Inclined surface mixed convection flow of viscous fluid with porous medium and Soret effects

Obulesu Mopuri; Charankumar Ganteda; Sarah A. Alsalhi; Sami Ullah Khan; Aruna Ganjikunta; Vediyappan Govindan; Faris Alqurashi; Mohamed Kchaou

doi:10.1515/phys-2023-0209

Article Open Access

Inclined surface mixed convection flow of viscous fluid with porous medium and Soret effects

Obulesu Mopuri , Charankumar Ganteda , Sarah A. Alsalhi , Sami Ullah Khan , Aruna Ganjikunta , Vediyappan Govindan , Faris Alqurashi and Mohamed Kchaou

Published/Copyright: April 15, 2024

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From the journal Open Physics Volume 22 Issue 1

Abstract

The combined heat and mass transfer phenomenon is a significant aspect of engineering and industrial processes. This phenomenon finds applications in various areas such as air conditioning, cooling and heating control of electronic devices, reactors, chemical systems, and emission processes. This research model focuses on the analysis of mixed convection flow of a viscous fluid with heat and mass transfer on an inclined surface with porous medium characteristics. The study also considers external heat transfer effects, radiation, Soret influence, and chemical reactions. A perturbation solution is derived in closed form, and the impact of various parameters on the thermal behavior is investigated. A comparative analysis of the heating and cooling regimes in plate flow is conducted, revealing a reduction in velocity in the heated plate regime with changes in the permeability parameter and an increase in concentration phase due to the Soret number.

Keywords: mixed convection flow; porous medium; Soret effects; thermal radiation; perturbation solution

Nomenclature

B 0: magnetic field strength
C ′: fluid concentration
C p: specific heat
C ∞ ′: ambient concentration
D: mass diffusivity
E: Eckert number
F: radiation parameter
Gr: Grashof number
g: gravity
K: permeability parameter
k: thermal conductivity
K C: reaction coefficient
K P: permeability of porous space
K 0: chemical reaction parameter
M: magnetic parameter
Pr: Prandtl number
Q: heat sink
Q 1: external heat source
q r ′: radiative heat flux
S 0: Soret parameter
Sc: Schmidt number
T ′: fluid temperature
T ∞ ′: ambient temperature
( u ′ , v ′ ): velocity components
ϑ: kinematic viscosity
α: inclined angle
σ: electric conductivity
ρ: density

1 Introduction

The magneto-hydro-dynamics (MHD) phenomenon is significant in plasma physics, with various applications in industrial and engineering processes. In cases of electrically conducting flow, MHD characteristics arise when magnetic forces act on the fluid flow. Important applications of MHD flows include cooling nuclear reactors, enhancing geothermal reservoirs, providing thermal protection, aiding in oil recovery, and facilitating packed-bed catalytic processes. Chamkha [1] studied MHD effects in accelerating flow with a heated source. Pal and Mondal [2] presented visualizations of convective phenomena influenced by dynamic MHD effects. Obulesu et al. [3] investigated different slip effects in MHD flow. Mahmoud [4] discussed vertically fluctuating surface flow under MHD influence. Turkyilmazoglu [5] analyzed MHD effects on a permeable moving surface with mixed convective transport. Nandalur [6] examined infinite surface flow with MHD phenomena. Khan et al. [7] visualized Carreau fluid flow with magnetized impact. Mansour et al. [8] examined the impact of MHD on bio-convection in a cavity with a moving surface flow. Kumar et al. [9] studied rotating MHD flow using Noumerov’s scheme. Lou et al. [10] analyzed rotating flow considering the effects of Lorentz force and Coriolis contribution. Ashraf et al. [11] discussed tangent hyperbolic flow in nanofluids with MHD applications. Ma et al. [12] investigated the flow in micro-tubes with MHD characteristics. Prabhakar Reddy and Jefta Sunzu [13] conducted a thermos-diffusion analysis of MHD-driven flow.

