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A Comparative Study on the Effects of Vertical and Horizontal Magnetic Fields on Double Diffusive Convection with Soret Effect in Couple-Stressed Nanofluids

Suman Sindhwani

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Zenodo
DOI
10.54105/ijam.B1215.05021025
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zenodo.org
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en

Abstract

Abstract: This study aims to conduct a comparative analysis of double diffusive convection in a Soret-induced couple-stressed nanofluid layer under the influence of uniform vertical and horizontal magnetic fields. The linear stability analysis is based on the standard mode technique. The Galerkin method has been applied to find the critical Rayleigh number and the corresponding wave number in terms of various parameters. The effects of the Soret parameter, magnetic field, Lewis number, modified diffusivity ratio, Concentration Rayleigh-Darcy number, and Solutal Rayleigh number on the system's stability have been investigated. It has been observed that Stationary convection remains unaffected by the relaxation parameter, and the critical wave number is a function of both the couple stress parameter and the magnetic field; however, stationary convection is stabilised by the couple stress. In double diffusive convection under a magnetic field, the Darcy number also comes into play and has been observed to provide a stabilising effect on stationary convection. In comparison to ordinary fluids, convection sets up earlier in nanofluids. A comparative graphical analysis has been conducted to depict the effects of couple stress and Soret parameter on the stability of a nanofluid in the presence of both vertical and horizontal magnetic fields. It has been observed that the magnetic field still has a stabilising effect in both cases. However, in the case of a vertical magnetic field, the critical value of the stationary Rayleigh number is obtained at a significantly larger wave number compared to that in the case of a horizontal magnetic field, with all parameters remaining the same. The literature survey indicates that no study has investigated the comparison of the effect of vertical and horizontal magnetic fields on double diffusive convection in a couple stress nanofluid layer with a Soret factor. The present study examines the comparative analysis of the effect of vertical and horizontal magnetic fields on Soret-induced double diffusive convection in a couple-stress nanofluid horizontal layer.

