# Influence of Temperature Gradients and Heat Source in a Combined Layer on Double Component-Magneto-Marangoni-Convection

Authors: N. Manjunatha; R. Sumithra; Nazek Alessa; K. Loganathan; C. Selvamani; Sonam Gyeltshen
Journal: Journal of Mathematics
 (2023)

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## Abstract

The present paper has enormous applications including fields of crystal growth, material processing, spacecraft, underground spread of chemical contaminants, petroleum reservoirs, waste dispersal and fertilizer migration in saturated soil, alloy solidification, and many more. The importance of double-diffusive convection has been recognized in various engineering applications, and it has thoroughly been investigated theoretically. In the presence of a constant heat source/sink on both layers, the double component-magneto-Marangoni-convection in a composite layer is examined. Due to heat, this combined layer is enclosed by adiabatic boundaries and exposed to basic temperature gradients. For the system of ordinary differential equations, the thermal surface-tension-driven (Marangoni) number, which also happens to be the Eigenvalue, is solved in closed form. For basic temperature gradients, the eigenvalues, or thermal Marangoni numbers are determined in closed form for lower rigid and higher free boundaries with surface tension, depending on both temperature and concentration. The effect of several parameters on the eigenvalue against depth ratio was studied. It is observed that the inverted parabolic temperature gradient on double component magneto-Marangoni-convection in a combined layer is the most stable of the three temperature gradients and for larger values of depth ratios, all physical parameters are nominal for the porous region dominant combined system.


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## Body

## 1. Introduction

Owing to its use in geophysics, astrophysics, engineering, and technology, MHD flow research is important. The influence of a magnetic field on the distribution of the heat by a heat source/sink where the liquid can produce and absorb hot radiation is influential when it comes to confined use and high operating temperatures. Hill [1] studied double-diffusive convection in a fluid-saturated porous medium with an internal heat source depending on concentration. Using linear stability analysis, Liu and Umavathi [2] investigated the double-diffusive convection of sparsely packed micropolar fluid-saturated porous media. Using the Runge–Kutta technique, Bhuvanavijaya and Mallikarjuna [3] investigated the effect of nonuniform heat sources on convective heat and mass transport over a vertical plate in a rotating system embedded in a non-Darcy porous medium. In a layered system, Köllner and Boeck [4] demonstrated a two-dimensional simulation of solutal Rayleigh–Bénard–Marangoni convection. Alizadeh et al. [5] investigate the possibility of Marangoni convection causing instability in porous medium statistically. They discovered that porous media’s damping forces postpone the commencement of convection. In the context of the Soret effect, Gaikwad and Dhanraj [6] used linear and weakly nonlinear stability calculations to investigate the beginning of Darcy–Brinkman double-diffusive convection in a binary viscoelastic fluid-saturated porous layer. Kumar et al. [7] explored the topic of triple-diffusive convection and the impact of the solute Rayleigh numbers, concentration Rayleigh number, Lewis number and modified diffusivity ratio on system stability. Mebarek–Oudina [8] investigated the numerical modeling of hydrodynamic stability in a vertical annulus with varying lengths of heat source. He discovered that as the heat source length ratio increases, the critical Rayleigh number decreases. Awasthi et al. [9] conduct a linear stability study for the beginning of triple-diffusive convection in the presence of an internal heat source in a Maxwell fluid-saturated porous layer. It has been discovered that Lewis numbers have a destabilizing impact, whereas solute Rayleigh numbers have a stabilizing effect. Using the shooting technique, Reddy and Dinesh [10] investigated coupled highly nonlinear partial differential equations for the effects of internal heat generation, combined effects of Soret and Dufour, and changeable fluid characteristics. Basha et al. [11] use the Crank-Nicolson technique to investigate the effects of Lorentz force on the fluid transport parameters of a chemically reactive nanofluid with two types of geometries. They discovered that the Hartman number has a significant impact on fluid flow and heat transfer properties. Athira et al. [12] investigated the effects of nonlinear convection and induced magnetic fields in a viscous fluid flow over a porous plate when chemical reactions and heat sources/sinks were present. K. H. Ahmed and W. M. Ahmed [13] used the finite volume approach to investigate the natural convection in a nanofluid including water-copper nanoparticles. Manjunatha and Sumithra [14] studied the triple diffusive convection in the absence of a heat source. Sumithra et al. [15, 16] discuss single components, heat sources, and with/without magnetic fields in the presence of temperature gradients. They developed the closed form of the thermal Marangoni number solution.Recently, Hemanthkumar et al. [17] looked into the impact of a horizontal pressure gradient on the beginning of Soret-driven thermosolutal porous convection. Using normal modes, Jayalatha and Suma [18] investigated the effect of heat sources on instability in rotating viscoelastic liquids. Supraja and Raju [19] give a numerical investigation of heat transmission by the natural convection of a fluid inside a square cavity with two inner bodies. Using the incompressible smoothed particle hydrodynamics approach, Raizah and Aly [20] investigated the suspension of a nanoencapsulated phase change material during a double-diffusive convection flow inside a compound cavity. Double component convection in rectangular hollows with varied factor ratios is investigated numerically by Yang and Zhao [21]. The effects of magnetic field and heat source on double-diffusive convection two layer system studied by Manjunatha and Sumithra [22]. The effects of waveforms of internal heat source modulation on triple diffusive convection in viscoelastic liquids were investigated by Arshika et al. [23]. Kumar Pundir et al. [24] investigated the linear stability of a fluid layer with nanoparticles in the presence of a solute gradient. Using linear and nonlinear stability approaches, Yadav et al. [25, 26] investigated the combined influence of heating and chemical reaction on the double-diffusive convective in a non-Newtonian Maxwell fluid and Kuvshiniski type saturated porous layer. They determined that oscillatory convection is only possible when the solute Rayleigh–Darcy number is negative and that it is also influenced by other physical factors. Animasaun et al. [27] studied the Ratio of Momentum Diffusivity to Thermal Diffusivity. Sumithra et al. [28] evaluated the role of thermal diffusion on the commencement of double-diffusive convection in a composite system and the influence of the Soret effect on the system’s stability. For adiabatic-adiabatic and adiabatic-isothermal boundary conditions, Manjunatha et al. [29] investigated the effect of a constant heat source/sink on double-diffusive convection in a combined layer system in the absence of a magnetic field. Some interesting and recent papers on nanofluid flows are [30–32]. Convective stability analysis has been studied by a few authors for porous layers [33–39].The current paper investigates the effect of fundamental temperature gradients on double component-magneto-Marangoni-convection (DCMC) in a combined layer with a constant heat source (sink) at both levels. The Eigen value problem was solved by using the exact method and the effect of several parameters on the eigenvalue against the depth ratio was studied. Numerous applications in astronomy, engineering, geophysics, climatology, and crystal formation [40] will surely benefit from this work.


