Effects of LTNE on Two-Component Convective Instability in a Composite System with Thermal Gradient and Heat Source

Abstract

An analytical study is conducted to examine the influence of thermal gradients and heat sources on the onset of two-component Rayleigh–Bènard (TCRB) convection using the Darcy model. The study takes into account the effects of local thermal non-equilibrium (LTNE), thermal profiles, and heat sources. The composite structure is horizontally constrained by adiabatic stiff boundaries, and the resulting solution to the problem is obtained using the perturbation approach. The various physical parameters have been thoroughly examined, revealing that the fluid layer exhibits dominance in the two-layer configuration. It has been observed that the parabolic profile demonstrates greater stability in comparison to the step function. Conversely, in the setup where the porous layer dominates, the step function plays a crucial role in maintaining stability. The porous layer, model (iv), exhibits greater stability in the predominant combined structure, while the linear configuration is characterized by higher instability.

1. Introduction

Double-component convection is very important for studying the evolution of systems with various causes of density fluctuations. Such systems occur in the sun, the oceans, and in the mantle of the Earth. Modeling of two-component convection in a composite layer can be used in many situations, including the storage of nuclear waste, thermal insulation, grain storage, chemical dispersion through water-saturated soil, and soil pollution. However, one more crucial use is for modeling the boundary conditions at interfaces. This matter has garnered significant theoretical interest due to its importance in understanding interface boundary conditions, as well as its aforementioned practical implications. The role of LTNE is significant in inflows with pronounced temperature differences between the solid phase and fluid phase. Wang et al. [1] studied solar air receiver and radiation effects using the LTNE model. Internal heat, using a local thermal non-equilibrium model, was analyzed analytically by Altawallbeh et al. [2,3] in a Maxwell fluid-porous layer with and without a magnetic field. Abidin et al. [4] conducted a study to examine the effects of viscosity in a binary fluid layer. In a two-component system, the closed form of the solution to Marangoni convection with a magnetic field and salinity gradients was studied by Komala and Sumithra [5]. Kuznetsov and Nield [6] studied the LTNE effects for a nanofluid in the presence of a porous domain using the Darcy model. Thirupathi Thumma and Mishra [7] studied nanofluid flow over a stretching sheet in the presence of a heat source/sink using the domain decomposition method. Heat transmission in a porous medium under LTNE conditions was studied by Shashi Prabha et al. [8]. Double-component convective flow over a chemically reactive plate with a heat source/sink was investigated numerically by Kannan and Pullepu [9]. Using the LTNE model, Astanina et al. [10] studied the heat source effect for porous cavities. Hema et al. [11] used the LTNE model to investigate how heat flow affects two-component convection. Pulkit et al. [12] analyzed the phenomenon of double-diffusive convection in a rotating couple-stress ferromagnetic fluid-porous medium, taking into consideration the influence of both gravitational and magnetic fields. Shukla and Gupta [13] studied three-component convection in nanofluids using the LTNE model. Two-component convection in a combined setup with a heated and salted subsurface was investigated by Mahajan and Tripathi [14]. Meften and Ali [15] studied two-component convection with variable viscosity. Capone et al. [16,17] investigated the convection in a rotating, viscous, anisotropic porous medium in the LTNE regime. Using the Brinkmann–Forchheimer model, Meften et al. [18] examined the LTNE effects for dual-component convection with rotation. Safia et al. [19] investigated the impact of magnetic field, temperature, and concentration on convection in a nanofluid. Double diffusion of a nanofluid was investigated by Tahar et al. [20], who focused on magneto-natural convection. A stability assessment of Brinkman–Bènard convection was studied by Siddabasappa et al. [21]. Thermosolutal convection in an LTNE porous medium was examined by Noon and Haddad [22]. Corcione and Quintino [23] examined the dual-component effects for water-based nanofluids using Rayleigh–Bènard convection. Gangadharaih et al. [24] studied the gravity fluctuation and throughflow for an anisotropic porous layer. Ahmed [25] conducted a numerical study on convection in a fluid layer within a square-packed bed enclosure using the LTNE model. Atul and Anand Kumar [26] studied the effects of the solute boundary conditions and the heat source on two-component convection. Yellamma et al. [27] investigated the third-component effect for a combined system with a magnetic field, heat source, and thermal profiles. They obtained the thermal Marangoni number analytically.

Inspired by the aforementioned literature review, the current study examines the effects of LTNE in a composite layer with six thermal gradients on the onset of SCRB convection in a two-layer configuration with an incompressible horizontal fluid flow. The problem has been solved by using the perturbation technique and it is noted that the parabolic profile demonstrates greater stability in comparison to the step function. Conversely, in the setup where the porous layer dominates, the step function plays a crucial role in maintaining stability. This work undoubtedly holds tremendous potential for a multitude of applications in the fields of pure crystal growth formation, as elucidated by Rudolph et al. [28].

2. Materials and Methods

Assume a continuous horizontal layer of incompressible fluid with a thickness $d_{fL}$ and a densely packed region-II $d_{pL}$ of the same fluid that lies behind region-I and is heated continuously by heat sources $Q_{fL}$ and $Q_{pL}$ The area above region-1 is expected to be affected by surface tension effects and concentration as functions of temperature and concentration, while the area below region-II is assumed to be stable. The coordinate system used is Cartesian, with the z-axis pointing upward and the origin located at the interface between the fluid and porous layers, as shown in Figure 1. Both the solid and liquid phases are assumed to be in the LTNE, and it is thought that a solid–fluid field model can characterize the temperature differences between the solid and liquid phases separately for the porous layer.

For the composite system Darcy model, the basic equations assume the Boussinesq hypothesis (see Sumithra and Shyamala [29]).

Fluid layer (region-I):

$${\nabla }_{fL} · {\overrightarrow{q}}_{fL}=0$$

(1)

$$\begin{array}{l}\frac{\partial {\overrightarrow{q}}_{fL}}{\partial t_{fL}}+\left({\overrightarrow{q}}_{fL} ·{\nabla }_{fL}\right){\overrightarrow{q}}_{fL}=-\frac{{\nabla }_{fL}{P}_{fL}}{{\rho }_0}+\frac{{\mu }_{fL}}{{\rho }_0}{\nabla }_{fL}^2{\overrightarrow{q}}_{fL}\\ -\left[1-{{\displaystyle \alpha }}_{fL}({{\displaystyle T}}_{fL}-{{\displaystyle T}}_{fL0})+{{\displaystyle \alpha }}_{fLs}({{\displaystyle C}}_{fL}-{{\displaystyle C}}_{fL0})\right]g \widehat{k}\end{array}$$

(2)

$$\frac{\partial {{\displaystyle T}}_{fL}}{\partial t}+\left({\overrightarrow{q}}_{fL}·{\nabla }_{fL}\right)T_{fL}={\kappa }_{fL}{\nabla }_{fL}^2{{\displaystyle T}}_{fL}+{{\displaystyle Q}}_{fL}$$

(3)

$$\frac{\partial {{\displaystyle C}}_{fL}}{\partial t}+\left({\overrightarrow{q}}_{fL} · {\nabla }_{fL}\right)C_{fL}={{\displaystyle \kappa }}_{fLs}{\nabla }_{fL}^2{{\displaystyle C}}_{fL}$$

(4)

The porous layer (region-II) (see Nield and Bejan [30]):

$${{\displaystyle \nabla }}_{pL}· {{\displaystyle \overrightarrow{q}}}_{pL}=0$$

(5)

$$\begin{array}{l}{\varphi }^{-1}\frac{\partial {\overrightarrow{q}}_{pL}}{\partial t_{pL}}+{\varphi }^{-2}\left({\overrightarrow{q}}_{pL}·{\nabla }_{pL}\right){\overrightarrow{q}}_{pL}=-\frac{{\nabla }_{pL}{P}_{pL}}{{\rho }_0}+\frac{{\mu }_{pL}}{K {\rho }_0}{\overrightarrow{q}}_{pL}\\ -\left[1-{{\displaystyle \alpha }}_{fpL}({{\displaystyle T}}_{fpL}-{{\displaystyle T}}_{pL0})-{{\displaystyle \alpha }}_{spL}({{\displaystyle T}}_{spL}-{{\displaystyle T}}_{pL0})+{{\displaystyle \alpha }}_{pL}({{\displaystyle C}}_{pL}-{{\displaystyle C}}_{pL0})\right]g \widehat{k}\end{array}$$

(6)

$${\left(\rho c_p\right)}_{fpL}\left[\varphi \frac{\partial T_{fpL}}{\partial t_{pL}}+\left({\overrightarrow{q}}_{pL} · {\nabla }_{pL}\right)T_{fpL}\right]=\varphi {\kappa }_{fpL}{\nabla }_{pL}^2{T}_{fpL}+h(T_{spL}-T_{fpL})+Q_{pL}$$

(7)

$$\left(1-\varphi \right){\left(\rho c_p\right)}_{spL}\frac{\partial {{\displaystyle T}}_{spL}}{\partial {{\displaystyle t}}_{pL}}=\left(1-\varphi \right){{\displaystyle \kappa }}_{spL}{\nabla }_{pL}^2{{\displaystyle T}}_{spL}-h\left({{\displaystyle T}}_{spL}-{{\displaystyle T}}_{fpL}\right)$$

(8)

$$\varphi \frac{\partial {{\displaystyle C}}_{pL}}{\partial t_{pL}}+\left({\overrightarrow{q}}_{pL} . {\nabla }_{pL}\right)C_{pL}={{\displaystyle \kappa }}_{pLs} {\nabla }_{pL}^2{{\displaystyle C}}_{pL}$$

(9)

