Double diffusion in a combined cavity occupied by a nanofluid and heterogeneous porous media

Abdelraheem M. Aly; Zehba Raizah; Hijaz Ahmed; Amal M. Al-Hanaya; Noura Alsedias

doi:10.1515/phys-2022-0189

Open Access Published by De Gruyter Open Access September 16, 2022

Double diffusion in a combined cavity occupied by a nanofluid and heterogeneous porous media

Abdelraheem M. Aly , Zehba Raizah , Hijaz Ahmed , Amal M. Al-Hanaya and Noura Alsedias

From the journal Open Physics

https://doi.org/10.1515/phys-2022-0189

Abstract

The aim of the present study is to simulate double diffusion in a circular cylinder over a rectangular cavity by utilizing incompressible smoothed particle hydrodynamics (ISPH) method. An originality of this study is adopting the ISPH method in simulating double diffusion in a novel domain of a circular cylinder over a rectangular shape occupied by Al 2 O 3 – H 2 O and heterogeneous porous media. The variations of Darcy parameter (Da) between 1 0 − 3 and 1 0 − 5 with two levels of porous media, ( 0 ≤ η 1 = η 2 ≤ 1.5 ) , Rayleigh number ( 1 0 3 ≤ Ra ≤ 1 0 5 ) with variable buoyancy ratio parameter ( 0 ≤ N ≤ 2 ) , solid volume fraction ϕ between 0 and 0.05, and Lewis number ( 10 ≤ Le ≤ 40 ) on the features of heat/mass transport as well as velocity field are discussed. It is found that the homogeneous porous medium reduces the temperature and concentration within a combined cavity. A decrease in Darcy parameter from 1 0 − 2 and 1 0 − 5 suppresses the maximum of a nanofluid velocity by 75% regardless the levels of porous media. An increase in parameters Ra and N enhances the heat and mass transmission, as well as the nanofluid velocity. Adding more concentration of nanoparticles until 5 % reduces the nanofluid velocity. The variations of boundary conditions are acting effectively in changing the temperature and concentration circulations within a combined cavity. Besides, the variations of boundary conditions change the maximum of the velocity field by 86.9%.

Keywords: circular cylinder; double-diffusion; ISPH method; homogeneous/heterogeneous porous media; nanofluid

1 Introduction

Numerical simulations for heat/mass transport in several cavities are studied by refs. [1,2,3]. There are several applications of double diffusion in the porous cavities such as the design of solar cells, petroleum reservoirs, and nuclear wastes [4,5,6, 7,8].

In recent years, there are many studies in thermal systems filled with porous media and nanofluid due to their applications in heat exchangers, copper foams in channels, and electronic cooling systems. Nguyen et al. [9] used the incompressible smoothed particle hydrodynamics (ISPH) method for examining the impacts of wavy interface for a porous-nanofluid layers on natural convection in an enclosure. Ataei-Dadavi et al. [10] reported an experimental examination on mixed convection in a vented cubical cavity. Nguyen et al. [11] studied ferro-nanofluid in a curved porous enclosure. Selimefendigil and Oztop [12] examined forced convection of a nanofluid in U-shaped cavity. The heat transfer for a ferrofluid with Fe 3 O 4 nanoparticles in a trapezoidal cavity was numerically examined by Khan et al. [13]. Raja et al. [14] used a stochastic numerical scheme for nanofluidics of nonlinear Jeffery–Hamel flow. Raja et al. [15] utilized neural networks with Levenberg Marquardt technique to endorse the biconvection nanofluidic flow and Cattaneo-christov heat & mass flux on a permeable medium with considering the impacts of Darcy-Forchheimer law. More studies are introduced in ref. [16,17,18, 19,20,21, 22,23]. Ismael [24] studied the heat-mass transmission during mixed convection in a partially stratified porous enclosure. Numerical studies of natural convection in a partially stratified porous enclosure including an embedded solid conductive shape are introduced by Ismael and Ghalib [25].