Heat and mass transport play a crucial role in a variety of industrial processes and engineering systems. The significance of understanding heat and mass transport patterns is evident in manufacturing processes, polymer solutions, rolling phenomena, metal casting, cooling processes, and heat transfer, among others. The study of heat transport is particularly important in chemical engineering and other engineering processes. Rashidi et al. [14] conducted an analysis of changes in mass and heat transfer influenced by buoyancy-driven flow. Shankar Goud and Dharmendar Reddy [15] investigated the Soret and Dufour effects in infinite plate flow, considering different mass and heat transfer characteristics. Seid et al. [16] studied heat and mass transfer in magnetized flow. Rauf et al. [17] examined the thermal transport phenomena in hybrid nanofluid flow. Siddique et al. [18] reported similar findings for viscoelastic fluids. Additionally, recent developments in fluid flow under various flow assumptions have been discussed in previous studies [19–23]. Previous literature [24–27] discuss various physical applications of highly dispersed transition metal oxide nanoclusters in mesoporous silica, pressure oscillation in low-velocity steam jet condensation, global sensitivity approaches that take into account model and parameter uncertainty, and closed-loop geothermal systems based on net present value. In a study by Bhatti et al. [28], the convective flow of Maxwell fluid was deduced using Lie transform analysis.

The current study explores the significance of mixed convection and thermos-diffusion phenomena in viscous fluid flow on an inclined surface. The investigation focuses on the permeability of porous media and the influence of inclined magnetic forces. Additionally, the impact of external heating sources and radiation on thermal transport processes is considered. The study also examines the effects of Soret diffusion and chemical reactions. The analysis involves varying parameters for both the cooler and heated plate, with perturbation simulations conducted to develop a highly accurate model. Physical analysis is presented graphically. Li et al. [29,30,31,32] recently studied fluid flow behavior under multi-flow assumptions with different boundary conditions and geometries. Yu et al. [33], Yang et al. [34], and Zhu et al. [35] explored hybrid empirical numerical techniques for tubing threaded connections, heat flow in 3D thermal metamaterials, and generalized micro-fluid rectifiers with anisotropic hollow channels. Sun et al. [36] and Kong et al. [37] investigated the shear-thickening materials and the use of hybrid CaAl₁₂O₁₉:Mn⁴⁺ doped with Ga³⁺ for plant growth lighting.

2 Mathematical formulation

The study is focused on the flow of a viscous fluid over an inclined surface. The fluid is assumed to be incompressible. The flow is induced by the motion of the inclined surface. Figure 1 depicts the schematic flow diagram with coordinate axes. The effects of magnetic force and permeability of the porous medium are taken into account. Let B 0 be the magnetic force strength. Assume u ′ is the velocity component, T ′ is the temperature while C ′ is the concentration. The surface endorsed temperature is T w , while C w denotes the plate surface concentration. Furthermore, the ambient concentration and temperature are expressed with C ∞ and T ∞ , respectively. In energy equation, the external heating relations and radiative applications are contributed. With such assumptions, the governing equations lead to [3].

(1) ∂ v ′ ∂ y ′ = 0 ,

(2) v ′ ∂ u ′ ∂ y ′ = ϑ ∂ 2 u ′ ∂ y ′ 2 + g β T ( T ′ − T ′ ∞ ) cos α + g β C ( C ′ − C ∞ ′ ) cos α − σ B 0 2 ρ sin 2 γ u ′ − ϑ u ′ K P ,

(3) v ′ ∂ T ′ ∂ y ′ = k ρ C p ∂ 2 T ′ ∂ y ′ 2 + ϑ C p ∂ u ′ ∂ y ′ 2 − 1 ρ C p ∂ q r ′ ∂ y ′ + σ B 0 2 ρ C P sin 2 γ u ′ 2 − Q 1 ρ C p ( T ′ − T ∞ ′ ) ,

(4) v ′ ∂ C ′ ∂ y ′ = D ∂ 2 C ′ ∂ y ′ 2 − K C ( C ′ − C ∞ ′ ) + D 1 ∂ 2 T ′ ∂ y ′ 2 .