For the two regions, the basic state is expressed as follows:

$$\begin{array}{l}\left[{{\displaystyle \overrightarrow{q}}}_{fL},{{\displaystyle P}}_{fL},{{\displaystyle T}}_{fL},{{\displaystyle T}}_{sfL},C_{fL}\right]\\ =\left[0,{{\displaystyle P}}_{fLb}(z_{fL}),{{\displaystyle T}}_{fLb}(z_{fL}),{{\displaystyle T}}_{sfLb}(z_{fL}),C_{fLb}({{\displaystyle z}}_{fL})\right],\frac{-d_{fL}}{\mathsf{\Delta }T} \frac{d{T}_{fLb}}{d{z}_{fL}}={{\displaystyle F}}_{fL}(z_{fL})\end{array}$$

(10)

$$\begin{array}{l}\left[{{\displaystyle \overrightarrow{q}}}_{pL},{{\displaystyle P}}_{pL},{{\displaystyle T}}_{fpL},{{\displaystyle T}}_{spL},C_{pL}\right]\\ =\left[0,{{\displaystyle P}}_{pLb}(z_{pL}),{{\displaystyle T}}_{fpLb}(z_{pL}),{{\displaystyle T}}_{spLb}(z_{pL}),C_{pLb}({{\displaystyle z}}_{pL})\right],\frac{-d_{pL}}{\mathsf{\Delta }T} \frac{d{T}_{pLb}}{d{z}_{pL}}={{\displaystyle F}}_{pL}(z_{pL})\end{array}$$

(11)

The basic state salinity and temperature distributions are, respectively, found to be:

$${{\displaystyle T}}_{fLb}{{\displaystyle (z}}_{fL})=\frac{Q_{fL}{z}_{fL}{{\displaystyle (z}}_{fL}-d_{fL})}{2{{\displaystyle \kappa }}_{fL}}+\left(\frac{{{\displaystyle T}}_U-{{\displaystyle T}}_0}{{{\displaystyle d}}_{fL}}\right)F_{fL}{{\displaystyle (z}}_{fL})+{{\displaystyle T}}_0 0\le z_{fL}\le {{\displaystyle d}}_{fL}$$

$${{\displaystyle T}}_{fpLb}{{\displaystyle (z}}_{pL})={{\displaystyle T}}_{spLb}(z_{pL})=\frac{{{\displaystyle Q}}_{pL}{z}_{pL}\left({{\displaystyle z}}_{pL}+{{\displaystyle d}}_{pL}\right)}{2\varphi {{\displaystyle \kappa }}_{pL}}+\left(\frac{{{\displaystyle T}}_L-{{\displaystyle T}}_0}{{{\displaystyle d}}_{pL}}\right)F_{pL}{{\displaystyle (z}}_{pL})+{{\displaystyle T}}_0 0\le z_{pL}\le d_{pL}$$

$${{\displaystyle C}}_{fLb}(z_{fL})=\left(\frac{{{\displaystyle C}}_U-{{\displaystyle C}}_0}{{{\displaystyle d}}_{fL}}\right){{\displaystyle z}}_{fL}+{{\displaystyle C}}_0 0\le z_{fL}\le {{\displaystyle d}}_{fL}$$

$${{\displaystyle C}}_{pLb}(z_{pL})=\left(\frac{{{\displaystyle C}}_L-{{\displaystyle C}}_0}{{{\displaystyle d}}_{pL}}\right){{\displaystyle z}}_{pL}+{{\displaystyle C}}_0 0\le z_{pL}\le d_{pL}$$

$$\mathrm{Here}, {{\displaystyle T}}_0=\frac{{\kappa }_{fL}{{\displaystyle d}}_{pL}{{\displaystyle T}}_u+{{\displaystyle \kappa }}_{pL}{{\displaystyle d}}_{fL}{{\displaystyle T}}_L}{{\kappa }_{fL}{{\displaystyle d}}_{pL}+{{\displaystyle \kappa }}_{pL}{{\displaystyle d}}_{fL}}+\frac{{{\displaystyle d}}_{fL}{{\displaystyle d}}_{pL}\left({{\displaystyle Q}}_{fL}{{\displaystyle d}}_{fL}+{{\displaystyle Q}}_{pL}{{\displaystyle d}}_{pL}\right)}{2\left({\kappa }_{fL}{{\displaystyle d}}_{pL}+{{\displaystyle \kappa }}_{pL}{{\displaystyle d}}_{fL}\right)}$$

$${{\displaystyle C}}_0=\frac{{{\displaystyle \kappa }}_{fLs}{{\displaystyle C}}_U{{\displaystyle d}}_{pL}+{{\displaystyle \kappa }}_{pLs}{{\displaystyle C}}_L{{\displaystyle d}}_{fL}}{{{\displaystyle \kappa }}_{pLs}{{\displaystyle d}}_{fL}+{{\displaystyle \kappa }}_{fLs}{{\displaystyle d}}_{pL}}.$$

where the $‘b’$ symbolizes the basic state, and ${{\displaystyle F}}_{fL}(z_{fL})$ and ${{\displaystyle F}}_{pL}(z_{pL})$ are the non-dimensionalized thermal gradient that fulfils the situation ${\displaystyle \underset{0}{\overset{1}{\int }}{F}_{fL}(z_{fL}) d{z}_{fL}}={\displaystyle \underset{0}{\overset{1}{\int }}{F}_{pL}(z_{pL}) d{z}_{pL}}=1$.

To study the basic solution’s stability, the following mild disturbances are introduced.

$$\left[{{\displaystyle \stackrel{\rightharpoonup }{q}}}_{fL},{{\displaystyle P}}_{fL},{{\displaystyle T}}_{fL},{{\displaystyle C}}_{fL}\right]=\left[0,{{\displaystyle P}}_{fLb}(z_{fL}),{{\displaystyle T}}_{fLb}(z_{fL}),{{\displaystyle C}}_{fLb}(z_{fL})\right]+\left[{\overrightarrow{q}}_{fL}^{\prime },{\overrightarrow{P}}_{fL}^{\prime },\theta ,S\right]$$

(12)

$$\begin{array}{l}\left[{{\displaystyle \stackrel{\rightharpoonup }{q}}}_{pL},{{\displaystyle P}}_{pL},{{\displaystyle T}}_{fpL},{{\displaystyle T}}_{spL},C_{pL}\right]=\left[0,{{\displaystyle P}}_{pLb}(z_{pL}),{{\displaystyle T}}_{pLb}(z_{pL}),{{\displaystyle T}}_{spLb}(z_{pL}),{{\displaystyle C}}_{pLsb}(z_{pL})\right]\\ +\left[{\overrightarrow{q}}_{pL}^{\prime },{\overrightarrow{P}}_{pL}^{\prime },{{\displaystyle \theta }}_{pL},{{\displaystyle \theta }}_{spL},S_{pL}\right]\end{array}$$

(13)

The prime symbol represents a perturbation from the corresponding values at equilibrium. The physical values in Equations (1)–(9) are now augmented with the introduction of (12) and (13), and subsequently linearized based on convection. The pressure term is subsequently eliminated from Equations (2) and (6) through the application of the curl operation twice, while retaining only the upright factor.

The variables are non-dimensionalized and normal mode study is applied (see Yellamma et al. [31]); the corresponding distinctive quantities are found to be:

In $0\le z_{fL}\le 1$,

$${{\displaystyle \left({{\displaystyle D}}_{fL}^2-{{\displaystyle a}}_{fL}^2\right)}}^2{{\displaystyle W}}_{fL}=R_{fL}{{\displaystyle a}}_{fL}^2{{\displaystyle \theta }}_{fL}-R_{fLs}{{\displaystyle a}}_{fL}^2{{\displaystyle S}}_{fL}$$

(14)

$${{\displaystyle \left({{\displaystyle D}}_{fL}^2-{{\displaystyle a}}_{fL}^2\right)}}^2{{\displaystyle \theta }}_{fL}+\left[{{\displaystyle F}}_{fL}(z_{fL})+{{\displaystyle R}}_I^{\ast }\left(2z_{fL}-1\right)\right]{{\displaystyle W}}_{fL}=0$$

(15)

$${\tau }_{fLs}{{\displaystyle \left({{\displaystyle D}}_{fL}^2-{{\displaystyle a}}_{fL}^2\right)}}^2{{\displaystyle S}}_{fL}+{{\displaystyle W}}_{fL}=0$$

(16)

In $0\le z_{pL}\le 1$,

$${{\displaystyle \left({{\displaystyle D}}_{pL}^2-{{\displaystyle a}}_{pL}^2\right)}}^2{{\displaystyle W}}_{pL}=-{{\displaystyle R}}_{fpL}{{\displaystyle \beta }}^2{{\displaystyle a}}_{pL}^2{{\displaystyle \theta }}_{fpL}-{\tau }_{spL}{{\displaystyle R}}_{spL}{{\displaystyle \beta }}^2{{\displaystyle a}}_{pL}^2{{\displaystyle \theta }}_{spL}+{\tau }_{pLs}{{\displaystyle R}}_{pLs}{{\displaystyle \beta }}^2{{\displaystyle a}}_{pL}^2{{\displaystyle S}}_{pL}$$

(17)

$$\varphi \left({{\displaystyle D}}_{pL}^2-{{\displaystyle a}}_{pL}^2\right){{\displaystyle \theta }}_{fpL}+\left[{{\displaystyle F}}_{pL}(z_{pL})+{{\displaystyle R}}_{IM}^{\ast }\left(2z_{pL}+1\right)\right]{{\displaystyle w}}_{pL}=-H\left({{\displaystyle \theta }}_{spL}-{{\displaystyle \theta }}_{fpL}\right)$$

(18)

$$\left(1-\varphi \right)\left({{\displaystyle D}}_{pL}^2-{{\displaystyle a}}_{pL}^2\right){{\displaystyle \theta }}_{spL}=H\left({{\displaystyle \theta }}_{spL}-{{\displaystyle \theta }}_{fpL}\right)$$

(19)

$${\tau }_{pLs}{{\displaystyle \left({{\displaystyle D}}_{pL}^2-{{\displaystyle a}}_{pL}^2\right)}}^2{{\displaystyle S}}_{pL}+{{\displaystyle W}}_{pL}=0$$

