Double Diffusion Mixed Convection in a Porous Cylindrical Cavity Filled with a Casson Nanofluid Driven by a Lid, Taking Into Account the Soret and Dufour Effects

1. Introduction

Understanding thermosolutal mixed convection is important for various scientific and engineering applications, as it enables the solution of complex issues and the advancement of more efficient and sustainable technologies in areas like solar ponds, materials processing, drug delivery systems, cancer treatments, solar air conditioning, electronics packaging, and food processing, etc. [ 1 - 2 ]. This effect is of great interest to scientific experimenters modeling double diffusion mixed convection phenomena. The exploration carried out to model the miracle of mixed convection or double diffusion is different and focuses on understanding the abecedarian mechanisms of heat and mass transfer. Thus, you can cite the ensuing works :Sivasankaran et al. [ 3 ] were fascinated by twofold dissemination blended convection in depth driven by a top with non-uniform warming on the side dividers. Bhuvaneswari et al. [ 4 ] carried out a numerical consideration on twofold dissemination blended convection with Soret impact in depth driven by a two-sided cover. Guimarãesa et al. [ 5 ] inspected blended convection in a ventilated square depression utilizing nanofluid. Hayat et al. [ 6 ] were curious about the stagnation point stream by blended convection of the Casson liquid with convection boundary conditions. Teamah et al. [ 7 ] carried out a numerical think about a blended convective stream with twofold dissemination in a rectangular walled in area with an protected moving cover. K. Khanafer et al. [ 8 ] examined the phenomenon of double diffusive mixed convection in a lidded enclosure filled with a fluid saturated permeable medium. Jamir et al. [ 9 ] carried out a consideration on the impacts of radiation absorption, Soret and Dufour on an unsteady MHD mixed convective stream past a vertical porous plate with the sliding conditions and viscous dissipation. El Hamma et al. [ 10 ] carried out a think about thermosolutal convection in a cylindrical permeable cavity filled with a nanofluid and taking into consideration the Soret and Dufour impacts. Mahdy [ 11 ] inspected the Soret and Dufour Impact on double diffusion mixed convection from a vertical surface in a porous medium immersed with a non-Newtonian liquid. Mondal et al. [ 12 ] carried out a think about mixed convection with double diffusion and entropy generation of nanofluid in a trapezoidal depth. Moolya et al. [ 13 ] carried out a think about the Part of magnetic field and cavity inclination on mixed double diffusion convection in a closed rectangular space. El Hamma et al. [ 14 ] were curious about thermosolutal convection in a permeable cylindrical cavity filled with a Casson nanofluid, taking into consideration the Soret and Dufour impacts. Guerroudj et al. [ 15 ] carried out an in-depth ponder on Ferrohydrodynamics Mixed convection of a ferrofluid in a vertical channel with permeable squares of different shapes. Wahid et al. [ 16 ] think about the stream of mixed convection magnetic nanofluids before a rotating vertical permeable cone. Bouras et al. [ 17 ] were fascinated by the 3D numerical recreation of turbulent mixed convection in a cubic cavity containing a hot square. Khan et al. [ 18 ] carried out Thermal treatment inside a partially heated triangular cavity filled with casson fluid with an inner cylindrical obstacle via FEM approach. Hamid et al. [ 19 ] were curious about Characterizing natural convection and thermal behavior in a square cavity with curvilinear corners and central circular obstacles. Bendrer [ 20 ] carried out a study on 3D magnetic buoyancy-driven flow of hybrid nanofluids confined wavy cubic enclosures, including multi-layers and heated obstacles. Ahmed [ 21 ] examined the Dissipated-radiative compressible flow of nanofluids over unsmoothed inclined surfaces with variable properties.

The originality of the display work is the modeling of mixed thermosolutal convection in a permeable cylindrical cavity filled with a Casson nanofluid driven by a cover beneath the impact of the thermo-diffusion phenomenon (Soret and Dufour impact).

2. Description of The Problem And Governing Equations

2.1. Detail the Problem

The phenomenon of mixed double diffusion convection was considered in a porous cylindrical enclosure filled and saturated by a Casson nanofluid driven by an upper cover. The enclosure in address (Fig. 1) has lower and upper bases subjected, individually, to uniform temperatures and concentrations. The sidewalls are accepted as adiabatic and impermeable.