Figure 1

Geometry of the flow model.

subject to

(5) u ′ = 0 T ′ = T w , C ′ = C w at y ′ = 0 , u ′ → 0 T ′ → T ′ ∞ , C ′ → C ′ ∞ as y ′ → ∞ .

Simplifying Eq. (1) gives that v ′ = − v 0 ′ ( v 0 ′ > 0 ) , where v 0 ′ is constant. Defining the radiative flux [13]

(7) ∂ q r ∂ y ′ = 4 ( T ′ − T ∞ ′ ) ∫ 0 ∞ K λ w d e b λ d T ′ d λ = 4 I 1 ( T ′ − T ′ ∞ ) .

The following non-dimensional quantities are introduced:

u = u ′ v 0 , y = v 0 y ′ ϑ , θ = T ′ − T ∞ ′ T w − T ∞ ′ , ϕ = C ′ − C ∞ ′ C w − C ∞ ′ , Pr = μ C p k , Sc = ϑ D , M = σ B 0 2 ϑ ρ v 0 2 ,

Gr = ϑ g β T ( T w − T ∞ ′ ) v 0 3 , Gm = ϑ g β C ( C w − C ∞ ′ ) v 0 3 , E = v 0 2 C P ( T w − T ∞ ′ ) , K = v 0 2 K P ϑ 2 ,

(8) K 0 = ϑ K C v 0 2 , F = 4 I 1 ϑ 2 k v 0 2 , Q = Q 1 v 2 k v 0 2 , S 0 = D 1 ( T w − T ∞ ′ ) ϑ ( C w − C ∞ ′ ) .

In view of the above suggested quantities, the developed system is

(9) u ″ + u ′ = − Gr cos α θ − Gm cos α ϕ + M 1 u ,

(10) θ ″ + Pr θ ′ − ( F + Q ) θ = − Pr u 1 2 − Pr M sin 2 γ u 2 ,

(11) ϕ ″ + Sc ϕ ′ − Sc K 0 ϕ = − S 0 Sc θ ″ ,

where M 1 = M sin 2 γ + 1 / K .

The transformed boundary conditions are:

(12) u = 0 , θ = 1 , ϕ = 1 at y = 0 , u → 0 , θ → 0 , ϕ → 0 as y → ∞ .

3 Solution of the problem

The perturbation method is imposed for solving the problem via analytical way. The motivations for implementation of perturbation procedure is due to fine accuracy. The initial expansion is

(13) u = u 0 + E u 1 + O ( E 2 ) , θ = θ 0 + E θ 1 + O ( E 2 ) , ϕ = ϕ 0 + E ϕ 1 + O ( E 2 ) .

Zeroth order system is

(14) u 0 ″ + u 0 ′ = − Gr cos α θ 0 − Gm cos α ϕ 0 + M 1 u 0 ,

(15) θ 0 ″ + Pr θ 0 ′ − ( F + Q ) θ 0 = 0,

(16) ϕ 0 ″ + Sc ϕ ′ 0 − Sc K 0 ϕ 0 = − S 0 Sc θ 0 ″ .

First order expressions are

(17) u 1 ″ + u 0 ′ = − Gr cos α θ 1 − Gm cos α ϕ 1 + M 1 u 1 ,

(18) θ 1 ″ + Pr θ 1 ′ − ( F + Q ) θ 1 = − Pr u 0 1 2 − Pr M sin 2 γ u 0 2 ,

(19) ϕ 1 ″ + Sc ϕ 1 ′ − Sc K 0 ϕ 1 = − S 0 Sc ϕ 1 ″ ,

with

(20) u 0 = 0 , u 1 = 0 , θ 0 = 1 , θ 1 = 0 , ϕ 0 = 1 , ϕ 1 = 0 at y = 0 u 0 → 0 , u 1 → 0 , θ 0 → 1 , θ 1 → 0 , ϕ 0 → 1 , ϕ 1 → 0 as y = ∞ .