(20)

where $R_{fL}=\frac{g{{\displaystyle \alpha }}_{fL}\left({{\displaystyle T}}_0-{{\displaystyle T}}_U\right)d_{fL}^3}{{\nu }_{fL}{\kappa }_{fL}},$ ${{\displaystyle R}}_{pL}=\frac{g{{\displaystyle \alpha }}_{pL}\left({{\displaystyle T}}_L-{{\displaystyle T}}_0\right){{\displaystyle d}}_{pL}^3}{{\nu }_{pL}{{\displaystyle \kappa }}_{pL}},$ $R_{fLs}=\frac{g{{\displaystyle \alpha }}_{fLs}\left({{\displaystyle C}}_0-{{\displaystyle C}}_U\right)d_{fL}^3}{{\nu }_{fL}{\kappa }_{fLs}},$ ${{\displaystyle R}}_{pLs}=\frac{g{{\displaystyle \alpha }}_{pLs}\left({{\displaystyle C}}_L-{{\displaystyle C}}_0\right){{\displaystyle d}}_{pL}^3}{{\nu }_{pL}{{\displaystyle \kappa }}_{pLs}},$ ${{\displaystyle R}}_{spL}=\frac{g{{\displaystyle \alpha }}_{spL}\left({{\displaystyle T}}_L-{{\displaystyle T}}_0\right){{\displaystyle d}}_{pL}^3}{{\nu }_{pL}{{\displaystyle \kappa }}_{spL}}$ are, respectively, the thermal, solute, and solid phase Rayleigh numbers, ${{\displaystyle R}}_I=\frac{Q_{fL}{d}_{fL}^2}{{\kappa }_{fL}},{{\displaystyle R}}_{IM}=\frac{{{\displaystyle Q}}_{pL}{{\displaystyle d}}_{pL}^2}{{{\displaystyle \kappa }}_{fpL}},$ ${{\displaystyle R}}_I^{\ast }=\frac{{{\displaystyle R}}_I}{2\left({{\displaystyle T}}_0-{{\displaystyle T}}_U\right)},$ ${{\displaystyle R}}_{IM}^{\ast }=\frac{{{\displaystyle R}}_{IM}}{2\left({{\displaystyle T}}_L-{{\displaystyle T}}_0\right)},$ ${{\displaystyle \beta }}^2=\frac{K}{{{\displaystyle d}}_{pL}^2}=Da$, $H=\frac{h{d}_{pL}^3}{{\kappa }_{fpL}}$ are the internal and corrected Rayleigh numbers, the Darcy number, and the inter-phase heat transfer coefficient, respectively; ${\alpha }_{fpL},$ ${\alpha }_{spL}$ are the fluid and solid phase thermal expansion coefficients; ${\kappa }_{fpL},$ ${\kappa }_{spL}$ are the fluid phase and solid phase thermal diffusivities; and ${\kappa }_{fLs},{\kappa }_{pLs}$ are the fluid and porous layer solute diffusivities (see nomenclature).

After non-dimensionalization, normal mode expansion is carried out on the proper boundary conditions.

At $z_{fL}=d_{fL}$, the situations are

$${{\displaystyle W}}_{fL}\left(1\right)=0,{{\displaystyle D}}_{fL}{{\displaystyle W}}_{fL}\left(1\right)=0,{{\displaystyle D}}_{fL}{{\displaystyle \theta }}_{fL}\left(1\right)=0,{{\displaystyle D}}_{fL}{{\displaystyle S}}_{fL}\left(1\right)=0$$

At $z_{pL}=d_{pL}$, the situations are

$${{\displaystyle W}}_{pL}(0)=0, D_{pL}{W}_{pL}(0)=0, {{\displaystyle D}}_{pL}{{\displaystyle \theta }}_{fpL}(0)=0, {{\displaystyle {{\displaystyle D}}_{pL}\theta }}_{spL}(0)=0, {{\displaystyle D}}_{pL}{{\displaystyle S}}_{pL}(0)=0$$

At $z_{fL}=0$, $z_{pL}=1$, the situations are

$$\widehat{T}{{\displaystyle W}}_{fL}\left(0\right)={\widehat{d}}^2{{\displaystyle W}}_{pL}\left(1\right), \widehat{T}{{\displaystyle D}}_{fL}{{\displaystyle W}}_{fL}\left(0\right)=\widehat{d}\widehat{\mu }{{\displaystyle D}}_{pL}{{\displaystyle W}}_{pL}\left(1\right),$$

$${{\displaystyle \theta }}_{fL}\left(0\right)=\widehat{T}{\widehat{d}}^2{{\displaystyle \theta }}_{fpL}\left(1\right), {{\displaystyle \theta }}_{fL}\left(0\right)=\widehat{T}{\widehat{d}}^2{{\displaystyle \theta }}_{spL}\left(1\right),$$

$${{\displaystyle D}}_{fL}{{\displaystyle \theta }}_{fL}\left(0\right)={\widehat{d}}^2{{\displaystyle D}}_{pL} {{\displaystyle \theta }}_{fpL}\left(1\right), {{\displaystyle D}}_{fL}{{\displaystyle \theta }}_{fL}\left(0\right)={\widehat{d}}^2{{\displaystyle D}}_{pL}{{\displaystyle \theta }}_{spL}\left(1\right),$$

$$S_{fL}(0)=\widehat{S} S_{pL}(1), D_{fL}{S}_{fL}(0)=D_{pL}{S}_{pL}(1),$$

$$\widehat{T}\widehat{d}{\beta }^2{{\displaystyle D}}_{fL}^3{{\displaystyle W}}_{fL}\left(0\right)=-{{\displaystyle D}}_{pL}{{\displaystyle W}}_{pL}\left(1\right)+\widehat{\mu } {\beta }^2{{\displaystyle D}}_{pL}^3{{\displaystyle W}}_{pL}\left(1\right),$$

$$\widehat{T}{{\displaystyle D}}_{fL}^2{{\displaystyle W}}_{fL}\left(0\right)=\widehat{\mu }{{\displaystyle D}}_{pL}^2{{\displaystyle W}}_{pL}\left(1\right).$$

where $\widehat{T}=\frac{\left({{\displaystyle T}}_L-{{\displaystyle T}}_0\right)}{\left({{\displaystyle T}}_0-{{\displaystyle T}}_U\right)},$ $\widehat{S}=\frac{C_L-C_0}{C_0-C_U},$ $\widehat{d}=\frac{{{\displaystyle d}}_{pL}}{d_{fL}}$, and $\widehat{\mu }=\frac{{{\displaystyle \mu }}_{pL}}{{\mu }_{fL}}$ are the thermal, salinity, depth, and viscosity ratio, respectively.

3. Method of Solution

The task at hand involves addressing the eigenvalue problem within the context of arbitrary boundary conditions related to temperature and concentration. This will be accomplished by employing the standard perturbation method, with the wave number $a_{fL}^2,a_{pL}^2$ serving as the perturbation parameter. The results of such a study can be used to assess the reliability of the method, and they can also be used to justify the method’s application to the solution of problems involving convective instability in which the critical stability parameter must be determined analytically.

The powers $a_{fL}^2,a_{pL}^2$ have been added to the dependent variables in both levels, as seen below:

$$\left[\begin{array}{c}{{\displaystyle W}}_{fL}\\ {{\displaystyle \theta }}_{fL}\\ {{\displaystyle S}}_{fL}\end{array}\right]={\displaystyle \sum _{j=0}^{\infty }{a}_{fL}^{2j}}\left[\begin{array}{c}{{\displaystyle W}}_{fLj}\\ {{\displaystyle \theta }}_{fLj}\\ {{\displaystyle S}}_{fLj}\end{array}\right] \mathrm{and} \left[\begin{array}{c}{{\displaystyle W}}_{pL}\\ {{\displaystyle \theta }}_{fpL}\\ {{\displaystyle \theta }}_{spL}\\ {{\displaystyle S}}_{pL}\end{array}\right]={\displaystyle \sum _{j=0}^{\infty }{a}_{pL}^{2j}}\left[\begin{array}{c}{{\displaystyle W}}_{pLj}\\ {{\displaystyle \theta }}_{fpLj}\\ {{\displaystyle \theta }}_{spLj}\\ {{\displaystyle S}}_{pLj}\end{array}\right]$$

(21)

When solving $a_{fL}^2,a_{pL}^2$ using the preceding equation in (14)–(20), we obtain zero-order expressions (see Appendix A).

The first-order equations for $a_{fL}^2,a_{pL}^2$ are as follows:

For region-I,

$${{\displaystyle D}}_{fL}^4{{\displaystyle W}}_{fL1}-R_{fL}\widehat{T}+R_{fLs}\widehat{S}=0$$

(22)

$${{\displaystyle D}}_{fL}^2{{\displaystyle \theta }}_{fL1}-\widehat{T}+\left[{{\displaystyle F}}_{fL}\right({{\displaystyle z}}_{fL})+{{\displaystyle R}}_I^{\ast }(2{{\displaystyle z}}_{fL}-1\left)\right]{{\displaystyle W}}_{fL1}=0$$

(23)

$${\tau }_{fLs}{{\displaystyle D}}_{fL}^2{{\displaystyle S}}_{fL1}-{\tau }_{fLs}\widehat{S}+{{\displaystyle W}}_{fL1}=0$$

(24)

For region-II,

$${{\displaystyle D}}_{pL}^2{{\displaystyle W}}_{pL1}+{{\displaystyle R}}_{fpL}{{\displaystyle \beta }}^2+{\tau }_{spL}{{\displaystyle R}}_{spL}{{\displaystyle \beta }}^2-{\tau }_{pLs}{{\displaystyle R}}_{pLs}{{\displaystyle \beta }}^2=0$$

(25)

$$\left(\varphi {{\displaystyle D}}_{pL}^2-H\right){{\displaystyle \theta }}_{fpL1}+H{{\displaystyle \theta }}_{spL1}+\left[{{\displaystyle F}}_{pL}(z_{pL})+{{\displaystyle R}}_{IM}^{\ast }\left(2z_{pL}+1\right)\right]W_{pL1}-\varphi =0$$