## 2. Research Methodology

Consider a two-component, subject to the Boussinesq approximation, electrically conducting fluid-saturated, incompressible, sparsely packed porous layer of thicknessdp underlying a two-component fluid layer of thickness df with an imposed magnetic field intensity H0 in the vertical Z-direction and with heat sources Φp and Φf, respectively. Let ΔT and ΔC are the temperature and concentration difference between lower and upper boundaries. The constant temperature and concentration T=T0 & C=C0. At the upper free region, surface tension force acts which varies linearly with temperature and concentration respectively in the form σt=σ0−σTΔT, σt=σ0−σCΔC where σT=−∂σt/∂T, σC=−∂σt/∂C, σ0 is the unperturbed value. The porous layer’s lower surface is hard, while the fluid layer’s upper surface is free, with surface tension effects depending on temperature and concentration as shown in Figure 1. The rectangular coordinate system’s origin is defined at the interface of the fluid-porous composite system and the magnetic field is imposed along z–axis upwards. The Darcy model predicts that at the interface, the velocity, shear stress, normal stress, heat, heat flux, mass, and mass flux are all continuous.Figure 1
Physical model of the problem.The basic equations are (see Venkatachalappa et al. [41] and Shivakumara et al. [42]).


### 2.1. Fluid Layer: Region-1

The fundamental equations are(1)∇f.q⟶f=0,(2)∇f.H⟶=0,(3)ρ0∂q⟶f∂t+q⟶f.∇fq⟶f=μf∇f2q⟶f−∇fPf+γfH⟶.∇fH⟶,(4)∂Tf∂t+q⟶f.∇fTf=κf∇f2Tf+Φf,(5)∂Cf∂t+q⟶f.∇fCf=κcf∇f2Cf,(6)∂H⟶∂t=∇f×q⟶f×H⟶+νf∇f2H⟶.