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Indian Journal of Advanced Mathematics (IJAM) ISSN: 2582-8932 (Online), Volume-5, Issue-2, October 2025 36 Retrieval Number:100.1/ijam.B121505021025 DOI: 10.54105/ijam.B1215.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) Β© Copyright: All rights reserved. A Comparative Study on the Effects of Vertical and Horizontal Magnetic Fields on Double Diffusive Convection with Soret Effect in Couple-Stressed Nanofluids Suman Sindhwani Abstract: This study aims to conduct a comparative analysis of double diffusive convection in a Soret-induced couple-stressed nanofluid layer under the influence of uniform vertical and horizontal magnetic fields. The linear stability analysis is based on the standard mode technique. The Galerkin method has been applied to find the critical Rayleigh number and the corresponding wave number in terms of various parameters. The effects of the Soret parameter, magnetic field, Lewis number, modified diffusivity ratio, Concentration Rayleigh-Darcy number, and Solutal Rayleigh number on the system's stability have been investigated. It has been observed that Stationary convection remains unaffected by the relaxation parameter, and the critical wave number is a function of both the couple stress parameter and the magnetic field; however, stationary convection is stabilised by the couple stress. In double diffusive convection under a magnetic field, the Darcy number also comes into play and has been observed to provide a stabilising effect on stationary convection. In comparison to ordinary fluids, convection sets up earlier in nanofluids. A comparative graphical analysis has been conducted to depict the effects of couple stress and Soret parameter on the stability of a nanofluid in the presence of both vertical and horizontal magnetic fields. It has been observed that the magnetic field still has a stabilising effect in both cases. However, in the case of a vertical magnetic field, the critical value of the stationary Rayleigh number is obtained at a significantly larger wave number compared to that in the case of a horizontal magnetic field, with all parameters remaining the same. The literature survey indicates that no study has investigated the comparison of the effect of vertical and horizontal magnetic fields on double diffusive convection in a couple stress nanofluid layer with a Soret factor. The present study examines the comparative analysis of the effect of vertical and horizontal magnetic fields on Soret-induced double diffusive convection in a couple-stress nanofluid horizontal layer. Keywords: Magnetic Field, Rayleigh Number, Couple-Stressed Nanofluid, Soret Factor I. INTRODUCTION Creating a cutting-edge cooling system to cool crystal silicon mirrors used in high-intensity X-ray sources is one of the earliest