(26)

$$\left[\left(1-\varphi \right){{\displaystyle D}}_{pL}^2-H\right]{{\displaystyle \theta }}_{spL1}+H{{\displaystyle \theta }}_{pL1}-\left(1-\varphi \right)=0$$

(27)

$${\tau }_{pLs}{{\displaystyle D}}_{pL}^2{{\displaystyle S}}_{pL}-{\tau }_{pLs}+{{\displaystyle W}}_{pL}=0$$

(28)

The related boundary conditions are as follows:

$${{\displaystyle W}}_{fL1}\left(1\right)=0,{{\displaystyle D}}_{fL1}{{\displaystyle W}}_{fL1}\left(1\right)=0,{{\displaystyle D}}_{fL1}{{\displaystyle \theta }}_{fL1}\left(1\right)=0,{{\displaystyle D}}_{fL1}{{\displaystyle S}}_{fL1}\left(1\right)=0,$$

$$\widehat{T}{{\displaystyle W}}_{fL1}\left(0\right)={\widehat{d}}^2{{\displaystyle W}}_{pL1}\left(1\right), \widehat{T}{{\displaystyle D}}_{fL}{{\displaystyle W}}_{fL1}\left(0\right)=\widehat{d}\widehat{\mu }{{\displaystyle D}}_{pL}{{\displaystyle W}}_{pL1}\left(1\right),$$

$${{\displaystyle \theta }}_{fL1}\left(0\right)=\widehat{T}{\widehat{d}}^2{{\displaystyle \theta }}_{pL1}\left(1\right), {{\displaystyle \theta }}_{fL1}\left(0\right)=\widehat{T}{\widehat{d}}^2{{\displaystyle \theta }}_{spL1}\left(1\right),{{\displaystyle S}}_{fL1}\left(0\right)=\widehat{S}{\widehat{d}}^2{{\displaystyle S}}_{pL1}\left(1\right),$$

$${{\displaystyle D}}_{fL}{{\displaystyle \theta }}_{fL1}\left(0\right)={\widehat{d}}^2{{\displaystyle D}}_{pL}{{\displaystyle \theta }}_{fpL1}\left(1\right),D_{fL}{\theta }_{fL1}(0)={\widehat{d}}^2{{\displaystyle D}}_{pL}{{\displaystyle \theta }}_{spL1}\left(1\right),{{\displaystyle D}}_{fL}{{\displaystyle S}}_{fL1}\left(0\right)={\widehat{d}}^2{{\displaystyle D}}_{pL}{{\displaystyle S}}_{pL1}\left(1\right),$$

$$\widehat{T}\widehat{d}{\beta }^2{{\displaystyle D}}_{fL}^3{{\displaystyle W}}_{fL1}\left(0\right)=-{{\displaystyle D}}_{pL}{{\displaystyle W}}_{pL1}\left(1\right)+\widehat{\mu }{\beta }^2{{\displaystyle D}}_{pL}^3{{\displaystyle W}}_{pL1}\left(1\right), \widehat{T}{{\displaystyle D}}_{fL}^2{{\displaystyle W}}_{fL1}\left(0\right)=\widehat{\mu }{{\displaystyle D}}_{pL}^2{{\displaystyle W}}_{pL1}\left(1\right),$$

$${{\displaystyle W}}_{pL1}\left(0\right)=0, {{\displaystyle D}}_{pL}{{\displaystyle W}}_{pL1}\left(0\right)=0, {{\displaystyle D}}_{pL}{{\displaystyle \theta }}_{fpL1}\left(0\right)=0, {{\displaystyle D}}_{pL}{{\displaystyle \theta }}_{spL1}\left(0\right)=0,{{\displaystyle D}}_{pL}{{\displaystyle S}}_{pL1}\left(0\right)=0.$$

The solutions of $W_{fL1}$ and $W_{pL1}$ are as follows:

$${{\displaystyle W}}_{fL1} (z_{fL})={{\displaystyle \delta }}_{fL1}+{{\displaystyle \delta }}_{fL2} z_{fL}+{{\displaystyle \delta }}_{fL3} z_{fL}^2+{{\displaystyle \delta }}_{fL4} z_{fL}^3+\frac{(R_{fL}\widehat{T}-R_{fLs}\widehat{S})}{24}{z}_{fL}^4$$

(29)

$${{\displaystyle W}}_{pL1}(z_{pL})={{\displaystyle \delta }}_{pL5}+{{\displaystyle \delta }}_{pL6} z_{pL}-\frac{{{\displaystyle \beta }}^2\left({{\displaystyle R}}_{fpL}+{{\displaystyle R}}_{spL} {\tau }_{spL}-{{\displaystyle R}}_{pLs} {\tau }_{pLs}\right)}{2}{z}_{pL}^2$$

(30)

where ${{\displaystyle \delta }}_{fL1},{{\displaystyle \delta }}_{fL2},{{\displaystyle \delta }}_{fL3},{{\displaystyle \delta }}_{fL4},{{\displaystyle \delta }}_{fL5},{{\displaystyle \delta }}_{fL6}$ are the constants obtained utilizing velocity boundary conditions; for the form they are in, see Appendix A.

4. Condition of Solvability

The solvability condition is derived by solving the differential equation with the necessary boundary conditions concerning heat.

$$\begin{array}{l}{{\displaystyle \widehat{d}}}^2\left(1+{{\displaystyle R}}_{IM}^{\ast }+{{\displaystyle \tau }}_{fLs}\right){\displaystyle \underset{0}{\overset{1}{\int }}{{\displaystyle W}}_{pL1}(z_{pL}) ·}{{\displaystyle F}}_{pL}(z_{pL}) d{z}_{pL}+2{{\displaystyle \widehat{d}}}^2{{\displaystyle R}}_{IM}^{\ast }{\displaystyle \underset{0}{\overset{1}{\int }}{{\displaystyle W}}_{pL1}}(z_{pL})·{{\displaystyle F}}_{pL}(z_{pL}) d{z}_{pL}\\ +\left(1-{{\displaystyle R}}_I^{\ast }+{{\displaystyle \tau }}_{pLs}\right){\displaystyle \underset{0}{\overset{1}{\int }}{{\displaystyle W}}_{fL1}(z_{fL}){{\displaystyle · F}}_{fL}(z_{fL}) d{{\displaystyle z}}_{fL}+2}{{\displaystyle R}}_I^{\ast }{\displaystyle \underset{0}{\overset{1}{\int }}{{\displaystyle W}}_{fL1}(z_{fL})·{{\displaystyle F}}_{fL}(z_{fL}) d{{\displaystyle z}}_{fL}}\\ =\left({\widehat{d}}^2+\widehat{T}\right)+\left({\widehat{d}}^2+\widehat{S}\right){\tau }_{fLs}{\tau }_{pLs}\end{array}$$

(31)

Computing ${{\displaystyle R}}_C$ for different profiles can be achieved by substituting expressions ${{\displaystyle W}}_{fL1}(z_{fL})$ and ${{\displaystyle W}}_{pL1}(z_{pL})$ from Equations (29) and (30) into Equation (31).

5. Thermal Gradients

The following thermal gradients are considered for the present study (refer to

Table 1. Thermal gradients for fluid and porous layer.

Model	Thermal Gradients	Fluid Layer (Region-I)	Porous Layer (Region-II)
Model (i)	Linear	${{\displaystyle F}}_{fL}(z_{fL})=1$	${{\displaystyle F}}_{pL}(z_{pL})=1$
Model (ii)	Parabolic	${{\displaystyle F}}_{fL}(z_{fL})=2z_{fL}$	${{\displaystyle F}}_{pL}(z_{pL})=2z_{pL}$
Model (iii)	Inverted parabolic	${{\displaystyle F}}_{fL}(z_{fL})=2(1-z_{fL})$	${{\displaystyle F}}_{pL}(z_{pL})=2(1-z_{pL})$
Model (iv)	Piecewise linear gradient heated from below (PLHB)	${{\displaystyle F}}_{fL}(z_{fL})=\{\begin{array}{cc}{\epsilon }_{fL}^{-1},& 0\le z_{fL}\le {\epsilon }_{fL}\\ 0,& {\epsilon }_{fL}\le z_{fL}\le 1\end{array}$	${{\displaystyle F}}_{pL}(z_{pL})=\{\begin{array}{cc}{\epsilon }_{pL}^{-1},& 0\le z_{pL}\le {\epsilon }_{pL}\\ 0,& {\epsilon }_{pL}\le z_{pL}\le 1\end{array}$
Model (v)	Piecewise linear gradient cooled from above (PLCA)	${{\displaystyle F}}_{fL}(z_{fL})=\{\begin{array}{cc}0,& 0\le z_{fL}\le (1-{\epsilon }_{fL})\\ {\epsilon }_{fL}^{-1},& (1-{\epsilon }_{fL})\le z_{fL}\le 1\end{array}$	${{\displaystyle F}}_{pL}(z_{pL})=\{\begin{array}{cc}0,& 0\le z_{pL}\le (1-{\epsilon }_{pL})\\ {\epsilon }_{pL}^{-1},& (1-{\epsilon }_{pL})\le z_{pL}\le 1\end{array}$
Model (vi)	Step function (SF)	${{\displaystyle F}}_{fL}(z_{fL})=\delta \left(z_{fL}-{\epsilon }_{fL}\right)$	${{\displaystyle F}}_{pL}(z_{pL})=\delta \left(z_{pL}-{\epsilon }_{pL}\right)$

; see Rudraiah et al. [32], Vasseur and Robillard [33], and Shivakumara [34]).