### 2.2. Porous Layer: Region-2

(7)∇p.q⟶p=0,(8)∇p.H⟶=0,(9)ρ01ϕp∂q⟶p∂t+1ϕp2q⟶p.∇pq⟶p=−∇pPp−μpKq⟶p+γpH⟶.∇pH⟶,(10)M∂Tp∂t+q⟶p.∇pTp=κp∇p2Tp+Φp,(11)ϕp∂Cp∂t+q⟶p.∇pCp=κcp∇p2Cp,(12)ϕp∂H⟶∂t=∇p×q⟶p×H⟶+νp∇p2H⟶,where subscript “f” and “p” denotes the fluid and porous layer, q⟶f,q⟶p are the velocities, ρ0 is the density, μf,μp are the viscosities, Pf,Pp are the hydromagnetic pressures, Tf,Tp are the temperatures, κf,κp are the thermal diffusivities, Φf, Φp are the heat sources, κcf,κcp are the salinity diffusivities, Cf,Cp are the concentrations, γf,γp are the magnetic permeability’s, νf,νp are the magnetic viscosities, H⟶ is the magnetic field, ϕp is the porosity, K is the permeability of the porous medium, M=ρ0Cνp/ρ0Cνf is the ratio heat capacities and Cν is the specific heat.The goal of this research is to see if a quiescent state can withstand tiny perturbations superimposed on the basic state.Region-1(13)q⟶f=0,Pf=Pfbzf,Tf=Tfbzf,Cf=Cfbzf,H⟶=H0zf,−dfΔTfdTfbdzf=χzf.Region-2(14)q⟶p=0,Pp=Ppbzp,Tp=Tpbzp,Cp=Cpbzp,H⟶=H0zp,−dpΔTpdTpbdzp=χzp.Temperature and concentration distributions in the basic state are derived, respectively.(15)Tfbzf=Φfzfdf−zf2κf+Tu−T0χzfdf+T00≤zf≤df,Tpbzp=−Φpzpzp+dp2κp+T0−Tlχzpdp+T0−dp≤zp≤0,Cfbzf=C0+Cu−C0zfdf0≤zf≤df,Cpbzp=C0+C0−Clzpdp−dp≤zp≤0,where the subscript “b” denotes the basic state and χzf & χzp are the nondimensional basic temperature gradients in region-1 and region-2, respectively.At the interfaceχzf=χzp and note that T0=2κfdpTu+κpdfTl+dfdpΦpdp+Φfdf/2κfdp+κpdf and C0=κcfdpCu+κcpdfCl/κcfdp+κcpdf.To investigate the stability of the basic state, we apply infinite perturbations to region-1 and region-2, respectively(16)q⟶f=0+q⟶f′=0,Pf=Pfb+Pf′,Tf=Tfbzf+θf′,Cf=Cfbzf+Sf′,H⟶=H0zf+H′,(17)q⟶p=0+q⟶p′=0,Pp=Ppb+Pp′,Tp=Tpbzp+θp′,Cp=Cpbzp+Sp′,H⟶=H0zp+H′,where q⟶f′,q⟶p′ are velocities, Pf′,Pp′ are the pressures, θf′,θp′ are the temperatures, Sf′,Sp′ are the salinities and H′ is the magnetic field respectively are perturbed quantities for region-1 and region-2. Substituting (16) and (17) in (1)–(12) and omitting the primes for simplicity, linearizing the equations, and taking curl twice on the momentum equations in both the regions to eliminate pressures. In the dimensionless formulation, scales for length, time, velocity, temperature, species concentration, magnetic field, and temperature gradient, respectively, df, df2/κf, κf/df, T0−Tu, C0−Cu, H0, ΔTf/afMt in the region-1 and dp, dp2/κp, κp/dp, Tl−T0, Cl−C0, H0, ΔTp/apMt are the corresponding quantities in the region-2. The scales for the two regions are determined separately (refer F. Chen and C. F. Chen [43], Chen [44] and Nield [45]), so that each region is of unit depth such that xf,yf,zf=dfxf′,yf′,zf′ and xp,yp,zp=dpxp′,yp′,zp′−1.The following perturbed equations areRegion-1(18)1Prf∂∇f2Vf∂t=∇f4Vf+Qfτf∂∇f2H∂zf,(19)∂θf∂t−χzfVf−RIf∗2zf−1Vf=∇f2θf,(20)∂Sf∂t=Vf+τf∇2Sf,(21)∂H∂t=∂Vf∂t+τf∇2H.Region-2(22)β2Prp∂∇p2Vp∂t=−∇p2Vp+Qpτpβ2∂∇p2H∂zp,(23)M∂θp∂t−χzpVp−RIp∗2zp+1Vp=∇p2θp,(24)ϕp∂Sp∂t=Vp+τp∇p2Sp,(25)ϕp∂H∂t=∂Vp∂t+τp∇p2H.Solutions for dependent variables are achieved in regions 1 and 2 utilizing normal mode analysis of the dimensionless equations for perturbed variables such as(26)wf′,θf′,Sf′,H′=Vf,θf,Sf,Hzfexpilfx+imfy+nft,(27)wp′,θp′,Sp′,H′=Vp,θp,Sp,Hzpexpilpx+impy+npt.With∇22f=−af2f and ∇2p2f=−ap2f such that af=lf2+mf2ap=lp2+mp2, here af, ap are horizontal wave numbers and lf,mf & lp,mp are wave numbers in x and y direction, respectively, and nf,np are frequencies in region-1 and region-2, respectively. Introducing (26) and (27) in (18)–(25), the following equations are obtained.Region-1:0≤zf≤1(28)Df2−af2+nfPrfDf2−af2Vfzf=−QfτfDfDf2−af2Hzf,Df2−af2+nfθfzf+χzf+RIf∗2zf−1Vfzf=0,τfDf2−af2+nfSfzf+Vfzf=0,τfDf2−af2+nfHzf+DfVfzf=0.Region-2:−1≤zp≤0(29)1−β2npPrpDp2−ap2Vpzp=Qpτpβ2DpDp2−ap2Hzp,Dp2−ap2+Mnpθpzp+χzp+RIp∗2zp+1Vpzp=0,τpDp2−ap2+npϕpSpzp+Vpzp=0,τpDp2−ap2+npϕpHzp+DpVpzp=0.In the above-given expressions,Prf=νf/κf, Prp=νp/κp are the Prandtl numbers, Qf=γfH02df2/μfκfτf, Qm=γpH02dp2/μpκpτp are the Chandrasekhar numbers and τf=κcf/κf, τp=κcp/κp are the diffusivity ratios, and β=K/dp2 is the square root of Darcy number.The linear stability analysis technique and the concept of exchange of stability are applicable for the current scenario, and the study is confined to stationary convection and takenf=0 and np=0 (see Shivakumara et al. [42], Nield [45] and Sumithra [46]). The eigenvalue problem is transformed into the following equations:Region-1(30)Df2−af22−QfDf2Vfzf=0,(31)Df2−af2θfzf+χzf+RIf∗2zf−1Vfzf=0,(32)τfDf2−af2Sfzf+Vfzf=0.Region-2(33)Dp2−ap2+Qpβ2Dp2Vpzp=0,(34)Dp2−ap2θpzp+χzp+RIp∗2zp+1Vpzp=0,(35)τpDp2−ap2Spzp+Vpzp=0.In the above equationsRIf∗=Rif/2T0−Tu, Rif=Φfdf2/κf, Vfzf, θfzf, Sfzf are, namely, the modified internal Rayleigh number, the internal Rayleigh number, the vertical velocity, the temperature distribution, and concentration distribution, respectively, for region-1 and RIp∗=Rip/2Tl−T0, Rip=Φpdp2/κp, Vpzp, θpzp, Spzp are the similar quantities in the region-2. Because the horizontal wave numbers for the combined layers must be the same so that we have ap=d^af here d^=dp/df is the depth ratio.