applications of nanofluid research. Manuscript received on 05 September 2025 | First Revised Manuscript received on 12 September 2025 | Second Revised Manuscript received on 30 September 2025 | Manuscript Accepted on 15 October 2025 | Manuscript published on 30 October 2025. *Correspondence Author(s) Suman Sindhwani*, Associate Professor, Department of Mathematics, Hindu Girls College, Sonipat (Haryana), India. Email ID: [email protected] Β© The Authors. Published by Lattice Science Publication (LSP). This is an open-access article under the CC-BY-NC-ND license http://creativecommons.org/licenses/by-nc-nd/4.0/ Because metallic nanoparticles enhance the effective thermal conductivity of coolants and microchannels increase the effective heat transfer area, this technique may offer more effective cooling through the use of microchannels filled with nanofluids. In MHD power generators, electronic device cooling systems and metal casting, the impact of a magnetic field on the movement of electrically conducting fluids through a vertical plate is crucial. To enhance the heat transfer performance of such devices, the working medium is a nanofluid with enhanced thermal conductivity. A magnetic field can be used to control the flow and temperature fields after an electrically conducting nanofluid has been poured into the area between the plates. When a magnetic field is applied, a Lorentz force is created, which interacts with buoyancy. Yadav investigated the effect of magnetic fields on the onset of nanofluid convection, assuming that the boundaries were imposed by temperature. In unsteady nanofluid flow, the magnetic field effect was studied by Sheikholeslami et.al. [4]. An analytical investigation was conducted for various governing parameters. A macroscopic filtration model for natural convection in a Darcy-Maxwell nanofluid-saturated porous layer with no nanoparticle flux at the boundary was studied by Jaimala et al. [1]. Shankar et al. [3] investigated the effect of magnetic fields in couple stress fluids. They discovered that a magnetic field slows down the onset of instability, but as the couple stress parameter increases, the opposite behaviour is observed. In the thermo-solutal convection issue for a couple of stresses, Kumar [2] found that both the magnetic field and the couple stress had both stabilising and destabilising effects. Sithole et. al. [6] investigated the issue of thermal radiation and heat generation in a pair stress nanofluid in the presence of a magnetic field. The author studied the linear stability graphical analysis for this case [5] in the presence of a vertical magnetic field. The present study examines the comparison of vertical and