5.1. Model (i): Linear Thermal Gradient

The ${{\displaystyle R}}_C$ is attained for the linear model ${{\displaystyle F}}_{fL}(z_{fL})=1$ and ${{\displaystyle F}}_{pL}(z_{pL})=1$ by utilizing (31):

$$R_C=\frac{\left({\widehat{d}}^2+\widehat{T}\right)+\left({\widehat{d}}^2+\widehat{S}\right){\tau }_{fLs}{\tau }_{pLs}-{\widehat{d}}^4{\kappa }_{pLs}^2{\tau }_{pLs}{\mathsf{\Psi }}_1-{{\displaystyle R}}_{pLs}\left\{{{\displaystyle \mathsf{\Phi }}}_{11}+{{\displaystyle \mathsf{\Phi }}}_{12}\right\}}{\widehat{T}\left\{{{\displaystyle \mathsf{\Phi }}}_{13}+{{\displaystyle \mathsf{\Phi }}}_{14}\right\}+\left[{{\displaystyle \alpha }}_{fpL} {\kappa }_{fpL}^2+{{\displaystyle \alpha }}_{spL}{\kappa }_{spL}^2{\tau }_{spL}\right]{\widehat{d}}^4 {\mathsf{\Psi }}_1}$$

(32)

where ${\mathsf{\Psi }}_1={{\displaystyle \mathsf{\Phi }}}_1+{{\displaystyle \mathsf{\Phi }}}_2+{{\displaystyle \mathsf{\Phi }}}_3+{{\displaystyle \mathsf{\Phi }}}_4+{{\displaystyle \mathsf{\Phi }}}_5+{{\displaystyle \mathsf{\Phi }}}_6+{{\displaystyle \mathsf{\Phi }}}_7+{{\displaystyle \mathsf{\Phi }}}_8+{{\displaystyle \mathsf{\Phi }}}_9+{{\displaystyle \mathsf{\Phi }}}_{10},$ ${{\displaystyle \mathsf{\Phi }}}_1=\frac{{{\displaystyle \widehat{d}}}^2{{\displaystyle \beta }}^2\left(1+{{\displaystyle R}}_{IM}^{\ast }+{{\displaystyle \tau }}_{fLs}\right)}{6},$ ${{\displaystyle \mathsf{\Phi }}}_2=\frac{{{\displaystyle \widehat{d}}}^2{{\displaystyle \beta }}^2{{\displaystyle R}}_{IM}^{\ast }}{4},$ ${{\displaystyle \mathsf{\Phi }}}_3=\frac{{\widehat{d}}^2{\beta }^2\left(1-{{\displaystyle R}}_I^{\ast }+{{\displaystyle \tau }}_{pLs}\right)}{2},$ ${{\displaystyle \mathsf{\Phi }}}_4=\frac{{{\displaystyle \widehat{d}\beta }}^2\widehat{\mu }\left(1-{{\displaystyle R}}_I^{\ast }+{{\displaystyle \tau }}_{pLs}\right)}{2\widehat{T}},$ ${{\displaystyle \mathsf{\Phi }}}_5=\frac{{{\displaystyle \beta }}^2\widehat{\mu }\left(1-{{\displaystyle R}}_I^{\ast }+{{\displaystyle \tau }}_{pLs}\right)}{6\widehat{T}},$ ${{\displaystyle \mathsf{\Phi }}}_6=\frac{\left(1-{{\displaystyle R}}_I^{\ast }+{{\displaystyle \tau }}_{pLs}\right)}{24\widehat{d}\widehat{T}},$ ${{\displaystyle \mathsf{\Phi }}}_7=\frac{{{\displaystyle \widehat{d}}}^2{{\displaystyle \beta }}^2{{\displaystyle R}}_I^{\ast }}{2\widehat{T}},$ ${{\displaystyle \mathsf{\Phi }}}_8=\frac{2\widehat{d}{{\displaystyle \beta }}^2\widehat{\mu }{{\displaystyle R}}_I^{\ast }}{3\widehat{T}},$ ${{\displaystyle \mathsf{\Phi }}}_9=\frac{{{\displaystyle \beta }}^2\widehat{\mu }{{\displaystyle R}}_I^{\ast }}{4\widehat{T}},$ ${{\displaystyle \mathsf{\Phi }}}_{10}=\frac{{{\displaystyle R}}_I^{\ast }}{15\widehat{d}\widehat{T}},$ ${{\displaystyle \mathsf{\Phi }}}_{11}=\frac{\widehat{S}\left(1-{{\displaystyle R}}_I^{\ast }+{{\displaystyle \tau }}_{pLs}\right)}{120},$ ${{\displaystyle \mathsf{\Phi }}}_{12}=\frac{2\widehat{S}{{\displaystyle R}}_I^{\ast }}{144},$ ${{\displaystyle \mathsf{\Phi }}}_{13}=\frac{\left(1-{{\displaystyle R}}_I^{\ast }+{{\displaystyle \tau }}_{pLs}\right)}{120},$ ${{\displaystyle \mathsf{\Phi }}}_{14}=\frac{2{{\displaystyle R}}_I^{\ast }}{144}$.

5.2. Model (ii): Parabolic Thermal Gradient

The ${{\displaystyle R}}_C$ is attained for the parabolic model ${{\displaystyle F}}_{fL}(z_{fL})=2{{\displaystyle z}}_{fL},{{\displaystyle F}}_{pL}(z_{pL})=2{{\displaystyle z}}_{pL}$ by utilizing (31):

$${{\displaystyle R}}_C=\frac{\left({\widehat{d}}^2+\widehat{T}\right)+\left({\widehat{d}}^2+\widehat{S}\right){\tau }_{fLs}{\tau }_{pLs}-{\widehat{d}}^4{\kappa }_{pLs}^2{\tau }_{pLs}{\mathsf{\Psi }}_2-{{\displaystyle R}}_{pLs}\left\{{{\displaystyle \mathsf{{\rm N}}}}_{11}+{{\displaystyle \mathsf{{\rm N}}}}_{12}\right\}}{\widehat{T}\left\{{{\displaystyle \mathsf{{\rm N}}}}_{13}+{{\displaystyle \mathsf{{\rm N}}}}_{14}\right\}+\left[{{\displaystyle \alpha }}_{fpL} {\kappa }_{fpL}^2+{{\displaystyle \alpha }}_{spL}{\kappa }_{spL}^2{\tau }_{spL}\right]{\widehat{d}}^4 {\mathsf{\Psi }}_2}$$

(33)

where ${\mathsf{\Psi }}_2={{\displaystyle \mathsf{{\rm N}}}}_1+{{\displaystyle \mathsf{{\rm N}}}}_2+{{\displaystyle \mathsf{{\rm N}}}}_3+{{\displaystyle \mathsf{{\rm N}}}}_4+{{\displaystyle \mathsf{{\rm N}}}}_5+{{\displaystyle \mathsf{{\rm N}}}}_6+{{\displaystyle \mathsf{{\rm N}}}}_7+{{\displaystyle \mathsf{{\rm N}}}}_8+{{\displaystyle \mathsf{{\rm N}}}}_9+{{\displaystyle \mathsf{{\rm N}}}}_{10},$ ${{\displaystyle \mathsf{{\rm N}}}}_1=\frac{{{\displaystyle \widehat{d}}}^2{{\displaystyle \beta }}^2\left(1+{{\displaystyle R}}_{IM}^{\ast }+{{\displaystyle \tau }}_{fLs}\right)}{4},{{\displaystyle \mathsf{{\rm N}}}}_2=\frac{{{\displaystyle 2\widehat{d}}}^2{{\displaystyle \beta }}^2{{\displaystyle R}}_{IM}^{\ast }}{5},{{\displaystyle \mathsf{{\rm N}}}}_3=\frac{{\widehat{d}}^2{{\displaystyle \beta }}^2\left(1-{{\displaystyle R}}_I^{\ast }+{{\displaystyle \tau }}_{pLs}\right)}{2\widehat{T}},{{\displaystyle \mathsf{{\rm N}}}}_4=\frac{2\widehat{d}\widehat{\mu }{{\displaystyle \beta }}^2\left(1-{{\displaystyle R}}_I^{\ast }+{{\displaystyle \tau }}_{pLs}\right)}{3\widehat{T}},$ ${{\displaystyle \mathsf{{\rm N}}}}_5=\frac{{{\displaystyle \beta }}^2\widehat{\mu }\left(1-{{\displaystyle R}}_I^{\ast }+{{\displaystyle \tau }}_{pLs}\right)}{4\widehat{T}},{{\displaystyle \mathsf{{\rm N}}}}_6=\frac{\left(1-{{\displaystyle R}}_I^{\ast }+{{\displaystyle \tau }}_{pLs}\right)}{15\widehat{d}\widehat{T}},{{\displaystyle \mathsf{{\rm N}}}}_7=\frac{{{\displaystyle 3\widehat{d}}}^2{{\displaystyle \beta }}^2{{\displaystyle R}}_I^{\ast }}{3\widehat{T}},{{\displaystyle \mathsf{{\rm N}}}}_8=\frac{\widehat{d}{{\displaystyle \beta }}^2\widehat{\mu }{{\displaystyle R}}_I^{\ast }}{\widehat{T}},$ ${{\displaystyle \mathsf{{\rm N}}}}_9=\frac{{{\displaystyle 2\beta }}^2\widehat{\mu }{{\displaystyle R}}_I^{\ast }}{5\widehat{T}},{{\displaystyle \mathsf{{\rm N}}}}_{10}=\frac{{{\displaystyle R}}_I^{\ast }}{9\widehat{d}\widehat{T}},{{\displaystyle \mathsf{{\rm N}}}}_{11}=\frac{\widehat{S}\left(1-{{\displaystyle R}}_I^{\ast }+{{\displaystyle \tau }}_{pLs}\right)}{72},{{\displaystyle \mathsf{{\rm N}}}}_{12}=\frac{\widehat{S}{{\displaystyle R}}_I^{\ast }}{42},{{\displaystyle \mathsf{{\rm N}}}}_{13}=\frac{\left(1-{{\displaystyle R}}_I^{\ast }+{{\displaystyle \tau }}_{pLs}\right)}{72},{{\displaystyle \mathsf{{\rm N}}}}_{14}=\frac{{{\displaystyle R}}_I^{\ast }}{42}.$

