 0.1, 1$ with fixed value at $Λ = 8$ , $M = 0.9$ , and $Q = 2$ . It is observed from Figure 4 that the critical Rayleigh number increases as $δ$ increases. So it is showing the stabilizing effect on the system. Therefore, the onset of convection is delayed because of the increasing amplitude of modulation of the constant internal heat source. It is seen that $R a_{c}$ increases as $D a$ decreases for three types of boundary conditions. So Darcy number $D a$ shows destabilizing effects on the system. One can find from Figure 4 that the system is more stable for the exponential case of gravity function and least stable for the cubic variation of gravity function.

Figure 4.

Variation of $R a_{c}$ with $δ$ for distinct values of $D a$ for four instances 1, 2, 3, and 4 when $Λ = 8, M = 0.9$ , and $Q = 2$ .

The relationship between $R a_{c}$ versus $δ$ for internal heat source $Q = 0, 2, 4$ and other fixed parameters are shown in Figure 5. From Figure 5, it can observe that when the values of $Q$ decrease, the value of $R a_{c}$ increases, therefore, the system becomes unstable and hence the onset of convection is enhanced. $R a_{c}$ increases as $δ$ increases, so $δ$ has a stabilizing effect on the system, therefore, the onset of convection is delayed.

Figure 5.

Variation of $R a_{c}$ with $δ$ for distinct values of $Q$ for four instances 1, 2, 3, and 4 when $Λ = 8, M = 0.9$ , and $D a = 0.01$ .

Figure 6 gives a visual representation between the critical value of Rayleigh number $(R a_{c})$ and $(Q)$ for different values of $Λ = 6, 8, 10$ and other parameters is fixed. From Figure 6 it is clear that as $Q$ increases, $R a_{c}$ decreases for three types of boundary conditions so $Q$ have destabilized the system, therefore, the onset of convection is enhanced. On the other hand, as $Λ$ increases, $R a_{c}$ increases so $Λ$ have stabilized the system, therefore, the onset of convection is delayed. It is also clear from Figure 6 that for rigid-rigid boundaries system is more stable and less stable in the case of free-free boundaries. It is also concluded from Figure 6 that the system is more stable for the exponential case of gravity function and least stable for the cubic variation of gravity function.

Figure 6.

Variation of $R a_{c}$ with $Q$ for distinct values of $Λ$ for four instances 1, 2, 3, and 4 when $δ = 0.2, M = 0.9$ , and $D a = 0.01$ .

Conclusions

The study of the variable gravity field and internal heat parameter for convection instability in a sparsely packed porous layer is examined. The behavior of many parameters like gravity variation parameter $δ$ , internal heat source parameter, $Q$ , and Darcy number, $D a$ has been analyzed for critical Rayleigh number, $R a_{c}$ in case of different boundary conditions, namely free-free, rigid-rigid, and rigid-free. The eigenvalue problem is solved using bvp4c in MATLAB R2020a. The investigation is accomplished for four categories of variable gravity field function and results can be summarized as below:

Internal heat source destabilizes the system.

Gravity variation parameter is to stabilize effect on the system.

When Darcy’s number is increasing, the system becomes unstable.

In the case of exponential gravity variation, the system is more stable, and the cubic gravity variation system is least stable in the presence of an internal heat source.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Ravi Ragoju

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The study of internal heat and variable gravity field on the onset of convection in a sparsely packed porous medium

Abstract

Keywords

Introduction

Mathematical formulation

Basic state

Linear stability analysis

Free-free boundary conditions

Rigid-rigid boundary conditions

Rigid-free boundary conditions

Solution methodology

Results and discussions

Conclusions

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

Appendix

References