horizontal magnetic fields on Soret-induced double diffusive convection in a couple-stress nanofluid horizontal layer. II. MATHEMATICAL FORMULATION An infinite isotropic couple-stressed nanofluid porous layer of width a ' between two horizontal planes, with temperatures at the l,ower and upper boundaries are π‘‡π‘™βˆ—and π‘‡π‘Ÿβˆ—(π‘‡π‘™βˆ— being greater than π‘‡π‘Ÿβˆ—) has been considered. A uniform horizontal magnetic field A Comparative Study on the Effects of Vertical and Horizontal Magnetic Fields on Double Diffusive Convection with Soret Effect in Couple-Stressed Nanofluids 37 Retrieval Number:100.1/ijam.B121505021025 DOI: 10.54105/ijam.B1215.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) Β© Copyright: All rights reserved. π‘€βˆ—=(𝑀0 βˆ—,0,0) acts on the system. The governing equations are as follows: π›»βˆ—.π‘žπ‘‘ βˆ—=0 … (1) 1 𝐾(πœ‡βˆ’πœ‡π‘π›»βˆ—2)π‘žπ‘‘ βˆ—=(1+πœ†βˆ—πœ• πœ•π‘‘βˆ—)[{βˆ’π›»βˆ—π‘βˆ—+ (πœ“βˆ—πœŒπ‘ƒ+(1βˆ’πœ“βˆ—){𝜌(1βˆ’π›½π‘‘(π‘‡βˆ—βˆ’π‘‡π‘Ÿβˆ—)βˆ’π›½π‘(π‘†βˆ—βˆ’ π‘†π‘Ÿ βˆ—)})𝑔]+πœ‡π‘’ 4πœ‹(π›»βˆ—Γ—π‘€βˆ—)Γ—π‘€βˆ—] … (2) (πœŒπ‘)π‘€πœ•π‘‡βˆ— πœ•π‘‘βˆ—+(πœŒπ‘)πΉπ‘žπ‘‘ βˆ—.π›»βˆ—π‘‡βˆ—= π‘˜π‘šπ›»βˆ—2π‘‡βˆ—+ ∈(πœŒπ‘)𝑃[π΅π‘‘π›»βˆ—πœ“βˆ—.π›»βˆ—π‘‡βˆ— +(𝐡𝑑 π‘‡π‘βˆ—)π›»βˆ—π‘‡βˆ—.π›»βˆ—π‘‡βˆ—] … (3) πœ•π‘†βˆ— πœ•π‘‘βˆ—+1 βˆˆπ‘žπ‘‘ βˆ—.π›»βˆ—π‘†βˆ—=π‘†π‘‘π›»βˆ—2π‘†βˆ—+π‘†π‘π‘‘π›»βˆ—2π‘‡βˆ— … (4) πœ•πœ“βˆ— πœ•π‘‘βˆ—+1 βˆˆπ‘žπ‘‘ βˆ—.π›»βˆ—πœ“βˆ—=π΅π‘‘π›»βˆ—2πœ“βˆ—+𝐡𝑑 π‘‡π‘Ÿ βˆ—π›»βˆ—2π‘‡βˆ— … (5) (πœ• πœ•π‘‘βˆ—+1 ∈(π‘žπ‘‘ βˆ—.π›»βˆ—))π‘€βˆ—=(π‘€βˆ—.π›»βˆ—)1 βˆˆπ‘žπ‘‘ βˆ—+πœ‚π›»βˆ—2π‘€βˆ— … (6) whereπ›»βˆ—β‹…π‘€βˆ—=0 , πœ‚ = 1 4πœ‹πœ‡π‘’πœŽβ€² and π‘žπ‘‘ βˆ—=(𝑒1𝑑 βˆ—,𝑒2𝑑 βˆ—,𝑒3𝑑 βˆ—)along with boundary conditions π‘žπ‘‘ βˆ—=0,π‘‡βˆ—=π‘‡π‘™βˆ—,π‘†βˆ—=π‘†π‘™βˆ—,π΅π‘‘πœ•πœ“βˆ— πœ•π‘§βˆ—+𝐡𝑑 π‘‡π‘Ÿβˆ—πœ•π‘‡βˆ— πœ•π‘§βˆ—=0 at π‘§βˆ— =0 … (7) π‘žπ‘‘ βˆ—=0,π‘‡βˆ—=π‘‡π‘Ÿβˆ—,π‘†βˆ—=π‘†π‘Ÿ βˆ—,π΅π‘‘πœ•πœ“βˆ— πœ•π‘§βˆ—+𝐡𝑑 π‘‡π‘Ÿβˆ—πœ•π‘‡βˆ— πœ•π‘§βˆ—=0 at π‘§βˆ— =π‘Ž … (8) Introducing non-dimensional parameters in taking * 0  as a reference scale for volumetric fraction of nanoparticles, π›Όπ‘š(= π‘˜π‘š (πœŒπ‘)𝐹) as the thermal diffusivity of the porous medium and 𝜎(=(πœŒπ‘)𝑀 (πœŒπ‘)𝐹) as the heat capacity ratio parameter, (𝑋,π‘Œ,𝑍)=(π‘₯βˆ—,π‘¦βˆ—,π‘§βˆ—) π‘Ž, 𝑑 = π‘‘βˆ—π›Όπ‘š πœŽπ‘Ž2, (𝑒1𝑑,𝑒2𝑑,𝑒3𝑑)= (𝑒1𝑑 βˆ—,𝑒2𝑑 βˆ—,𝑒3𝑑 βˆ—)π‘Ž π›Όπ‘š,𝑝 = π‘βˆ—πΎ πœ‡π›Όπ‘š, πœ“=πœ“βˆ—βˆ’πœ“0 βˆ— πœ“0 βˆ—,𝑇= π‘‡βˆ—βˆ’π‘‡π‘Ÿ βˆ— π‘‡π‘™βˆ—βˆ’π‘‡π‘Ÿ βˆ—, 𝑆 = π‘†βˆ—βˆ’π‘†π‘Ÿ βˆ— π‘†π‘™βˆ—βˆ’π‘†π‘Ÿ βˆ—,πœ†= πœ†βˆ—π›Όπ‘š π‘Ž2,(𝑀𝑋,π‘€π‘Œ,𝑀𝑍)=(𝑀𝑋 βˆ—,π‘€π‘Œ βˆ—,𝑀𝑍 βˆ—) 𝑀0 βˆ—. Non-dimensional form of equations is given by 𝛻.π‘ž =0 … (9) (π‘žβˆ’β„‚π›»2π‘ž)=(1 +πœ† πœŽπœ• πœ•π‘‘)[(βˆ’π›»π‘βˆ’π‘…π‘šπ‘’ξžΈπ‘§βˆ’π‘…π‘›πœ“π‘’ξžΈπ‘§ +π‘…π‘Žπ‘‡π‘’ξžΈπ‘§+𝑅𝑠 πΏπ‘›π‘†π‘’ξžΈπ‘§) +𝑃1 𝑃1𝑀 π‘„π·π‘Ž(𝛻×𝑀)×𝑀] … (10) πœ•π‘‡ πœ•π‘‘ +(π‘ž.