5.3. Model (iii): Inverted Parabolic Thermal Gradient

The $R_C$ is attained for this model ${{\displaystyle F}}_{fL}(z_{fL})=2(1-z_{fL})$ and ${{\displaystyle F}}_{pL}(z_{pL})=2(1-z_{pL})$ by utilizing (31):

$${{\displaystyle R}}_C=\frac{\left({\widehat{d}}^2+\widehat{T}\right)+\left({\widehat{d}}^2+\widehat{S}\right){\tau }_{fLs}{\tau }_{pLs}-{\widehat{d}}^4{\kappa }_{pLs}^2{\tau }_{pLs}{\mathsf{\Psi }}_3-{{\displaystyle R}}_{pLs}\left\{{{\displaystyle \mathrm{A}}}_{11}+{{\displaystyle \mathrm{A}}}_{12}\right\}}{\widehat{T}\left\{{{\displaystyle \mathrm{A}}}_{13}+{{\displaystyle \mathrm{A}}}_{14}\right\}+\left[{{\displaystyle \alpha }}_{fpL} {\kappa }_{fpL}^2+{{\displaystyle \alpha }}_{spL}{\kappa }_{spL}^2{\tau }_{spL}\right]{\widehat{d}}^4 {\mathsf{\Psi }}_3}$$

(34)

where ${\mathsf{\Psi }}_3={{\displaystyle \mathrm{A}}}_1+{{\displaystyle \mathrm{A}}}_2+{{\displaystyle \mathrm{A}}}_3+{{\displaystyle \mathrm{A}}}_4+{{\displaystyle \mathrm{A}}}_5+{{\displaystyle \mathrm{A}}}_6+{{\displaystyle \mathrm{A}}}_7+{{\displaystyle \mathrm{A}}}_8+{{\displaystyle \mathrm{A}}}_9+{{\displaystyle \mathrm{A}}}_{10},$ ${{\displaystyle \mathrm{A}}}_1=\frac{{{\displaystyle \widehat{d}}}^2{{\displaystyle \beta }}^2\left(1+{{\displaystyle R}}_{IM}^{\ast }+{{\displaystyle \tau }}_{fLs}\right)}{12},{{\displaystyle \mathrm{A}}}_2=\frac{{{\displaystyle \widehat{d}}}^2{{\displaystyle \beta }}^2{{\displaystyle R}}_{IM}^{\ast }}{10},{{\displaystyle \mathrm{A}}}_3=\frac{{{\displaystyle {{\displaystyle \widehat{d}}}^2\beta }}^2\left(1-{{\displaystyle R}}_I^{\ast }+{{\displaystyle \tau }}_{pLs}\right)}{2\widehat{T}},{{\displaystyle \mathrm{A}}}_4=\frac{{{\displaystyle \widehat{d}\beta }}^2\widehat{\mu }\left(1-{{\displaystyle R}}_I^{\ast }+{{\displaystyle \tau }}_{pLs}\right)}{3\widehat{T}}$ ${{\displaystyle \mathrm{A}}}_5=\frac{{{\displaystyle \beta }}^2\widehat{\mu }\left(1-{{\displaystyle R}}_I^{\ast }+{{\displaystyle \tau }}_{pLs}\right)}{12\widehat{T}},{{\displaystyle \mathrm{A}}}_6=\frac{\left(1-{{\displaystyle R}}_I^{\ast }+{{\displaystyle \tau }}_{pLs}\right)}{60\widehat{d}\widehat{T}},{{\displaystyle \mathrm{A}}}_7=\frac{{{\displaystyle \widehat{d}}}^2{{\displaystyle \beta }}^2{{\displaystyle R}}_I^{\ast }}{3\widehat{T}},{{\displaystyle \mathrm{A}}}_8=\frac{\widehat{d}{{\displaystyle \beta }}^2\widehat{\mu }{{\displaystyle R}}_I^{\ast }}{3\widehat{T}}$, ${{\displaystyle \mathrm{A}}}_9=\frac{{{\displaystyle \beta }}^2\widehat{\mu }{{\displaystyle R}}_I^{\ast }}{10\widehat{T}},{{\displaystyle \mathrm{A}}}_{10}=\frac{{{\displaystyle R}}_I^{\ast }}{45\widehat{d}\widehat{T}},{{\displaystyle \mathrm{A}}}_{11}=\frac{\widehat{S}\left(1-{{\displaystyle R}}_I^{\ast }+{{\displaystyle \tau }}_{pLs}\right)}{360},{{\displaystyle \mathrm{A}}}_{12}=\frac{\widehat{S}{{\displaystyle R}}_I^{\ast }}{252},{{\displaystyle \mathrm{A}}}_{13}=\frac{\left(1-{{\displaystyle R}}_I^{\ast }+{{\displaystyle \tau }}_{pLs}\right)}{360},{{\displaystyle \mathrm{A}}}_{14}=\frac{{{\displaystyle R}}_I^{\ast }}{252}.$

5.4. Model (iv): Piecewise Linear Gradient Heated from Below

$${{\displaystyle F}}_{fL}(z_{fL})=\{\begin{array}{cc}{\epsilon }_{fL}^{-1},& 0\le z_{fL}\le {\epsilon }_{fL}\\ 0,& {\epsilon }_{fL}\le z_{fL}\le 1\end{array} \mathrm{and} {{\displaystyle F}}_{pL}(z_{pL})=\{\begin{array}{cc}{\epsilon }_{pL}^{-1},& 0\le z_{pL}\le {\epsilon }_{pL}\\ 0,& {\epsilon }_{pL}\le z_{pL}\le 1\end{array}$$

(35)

The R_c is attained for this model by utilizing (35) in (31) and is calculated as follows:

$${{\displaystyle R}}_C=\frac{\left({\widehat{d}}^2+\widehat{T}\right)+\left({\widehat{d}}^2+\widehat{S}\right){\tau }_{fLs}{\tau }_{pLs}-{\widehat{d}}^4{\kappa }_{pLs}^2{\tau }_{pLs}{\mathsf{\Psi }}_4-{{\displaystyle R}}_{pLs}\left\{{{\displaystyle \mathrm{B}}}_{11}+{{\displaystyle \mathrm{B}}}_{12}\right\}}{\widehat{T}\left\{{{\displaystyle \mathrm{B}}}_{13}+{{\displaystyle \mathrm{B}}}_{14}\right\}+\left[{{\displaystyle \alpha }}_{fpL} {\kappa }_{fpL}^2+{{\displaystyle \alpha }}_{spL}{\kappa }_{spL}^2{\tau }_{spL}\right]{\widehat{d}}^4 {\mathsf{\Psi }}_4}$$

(36)

where ${\mathsf{\Psi }}_4={{\displaystyle \mathrm{B}}}_1+{{\displaystyle \mathrm{B}}}_2+{{\displaystyle \mathrm{B}}}_3+{{\displaystyle \mathrm{B}}}_4+{{\displaystyle \mathrm{B}}}_5+{{\displaystyle \mathrm{B}}}_6+{{\displaystyle \mathrm{B}}}_7+{{\displaystyle \mathrm{B}}}_8+{{\displaystyle \mathrm{B}}}_9+{{\displaystyle \mathrm{B}}}_{10},$ ${{\displaystyle \mathrm{B}}}_1=\frac{{{\displaystyle \widehat{d}}}^2{{\displaystyle \beta }}^2\left(1+{{\displaystyle R}}_{IM}^{\ast }+{{\displaystyle \tau }}_{fLs}\right){{\displaystyle \epsilon }}_{pL}^2}{6},{{\displaystyle \mathrm{B}}}_2=\frac{{{\displaystyle \widehat{d}}}^2{{\displaystyle \beta }}^2{{\displaystyle R}}_{IM}^{\ast }{{\displaystyle \epsilon }}_{pL}^3}{4},{{\displaystyle \mathrm{B}}}_3=\frac{{{\displaystyle {{\displaystyle \widehat{d}}}^2\beta }}^2\left(1-{{\displaystyle R}}_I^{\ast }+{{\displaystyle \tau }}_{pLs}\right)}{2\widehat{T}},{{\displaystyle \mathrm{B}}}_4=\frac{{{\displaystyle \widehat{d}\beta }}^2{\mu }_{fL}(1+{{\displaystyle R}}_I^{\ast }+{{\displaystyle \tau }}_{pLs}){{\displaystyle \epsilon }}_{fL}}{2\widehat{T}}$ ${{\displaystyle \mathrm{B}}}_5=\frac{{{\displaystyle \beta }}^2\widehat{\mu }(1-{{\displaystyle R}}_I^{\ast }+{{\displaystyle \tau }}_{pLs}){\epsilon }_{fL}^2}{6\widehat{T}},{{\displaystyle \mathrm{B}}}_6=\frac{(1-{{\displaystyle R}}_I^{\ast }+{{\displaystyle \tau }}_{pLs}){\epsilon }_{fL}^3}{24\widehat{d}\widehat{T}},{{\displaystyle \mathrm{B}}}_7=\frac{{{\displaystyle \widehat{d}}}^2{{\displaystyle \beta }}^2{{\displaystyle R}}_I^{\ast }{\epsilon }_{fL}}{2\widehat{T}},{{\displaystyle \mathrm{B}}}_8=\frac{2\widehat{d}{{\displaystyle \beta }}^2\widehat{\mu }{{\displaystyle R}}_I^{\ast }{\epsilon }_{fL}^2}{3\widehat{T}}$, ${{\displaystyle \mathrm{B}}}_9=\frac{{{\displaystyle \beta }}^2\widehat{\mu }{{\displaystyle R}}_I^{\ast }{\epsilon }_{fL}^3}{4\widehat{T}},{{\displaystyle \mathrm{B}}}_{10}=\frac{{{\displaystyle R}}_I^{\ast }{\epsilon }_{fL}^4}{15\widehat{d}\widehat{T}},{{\displaystyle \mathrm{B}}}_{11}=\frac{\widehat{S}\left(1-{{\displaystyle R}}_I^{\ast }+{{\displaystyle \tau }}_{pLs}\right){\epsilon }_{fL}^4}{120},{{\displaystyle \mathrm{B}}}_{12}=\frac{\widehat{S}{{\displaystyle R}}_I^{\ast }{\epsilon }_{fL}^5}{72},{{\displaystyle \mathrm{B}}}_{13}=\frac{\left(1-{{\displaystyle R}}_I^{\ast }+{{\displaystyle \tau }}_{pLs}\right){\epsilon }_{fL}^4}{120},{{\displaystyle \mathrm{B}}}_{14}=\frac{{{\displaystyle R}}_I^{\ast }{\epsilon }_{fL}^5}{72}.$