Fig. 1. Geometry of the studied problem

2.2. Governing Equations

The Governing Equations deciphering the phenomenon examined are based on the conservation equations of continuum mechanics. (Equation of continuity, quantity of movement, conservation of energy and mass). Hence, the Darcy-Brinkman-Forchheimer laws depict the stream of the nanofluid within the porous layers. The thermo-physical properties of the nanofluid are expected constant. Conditions in dimensional shape are composed.

$\begin{array}{cc}\frac{1}{r^{*}}\frac{\partial \left(r^{*}{u}^{*}\right)}{\partial r^{*}}+\frac{\partial w^{*}}{\partial z^{*}}=0& \mathrm{(1)}\end{array}$

$\begin{array}{c}\frac{1}{{\epsilon }^2}(\frac{1}{r^{*}}\frac{\partial \left(r^{*}{u}^{*}{\Omega }^{*}\right)}{\partial r^{*}}+\frac{\partial \left(w^{*}{\Omega }^{*}\right)}{\partial z^{*}}-\frac{u^{*}{\Omega }^{*}}{r^{*}})=\\ -\frac{{\mu }_{\mathrm{nf}}}{K{\rho }_{\mathrm{0nf}}}{\Omega }^{*}-\frac{{\rho }_0(1-\phi )}{{\rho }_{\mathrm{0nf}}}{\beta }_{T}g\frac{\partial T^{*}}{\partial r^{*}}-\frac{{\rho }_0(1-\phi )}{{\rho }_{\mathrm{0nf}}}{\beta }_{s}g\frac{\partial C^{*}}{\partial r^{*}}\\ +\frac{{\mu }_{\mathrm{nf}}}{\epsilon {\rho }_{\mathrm{0nf}}}(1+\frac{1}{\beta })(\frac{{\partial }^2{\Omega }^{*}}{\partial r^{{*}^2}}+\frac{1}{r^{*}}\frac{\partial {\Omega }^{*}}{\partial r^{*}}+\frac{{\partial }^2{\Omega }^{*}}{\partial z^{{*}^2}}-\frac{{\Omega }^{*}}{r^{{*}^2}})\\ -\frac{C_F}{K^{\frac{1}{2}}}\left|{\overrightarrow{V}}^{*}\right|{\Omega }^{*}+\frac{C_F}{K^{\frac{1}{2}}}(w^{*}\frac{\partial \left|{\overrightarrow{V}}^{*}\right|}{\partial r^{*}}-u^{*}\frac{\partial \left|{\overrightarrow{V}}^{*}\right|}{\partial z^{*}})& \mathrm{(2)}\end{array}$

$\begin{array}{c}(\frac{\partial \left(u^{*}{T}^{*}\right)}{\partial r^{*}}+\frac{\partial \left(w^{*}{T}^{*}\right)}{\partial z^{*}}+\frac{u^{*}{T}^{*}}{r^{*}})=\alpha (\frac{{\partial }^2{T}^{*}}{\partial r^{{*}^2}}+\frac{{\partial }^2{T}^{*}}{\partial z^{{*}^2}}+\frac{1}{r^{*}}\frac{\partial T^{*}}{\partial r^{*}})\\ +\epsilon N_{\mathrm{TC}}(\frac{{\partial }^2{C}^{*}}{\partial r^{{*}^2}}+\frac{{\partial }^2{C}^{*}}{\partial z^{{*}^2}}+\frac{1}{r^{*}}\frac{\partial C^{*}}{\partial r^{*}})& \mathrm{(3)}\end{array}$

$\begin{array}{c}(\frac{\partial \left(u^{*}{C}^{*}\right)}{\partial r^{*}}+\frac{\partial \left(w^{*}{C}^{*}\right)}{\partial z^{*}}+\frac{u^{*}{C}^{*}}{r^{*}})=D\epsilon (\frac{{\partial }^2{C}^{*}}{\partial r^{{*}^2}}+\frac{{\partial }^2{C}^{*}}{\partial z^{{*}^2}}+\frac{1}{r^{*}}\frac{\partial C^{*}}{\partial r^{*}})\\ +\epsilon N_{\mathrm{CT}}(\frac{{\partial }^2{T}^{*}}{\partial r^{{*}^2}}+\frac{{\partial }^2{T}^{*}}{\partial z^{{*}^2}}+\frac{1}{r^{*}}\frac{\partial T^{*}}{\partial r^{*}})& \mathrm{(4)}\end{array}$