The initial solution are

(21) θ 0 = e − m 1 y ,

(22) ϕ 0 = − A 1 e − m 1 y + A 2 e − m 2 y ,

(23) u 0 = A 3 e − m 1 y − A 4 e − m 2 y + A 5 e − m 3 y ,

(24) θ 1 = A 18 e − 2 m 1 y + A 19 e − 2 m 2 y + A 20 e − 2 m 3 y + A 21 e − δ 1 y + A 22 e − δ 2 y + A 23 e − δ 3 y + A 24 e − m 1 y ,

(25) ϕ 1 = A 25 e − m 1 y + A 26 e − 2 m 1 y + A 27 e − 2 m 2 y + A 28 e − 2 m 3 y + A 29 e − δ 1 y + A 30 e − δ 2 y + A 31 e − δ 3 y + A 32 e − m 2 y ,

(26) u 1 = A 33 e − m 1 y + A 34 e − m 2 y + A 35 e − 2 m 1 y + A 36 e − 2 m 2 y + A 37 e − 2 m 3 y + A 38 e − δ 1 y + A 39 e − δ 2 y + A 40 e − δ 3 y + A 41 e − m 3 y .

The requested solution is

(27) u = A 3 e − m 1 y − A 4 e − m 2 y + A 5 e − m 3 y + E [ A 33 e − m 1 y + A 34 e − m 2 y + A 35 e − 2 m 1 y + A 36 e − 2 m 2 y + A 37 e − 2 m 3 y + A 38 e − δ 1 y + A 39 e − δ 2 y + A 40 e − δ 3 y + A 41 e − m 3 y ] .

(28) θ = e − m 1 y + E [ A 18 e − 2 m 1 y + A 19 e − 2 m 2 y + A 20 e − 2 m 3 y + A 21 e − δ 1 y + A 22 e − δ 2 y + A 23 e − δ 3 y + A 24 e − m 1 y ] ,

(29) ϕ = − A 1 e − m 1 y + A 2 e − m 2 y + E [ A 25 e − m 1 y + A 26 e − 2 m 1 y + A 27 e − 2 m 2 y + A 28 e − 2 m 3 y + A 29 e − δ 1 y + A 30 e − δ 2 y + A 31 e − δ 3 y + A 32 e − m 2 y ] .

Defining the surface skin force [3]

τ = ∂ u ∂ y y = 0 ,

(30) τ = [ − m 3 A 5 + m 2 A 4 − m 1 A 3 ] + E [ m 1 A 33 − m 2 A 34 − 2 m 1 A 35 − 2 m 2 A 36 − 2 m 3 A 37 − δ 1 A 38 − δ 2 A 39 − δ 3 A 40 − m 3 A 41 ] .

The Nusselt number is expressed as [3]

Nu = − ∂ θ ∂ y y = 0 ,

(31) Nu = m 1 + E [ m 1 A 24 + 2 m 1 A 18 + 2 m 2 A 19 + 2 m 3 A 20 + δ 1 A 21 + δ 2 A 22 + δ 3 A 23 ] .

Defining the Sherwood number [3]

Sh = − ∂ ϕ ∂ y y = 0 ,

(32) Sh = [ m 2 A 2 − m 1 A 1 ] + E [ m 1 A 25 + 2 m 1 A 26 + 2 m 2 A 27 + 2 m 3 A 28 + δ 1 A 29 + δ 2 A 30 + δ 3 A 31 + m 2 A 32 ] .