5.5. Model (v): Piecewise Linear Gradient Cooled from Above

$${{\displaystyle F}}_{fL}(z_{fL})=\{\begin{array}{cc}0,& 0\le z_{fL}\le (1-{\epsilon }_{fL})\\ {\epsilon }_{fL}^{-1},& (1-{\epsilon }_{fL})\le z_{fL}\le 1\end{array} \mathrm{and} {{\displaystyle F}}_{pL}(z_{pL})=\{\begin{array}{cc}0,& 0\le z_{pL}\le (1-{\epsilon }_{pL})\\ {\epsilon }_{pL}^{-1},& (1-{\epsilon }_{pL})\le z_{pL}\le 1\end{array}$$

(37)

The R_c is attained for this model by utilizing (37) in (31) and is calculated as follows:

$${{\displaystyle R}}_C=\frac{\left({\widehat{d}}^2+\widehat{T}\right)+\left({\widehat{d}}^2+\widehat{S}\right){\tau }_{fLs}{\tau }_{pLs}-{\widehat{d}}^4{\kappa }_{pLs}^2{\tau }_{pLs}{\mathsf{\Psi }}_5-{{\displaystyle R}}_{pLs}\left\{{{\displaystyle ℝ}}_{11}+{{\displaystyle ℝ}}_{12}\right\}}{\widehat{T}\left\{{{\displaystyle ℝ}}_{13}+{{\displaystyle ℝ}}_{14}\right\}+\left[{{\displaystyle \alpha }}_{fpL} {\kappa }_{fpL}^2+{{\displaystyle \alpha }}_{spL}{\kappa }_{spL}^2{\tau }_{spL}\right]{\widehat{d}}^4 {\mathsf{\Psi }}_5}$$

(38)

where ${\mathsf{\Psi }}_5={{\displaystyle ℝ}}_1+{{\displaystyle ℝ}}_2+{{\displaystyle ℝ}}_3+{{\displaystyle ℝ}}_4+{{\displaystyle ℝ}}_5+{{\displaystyle ℝ}}_6+{{\displaystyle ℝ}}_7+{{\displaystyle ℝ}}_8+{{\displaystyle ℝ}}_9+{{\displaystyle ℝ}}_{10},$ ${{\displaystyle ℝ}}_1=\frac{{\widehat{d}}^2{{\displaystyle \beta }}^2(1+{{\displaystyle R}}_{IM}^{\ast }+{\tau }_{fLs})}{2{\epsilon }_{pL}}\frac{\left(1-{(1-{\epsilon }_{fL})}^3\right)}{3},$ ${{\displaystyle ℝ}}_2=\frac{{\widehat{d}}^2{{\displaystyle \beta }}^2{{\displaystyle R}}_{IM}^{\ast }}{{\epsilon }_{pL}}\frac{\left(1-{(1-{\epsilon }_{pL})}^4\right)}{4},$ ${{\displaystyle ℝ}}_3=\frac{{\widehat{d}}^2{{\displaystyle \beta }}^2(1-{{\displaystyle R}}_I^{\ast }+{\tau }_{pLs})}{2},$ ${{\displaystyle ℝ}}_4=\frac{\widehat{d}{{\displaystyle \beta }}^2\widehat{\mu }(1+{{\displaystyle R}}_I^{\ast }+{\tau }_{pLs})}{2\widehat{T}{\epsilon }_{fL}},$ ${\mathbb{R}}_5=\frac{{\beta }^2\widehat{\mu }\left(1-R_I^{\ast }+{\tau }_{pLs}\right)\left(1-{\left(1-{\epsilon }_{fL}\right)}^3\right)}{6\widehat{T}{\epsilon }_{fL}},$ ${\mathbb{R}}_6=\frac{\left(1-R_I^{\ast }+{\tau }_{pLs}\right)\left(1-{\left(1-{\epsilon }_{fL}\right)}^4\right)}{24\widehat{d}\widehat{T}{\epsilon }_{fL}},$ ${\mathbb{R}}_7=\frac{{\widehat{d}}^2{\beta }^2{R}_I^{\ast }\left(1-{\left(1-{\epsilon }_{fL}\right)}^2\right)}{2{\epsilon }_{fL}\widehat{T}},$ ${\mathbb{R}}_8=\frac{2\widehat{d}{\beta }^2\widehat{\mu }{R}_I^{\ast }\left(1-{\left(1-{\epsilon }_{fL}\right)}^3\right)}{3{\widehat{T}}_{\epsilon fL}},$ ${{\displaystyle ℝ}}_9=\frac{{{\displaystyle \beta }}^2\widehat{\mu }{{\displaystyle R}}_I^{\ast }\left(1-{(1-{\epsilon }_{fL})}^4\right)}{4\widehat{T}{\epsilon }_{fL}},$ ${{\displaystyle ℝ}}_{10}=\frac{{{\displaystyle R}}_I^{\ast }\left(1-{(1-{\epsilon }_{fL})}^5\right)}{15\widehat{d}\widehat{T}{\epsilon }_{fL}},$ ${{\displaystyle ℝ}}_{11}=\frac{\widehat{S}\left(1-{{\displaystyle R}}_I^{\ast }+{{\displaystyle \tau }}_{pLs}\right)\left(1-{(1-{\epsilon }_{fL})}^5\right)}{120{\epsilon }_{fL}},$ ${{\displaystyle ℝ}}_{12}=\frac{{{\displaystyle \widehat{S}R}}_I^{\ast }\left(1-{(1-{\epsilon }_{fL})}^6\right)}{72{\epsilon }_{fL}}, {{\displaystyle ℝ}}_{13}=\frac{\left(1-{{\displaystyle R}}_I^{\ast }+{{\displaystyle \tau }}_{pLs}\right)\left(1-{(1-{\epsilon }_{fL})}^5\right)}{120{\epsilon }_{fL}}, {{\displaystyle ℝ}}_{14}=\frac{{{\displaystyle R}}_I^{\ast }\left(1-{(1-{\epsilon }_{fL})}^6\right)}{72{\epsilon }_{fL}}.$

5.6. Model (vi): Step Function

The basic temperature in this profile drops rapidly by a certain amount $\mathsf{\Delta }{{\displaystyle T}}_{fL}$ at ${{\displaystyle z}}_{fL}={{\displaystyle \epsilon }}_{fL}$ and $\mathsf{\Delta }{{\displaystyle T}}_{pL}$ at ${{\displaystyle z}}_{pL}={{\displaystyle \epsilon }}_{pL}$, otherwise it is uniform. Accordingly,

$${{\displaystyle F}}_{fL}(z_{fL})=\delta \left(z_{fL}-{\epsilon }_{fL}\right) \mathrm{and} {{\displaystyle F}}_{pL}\left({{\displaystyle z}}_{pL}\right)=\delta \left({{\displaystyle z}}_{pL}-{{\displaystyle \epsilon }}_{pL}\right)$$

(39)

The ${{\displaystyle R}}_C$ is attained for this model by utilizing (39) in (31) and is calculated as follows:

$${{\displaystyle R}}_C=\frac{\left({\widehat{d}}^2+\widehat{T}\right)+\left({\widehat{d}}^2+\widehat{S}\right){\tau }_{fLs}{\tau }_{pLs}-{\widehat{d}}^4{\kappa }_{pLs}^2{\tau }_{pLs}{\mathsf{\Psi }}_6-{{\displaystyle R}}_{pLs}\left\{{{\displaystyle ℝ}}_{25}+{{\displaystyle ℝ}}_{26}\right\}}{\widehat{T}\left\{{{\displaystyle ℝ}}_{27}+{{\displaystyle ℝ}}_{28}\right\}+\left[{{\displaystyle \alpha }}_{fpL} {\kappa }_{fpL}^2+{{\displaystyle \alpha }}_{spL}{\kappa }_{spL}^2{\tau }_{spL}\right]{\widehat{d}}^4 {\mathsf{\Psi }}_6}$$