$\begin{array}{cc}{\Omega }^{*}=(\frac{\partial u^{*}}{\partial z^{*}}-\frac{\partial w^{*}}{\partial r^{*}})& \mathrm{(5)}\end{array}$

By introducing the following dimensionless quantities:

$\begin{array}{cccc}r=\frac{r^{*}}{R}& z=\frac{z^{*}}{R},& U=\frac{u^{*}}{u_0},& W=\frac{w^{*}}{u_0},\\ \Omega =\frac{{\Omega }^{*}R}{u_0},& \Omega =\frac{{\psi }^{*}}{R^2{u}_0},& T=\frac{T^{*}-T_F^{*}}{T_C^{*}-T_F^{*}},& C=\frac{C^{*}-C_{\sup }^{*}}{C_{\inf }^{*}-C_{\sup }^{*}},\end{array}$

The previously established equations are written in dimensionless form:

$\begin{array}{cc}\frac{1}{r}\frac{\partial \left(rU\right)}{\partial r}+\frac{\partial W}{\partial z}=0& \mathrm{(6)}\end{array}$

$\begin{array}{c}\frac{1}{{\epsilon }^2}(\frac{1}{r}\frac{\partial \left(rU\Omega \right)}{\partial r}+\frac{\partial \left(W\Omega \right)}{\partial z}-\frac{U\Omega }{r})=-\frac{1}{R_{\mathrm{e,nf}}\mathrm{Da}}\Omega -R_{\mathrm{i,nf}}(\frac{\partial T}{\partial r}+N\frac{\partial C}{\partial r})+\frac{1}{\epsilon R_{\mathrm{e,nf}}}\\ (1+\frac{1}{\beta })(\frac{{\partial }^2\Omega }{\partial r^2}+\frac{1}{r}\frac{\partial \Omega }{\partial r}+\frac{{\partial }^2\Omega }{\partial z^2}-\frac{\Omega }{r^2})-\frac{C_F}{\sqrt{\mathrm{Da}}}\left|\overrightarrow{V}\right|\Omega \frac{C_F}{\sqrt{\mathrm{Da}}}(W\frac{\partial \left|\overrightarrow{V}\right|}{\partial \stackrel{-}{r}}-U\frac{\partial \left|\overrightarrow{V}\right|}{\partial \stackrel{-}{z}})& \mathrm{(7)}\end{array}$

$\begin{array}{c}\frac{\partial \left(UT\right)}{\partial r}+\frac{\partial \left(WT\right)}{\partial z}+\frac{UT}{r}=\frac{1}{{\mathrm{pr}}_{\mathrm{nf}}{R}_{\mathrm{e,nf}}}(\frac{{\partial }^{2}T}{\partial r^2}+\frac{{\partial }^{2}T}{\partial z^2}+\frac{1}{r}\frac{\partial T}{\partial r})\\ +\frac{\epsilon \mathrm{Du}}{{\mathrm{pr}}_{\mathrm{nf}}{R}_{\mathrm{e,nf}}}(\frac{{\partial }^{2}C}{\partial r^2}+\frac{{\partial }^{2}C}{\partial z^2}+\frac{1}{r}\frac{\partial C}{\partial r})& \mathrm{(8)}\end{array}$

$\begin{array}{c}\frac{\partial \left(UC\right)}{\partial r}+\frac{\partial \left(WC\right)}{\partial z}+\frac{UC}{r}=\frac{\epsilon }{{\mathrm{Sc}}_{\mathrm{nf}}{R}_{\mathrm{e,nf}}}(\frac{{\partial }^{2}C}{\partial r^2}+\frac{{\partial }^{2}C}{\partial z^2}+\frac{1}{r}\frac{\partial C}{\partial r})\\ +\frac{\epsilon \mathrm{Sr}}{{\mathrm{Sc}}_{\mathrm{nf}}{R}_{\mathrm{e,nf}}}(\frac{{\partial }^{2}T}{\partial r^2}+\frac{{\partial }^{2}T}{\partial z^2}+\frac{1}{r}\frac{\partial T}{\partial r})& \mathrm{(9)}\end{array}$