4 Results and discussion

The physical significance of parameters for velocity profile u , temperature θ , and concentration ϕ is visualized in this section. The results and observations are supported with graphs. Figure 2(a)–(c) pronounced the variation in velocity u subject to variation in permeability parameter K , magnetic parameter M , and inclined angle γ . The analysis is subject to two regimes of plate like cooler and heated surface. Figure 2(a) shows the assessment of u for permeability parameter K . A boosted effect in determination of u is noticed for K . Such effects are physically justified due to permeability of porous regime. Figure 2(b) addresses the onset of magnetic parameter M on u . More slower rate of velocity has been exhibited for M in both heated and cooler surface. Such decrement appearing in u is due to Lorentz force which exclusively presents resistivity in motion. Figure 2(c) justified the fluctuated pattern in the behavior of u against inclination angle γ . A control of the changing u due to γ is observed for both surface constraints. Figure 3(a) and (b) addresses the prediction for u due to Grashof number ( Gr ) and Mass Grashof number ( Gm ). Both parameters are physically involved in the role of buoyancy forces. An increment is presented in change of u for both parameters.

$Figure 2 Velocity profile for (a) K K , (b) M M , and (c) γ \gamma .$

Figure 2

Velocity profile for (a) K , (b) M , and (c) γ .

$Figure 3 Velocity profile for (a) Gr and (b) Gm \text{Gm} .$

Figure 3

Velocity profile for (a) Gr and (b) Gm .

Figure 4(a) and (b) is prepared in order to justify the aim of Prandtl number Pr on temperature profile θ . The decreasing analysis in the rate of θ is predicted for Pr . Physically, the slower rate of heat transfer is pronounced for less thermal diffusivity. Figure 5(a) and (b) expresses the contribution of γ on θ . The decreasing change is exhibited in θ for increasing γ . The decreasing change is more progressive for cooler surface. The analysis for observing fluctuation in θ due to heat absorption parameter Q is shown in Figure 6(a) and (b). The slower heat fluctuation is inspected when heat absorption phenomenon appears. The removal of heat transfer results in a slower temperature profile of both cooler and heated plates. Figure 7(a) and (b) shows the change in concentration field ϕ due to Soret number S 0 . An improvement in concentration is noticed for S 0 . Figure 8(a) and (b) shows the contribution of reaction parameter K 0 in the field of ϕ . Decreasing outcomes are observed for ϕ due to K 0 .

$Figure 4 Variation in temperature profile due to Pr \text{Pr} : (a) flow on cooled plate and (b) flow on heated plate.$

Figure 4

Variation in temperature profile due to Pr : (a) flow on cooled plate and (b) flow on heated plate.

$Figure 5 Variation in temperature profile due to γ \gamma : (a) flow on cooled plate and (b) flow on heated plate.$

Figure 5

Variation in temperature profile due to γ : (a) flow on cooled plate and (b) flow on heated plate.

Figure 6

Variation in temperature profile due to Q : (a) flow on cooled plate and (b) flow on heated plate.

$Figure 7 Variation in concentration profile due to S 0 {S}_{0} : (a) flow on cooled plate and (b) flow on heated plate.$

Figure 7

Variation in concentration profile due to S 0 : (a) flow on cooled plate and (b) flow on heated plate.

$Figure 8 Variation in concentration profile due to K 0 {K}_{0} : (a) flow on cooled plate and (b) flow on heated plate.$

Figure 8

Variation in concentration profile due to K 0 : (a) flow on cooled plate and (b) flow on heated plate.

Table 1 presents the variation in wall shear force due to different parameters like Gm , M , and K . The analysis is subject to various values of Prandtl number. Note that these values are associated with different liquids like mercury, electrolytic solution, air, and water. A peak variation in wall shear force is noted when Gm is maximum. The increasing change is more reflective for mercury. Similar effects are noted for K . Tables 2 and 3 present the variation in involved parameters for Nusselt number and Sherwood number, respectively. The analysis is subject to prediction of decomposition of mercury, electrolytic solution, air, and water.