𝛻)𝑇 =𝛻2𝑇+𝑁𝑏 πΏπ‘’π›»πœ“.𝛻𝑇 +π‘π‘Žπ‘π‘ 𝐿𝑒 𝛻𝑇.𝛻𝑇 … (11) 1 πœŽπœ•π‘† πœ•π‘‘ +1 βˆˆπ‘ž.𝛻𝑆 =1 𝐿𝑛𝛻2𝑆+𝑁𝑐𝑑𝛻2𝑇 … (12) 1 πœŽπœ•πœ“ πœ•π‘‘ +1 ∈(π‘ž.𝛻)πœ“= 1 𝐿𝑒𝛻2πœ“+π‘π‘Ž 𝐿𝑒𝛻2𝑇 … (13) 1 πœŽπœ•π‘€ πœ•π‘‘ +1 ∈(π‘ž.𝛻)𝑀 = 1 ∈(𝑀.𝛻)π‘ž+ P1 𝑃1𝑀 𝛻2𝑀 … (14) Here π‘…π‘Ž(=πœŒπ‘”π›½πΎπ‘Ž(π‘‡π‘™βˆ—βˆ’π‘‡π‘’ βˆ—) πœ‡π›Όπ‘š), 𝑅𝑛(=(πœŒπ‘ƒβˆ’πœŒ)πœ“0 βˆ—π‘”πΎπ‘Ž πœ‡π›Όπ‘š), π‘…π‘š(= πœŒπ‘ƒπœ“0 βˆ—+𝜌(1βˆ’πœ“0 βˆ—)π‘”πΎπ‘Ž πœ‡π›Όπ‘š) ,𝑅𝑠(=πœŒπ›½π‘π‘”π‘ŽπΎ(π‘†π‘™βˆ—βˆ’π‘†π‘’ βˆ—) πœ‡π‘†π‘‘)They are thermal, concentration, basic density, and solutal Rayleigh-Darcy numbers, respectively. β„‚(= πœ‡π‘π‘  πœ‡π‘Ž2)is a couple of stress parameters,𝑃1(= πœ‡ πœŒπ›Όπ‘š)and 𝑃1π‘š(= πœ‡ πœŒπœ‚) are Prandtl numbers, 𝑄(=πœ‡π‘’π‘€0 βˆ—2π‘Ž2 4πœ‹πœ‡πœ‚ ) is the Magnetic Chandrasekhar number,π·π‘Ž(= 𝐾 π‘Ž2) is the Darcy number, 𝑁𝑐𝑑(= 𝑆𝑐𝑑(π‘‡π‘™βˆ—βˆ’π‘‡π‘’ βˆ—) π›Όπ‘š(π‘†π‘™βˆ—βˆ’π‘†π‘’ βˆ—))is Soret parameter, π‘π‘Ž(=𝐡𝑑(π‘‡π‘™βˆ—βˆ’π‘‡π‘Ÿ βˆ—) π΅π‘‘π‘‡π‘Ÿ βˆ—π‘„0 βˆ—)and 𝑁𝑏(=(πœŒπ‘)π‘ƒβˆˆπ‘„0 βˆ— (πœŒπ‘)𝐹) The modified diffusivity ratio and modified particle density increment, respectively. 𝐿𝑒(=π›Όπ‘š 𝐡𝑑) and 𝐿𝑛 =π›Όπ‘š 𝑆𝑑Are Lewis numbers for the nanofluid and the salt, respectively? A. Basic State and Perturbed State The time-independent basic state of the nanofluid is described as π‘ž =0,𝑝 =𝑝𝑏𝑠(𝑍),𝑇 = 𝑇𝑏𝑠(𝑍),πœ“ =πœ“π‘π‘ (𝑍),𝑆 = 𝑆𝑏𝑠(𝑍) ,𝑀 =π‘’ξžΈπ‘‹ … (15) where the suffix β€œbs” refers to the basic flow. Following Chandrasekhar [3], the basic volume fraction and temperature of nanoparticles are given as 𝑇𝑏𝑠 =1βˆ’π‘ , πœ“π‘π‘  =πœ“0+ π‘π‘Žπ‘, and 𝑆𝑏𝑠 =1βˆ’π‘ . In the basic state, we superimpose perturbations in the form. π‘ž =π‘žβ€², 𝑝=𝑝𝑏𝑠 +𝑝′, 𝑆 =𝑆𝑏𝑠 + 𝑆′ , πœ“ =πœ“π‘π‘  +πœ“β€², 𝑀 =π‘’ξžΈπ‘‹+𝑀′ Linearised perturbation equations of Couple Stress nanofluid are obtained as (1 πœŽπœ• πœ•π‘‘ βˆ’π‘ƒ1 𝑃1𝑀 𝛻2)[(𝛻2βˆ’β„‚π›»4)𝑒3𝑑 β€²βˆ’ (1+πœ† πœŽπœ• πœ•π‘‘)(π‘…π‘Žπ›»π» 2π‘‡β€²βˆ’π‘…π‘›π›»π» 2πœ“β€²+𝑅𝑠 𝐿𝑛𝛻𝐻 2𝑆′)] =(1+πœ† πœŽπœ• πœ•π‘‘)𝑄 𝑃1 𝑃1𝑀 π·π‘Ž βˆˆπ›»2πœ•2𝑒3𝑑 β€² πœ•π‘‹2 … (16) πœ•π‘‡β€² πœ•π‘‘ βˆ’π‘’3𝑑 β€²=𝛻2π‘‡β€²βˆ’π‘π‘Žπ‘π‘ 𝐿𝑒 πœ•π‘‡β€² πœ•π‘ βˆ’π‘π‘ 𝐿𝑒 πœ•πœ“β€² πœ•π‘ … (17) Indian Journal of Advanced Mathematics (IJAM) ISSN: 2582-8932 (Online), Volume-5, Issue-2, October 2025 38 Retrieval Number:100.1/ijam.B121505021025 DOI: 10.54105/ijam.B1215.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) Β© Copyright: All rights reserved. 