$\begin{array}{cc}\Omega =(\frac{\partial U}{\partial z}-\frac{\partial W}{\partial r})& \mathrm{(10)}\end{array}$

The following dimensionless numbers are introduced:

$R_{\mathrm{e,nf}}=\frac{u_{0}R}{u_{\mathrm{nf}}}$ (Reynolds number),

$\mathrm{Da}=\frac{K}{R^2}$ (Darcy number),

$R_{\mathrm{i,nf}}=\frac{G_{\mathrm{T,nf}}}{R_{\mathrm{e,nf}}^2}$ (Richardson number),

$N=\frac{{\beta }_s\mathrm{∆C}}{{\beta }_T\mathrm{∆T}}=\frac{G_{\mathrm{C,nf}}}{G_{\mathrm{T,nf}}}$ (Buoyancy ratio),

${\mathrm{Sc}}_{\mathrm{nf}}=\frac{{\upsilon }_{\mathrm{nf}}}{D}$ (Schmidt number),

$\mathrm{Du}=\frac{N_{\mathrm{TC}}\mathrm{∆C}}{\mathrm{\alpha ∆T}}$ (Dufour number),

$\mathrm{Sr}=\frac{N_{\mathrm{CT}}\mathrm{∆T}}{\mathrm{D∆C}}$ (Soret number),

${\mathrm{Pr}}_{\mathrm{nf}}=\frac{{ʋ}_{\mathrm{nf}}}{{\alpha }_{\mathrm{nf}}}$ (Prandtl number).

Here are the dimensionless boundary conditions of the selected problem

$\begin{array}{cc}r=0,U=W=0,& \frac{\mathrm{\partial T}}{\mathrm{\partial r}}{|}_{\mathrm{r=0}}=0,\\ \frac{\mathrm{\partial C}}{\mathrm{\partial r}}{|}_{\mathrm{r=0}}=0,& \Omega {|}_{\mathrm{r=0}}=0,r=1,U=W=0,\\ \frac{\mathrm{\partial T}}{\mathrm{\partial r}}{|}_{\mathrm{r=1}}=0,& \frac{\mathrm{\partial C}}{\mathrm{\partial r}}{|}_{\mathrm{r=1}}=0,\Omega {|}_{\mathrm{r=1}}=0,\\ z=0,U=W=0,& T=1,C=1,\Omega {|}_{\mathrm{z=0}}=0,\\ z=A,U=1,W=0,& T=0,C=0,\Omega {|}_{\mathrm{z=A}}=0\end{array}$

The average Nusselt (Nu) and Sherwood (Sh) numbers represent the heat and mass transfer on the active wall horizontally with the defined relationships.

$\begin{array}{c}\mathrm{Nu}=-\mathrm{2\; A}{\int }_0^1\frac{\mathrm{\partial T}}{\mathrm{\partial z}}{|}_{\mathrm{z=0}}=0,\\ \mathrm{Sh}=-\mathrm{2\; A}{\int }_0^1\frac{\mathrm{\partial C}}{\mathrm{\partial z}}{|}_{\mathrm{z=0}}=\mathrm{dr}\end{array}$

3. Resolution Process

The discretization of the governing equations was carried out by FVM utilizing uniform network sizes [ 22 , 23 ]. The algebraic form of the equations, to which we include boundary conditions, are fathomed utilizing the double sweep method. The equations obtained after discretization were illuminated utilizing the FORTRAN code. The convergence basis is given by:

$\frac{{\sum }_i{\sum }_j|f_{\mathrm{i,j}}^{\mathrm{n+1}}-f_{\mathrm{i,j}}^n|}{{\sum }_i{\sum }_j\left|f_{\mathrm{i,j}}^{\mathrm{n+1}}\right|}\le {\mathrm{10}}^{\mathrm{-5}}$

To confirm the good behavior of the numerical model, the grid convergence test is carried out at different grid levels and by restricting A=2 , ε=0.4 , N=1, Pr_nf=6.8, β =1, R_e,nf =1 , φ=0.05, Sc_nf=1, Da=0.001, Sr=1, Du=1 and are presented in Table 1. For this purpose, the average heat fluxes are calculated. We see that the Sizes of the grid converge in the 141 × 141 grid, for this we Avon works in this study on the 141 × 141 grid.