Computational fluid dynamics method is significant in handling the thermal manner of the nanofluid flow through numerous simulations. Different numerical methods were used to solve several fluid dynamic problems. However, some new numerical technique like the ISPH have likewise been utilized to accomplish accurate engineering predictions. Utilizing the Galerkin finite element method (FEM), Kadhim et al. [26] investigated heat transfer of a hybrid nanofluid in cavity having opposing sinusoidal walls. The Lattice Boltzmann method was utilized by several researchers including Sheikholeslami et al. [27] to examine magnetohydrodynamic-free convection in a cubic cavity including a hot sphere. Seyyedi et al. [28] adopted the control volume FEM (CVFEM) to study the entropy generation and free convection flow of nanoencapsulated phase change materials (NEPCMs) within a porous annulus. Also, the NEPCMs inside n elliptical-shaped enclosure under magnetic field effects are studied numerically using CVFEM by Seyyedi et al. [29]. Hashemi-Tilehnoee et al. [30] adopted finite volume method using ANSYS Fluent to examine the diluted NEPCM inside a water-filled a hot enclosure. Seyyedi [31] utilized CVFEM for entropy generation analysis and free convection within a cardioid shaped porous cavity. The ISPH method was introduced by Aly Abdelraheem [32] to examine natural convection of a heated obstacle within a circular domain. Further studies of the ISPH method in several problems of the CFD can be found in refs. [33,34,35, 36,37]. Raizah et al. [38] utilized ISPH method for studying impacts of thermal diffusion and diffusion-thermo on the transport of heat and mass of a nanofluid inside an annulus including three circular cylinders.

The novelty of this study is checking thermosolutal convection in a novel shape of a circular cylinder over rectangular domain by using the ISPH method. The problem of heat transfer and fluid flow within a porous combined domain of a circular cylinder over a rectangular shape can be applied in air crafting, electronic devices, cooling of a nuclear and power plant, and heat exchangers tubes. The combined domain is mobilized by a nanofluid ( Al 2 O 3 – H 2 O ) and homogeneous/heterogeneous porous media. By comparison between the heterogeneous and homogeneous porous media, it is found that a homogeneous porous medium reduces heat and mass transport and nanofluid velocity inside a combined cavity. Further, the homogeneous porous level provides more stable movements of the nanofluid flow within a combined domain of a circular cylinder over a rectangular shape compared to a heterogeneous porous level. Regardless of the porous level, a reduction in Da reduces the nanofluid movements due to the low permeability. Increasing Ra and N strengthens the buoyancy forces that support the mass/heat transfer and augments the nanofluid movements within a combined cavity. Average Nusselt number Nu ¯ is enhanced at a heterogeneous porous medium, while average Sherwood number Sh ¯ is enhanced at a homogeneous porous medium. We hope that this work serves in the forthcoming investigation of adjusting thermal transport within closed complex domains.

2 Mathematical formulations

Figure 1 indicates the primary physical and mesh generation diagrams. The model contains a circular cylinder over a rectangular-shaped cavity. The length of a rectangular shape is L = 1.0 and width 0.4, and a radius of upper circular cylinder is R c = 0.5 . The surface of the circular cylinder has T h and C h . The normal walls of a rectangular shape have adiabatic boundary conditions, and the plane wall has T c and C c . The combined cavity of a circular cylinder over a rectangular shape is occupied with Al 2 O 3 – H 2 O and heterogeneous/homogeneous porous media. The time-dependent flow is laminar and incompressible. Table 1 introduces the characteristics of Al 2 O 3 –water.

Figure 1

Primary physical diagram and mesh generation models.