1 πœŽπœ•π‘†β€² πœ•π‘‘ βˆ’π‘’3𝑑 β€² ∈=1 𝐿𝑛𝛻2𝑆′+𝑁𝑐𝑑𝛻2𝑇′ … (18) 1 πœŽπœ•πœ“β€² πœ•π‘‘ +1 βˆˆπ‘π‘Žπ‘’3𝑑 β€²=1 𝐿𝑒𝛻2πœ“β€²+π‘π‘Ž 𝐿𝑒𝛻2𝑇′ … (19) with the boundary conditions 𝑒3𝑑 β€²=0,𝑇′=0,𝑆′=0,πœ•πœ“β€² πœ•π‘ +π‘π‘Žπœ•π‘‡β€² πœ•π‘ =0 at 𝑍 =0 and 𝑍 =1 … (20) B. Linear Stability Analysis Following the linear stability theory by Chandrasekhar [3], the perturbations are taken in the form (πœ“β€²,𝑇′,𝑒3𝑑 β€²,𝑆′) =[𝛷(𝑍),𝛩(𝑍),𝛺(𝑍),𝛹(𝑍)]𝑒𝑠𝑑+𝑖𝐿𝑋+π‘–π‘€π‘Œ … (21) where L and M are dimensionless wave numbers in X and Y Directions respectively. Application of the Galerkin-type weighted residuals method with first approximation (N=1) gives 𝛺 =𝐴1sinπœ‹π‘ , 𝛩 =𝐡1sinπœ‹π‘, 𝛷 =βˆ’π‘π‘ŽπΆ1sinπœ‹π‘ , 𝛹 =𝐷1sinπœ‹π‘ and leads to the following Rayleigh number π‘…π‘Ž=𝜎 βˆˆπ›Ό2 [ 𝑅𝑠𝛼2(πœ†π‘ +𝜎)(πœŽπ΄π›Ώ2+𝑠)(πœŽπ›Ώ2+𝑠𝐿𝑒)(𝛿2+𝑠){𝛿2(βˆˆπ‘π‘π‘‘βˆ’1)βˆ’π‘ } βˆ’π‘…π‘›π‘π‘Žπ›Ό2(πœ†π‘ +𝜎)(𝛿2𝜎+𝑠𝐿𝑛)(𝐴𝛿2𝜎+𝑠){𝛿2(∈+𝐿𝑒)+𝑠𝐿𝑒} +∈(𝛿2𝜎+𝑠𝐿𝑛)(𝛿2𝜎+𝑠𝐿𝑒)(𝛿2+𝑠){π΄πœŽπ›Ώ4+𝐡𝑙2𝛿2(𝜎+πœ†π›Ώ)+𝑠𝛿2+𝑠ℂ𝛿4+π΄β„‚πœŽπ›Ώ6} (πœŽπ›Ώ2+𝑠𝐿𝑛)(πœŽπ›Ώ2+𝑠𝐿𝑒)(πœ†π‘ +𝜎)(𝐴𝛿2𝜎+𝑠) ] … (22) where 𝛿2= πœ‹2+𝛼2.𝛼 = (𝐿2+𝑀2)1 2 ⁄ Equation (22), on taking s=0, results in the following Rayleigh number π‘…π‘Ž 𝑠𝑑 =𝛿4 𝛼2βˆ’(1+𝐿𝑒 ∈)π‘…π‘›π‘π‘Ž+π‘„π·π‘ŽπΏ2𝛿2 βˆˆπ›Ό2βˆ’π‘…π‘  ∈(1βˆ’βˆˆπ‘π‘π‘‘) +ℂ𝛿6 𝛼2 … (23) To obtain the critical value of the Rayleigh number π‘‘π‘…π‘Ž 𝑠𝑑 𝑑𝛼 =0 And the following critical wave number equation is obtained. 2β„‚(𝛼2)3+(3πœ‹2β„‚+1)(𝛼2)2βˆ’(πœ‹4+β„‚πœ‹6+π‘„π·π‘ŽπΏ2 βˆˆπœ‹2) =0 … (24) Equations (23) and (24), which give the stationary Rayleigh number and critical wave equation, are similar to the following equations were obtained by the author [5] in the case of a vertical magnetic field. π‘…π‘Ž 𝑠𝑑 =𝛿4 𝛼2βˆ’(1+𝐿𝑒 ∈)π‘…π‘›π‘π‘Ž+π‘„π·π‘Žπœ‹2𝛿2 βˆˆπ›Ό2βˆ’π‘…π‘  ∈(1βˆ’βˆˆπ‘π‘π‘‘) +ℂ𝛿6 𝛼2 … (25) 2β„‚(𝛼2)3+(3πœ‹2β„‚+1)(𝛼2)2βˆ’(πœ‹4+β„‚πœ‹6+π‘„π·π‘Ž βˆˆπœ‹4) =0 … (26) III. RESULTS AND DISCUSSION The stationary convection curves for the Rayleigh number a R versus the wave number ()L  = are in Fig 1(a)-(f), where = 5,π‘π‘Ž= 4, π·π‘Ž = 0.2, 𝐿𝑒 = 10, 𝑅𝑛 = 4, ∈ = 0.4, 𝑄 = 800, 𝑅𝑠=5, 𝑁𝑐𝑑=0.1. A B A Comparative Study on the Effects of Vertical and Horizontal Magnetic Fields on Double Diffusive Convection with Soret Effect in Couple-Stressed Nanofluids 39 Retrieval Number:100.1/ijam.B121505021025 DOI: 10.54105/ijam.B1215.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) Β© Copyright: All rights reserved. C D E F g h [Fig.1: Linear Stationary Convection with Wave Number in the Presence of a Horizontal Magnetic Field for Different Values of (a) β„‚ (b) Da (c) ο₯ (d) Le (e) n R (f)Q (g) s R (h) ct N ] Indian Journal of Advanced Mathematics (IJAM) ISSN: 2582-8932 (Online), Volume-5, Issue-2, October 2025 40 Retrieval Number:100.1/ijam.B121505021025 DOI: 10.54105/ijam.B1215.05021025 Journal Website: www.ijam.latticescipub.com Published By: Lattice Science Publication (LSP) Β© Copyright: All rights reserved. IV. CONCLUSION In this analysis, the following results have been obtained: β–ͺ Stationary convection remains unaffected by the relaxation parameter. β–ͺ Critical wave number is a function