To validate this work we take values in this porange in order to adapt it to the work [ 14 ], $\frac{1}{R_{\mathrm{e,nf}}}=\stackrel{-}{\Lambda }\frac{{\mathrm{Pr}}_{\mathrm{nf}}}{\mathrm{Da}}=\frac{\mathrm{6.8}}{\mathrm{0.001}},R_{\mathrm{i,nf}}=\stackrel{-}{\Lambda }{R}_{\mathrm{T,nf}}{\mathrm{Pr}}_{\mathrm{nf}}={\mathrm{10}}^4\times \mathrm{6.8},\frac{1}{R_{\mathrm{Te,nf}}}=\stackrel{-}{\Lambda }{\mathrm{Pr}}_{\mathrm{nf}}=\mathrm{6.8},\frac{1}{{\mathrm{Pr}}_{\mathrm{nf}}{R}_{\mathrm{Te,nf}}}=1,\frac{1}{{\mathrm{Sc}}_{\mathrm{nf}}{R}_{\mathrm{Te,nf}}}=\frac{1}{\mathrm{Le}}=\frac{1}{\mathrm{2.5}}$ . When we solve these equations based on the values of item [ 14 ], we get the following values R_e,nf=0.147, R_i,nf=68 ×10⁴, Sc_nf=0.059, Da=0.001.

Table2 gives a comparison between this research and the article [ 14 ]:

4. Results and discussion

4.1. Effect of Variation of Richardson, Reynolds Numbers and Casson Fluid Parameter

Figures 2 and 3 show the influence of the Richardson number, Reynolds number and the Casson fluid parameter on the heat and mass transfer for A=2 ,ε=0.4 , N=1, Pr_nf=6.8, φ=0.05, Sc_nf=1, Da=0.001, Sr=1, Du=1

The average Nusselt number, which remains unchanged for:

■ The first two curves β =0.1, R_e,nf =1 and et β =0.5, R_e,nf = 1 in the interval R_i,nf<6.

■ The second two curves β=0.1, R_e,nf = 5 β =0.5, R_e,nf = 5 in the interval R_i,nf <8

■ The third two curves β =0.1, R_e,nf = 9 and β =0.5, R_e,nf = 9 in the interval R_i,nf <10

Fig. 2. The impact of R_i,nf, β and R_e,nf on Nu

Fig. 3. The impact of R_i,nf, β and R_e,nf on Sh

When the Richardson number exceeds the threshold, we see a decrease in the average Nusselt number. The results show that with an increase in the Richardson number from 0.1 to 100, the heat transfer decreases; this decrease is different for different Reynolds value sand Casson fluid parameters that the Reynolds value and β increases the rate of decreased heat transfer increases see table 3. We also see in the same figures that the heat and mass transfer increase with the increase in β. This increase results from the viscosity of the fluid, which decreases as the amount of the Casson parameter increases. This reduction in viscosity leads to better thermosolutal transfer. Do not forget that heat and mass transfer increase with increasing Reynold's number, causing an increase in the kinetic energy of the particles, leading to a better thermosolutal transfer.

On the other hand, the average Sherwood number remains unchanged for R_i,nf <10 for the cases R_e,nf = 1; for the other cases of Reynolds number, the variation threshold R_i,nf =8. When the Richardson number exceeds its threshold, the average Sherwood number decreases. The results show that when the Richardson number increases from 0.1 to 100, the mass transfer decreases. This decrease is different from the Reynolds value and the Casson flow parameter, and β this latter parameter increases the transfer rate of mass decreases, see Table 4.

4.2. Effect of Variation in Porosity Numbers and Prandtl Number

Figures 4 and 5, respectively, show the effect of Porosity and Prandtl numbers on heat and mass transfer. We notice in Figure 4 that the average Nusselt evolves monotonically with increasing Porosity. This increase results from the argument that thermal conductivity has with the increase in Porosity. We see in the same figures that the average Nusselt values decrease when the Prandtl number increases; this decrease results in an increase in the Prandtl number improving, the resistive force more than the thermal and solutal volume forces. In Figure 5, we represent the Porosity effect on mass transfer. We see in this figure that the average Sherwood value decreases with the increase in Porosity, this increase writing down the Permeability and decrease in Mass Diffusivity.