Table 1

Al 2 O 3 – H 2 O characteristics

	ρ(Kg ∕ m³)	K(W ∕ m K)	Cp(J ∕ kg K)	β × 10^{−5 (1 ∕ K)}	d (nm)
Al 2 O 3	3,970	40	765	0.85	33
Water	993	0.628	4,178	36.2	0.385

Time-dependent thermosolutal convection of nanofluid in a combined cavity suspended by a heterogeneous porous medium according to ref. [39] is expressed as follows:

(2.1) ∂ V ∂ Y = − ∂ U ∂ X ,

(2.2) 1 ε d U d τ = − ∂ P ∂ X + ρ f ρ n f 1 ε ∂ ∂ X Pr μ n f μ f ∂ U ∂ X + 1 ε ∂ ∂ Y Pr μ n f μ f ∂ U ∂ Y − Pr μ eff μ n f K ∗ ( X , Y ) Da U ,

(2.3) 1 ε d V d τ = ρ f ρ n f 1 ε ∂ ∂ X Pr μ n f μ f ∂ V ∂ X + 1 ε ∂ ∂ Y Pr μ n f μ f ∂ V ∂ Y − Pr μ eff μ n f K ∗ ( X , Y ) Da V + ( ρ β ) n f ( ρ β ) f Pr Ra θ − ∂ P ∂ Y ,

(2.4) γ d θ d τ = ( ρ c p ) f ( ρ c p ) n f ∂ ∂ X ζ ( X , Y ) ∂ θ ∂ X + ∂ ∂ Y ζ ( X , Y ) ∂ θ ∂ Y ,

(2.5) d Φ d τ = 1 Le ∂ ∂ X Π ( X , Y ) ∂ Φ ∂ X + ∂ ∂ Y Π ( X , Y ) ∂ Φ ∂ Y .

Here, the definitions of a porous matrix are expressed as follows:

(2.6) K ∗ = e η 1 H X + η 2 H Y ,

(2.7) ζ = ε k n f k f + ( 1 − ε ) e η 1 H X + η 2 H Y ,

(2.8) Π = D eff D 0 = D o e η 2 H Y + η 1 H X D 0 ,

(2.9) γ = ( ε ( ρ c p ) n f + ( 1 − ε ) ( ρ c p ) p ) ( ( ρ c p ) n f ) − 1 .

The physical definitions of a nanofluid according to refs. [40,41,42] are given as follows:

(2.10) ρ n f = ( 1 − ϕ ) ρ f + ϕ ρ p ,

(2.11) α n f = k n f ( ( ρ c p ) n f ) − 1 ,

(2.12) ( ρ β ) n f = ( 1 − ϕ ) ( ρ β ) f + ϕ ( ρ β ) p ,

(2.13) ( ρ c p ) n f = ( 1 − ϕ ) ( ρ c p ) f + ϕ ( ρ c p ) p ,

(2.14) μ n f = μ f − 34.87 d p d f − 0.3 φ 1.03 + 1 ,

(2.15) k n f = 4.4 ( Re B ) 0.4 Pr 0.66 T T r f 10 k p k f 0.03 φ 0.66 + 1 k f ,

(2.16) Re B = ρ f u B d p μ f ,

(2.17) u B = 2 K B T ( π μ f d p 2 ) ,

where T r f is a freezing point. Boltzmann’s factor K B = 1.380648 × 1 0 − 23 J/K .

Average Sherwood Sh ¯ and Nusselt Nu ¯ numbers on a circular cylinder wall are defined as follows:

(2.18) Sh ¯ = − 1 s ∫ 0 s ∂ Φ ∂ n d Y ,

(2.19) Nu ¯ = − 1 s ∫ 0 s ∂ ( ζ ( X , Y ) θ ) ∂ n d Y .