## 2.1. Fluid Layer: Region-1

The fundamental equations are(1)∇f.q⟶f=0,(2)∇f.H⟶=0,(3)ρ0∂q⟶f∂t+q⟶f.∇fq⟶f=μf∇f2q⟶f−∇fPf+γfH⟶.∇fH⟶,(4)∂Tf∂t+q⟶f.∇fTf=κf∇f2Tf+Φf,(5)∂Cf∂t+q⟶f.∇fCf=κcf∇f2Cf,(6)∂H⟶∂t=∇f×q⟶f×H⟶+νf∇f2H⟶.


## 2.2. Porous Layer: Region-2

(7)∇p.q⟶p=0,(8)∇p.H⟶=0,(9)ρ01ϕp∂q⟶p∂t+1ϕp2q⟶p.∇pq⟶p=−∇pPp−μpKq⟶p+γpH⟶.∇pH⟶,(10)M∂Tp∂t+q⟶p.∇pTp=κp∇p2Tp+Φp,(11)ϕp∂Cp∂t+q⟶p.∇pCp=κcp∇p2Cp,(12)ϕp∂H⟶∂t=∇p×q⟶p×H⟶+νp∇p2H⟶,where subscript “f” and “p” denotes the fluid and porous layer, q⟶f,q⟶p are the velocities, ρ0 is the density, μf,μp are the viscosities, Pf,Pp are the hydromagnetic pressures, Tf,Tp are the temperatures, κf,κp are the thermal diffusivities, Φf, Φp are the heat sources, κcf,κcp are the salinity diffusivities, Cf,Cp are the concentrations, γf,γp are the magnetic permeability’s, νf,νp are the magnetic viscosities, H⟶ is the magnetic field, ϕp is the porosity, K is the permeability of the porous medium, M=ρ0Cνp/ρ0Cνf is the ratio heat capacities and Cν is the specific heat.The goal of this research is to see if a quiescent state can withstand tiny perturbations superimposed on the basic state.Region-1(13)q⟶f=0,Pf=Pfbzf,Tf=Tfbzf,Cf=Cfbzf,H⟶=H0zf,−dfΔTfdTfbdzf=χzf.Region-2(14)q⟶p=0,Pp=Ppbzp,Tp=Tpbzp,Cp=Cpbzp,H⟶=H0zp,−dpΔTpdTpbdzp=χzp.Temperature and concentration distributions in the basic state are derived, respectively.(15)Tfbzf=Φfzfdf−zf2κf+Tu−T0χzfdf+T00≤zf≤df,Tpbzp=−Φpzpzp+dp2κp+T0−Tlχzpdp+T0−dp≤zp≤0,Cfbzf=C0+Cu−C0zfdf0≤zf≤df,Cpbzp=C0+C0−Clzpdp−dp≤zp≤0,where the subscript “b” denotes the basic state and χzf & χzp are the nondimensional basic temperature gradients in region-1 and region-2, respectively.At the interfaceχzf=χzp and note that T0=2κfdpTu+κpdfTl+dfdpΦpdp+Φfdf/2κfdp+κpdf and C0=κcfdpCu+κcpdfCl/κcfdp+κcpdf.To investigate the stability of the basic state, we apply infinite perturbations to region-1 and region-2, respectively(16)q⟶f=0+q⟶f′=0,Pf=Pfb+Pf′,Tf=Tfbzf+θf′,Cf=Cfbzf+Sf′,H⟶=H0zf+H′,(17)q⟶p=0+q⟶p′=0,Pp=Ppb+Pp′,Tp=Tpbzp+θp′,Cp=Cpbzp+Sp′,H⟶=H0zp+H′,where q⟶f′,q⟶p′ are velocities, Pf′,Pp′ are the pressures, θf′,θp′ are the temperatures, Sf′,Sp′ are the salinities and H′ is the magnetic field respectively are perturbed quantities for region-1 and region-2. Substituting (16) and (17) in (1)–(12) and omitting the primes for simplicity, linearizing the equations, and taking curl twice on the momentum equations in both the regions to eliminate pressures. In the dimensionless formulation, scales for length, time, velocity, temperature, species concentration, magnetic field, and temperature gradient, respectively, df, df2/κf, κf/df, T0−Tu, C0−Cu, H0, ΔTf/afMt in the region-1 and dp, dp2/κp, κp/dp, Tl−T0, Cl−C0, H0, ΔTp/apMt