of the couple stress parameter as well as the magnetic field. β–ͺ Stationary convection gets stabilised by the couple stress. β–ͺ As compared to ordinary fluids, convection sets up earlier in nanofluid. β–ͺ Magnetic field stabilises the nanofluid layer. β–ͺ Porosity has a stabilising as well as a destabilising effect on convection. β–ͺ Soret parameter promotes the stability of the flow. β–ͺ Behaviour of solutal Rayleigh number is destabilising, thus resulting in early convection. Furthermore, a comparison has been made to observe the stability under both vertical and horizontal magnetic fields. It is interesting to observe that the magnetic field still has a stabilising effect in both cases. However, in the case of a vertical magnetic field, the critical value of the stationary Rayleigh number is obtained at a significantly larger wave number compared to the case of a horizontal magnetic field, with all parameters remaining the same. DECLARATION STATEMENT I must verify the accuracy of the following information as the article's author. β–ͺ Conflicts of Interest/ Competing Interests: Based on my understanding, this article has no conflicts of interest. β–ͺ Funding Support: This article has not been funded by any organizations or agencies. This independence ensures that the research is conducted with objectivity and without any external influence. β–ͺ Ethical Approval and Consent to Participate: The content of this article does not necessitate ethical approval or consent to participate with supporting documentation. β–ͺ Data Access Statement and Material Availability: The adequate resources of this article are publicly accessible. β–ͺ Author’s Contributions: The authorship of this article is contributed solely. REFERENCES 1. Jaimala, Singh, R., Tyagi, V. K. (2017): A macroscopic filtration model for natural convection in a Darcy Maxwell nanofluid saturated porous layer with no nanoparticle flux at the boundary, International Journal of Heat and Mass Transfer, vol. 111, pp. 451-466. DOI: https://doi.org/10.1016/j.ijheatmasstransfer.2017.04.003 2. 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Sithole, H; Mondal, H; Goqo, S; Sibanda, P; Motsa, S (2018): Numerical simulation of couple stress nanofluid flow in magneto-porous medium with thermal radiation and a chemical reaction, Journal of Applied Mathematics and Computation, vol. 339, 820-836, DOI: https://doi.org/10.1016/j.amc.2018.07.042. Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of the Lattice Science Publication (LSP)/ journal and/ or the editor(s). The Lattice Science Publication (LSP)/ journal and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions, or products referred to in the content.