Fig. 4. The impact of ε and Pr_nf on Nu

Fig. 5. The impact of ε and Pr_nf on Sh

Table 5 shows the heat and mass transfer rate for ε (between 0.1 and 1).

4.3. Effect of Variation in Darcy, Soret and Dufour Numbers

From Figure 6, it can be seen that the average Sherwood number decreases with the increase in the Soret number, and the Soret number is estimated to be between 0.1 and 1. In contrast, the increase in the Soret coefficient, which represents thermal diffusion phenomena, leads to a reduction in mass transfer. On the other hand, the average Sherwood number remains almost unchanged for <10^-8 . When the Darcy number exceeds this threshold Da= 10^-8 we see a decrease in the average Sherwood number with the increase in Da, which results from an increase in permeability and improves the thermosolutal resistive force . Table 6 shows the mass transfer rate for Da (between Da= 10^-8 and 1).

Fig. 6. The impact of Da and Sr on Sh

Figure 7 shows the Dufour effect on heat transfer. From this figure, we can see that the average value of Nusselt decreases as the Dufour number increases. This represents the phenomenon of thermal diffusion. As for the number of Darcy increases, the number of Nusselt decreases. Table 7 shows the heat transfer rate for Da (between 10^-10 and 1).

Fig. 7.The impact of Da and Sr on Nu

4.4. Effect of Varying Reynolds Numbers, Buoyancy Ratio and Geometric Aspect Ratio

Figures 8 and 9 show the influence of Reynolds numbers, Buoyancy ratio and Geometric aspect ratio on heat and mass transfer pour ε=0.4 ,R_i,nf=1, Pr_nf=6.8, φ=0.05, Sc_nf=1, Da=0.001, Sr=1, Du=1.

Fig. 8. The impact of R_e,nf, N and A on Nu

Fig. 9. The impact of R_e,nf, N, and A on Sh

In this study, we focused our attention on the flow in the Darcy regime for R_e,nf<10.

We see in these figures that the heat and mass transfer increase with the increase in Re,nf; this fact is proven by the fact that the kinetic energy of the particles improves with the increase in Reynolds number. On the contrary behavior, we see in the same figures that the transformation of heat and mass decreases with the increase of Buoyancy ratio and Geometric aspect ratio. This decrease results in the increase in the numbers Buoyancy ratio and Aspect ratio geometric resistive force improvement more than the thermal and solutal volume forces. Table 8 shows the heat and mass transfer rate for R_e,nf (between 1 and 9).

5. Conclusions

This study focused on mixed thermosolutal convection in a cylindrical cavity driven by a lid filled with a casson nanofluid, leading to the following conclusions:

Heat and mass transfers remain unchanged. When the Richardson number exceeds the threshold, we see a decrease in the average Nusselt and average Sherwood numbers. We also see that the increase in the Casson parameter causes an increase in heat and mass transfer. This is explained by the importance of the flow intensity with increasing Casson fluid parameter. We notice that the heat transfer evolves monotonically with increasing Porosity. This increase results from the argumentation of thermal conductivity with the increase in Porosity. On the contrary the decrease in heat and mass transfer with the increase in Prandtl number. This decrease increases the resistive force more than the thermal and solute volume forces. We see that the average Shrewood number decreases with increasing Porosity. This decrease results from the increase in

Permeability and decrease in Mass Diffusivity with the increase in Porosity. It can be seen that the average Sherwood number remains almost unchanged for Da<10^-8. When the Darcy number exceeds this threshold, Da=10^-8 we see a decrease in the average Sherwood number with the increase in Da, which results from an increase in Permeability, which improves the thermosolutal resistive force. On the other hand, the average Sherwood number decreases with the increase in the Soret number. On the other hand, the increase in the Soret coefficient, which represents thermal diffusion phenomena, leads to a decrease in mass transfer. We also notice that Nusselt's average decreases with the increase in the Dufour and Darcy numbers. We noticed that the heat transfer number and mass increased with the increase in Reynolds number. This increase results from the increase in the kinetic energy of the particles, which improved with the increase in Reynolds number. On the contrary behavior, it is observed that heat and mass transformation decreases with the increase of Buoyancy ratio and Geometric aspect ratio. This decrease results in the increase in the numbers Buoyancy ratio and Geometric aspect ratio improvement of resistive force more than the thermal and solutal volume forces.