are the corresponding quantities in the region-2. The scales for the two regions are determined separately (refer F. Chen and C. F. Chen [43], Chen [44] and Nield [45]), so that each region is of unit depth such that xf,yf,zf=dfxf′,yf′,zf′ and xp,yp,zp=dpxp′,yp′,zp′−1.The following perturbed equations areRegion-1(18)1Prf∂∇f2Vf∂t=∇f4Vf+Qfτf∂∇f2H∂zf,(19)∂θf∂t−χzfVf−RIf∗2zf−1Vf=∇f2θf,(20)∂Sf∂t=Vf+τf∇2Sf,(21)∂H∂t=∂Vf∂t+τf∇2H.Region-2(22)β2Prp∂∇p2Vp∂t=−∇p2Vp+Qpτpβ2∂∇p2H∂zp,(23)M∂θp∂t−χzpVp−RIp∗2zp+1Vp=∇p2θp,(24)ϕp∂Sp∂t=Vp+τp∇p2Sp,(25)ϕp∂H∂t=∂Vp∂t+τp∇p2H.Solutions for dependent variables are achieved in regions 1 and 2 utilizing normal mode analysis of the dimensionless equations for perturbed variables such as(26)wf′,θf′,Sf′,H′=Vf,θf,Sf,Hzfexpilfx+imfy+nft,(27)wp′,θp′,Sp′,H′=Vp,θp,Sp,Hzpexpilpx+impy+npt.With∇22f=−af2f and ∇2p2f=−ap2f such that af=lf2+mf2ap=lp2+mp2, here af, ap are horizontal wave numbers and lf,mf & lp,mp are wave numbers in x and y direction, respectively, and nf,np are frequencies in region-1 and region-2, respectively. Introducing (26) and (27) in (18)–(25), the following equations are obtained.Region-1:0≤zf≤1(28)Df2−af2+nfPrfDf2−af2Vfzf=−QfτfDfDf2−af2Hzf,Df2−af2+nfθfzf+χzf+RIf∗2zf−1Vfzf=0,τfDf2−af2+nfSfzf+Vfzf=0,τfDf2−af2+nfHzf+DfVfzf=0.Region-2:−1≤zp≤0(29)1−β2npPrpDp2−ap2Vpzp=Qpτpβ2DpDp2−ap2Hzp,Dp2−ap2+Mnpθpzp+χzp+RIp∗2zp+1Vpzp=0,τpDp2−ap2+npϕpSpzp+Vpzp=0,τpDp2−ap2+npϕpHzp+DpVpzp=0.In the above-given expressions,Prf=νf/κf, Prp=νp/κp are the Prandtl numbers, Qf=γfH02df2/μfκfτf, Qm=γpH02dp2/μpκpτp are the Chandrasekhar numbers and τf=κcf/κf, τp=κcp/κp are the diffusivity ratios, and β=K/dp2 is the square root of Darcy number.The linear stability analysis technique and the concept of exchange of stability are applicable for the current scenario, and the study is confined to stationary convection and takenf=0 and np=0 (see Shivakumara et al. [42], Nield [45] and Sumithra [46]). The eigenvalue problem is transformed into the following equations:Region-1(30)Df2−af22−QfDf2Vfzf=0,(31)Df2−af2θfzf+χzf+RIf∗2zf−1Vfzf=0,(32)τfDf2−af2Sfzf+Vfzf=0.Region-2(33)Dp2−ap2+Qpβ2Dp2Vpzp=0,(34)Dp2−ap2θpzp+χzp+RIp∗2zp+1Vpzp=0,(35)τpDp2−ap2Spzp+Vpzp=0.In the above equationsRIf∗=Rif/2T0−Tu, Rif=Φfdf2/κf, Vfzf, θfzf, Sfzf are, namely, the modified internal Rayleigh number, the internal Rayleigh number, the vertical velocity, the temperature distribution, and concentration distribution, respectively, for region-1 and RIp∗=Rip/2Tl−T0, Rip=Φpdp2/κp, Vpzp, θpzp, Spzp are the similar quantities in the region-2. Because the horizontal wave numbers for the combined layers must be the same so that we have ap=d^af here d^=dp/df is the depth ratio.