3 Proposed methodology

The solving steps:

Predictor step:

(3.1) U ∗ = ε ρ f ρ n f Δ τ 1 ε ∂ ∂ X Pr μ n f μ f ∂ U ∂ X + 1 ε ∂ ∂ Y Pr μ n f μ f ∂ U ∂ Y − Pr μ eff μ n f K ∗ ( X , Y ) Da U n + U n ,

(3.2) V ∗ = ε ρ f ρ n f Δ τ 1 ε ∂ ∂ X Pr μ n f μ f ∂ V ∂ X + 1 ε ∂ ∂ Y Pr μ n f μ f ∂ V ∂ Y − Pr μ eff μ n f K ∗ ( X , Y ) Da V + ( ρ β ) n f ( ρ β ) f Pr Ra θ n + V n .

Pressure Poisson equation:

(3.3) ∇ 2 P n + 1 = α ρ − ρ n u m Δ τ 2 + ρ Δ τ ∂ U ∗ ∂ X + ∂ V ∗ ∂ Y .

Relaxation factor α is between 0 and 1.

Corrected velocities:

(3.4) U n + 1 = − Δ τ ρ ( ∇ P n + 1 ) + U ∗ ,

(3.5) V n + 1 = − Δ τ ρ ( ∇ P n + 1 ) + V ∗ .

Thermal energy equation:

(3.6) θ n + 1 = Δ τ ( ρ c p ) f ( ρ c p ) n f × ∂ ∂ X ζ ( X , Y ) ∂ θ ∂ X + ∂ ∂ Y ζ ( X , Y ) ∂ θ ∂ Y n + θ n

Concentration equation:

(3.7) Φ n + 1 = Δ τ 1 Le × ∂ ∂ X Π ( X , Y ) ∂ Φ ∂ X + ∂ ∂ Y Π ( X , Y ) ∂ Φ ∂ Y n + Φ n .

The positions:

(3.8) X n + 1 = Δ τ U n + 1 + X n

(3.9) Y n + 1 = Δ τ V n + 1 + Y n .

The shifting technique:

(3.10) ϒ i ′ = ( ∇ ϒ ) i δ R i i ′ + ϒ i + o ( δ R i i ′ 2 ) .

3.1 Boundary treatment

The dummy particles are utilized for treating the rigid walls. Dummy particles are settled uniformly around the cavity walls. Figure 2 displays the initial distributions of dummy particles. Dummy particles are utilized in preventing the error of a cut kernel.

Figure 2

Dummy boundary particles of the ISPH method.

3.2 Validation tests

To examine the grid independence of the ISPH method, Table 2 presents Nu ¯ for various particle sizes d 0 at Ra = 1 0 4 , Le = 20 , ε = 0.6 , N = 2 , and ϕ = 0.01 . In Table 2, there is a slight difference on Nu ¯ when changing d 0 . Therefore, the medium size d 0 = 0.01 is selected.

Table 2

Nu ¯ for various particle sizes d 0 at Le = 20 , N = 2 , ε = 0.6 , Ra = 1 0 4 , and ϕ = 0.01

Grid	d 0 = 0.02	d 0 = 0.01	d 0 = 0.005
Nu ¯	0.1124	0.1138	0.1143

To present the efficiency of the ISPH method, the numerical test of natural convection of an inserted hot source is introduced for the streamlines and isotherms by Kim et al. [43]. Figure 3 presents the streamlines and isotherms of an inserted hot source in a cavity. The ISPH method agrees well with the reference data of ref. [43].

Figure 3

Streamlines and isotherms of Kim et al. [43] (left) and the ISPH data (right).