Table 1

Numerical change in wall shear force due to different parameters

			Wall shear force
Gm	M	K	Mercury ( Pr = 0.025 )	Electrolytic solution ( Pr = 1.0 )	Air ( Pr = 0.71 )	Water ( Pr = 7.0 )
5.0			6.6185	6.3490	7.3516	5.6751
10			10.5656	10.4235	12.1723	10.2210
15			14.5238	14.5136	17.0227	14.7317
	2.0		6.5075	6.3490	7.3516	5.6751
	3.0		5.7299	5.5649	6.1709	4.9709
	4.0		5.7984	5.9447	7.4704	5.4962
		2.0	7.0617	7.0966	6.9600	6.2316
		3.0	7.2847	7.6694	7.2098	6.4633
		4.0	7.4052	8.2918	7.3516	6.5907

Table 2

Variations in Nusselt number for different parameters

S 0	F	Q	Nusselt number
S 0	F	Q	Mercury ( Pr = 0.025 )	Electrolytic solution ( Pr = 1.0 )	Air ( Pr = 0.71 )	Water ( Pr = 7.0 )
1			1.4091	1.5740	1.6245	6.1734
2			1.4098	1.6582	1.6152	5.7578
3			1.4099	1.6925	1.6071	5.2751
	4		1.8510	2.6302	2.4460	6.7311
	6		2.6516	3.0822	2.9409	7.0466
		3	1.9850	2.2331	0.6403	6.5589
		5	2.4299	2.8740	2.7269	6.8932
		6	2.6516	3.0822	2.9409	7.0466

Table 3

Variations in Sherwood number for different parameters

Sc	K 0	S 0	Sherwood number
Sc	K 0	S 0	Mercury ( Pr = 0.025 )	Electrolytic solution ( Pr = 1.0 )	Air ( Pr = 0.71 )	Water ( Pr = 7.0 )
0.22			0.4054	0.3667	0.3608	−0.6481
0.30			0.4959	0.4202	0.4265	−0.9395
0.60			0.9047	0.6327	0.7222	−1.8236
	0.5		0.2469	0.2182	0.2080	−0.7642
	1.5		0.5289	0.4830	0.4803	−0.5474
	3		0.8116	0.7498	0.7542	−0.2979
		2	0.2188	0.1046	0.1250	−1.7067
		4	−0.1539	−0.4007	−0.3128	−3.1048
		6	−0.5231	−0.6650	−0.6241	−3.2517

5 Conclusion

The study investigates the heat and mass transfer in mixed convection flow over a porous inclined plate. A perturbation solution is derived in a closed form. The analysis reveals that the velocity profile is influenced by the Grashof number, leading to an increase in velocity. Changes in the inclined angle result in a decrease in the velocity magnitude. Heat transfer is found to decrease with the Prandtl number, with a more pronounced effect on the heated surface. Slower heat transfer is observed due to heat absorption. The concentration field is affected by the Soret number, with a greater impact on the cooler surface. A decrease in the concentration profile is noted. The skin friction coefficient increases with the Grashof number for mercury, electrolytic solution, air, and water decomposition.

Acknowledgments

The authors are thankful to the Deanship of Graduate Studies and Scientific Research at University of Bisha for supporting this work through the Fast-Track Research Support Program. The authors express their gratitude to Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2024R453), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

Funding information: The University of Bisha for supporting this work through the fast-track research support program. The authors express their gratitude to Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2024R453), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.

Appendix

M 1 = M sin 2 γ + 1 K M 2 = F + Q ,

m 1 = Pr + Pr 2 + 4 M 2 2 m 2 = Sc + Sc 2 + 4 K 0 Sc 2 m 3 = 1 + 1 + 4 M 1 2

A 1 = m 1 2 S 0 Sc m 1 2 − Sc m 1 − K 0 Sc A 2 = 1 + A 1 A 3 = A 1 Gm cos α − Gr cos α m 1 2 − m 1 − M 1 A 4 = A 2 Gm cos α m 2 2 − m 2 − M 1 A 5 = A 4 − A 3

λ 1 = m 1 A 3 λ 2 = m 2 A 4 λ 3 = m 3 A 5

A 6 = − Pr λ 1 2 4 m 1 2 − 2 Pr m 1 − M 2 A 7 = − Pr λ 2 2 4 m 2 2 − 2 Pr m 2 − M 2 A 8 = − Pr λ 3 2 4 m 3 2 − 2 Pr m 3 − M 2