## 3. Boundary Conditions

The following boundary conditions are nondimensional and are subject to the normal mode analysis:(36)Df2Vf1+Mtθf1+MsSf1af2=0.The velocity conditions are(37)Vf1=0,Vp−1=0,DpVp−1=0,T^Vf0=Vp0,T^d^2Df2+af2Vf0=μ^Dp2+ap2Vp0,T^d^DfVf0=DpVp0,T^d^3β2Df3−3af2DfVf0=−Dp+μ^β2Dp3−3ap2DpVp0.The conditions for adiabatic-adiabatic boundary conditions are(38)Dfθf1=0,θf0=T^θp0,Dfθf0=Dpθp0,Dpθp−1=0.The salinity conditions are(39)DfSf1=0,Sf0=S^Sp0,DfSf0=DpSp0,DpSp−1=0.In the above-given expressions,S^=Cl−C0/C0−Cu is the solute diffusivity ratio, T^=Tl−T0/T0−Tu is the thermal ratio, μ^=μp/μf is the viscosity ratio, Mt=−∂σt/∂TfΔTfdf/μfκf is the thermal Marangoni number (tMn), Ms=−∂σt/∂CfΔCfdf/μfκf is the solute Marangoni number sMn, and σt is the surface tension.


## 4. Methods of Solution

The eigenvalue problem with an eigenvalue ofMt is formed by the boundary conditions of (36) and the equations (30)–(35). We address this problem using the exact technique procedure, which produces acceptable results when dealing with difficulties of this nature (see Manjunatha et al. [29]).


### 4.1. Velocity Profile

Using the velocity conditions (37), the solutions of Vfzf and Vpzp are obtained by solving (30)–(33) and appropriately written as follows:(40)Vfzf=B1coshψfzf+b1sinhψfzf+b2coshφfzf+b3sinhφfzf,Vpzp=B1b4coshδpzp+b5sinhδpzp,where ψf=Qf+Qf+4af2/2,φf=Qf−Qf+4af2/2,δp=ap2/1+Qpβ2,(41)b1=−Δ2b3Δ1,b2=Δ5Δ7−Δ8Δ4Δ3Δ7−Δ6Δ4,b3=Δ5Δ6−Δ8Δ3Δ4Δ6−Δ7Δ3,b4=T^1+b2,b5=1δpT^d^b1ψf+b3φf.(Refer Appendices).


### 4.2. Salinity Profile

From (32)–(35), the Sfzf & Spzp we get using conditions (39), as follows:(42)Sfzf=A1c1coshafzf+c2sinhafzf+Σf1zf,Spzp=A1c3coshapzp+c4sinhapzp+Σp1zp.


## 4.1. Velocity Profile

Using the velocity conditions (37), the solutions of Vfzf and Vpzp are obtained by solving (30)–(33) and appropriately written as follows:(40)Vfzf=B1coshψfzf+b1sinhψfzf+b2coshφfzf+b3sinhφfzf,Vpzp=B1b4coshδpzp+b5sinhδpzp,where ψf=Qf+Qf+4af2/2,φf=Qf−Qf+4af2/2,δp=ap2/1+Qpβ2,(41)b1=−Δ2b3Δ1,b2=Δ5Δ7−Δ8Δ4Δ3Δ7−Δ6Δ4,b3=Δ5Δ6−Δ8Δ3Δ4Δ6−Δ7Δ3,b4=T^1+b2,b5=1δpT^d^b1ψf+b3φf.(Refer Appendices).


## 4.2. Salinity Profile

From (32)–(35), the Sfzf & Spzp we get using conditions (39), as follows:(42)Sfzf=A1c1coshafzf+c2sinhafzf+Σf1zf,Spzp=A1c3coshapzp+c4sinhapzp+Σp1zp.


## 5. Thermal Marangoni Number

The Eigen value problem is solved for the thermal Marangoni number using the exact technique. The three temperature profiles are considered and the Marangoni numbers are calculated analytically.


### 5.1. Linear Temperature Profile Model

Consider the linear temperature profile(43)χzf=1,χzP=1.Introducing equation (43) into equations (31)–(34), we obtained using (38) as follows:(44)θfzf=B1b6coshafzf+b7sinhafzf+∑f1zf,θpzp=B1b8coshapzp+b9sinhapzp+∑p1zp.For the linear model, the thermal Marangoni number from (36), is as follows:(45)Mt1=−χ1−Msaf2℧af2b6coshaf+b7sinhaf+χ2+χ3,where(46)χ1=ψf2coshψf+b1sinhψf+φf2b2coshφf+b3sinhφf,℧=b18coshaf+b19sinhaf−1τfcoshψf+b1sinhψfψf2−af2+b2coshφf+b3sinhφfφf2−af2,χ2=ψf2−af2ϖ2+ϖ1coshψf+b1sinhψf−2ψfϖ2b1coshψf+sinhψfψf2−af22,χ3=ϖ2+ϖ1φf2−af2b2coshφf+b3sinhφf−2φfϖ2b3coshφf+b2sinhφfφf2−af22.