4 Results and discussions

The numerical simulations represented in temperature, concentration, and velocity field below the influences of pertinent parameters are discussed in this section. The porosity of a porous medium is taken as ε = 0.6 . Figures 4, 5, and 6 present the effects of ( D a ) with two levels of porous media in the temperature, concentration, and velocity field. In Figure 4, it is seen that a decrease in Da augments the temperature allotments inside a combined cavity of a circular cylinder over a rectangular shape. A heterogeneous porous medium, η 1 = η 2 = 1.5 , gives little higher temperature allotments compared to a homogeneous porous medium η 1 = η 2 = 0 . In Figure 5, the concentration allotments are higher at a heterogeneous porous medium compared to a homogeneous porous medium. Moreover, a decrease in Da reduces the concentration allotments inside a combined cylinder-rectangular shaped cavity. In Figure 6, the maximum of velocity field decreases by 75 % as Da reduces from 1 0 − 2 to 1 0 − 5 regardless the level of porous media. Besides, there are variations on the nanofluid movements between the homogeneous and heterogeneous porous media. The reason returns to the difference on the porosity of the porous media related to their level as a homogeneous or heterogeneous. It is found that the homogeneous porous media supports much stable nanofluid movements inside a combined domain compared to the heterogeneous porous media.

$Figure 4 Varied Da with changing porous levels on temperature at N = 1 N=1 , R c = 0.1 {R}_{c}=0.1 , Ra = 1 0 4 {\rm{Ra}}=1{0}^{4} , L h = 0.25 {L}_{h}=0.25 , ϕ = 0.01 \phi =0.01 , and Le = 20 {\rm{Le}}=20 . (a) Homogeneous porous media ( η 1 = η 2 = 0 {\eta }_{1}={\eta }_{2}=0 ). (b) Heterogeneous porous media ( η 1 = η 2 = 1.5 {\eta }_{1}={\eta }_{2}=1.5 ).$

Figure 4

Varied Da with changing porous levels on temperature at N = 1 , R c = 0.1 , Ra = 1 0 4 , L h = 0.25 , ϕ = 0.01 , and Le = 20 . (a) Homogeneous porous media ( η 1 = η 2 = 0 ). (b) Heterogeneous porous media ( η 1 = η 2 = 1.5 ).

$Figure 5 Varied Da with changing porous levels on concentration at N = 1 N=1 , R c = 0.1 {R}_{c}=0.1 , Ra = 1 0 4 {\rm{Ra}}=1{0}^{4} , L h = 0.25 {L}_{h}=0.25 , ϕ = 0.01 \phi =0.01 , and Le = 20 {\rm{Le}}=20 . (a) Homogeneous porous media ( η 1 = η 2 = 0 {\eta }_{1}={\eta }_{2}=0 ). (b) Heterogeneous porous media ( η 1 = η 2 = 1.5 {\eta }_{1}={\eta }_{2}=1.5 ).$

Figure 5

Varied Da with changing porous levels on concentration at N = 1 , R c = 0.1 , Ra = 1 0 4 , L h = 0.25 , ϕ = 0.01 , and Le = 20 . (a) Homogeneous porous media ( η 1 = η 2 = 0 ). (b) Heterogeneous porous media ( η 1 = η 2 = 1.5 ).

$Figure 6 Varied Da with changing porous levels on velocity fields at N = 1 N=1 , Ra = 1 0 4 {\rm{Ra}}=1{0}^{4} , R c = 0.1 {R}_{c}=0.1 , L h = 0.25 {L}_{h}=0.25 , ϕ = 0.01 \phi =0.01 , and Le = 20 {\rm{Le}}=20 . (a) Homogeneous porous media ( η 1 = η 2 = 0 {\eta }_{1}={\eta }_{2}=0 ). (b) Heterogeneous porous media ( η 1 = η 2 = 1.5 {\eta }_{1}={\eta }_{2}=1.5 ).$

Figure 6

Varied Da with changing porous levels on velocity fields at N = 1 , Ra = 1 0 4 , R c = 0.1 , L h = 0.25 , ϕ = 0.01 , and Le = 20 . (a) Homogeneous porous media ( η 1 = η 2 = 0 ). (b) Heterogeneous porous media ( η 1 = η 2 = 1.5 ).