δ 1 = m 1 + m 2 δ 2 = m 2 + m 3 δ 3 = m 3 + m 1

A 9 = 2 λ 1 λ 2 Pr δ 1 2 − Pr δ 1 − M 2 A 10 = 2 λ 2 λ 3 Pr δ 2 2 − Pr δ 2 − M 2

A 11 = − 2 λ 1 λ 3 Pr δ 3 2 − Pr δ 3 − M 2 A 12 = − Pr M sin 2 γ A 3 2 4 m 1 2 − 2 Pr m 1 − M 2

A 13 = − Pr M sin 2 γ A 4 2 4 m 2 2 − 2 Pr m 2 − M 2 A 14 = − Pr M sin 2 γ A 5 2 4 m 3 2 − 2 Pr m 3 − M 2

A 15 = 2 A 3 A 4 Pr M sin 2 γ δ 1 2 − Pr δ 1 − M 2 A 16 = 2 A 4 A 5 Pr M sin 2 γ δ 2 2 − Pr δ 2 − M 2

A 17 = − 2 A 5 A 3 Pr M sin 2 γ δ 3 2 − Pr δ 3 − M 2 A 18 = A 6 + A 12 A 19 = A 7 + A 13

A 20 = A 8 + A 14 A 21 = A 9 + A 15 A 22 = A 10 + A 16

A 23 = A 11 + A 17 A 24 = − ( A 18 + A 19 + A 20 + A 21 + A 22 + A 23 )

A 25 = − m 1 2 A 24 S 0 Sc m 1 2 − Sc m 1 − K 0 Sc A 26 = − 4 m 1 2 A 18 S 0 Sc 4 m 1 2 − 2 Sc m 1 − K 0 Sc

A 27 = − 4 m 2 2 A 19 S 0 Sc 4 m 2 2 − 2 Sc m 2 − K 0 Sc A 28 = − 4 m 3 2 A 20 S 0 Sc 4 m 3 2 − 2 Sc m 3 − K 0 Sc

A 29 = − δ 1 2 A 21 S 0 Sc δ 1 2 − Sc δ 1 − Sc K 0 A 30 = − δ 2 2 A 22 S 0 Sc δ 2 2 − Sc δ 2 − Sc K 0

A 31 = − δ 3 2 A 23 S 0 Sc δ 3 2 − Sc δ 3 − Sc K 0

A 32 = − ( A 25 + A 26 + A 27 + A 28 + A 29 + A 30 + A 31 )

A 33 = − A 24 Gr cos α − A 25 Gm cos α m 1 2 − m 1 − M 1 A 34 = − A 32 Gm cos α m 2 2 − m 2 − M 1

A 35 = − A 18 Gr cos α − A 26 Gm cos α 4 m 1 2 − 2 m 1 − M 1 A 36 = − A 19 Gr cos α − A 27 Gm cos α 4 m 2 2 − 2 m 2 − M 1

A 37 = − A 20 Gr cos α − A 28 Gm cos α 4 m 3 2 − 2 m 3 − M 1 A 38 = − A 21 Gr cos α − A 29 Gm cos α δ 1 2 − δ 1 − M 1

A 39 = − A 22 Gr cos α − A 30 Gm cos α δ 2 2 − δ 2 − M 1 A 40 = − A 23 Gr cos α − A 31 Gm cos α δ 3 2 − δ 3 − M 1

A 41 = − ( A 33 + A 34 + A 35 + A 36 + A 37 + A 38 + A 39 + A 40 ) .

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Received: 2023-12-30

Revised: 2024-02-04

Accepted: 2024-02-20

Published Online: 2024-04-15

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/phys-2023-0209

Keywords for this article

mixed convection flow; porous medium; Soret effects; thermal radiation; perturbation solution

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