### 5.2. Parabolic Temperature Profile Model

Consider the (see Sparrow et al. [47]) parabolic temperature profile of the form(47)χzf=2zf,χzp=2zp.The temperature distributions are determined using the temperature boundary conditions (38) by substituting (47) into (31)–(34), as follows:(48)θfzf=B1b10coshafzf+b11sinhafzf+∑f2zf,θpzp=A1b12coshapzp+b13sinhapzp+∑p2zp.For the parabolic model, the thermal Marangoni number from (36), is as follows:(49)Mt2=−χ1−MSaf2℧af2b10coshaf+b11sinhaf+χ4+χ5,where χ4=ψf2−af2ϖ4+ϖ3coshψf+b1sinhψf−2ψfϖ4b1coshψf+sinhψf/ψf2−af22,(50)χ5=φf2−af2ϖ4+ϖ3b2coshφf+b3sinhφf−2φfϖ4b3coshφf+b2sinhφfφf2−af22,


### 5.3. Inverted Parabolic Temperature Profile Model

Consider the inverted parabolic temperature profile of the form(51)χzf=21−zf,χzp=21−zp.The temperature distributions are determined using the temperature boundary conditions (38) by substituting equations (51) into equations (31)–(34) as follows:(52)θfzf=B1b14coshafzf+b15sinhafzf+∑f3zf,θpzp=B1b16coshapzp+b17sinhapzp+∑p3zp.For an inverse parabolic temperature profile model, the thermal Marangoni number of (36) has the form:(53)Mt3=−χ1−MSaf2℧af2b14coshaf+b15sinhaf+χ6+χ7,where χ6=ψf2−af2ϖ6+ϖ5coshψf+b1sinhψf−2ψfϖ6b1coshψf+sinhψf/ψf2−af22,(54)χ7=φf2−af2ϖ6+ϖ5b2coshφf+b3sinhφf−2φfϖ6b3coshφf+b2sinhφfφf2−af22.


## 5.1. Linear Temperature Profile Model

Consider the linear temperature profile(43)χzf=1,χzP=1.Introducing equation (43) into equations (31)–(34), we obtained using (38) as follows:(44)θfzf=B1b6coshafzf+b7sinhafzf+∑f1zf,θpzp=B1b8coshapzp+b9sinhapzp+∑p1zp.For the linear model, the thermal Marangoni number from (36), is as follows:(45)Mt1=−χ1−Msaf2℧af2b6coshaf+b7sinhaf+χ2+χ3,where(46)χ1=ψf2coshψf+b1sinhψf+φf2b2coshφf+b3sinhφf,℧=b18coshaf+b19sinhaf−1τfcoshψf+b1sinhψfψf2−af2+b2coshφf+b3sinhφfφf2−af2,χ2=ψf2−af2ϖ2+ϖ1coshψf+b1sinhψf−2ψfϖ2b1coshψf+sinhψfψf2−af22,χ3=ϖ2+ϖ1φf2−af2b2coshφf+b3sinhφf−2φfϖ2b3coshφf+b2sinhφfφf2−af22.


## 5.2. Parabolic Temperature Profile Model

Consider the (see Sparrow et al. [47]) parabolic temperature profile of the form(47)χzf=2zf,χzp=2zp.The temperature distributions are determined using the temperature boundary conditions (38) by substituting (47) into (31)–(34), as follows:(48)θfzf=B1b10coshafzf+b11sinhafzf+∑f2zf,θpzp=A1b12coshapzp+b13sinhapzp+∑p2zp.For the parabolic model, the thermal Marangoni number from (36), is as follows:(49)Mt2=−χ1−MSaf2℧af2b10coshaf+b11sinhaf+χ4+χ5,where χ4=ψf2−af2ϖ4+ϖ3coshψf+b1sinhψf−2ψfϖ4b1coshψf+sinhψf/ψf2−af22,(50)χ5=φf2−af2ϖ4+ϖ3b2coshφf+b3sinhφf−2φfϖ4b3coshφf+b2sinhφfφf2−af22,


## 5.3. Inverted Parabolic Temperature Profile Model

Consider the inverted parabolic temperature profile of the form(51)χzf=21−zf,χzp=21−zp.The temperature distributions are determined using the temperature boundary conditions (38) by substituting equations (51) into equations (31)–(34) as follows:(52)θfzf=B1b14coshafzf+b15sinhafzf+∑f3zf,θpzp=B1b16coshapzp+b17sinhapzp+∑p3zp.For an inverse parabolic temperature profile model, the thermal Marangoni number of (36) has the form:(53)Mt3=−χ1−MSaf2℧af2b14coshaf+b15sinhaf+χ6+χ7,where χ6=ψf2−af2ϖ6+ϖ5coshψf+b1sinhψf−2ψfϖ6b1coshψf+sinhψf/ψf2−af22,(54)χ7=φf2−af2ϖ6+ϖ5b2coshφf+b3sinhφf−2φfϖ6b3coshφf+b2sinhφfφf2−af22.