Funding Statement

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Conflicts of Interest

The author declares that there is no conflict of interest regarding the publication of this article.

References

1. Moolya, S., Satheesh, A., Rajan, D. and Moolya, R., 2022. Magnetohydrodynamics and Aspect Ratio Effects on Double Diffusive Mixed Convection and Their Prediction: Linear Regression Model. Journal of Heat and Mass Transfer Research, 9(2), pp. 169-188. https://doi.org./10.22075/JHMTR.2023.28029.1387.DOI
2. El Hamma, M., Taibi, M., Rtibi, A., Gueraoui, K. and Bernatchou, M., 2022. Effect of magnetic field on thermosolutal convection in a cylindrical cavity filled with nanofluid, taking into account Soret and Dufour effects. JP Journal of Heat and Mass Transfer, 26, pp.  1-26. http://dx.doi.org/10.17654/0973576322009.DOI
3. Sivasankaran, S., Ananthan, S.S., Bhuvaneswari, M. and Hakeem, A.A., 2017. Double-diffusive mixed convection in a lid-driven cavity with non-uniform heating on sidewalls. Sādhanā, 42, pp. 1929-1941. 10.1007/s12046-017-0735-4.DOI
4. Bhuvaneswari, M., Sivasankaran, S. and Kim, Y.J., 2011. Numerical study on double diffusive mixed convection with a Soret effect in a two-sided lid-driven cavity. Numerical Heat Transfer, Part A: Applications, 59(7), pp. 543-560. 10.1080/10407782.2011.561077.DOI
5. Guimarães, P.M., Ramos, M.D. and Menon, G.J., 2019. Mixed convection study in a ventilated square cavity using nanofluids. Journal of heat and mass transfer research, 6(2), pp. 143-153.10.22075/jhmtr.2018.14640.1208.DOI
6. Hayat, T., Shehzad, S.A., Alsaedi, A. and Alhothuali, M.S., 2012. Mixed convection stagnation point flow of Casson fluid with convective boundary conditions. Chinese Physics Letters, 29(11), p.114704. 10.1088/0256-307X/29/11/114704.DOI
7. Teamah, M.A. and El-Maghlany, W.M., 2010. Numerical simulation of double-diffusive mixed convective flow in rectangular enclosure with insulated moving lid. International Journal of Thermal Sciences, 49(9), pp.1625-1638.https://doi.org/10.1016/j.ijthermalsci.2010.04.023.DOI
8. Khanafer, K. and Vafai, K., 2002. Double-Diffusive Mixed Convection in a Lid-Driven Enclosure Filled With A Fluid Saturated Porous Medium, Numerical Heat Transfer, Part A, 42, pp. 465-486. https://doi.org/10.1080/10407780290059657.DOI
9. Jamir, T. and Konwar, H., 2022. Effects of Radiation Absorption, Soret and Dufour on Unsteady MHD Mixed Convective Flow past a Vertical Permeable Plate with Slip Condition and Viscous Dissipation. Journal of Heat and Mass Transfer Research ,9(2), pp. 155-168. 10.22075/JHMTR.2023.28693.1399.DOI
10. El Hamma, M., Rtibi, A., Taibi, M., Gueraoui, K. and Bernatchou, M., 2022. Theoretical and Numerical Study of Thermosolutal Convection in a Cylindrical Porous Cavity Filled with a Nanofluid and Taking into Account Soret and Dufour Effects. International Journal on Engineering Applications (I.R.E.A.), 10(1), pp. 56-65. DOI: https://doi.org/10.15866/irea.v10i1.20809.DOI
11. Mahdy, A., 2010. Soret and Dufour effect on double diffusion mixed convection from a vertical surface in a porous medium saturated with a non-Newtonian fluid. Journal of Non-Newtonian Fluid Mechanics, 165 (11-12), pp. 568-575. https://doi.org/10.1016/j.jnnfm.2010.02.013.DOI