Figures 7 and 8 depict the influences of Ra with changing N on concentration and nanofluid motion. Physically, the power of Ra and N boosts the convection flows that accelerate the nanofluid movements and double diffusion. Besides, the direction of the nanofluid flow is dramatically influenced by the variations of N . In Figure 7, an augmentation in Ra provides an enhancement in the concentration distributions. Besides, an increase in N augments concentration distributions. In Figure 8, as a result of a high buoyancy force at extra Ra, the nanofluid velocity accelerates as Ra boosts. An increase in N augments the velocity field regardless of the values of Ra.

$Figure 7 Variation of Ra with N N on concentration at η 1 = η 2 = 1.5 {\eta }_{1}={\eta }_{2}=1.5 , L h = 0.25 {L}_{h}=0.25 , ϕ = 0.01 \phi =0.01 , Da = 1 0 − 3 {\rm{Da}}=1{0}^{-3} , and Le = 20 {\rm{Le}}=20 .$

Figure 7

Variation of Ra with N on concentration at η 1 = η 2 = 1.5 , L h = 0.25 , ϕ = 0.01 , Da = 1 0 − 3 , and Le = 20 .

$Figure 8 Variation of Ra with N N on velocity field at η 1 = η 2 = 1.5 {\eta }_{1}={\eta }_{2}=1.5 , Da = 1 0 − 3 {\rm{Da}}=1{0}^{-3} , L h = 0.25 {L}_{h}=0.25 , ϕ = 0.01 \phi =0.01 , and Le = 20 {\rm{Le}}=20 .$

Figure 8

Variation of Ra with N on velocity field at η 1 = η 2 = 1.5 , Da = 1 0 − 3 , L h = 0.25 , ϕ = 0.01 , and Le = 20 .

The effects of adding nanoparticles in features of heat/mass transport and velocity fields have been shown in Figures 9, 10, and 11. In Figures 9 and 10, an increase in ϕ has a slight enhancement in the temperature and concentration circulations. The physical meaning is due to the high temperature/concentration in the top cylinder walls, which shrinks the contributions of ϕ . In Figure 11, as ϕ augments the viscosity of the mixture fluid and hence, the velocity of a nanofluid decreases as ϕ increases. To prevent the solidifications between the porous media and nanofluid, the value of ϕ is limited to 5 % .

$Figure 9 Variation of ϕ \phi on temperature at N = 1 N=1 , η 1 = η 2 = 1.5 {\eta }_{1}={\eta }_{2}=1.5 , R c = 0.1 {R}_{c}=0.1 , Da = 1 0 − 3 {\rm{Da}}=1{0}^{-3} , Ra = 1 0 4 {\rm{Ra}}=1{0}^{4} , and Le = 20 {\rm{Le}}=20 .$

Figure 9

Variation of ϕ on temperature at N = 1 , η 1 = η 2 = 1.5 , R c = 0.1 , Da = 1 0 − 3 , Ra = 1 0 4 , and Le = 20 .

$Figure 10 Variation of ϕ \phi on concentration at N = 1 N=1 , η 1 = η 2 = 1.5 {\eta }_{1}={\eta }_{2}=1.5 , R c = 0.1 {R}_{c}=0.1 , Da = 1 0 − 3 {\rm{Da}}=1{0}^{-3} , Ra = 1 0 4 {\rm{Ra}}=1{0}^{4} , and Le = 20 {\rm{Le}}=20 .$

Figure 10

Variation of ϕ on concentration at N = 1 , η 1 = η 2 = 1.5 , R c = 0.1 , Da = 1 0 − 3 , Ra = 1 0 4 , and Le = 20 .

$Figure 11 Variation of ϕ \phi on velocity fields at N = 1 N=1 , η 1 = η 2 = 1.5 {\eta }_{1}={\eta }_{2}=1.5 , R c = 0.1 {R}_{c}=0.1 , Da = 1 0 − 3 {\rm{Da}}=1{0}^{-3} , Ra = 1 0 4 {\rm{Ra}}=1{0}^{4} , and Le = 20 {\rm{Le}}=20 .$

Figure 11

Variation of ϕ on velocity fields at N = 1 , η 1 = η 2 = 1.5 , R c = 0.1 , Da = 1 0 − 3 , Ra = 1 0 4 , and Le = 20 .