## 6. Analysis and Discussion of Results

The present study aims at solving exactly the problem of Marangoni convection in presence of a heat source and temperature gradients. The exact method provides useful results and also general basic temperature profiles can be readily treated with minimum mathematical computations. The graphs are plotted using MATHEMATICA version 11. The thermal Marangoni numbers (tMn’s), linear profileMt1, parabolic profile Mt2, and inverted parabolic profile Mt3. These tMn’sMt1,Mt2 & Mt3 are drawn versus the depth ratio d^. The Chandrasekhar number, modified internal Rayleigh numbers for region-1 & region-2, solute Marangoni number, and diffusivity ratio impacts on the tMn are demonstrated in the following figures for various given parameters are af=0.5,β=10,Qf=10ϕp=0.3,T^=0.3, S^=1, τf=0.5, τp=0.5, RIf∗=1, RIp∗=1 and MS=10.The implications of the Chandrasekhar numberQf for linear, parabolic, and inverted parabolic temperature profiles are shown in Figure 2. The curves for all three profiles are slightly diverging, showing that bigger depth ratio values, i.e., for the Porous layer Dominant Composite System (PDCS), are more prominent. The value of raises the tMn’s for all three temperature gradients for a fixed depth ratio, hence the Double Component-Magneto-Marangoni-Convection (DCMC) may be advanced by decreasing the values of as well as the possibility of the system becoming unstable. A modified internal Rayleigh number RIf∗ for fluid region influence on the tMn’s are shown in Figure 3. The tMn’s increases as the value RIf∗ increases, indicating that the DCMC can be delayed by increasing the value of from sink to source. Hence, the heat absorption renders the system steady. This parameter’s influence is also stronger for larger depth ratios, implying that the modified internal Rayleigh number is useful for PDCS. The effect of the modified internal Rayleigh number RIp∗ for the porous region on the tMn’s is seen in Figure 4 for all three temperature gradients. The Marangoni number grows as the value increases from sink to source, hence the DCMC can be postponed by increasing the values of the modified internal Rayleigh number. As a result, heat absorption stabilizes the system. This parameter is also effective for a modest value of the depth ratios for the three temperature gradients, as shown in the figure. For all three temperature profiles shown in Figure 5, the influence of sMn on the Marangoni number is comparable. The tMn for DCMC grows as the value increases, hence the DCMC can be delayed by raising the value of Ms. The diverging curves show that the effect of the sMn is more intense for larger depth ratios, i.e., for PDCS. This demonstrates that there is a second diffusing component, at which point the DCMC influences the stability of the system. The tMn for DCMC is versus for different values of the diffusivity ratio τf, as shown in Figure 6. According to the diverging curves, the diffusivity ratio is only important for larger values of depth ratio, i.e., for PDCS. Also, for all three temperature profiles, the consequence is a drop in tMn’s. As a result, the diffusivity ratio has the effect of preplanning the DCMC and therefore destabilizing the composite layer system.Figure 2
The influence of Chandrasekhar’s numberQf=1,10,100 when af=0.5,β=10, ϕp=0.3,T^=0.3, S^=1, τf=τp=0.5RIf∗=RIp∗=1, MS=10.
(a)(b)(c)Figure 3
Effects of modified Rayleigh numberRIf∗=−1,0,1 in region-1 when af=0.5,β=10,Qf=10, ϕp=0.3,T^=0.3, S^=1, τf=τp=0.5RIp∗=1, MS=10.
(a)(b)(c)Figure 4
Modified Rayleigh numberRIp∗=−1,0,1 effects in region-2 when af=0.5,β=10,Qf=10, ϕp=0.3,T^=0.3, S^=1, τf=τp=0.5, RIf∗=1, MS=10.
(a)(b)(c)Figure 5
The influence of the solute Marangoni numberMS=5,10,50 when af=0.5,β=10,Qf=10, ϕp=0.3,T^=0.3,S^=1, τf=τp=0.5, RIf∗=RIp∗=1.
(a)(b)(c)Figure 6
The impact of the diffusivity ratioτf=0.1,0.5,1 when af=0.5,β=10,Qf=10,ϕp=0.3,T^=0.3, S^=1,τp=0.5, RIf∗=RIp∗=1, MS=10.
(a)(b)(c)


## 7. Conclusion

The influence of various physical parameters on the onset of double-component magneto convection are shown graphically in section 6 based on which the following conclusions may be drawn. The core findings from this study can be given as follows:(1)
In DCMC of the composite layer, the inverse parabolic temperature gradient is the most stable of the three temperature gradients.(2)
For combined systems where large values of the depth ratio, i.e., the porous layer predominates, all physical parameters apply.(3)
One can postpone the DCMC in a combined layer by increasing the values of Chandrasekhar number, modified internal Rayleigh numbers, and solute Marangoni number for the set of physical parameters chosen in the study, and one can prepone the DCMC in a composite layer by increasing the values of diffusivity ratio.(4)
By managing these parameters, stability-demanding conditions such as crystal growth, and material processing can be effectively achieved.(5)
In the process of manufacturing of permanent magnetic material, the destabilizing effects under normal gravity conditions are more efficient. Thus, by increasing the diffusivity ratio of the porous medium, a high-quality permanent magnetic material can be made.



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*Source: 1537674-2023-01-06.xml*