12. Mondal, P. and Mahapatra, T.R., 2021. MHD double-diffusive mixed convection and entropy generation of nanofluid in a trapezoidal cavity. International Journal of Mechanical Sciences, 208, p. 106665.https://doi.org/10.1016/j.ijmecsci.2021.106665.DOI
13. Moolya, S. and Satheesh, A., 2020. Role of magnetic field and cavity inclination on double diffusive mixed convection in rectangular enclosed domain. International Communications in Heat and Mass Transfer, 118, p.  104814. https://doi.org/10.1016/j.icheatmasstransfer.2020.104814.DOI
14. El Hamma, M., Aberdane, I., Taibi, M., Rtibi, A. and Gueraoui, K., 2023. Analysis of MHD Thermosolutal Convection in a Porous Cylindrical Cavity Filled with a Casson Nanofluid, Considering Soret and Dufour Effects. Journal of Heat and Mass Transfer Research, 10(2), pp. 197-206. doi.org/10.22075/JHMTR.2023.30532.1439.DOI
15. Guerroudj, N., Fersadou, B., Mouaici, K. and Kahalerras, H., 2023. Ferrohydrodynamics mixed convection of a ferrofluid in a vertical channel with porous blocks of various shapes. Journal of Applied Fluid Mechanics, 16(1), pp. 131-145. 10.47176/JAFM.16.01.1314.DOI
16. Wahid, N.S., Md Arifin, N., Khashi’ie, N.S., Pop, I., Bachok, N. and Hafidz Hafidzuddin, M.E., 2022. Mixed convection magnetic nanofluid flow past a rotating vertical porous cone. Journal of Applied Fluid Mechanics, 15(4), pp. 1207-1220. 10.47176/JAFM.15.04.1063.DOI
17. Bouras, A., Bouabdallah, S., Ghernaout, B., Arıcı, M., Cherif, Y. and Sassine, E., 2021. 3D numerical simulation of turbulent mixed convection in a cubical cavity containing a hot block. Journal of Applied Fluid Mechanics, 14(6), pp. 1869-1880. 10.47176/JAFM.14.06.32604.DOI
18. Khan, Z.H., Usman, M., Khan, W.A., Hamid, M. and Haq, R.U., 2022. Thermal treatment inside a partially heated triangular cavity filled with casson fluid with an inner cylindrical obstacle via FEM approach. The European Physical Journal Special Topics, 231(13), pp. 2683-2694. https://doi.org/10.1140/epjs/s11734-022-00587-6.DOI
19. Hamid, M., Usman, M., Khan, W.A., Haq, R.U. and Tian, Z., 2024. Characterizing natural convection and thermal behavior in a square cavity with curvilinear corners and central circular obstacles. Applied Thermal Engineering, 248, p. 123133. doi.org/10.1016/j.applthermaleng.2024.123133.DOI
20. Bendrer, B.A.I., Abderrahmane, A., Ahmed, S.E. and Raizah, Z.A., 2021. 3D magnetic buoyancy-driven flow of hybrid nanofluids confined wavy cubic enclosures including multi-layers and heated obstacle. International Communications in Heat and Mass Transfer, 126, p. 105431. doi.org/10.1016/j.icheatmasstransfer.2021.105431.DOI
21. Ahmed, S.E., Arafa, A.A. and Hussein, S.A., 2023. Dissipated-radiative compressible flow of nanofluids over unsmoothed inclined surfaces with variable properties. Numerical Heat Transfer, Part A: Applications, 84(5), pp.507-528. https://doi.org/10.1080/10407782.2022.2141389.DOI
22. Patankar, S.V., 1980. Numerical heat transfer and fluid flow, Hemisphere, New York, 1980. Google Scholar, pp.41-135.
23. Bounouar, A., Gueraoui, K., Taibi, M., Lahlou, A., Driouich, M., Sammouda, M., Men-La-Yakhaf, S. and Belcadi, M., 2016. Numerical and mathematical modeling of unsteady heat transfer within a spherical cavity: Applications laser in medicine. Contemporary Engineering Sciences, 9, pp. 1183-1199..