(40)

where ${\mathsf{\Psi }}_6={{\displaystyle ℝ}}_{15}+{{\displaystyle ℝ}}_{16}+{{\displaystyle ℝ}}_{17}+{{\displaystyle ℝ}}_{18}+{{\displaystyle ℝ}}_{19}+{{\displaystyle ℝ}}_{20}+{{\displaystyle ℝ}}_{21}+{{\displaystyle ℝ}}_{22}+{{\displaystyle ℝ}}_{23}+{{\displaystyle ℝ}}_{24},$ ${{\displaystyle ℝ}}_{15}=\frac{{{\displaystyle \widehat{d}}}^2{{\displaystyle \beta }}^2\left(1+{{\displaystyle R}}_{IM}^{\ast }+{{\displaystyle \tau }}_{fLs}\right){{\displaystyle \epsilon }}_{pL}^2}{2}, {{\displaystyle ℝ}}_{16}=\frac{{{\displaystyle \widehat{d}}}^2{{\displaystyle \beta }}^2{{\displaystyle R}}_{IM}^{\ast }{{\displaystyle \epsilon }}_{pL}^3}{4},$ ${{\displaystyle ℝ}}_{17}=\frac{{{\displaystyle \widehat{d}}}^2{{\displaystyle \beta }}^2\left(1-{{\displaystyle R}}_I^{\ast }+{{\displaystyle \tau }}_{pLs}\right)}{2\widehat{T}}, {{\displaystyle ℝ}}_{18}=\frac{\widehat{d}{{\displaystyle \beta }}^2\widehat{\mu }\left(1+{{\displaystyle R}}_I^{\ast }+{{\displaystyle \tau }}_{pLs}\right){{\displaystyle \epsilon }}_{fL}}{\widehat{T}},$ ${{\displaystyle ℝ}}_{19}=\frac{{{\displaystyle \beta }}^2\widehat{\mu }\left(1-{{\displaystyle R}}_I^{\ast }+{{\displaystyle \tau }}_{pLs}\right){{\displaystyle \epsilon }}_{fL}^2}{2\widehat{T}}, {{\displaystyle ℝ}}_{20}=\frac{\left(1-{{\displaystyle R}}_I^{\ast }+{{\displaystyle \tau }}_{pLs}\right){{\displaystyle \epsilon }}_{fL}^3}{6\widehat{d}\widehat{T}}, {{\displaystyle ℝ}}_{21}=\frac{{{\displaystyle \widehat{d}}}^2{{\displaystyle \beta }}^2{{\displaystyle R}}_I^{\ast }{{\displaystyle \epsilon }}_{fL}}{\widehat{T}},$ ${{\displaystyle ℝ}}_{22}=\frac{2\widehat{d}{{\displaystyle \beta }}^2\widehat{\mu }{{\displaystyle R}}_I^{\ast }{{\displaystyle \epsilon }}_{fL}^2}{\widehat{T}},$ ${{\displaystyle ℝ}}_{23}=\frac{{{\displaystyle \beta }}^2\widehat{\mu }{{\displaystyle R}}_I^{\ast }{{\displaystyle \epsilon }}_{fL}^3}{\widehat{T}}, {{\displaystyle ℝ}}_{24}=\frac{{{\displaystyle R}}_I^{\ast }{{\displaystyle \epsilon }}_{fL}^4}{6\widehat{d}\widehat{T}}, {{\displaystyle ℝ}}_{25}=\frac{\widehat{S}\left(1-{{\displaystyle R}}_I^{\ast }+{{\displaystyle \tau }}_{pLs}\right){{\displaystyle \epsilon }}_{fL}^4}{24},$ ${{\displaystyle ℝ}}_{26}=\frac{{{\displaystyle \widehat{S}R}}_I^{\ast }{{\displaystyle \epsilon }}_{fL}^5}{12}, {{\displaystyle ℝ}}_{27}=\frac{\left(1-{{\displaystyle R}}_I^{\ast }+{{\displaystyle \tau }}_{pLs}\right){{\displaystyle \epsilon }}_{fL}^4}{24}, {{\displaystyle ℝ}}_{28}=\frac{{{\displaystyle R}}_I^{\ast }{{\displaystyle \epsilon }}_{fL}^5}{12}.$

6. Results and Discussion

The theoretical investigation focuses on the initiation of TCRB convection in a combined structure. This system comprises a region-I positioned above a region-II that is saturated with the same fluid. The analysis is conducted in accordance with the LTNE model. The analytical calculations and graphical representations are performed utilizing the advanced computational software known as MATHEMATICA, version 11. The eigenvalue issue is resolved by the utilization of the perturbation method, resulting in the derivation of an analytical expression for the critical Rayleigh number. This expression is obtained for six distinct thermal gradients, as indicated in Table 1. The graph depicts the relationship between the critical Rayleigh number (${{\displaystyle R}}_C$) and the depth ratio ($\widehat{d}$) for all the cases examined in the current study, specifically in the context of adiabatic rigid boundaries. Also, there is a noticeable pattern in the profiles’ stability. The range of values under consideration in the present analysis is as follows: ${{\displaystyle \alpha }}_{spL}=2, {{\displaystyle \kappa }}_{fpL}=0.2, {{\displaystyle \kappa }}_{spL}=0.3,$ ${{\displaystyle \tau }}_{fLs}=0.2,$ ${{\displaystyle \tau }}_{pLs}=0.3,$ ${{\displaystyle R}}_{fLs}=10,$ ${{\displaystyle R}}_{pLs}=10,$ $\widehat{T}=1.0, {{\displaystyle R}}_I^{\ast }=0.1, {{\displaystyle R}}_{IM}^{\ast }=1.0, \beta =1, {{\displaystyle \kappa }}_{pLs}=0.5, {{\displaystyle \kappa }}_{fLs}=0.3$ and ${{\displaystyle \alpha }}_{fpL}=1$.

The impact of the thermal expansion ratio ${\alpha }_{fpL}$ on ${{\displaystyle R}}_C$ is illustrated in Figure 2. The curves exhibit divergence when considering the values ${\alpha }_{fpL}=1, 5$, and 10. The figure clearly demonstrates a positive correlation between ${\alpha }_{fpL}$ and $R_C$, indicating that an increase in ${\alpha }_{fpL}$ results in a corresponding increase in ${{\displaystyle R}}_C$. Consequently, the setup can achieve stability, thereby delaying the onset of TCRB convection. Based on the graph, it can be observed that the linear profile exhibits greater stability compared to the parabolic profile. In comparison, PLHB exhibits greater stability, while SF demonstrates a higher degree of instability. The impact of the parameter is minimal in the parabolic profiles, while it is more pronounced in the linear profiles. According to the data presented in Figure 3, an increase in the variable solid phase expansion ratio ${\alpha }_{spL}$ leads to a corresponding increase in the ${{\displaystyle R}}_C$. Consequently, this adjustment in the setup has the potential to enhance the stability and delay the onset of TCRB convection. Based on the graph, it can be observed that the linear profile exhibits the highest level of stability, while the parabolic profile demonstrates instability. In a similar vein, it can be observed that PLHB exhibits a higher degree of stability compared to SF, which is relatively less stable. The impact of the parameter is minimal in parabolic profiles, whereas it is more significant in linear profiles. It is observed that nearly equivalent outcomes are achieved for the variables ${\alpha }_{fpL}$ and ${\alpha }_{spL}$. It is evident from Figure 4 that the growth in the porous parameters $\beta$ leads to a rise in the ${{\displaystyle R}}_C$. Consequently, this allows for the stabilization of the setup, resulting in a delayed onset of TCRB convection. As a result, the system is postponed. The graphs indicate that the linear and PLHB models exhibit greater stability, while the parabolic and SF models demonstrate higher levels of instability. Additionally, the impact of the porous parameter is significantly more pronounced for the linear and PLHB models, moderately significant for the inverted parabolic and PLCA models, and detrimental for the parabolic and SF thermal gradient models. Figure 5 and Figure 6 exhibit the effects of the fluid and solid phase thermal diffusivity ratios, respectively, ${\kappa }_{fpL}$ and ${\kappa }_{spL}$. As the ratios increase, ${{\displaystyle R}}_C$ increases. As a result, the setup is stable, delaying the onset of TCRB convection. Hence the system is postponed. The effects of the corrected internal Rayleigh numbers ${{\displaystyle R}}_I^{\ast }$ and ${{\displaystyle R}}_{IM}^{\ast }$ are illustrated in Figure 7 and Figure 8. The system experiences destabilization as the ${{\displaystyle R}}_I^{\ast }$ parameter increases, resulting in the earlier occurrence of TCRB convection for higher values of this parameter. The TCRB convection effectively regulates smaller values of the ${{\displaystyle R}}_I^{\ast }$. It is observed in Figure 8 that there is an inverse response from the porous layer ${{\displaystyle R}}_{IM}^{\ast }$. This enhances the system stability by increasing the value of ${{\displaystyle R}}_C$. The observed curves exhibit a divergence pattern, suggesting that the sensitivity arises when the porous layer predominates over the composite layer across all thermal gradients. Therefore, the TCRB convection is delayed. The influence of the solute Rayleigh numbers ${{\displaystyle R}}_{fLs}$ and ${{\displaystyle R}}_{pLs}$ over the ${{\displaystyle R}}_C$ are depicted in Figure 9 when ${{\displaystyle R}}_{fLs}={{\displaystyle R}}_{pLs}=10,20,30$, and $30$. With the increase in ${{\displaystyle R}}_{fLs}$ and ${{\displaystyle R}}_{pLs}$, ${{\displaystyle R}}_C$ decreases, destabilizing the setup. TCRB convection, hence, starts up quickly. The observed phenomenon could potentially arise from the existence of a secondary constituent.

7. Conclusions

The primary focus of this theoretical investigation is the initiation of the two-component Rayleigh–Bènard (TCRB) convection in a combined structure. The system consists of a region-I situated above a region-II that is fully saturated with the same fluid. The analysis is performed following the LTNE model. The present endeavor necessitates the examination of the situation within the framework of arbitrary boundary conditions for salinity and temperature. The eigenvalue problem is solved using the perturbation method, which leads to the derivation of an analytical expression for the critical Rayleigh number. The aforementioned expression is derived from six unique thermal gradients, as outlined in Table 1. The present study has yielded the following outcomes.

(i)

The onset of TCRB convection is supported by a corrected internal Rayleigh number in region-I and the solute Rayleigh number. By increasing the system’s instability, the six profiles speed up the beginning of TCRB convection with the corrected internal Rayleigh number in region-I and the solute Rayleigh number.

(ii)

When applied to each of the six profiles, the thermal ratio, the thermal diffusivity ratio, the porous parameter, and the corrected internal Rayleigh number all work together to postpone the onset of TCRB convection.

(iii)

In a combined structure arrangement with region-I on top, the parabolic profile is more stable than the step function, but in a setup with region-II on top, the step function takes the lead.

(iv)

The PLHB structure exhibits the highest stability among the combined structures in region-II, whereas the linear structure demonstrates the lowest stability.

(v)

The commencement of TCRB convection in a composite fluid and porous layer system may be efficiently controlled by selecting the proper parameters.