The impacts of Lewis number Le on the heat/mass transfer and velocity field have been shown in Figures 12,13, and 14. In Figure 12, an increment in Le has a slight decrease in the temperature circulations. In Figure 13, increasing Le reduces the distributions of the concentration within a combined cylinder-rectangular cavity. In Figure 14, an increase in Le augments the nanofluid’s velocity inside a combined cylinder-rectangular cavity.

$Figure 12 Variation of Le on temperature at N = 1 N=1 , η 1 = η 2 = 1.5 {\eta }_{1}={\eta }_{2}=1.5 , L h = 0.25 {L}_{h}=0.25 , R c = 0.1 {R}_{c}=0.1 , Da = 1 0 − 3 {\rm{Da}}=1{0}^{-3} , Ra = 1 0 4 {\rm{Ra}}=1{0}^{4} , and ϕ = 0.01 \phi =0.01 .$

Figure 12

Variation of Le on temperature at N = 1 , η 1 = η 2 = 1.5 , L h = 0.25 , R c = 0.1 , Da = 1 0 − 3 , Ra = 1 0 4 , and ϕ = 0.01 .

$Figure 13 Variation of Le on concentration at N = 1 N=1 , L h = 0.25 {L}_{h}=0.25 , η 1 = η 2 = 1.5 {\eta }_{1}={\eta }_{2}=1.5 , R c = 0.1 {R}_{c}=0.1 , Da = 1 0 − 3 {\rm{Da}}=1{0}^{-3} , Ra = 1 0 4 {\rm{Ra}}=1{0}^{4} , and ϕ = 0.01 \phi =0.01 .$

Figure 13

Variation of Le on concentration at N = 1 , L h = 0.25 , η 1 = η 2 = 1.5 , R c = 0.1 , Da = 1 0 − 3 , Ra = 1 0 4 , and ϕ = 0.01 .

$Figure 14 Variation of Le on velocity fields at N = 1 N=1 , L h = 0.25 {L}_{h}=0.25 , η 1 = η 2 = 1.5 {\eta }_{1}={\eta }_{2}=1.5 , R c = 0.1 {R}_{c}=0.1 , Da = 1 0 − 3 {\rm{Da}}=1{0}^{-3} , Ra = 1 0 4 {\rm{Ra}}=1{0}^{4} , and ϕ = 0.01 \phi =0.01 .$

Figure 14

Variation of Le on velocity fields at N = 1 , L h = 0.25 , η 1 = η 2 = 1.5 , R c = 0.1 , Da = 1 0 − 3 , Ra = 1 0 4 , and ϕ = 0.01 .

Figures 15, 16, and 17 present the thermal, mass patterns, and velocity field under changing boundary conditions of a circular cylinder. In Figures 15 and 16, there are clear differences in the temperature and concentration patterns according to a change in circular cylinder boundary conditions. At case 1, a cylinder-wall has T h , C h , the thermal-solutal convection is highly increasing through the whole cavity. When the boundary condition of a circular cylinder became half T h , C h , and half T c , C c , (case 2), the thermal/solutal convection is shrinking into a half of a cavity. In Figure 17, the maximum of the velocity field reduces by 86.9% as the boundary condition of a circular cylinder changes from case 2 into case 1.

$Figure 15 Variation of wall-boundary conditions on temperature at N = 1 N=1 , η 1 = η 2 = 1.5 {\eta }_{1}={\eta }_{2}=1.5 , L h = 0.25 {L}_{h}=0.25 , R c = 0.1 {R}_{c}=0.1 , Da = 1 0 − 3 {\rm{Da}}=1{0}^{-3} , Ra = 1 0 4 {\rm{Ra}}=1{0}^{4} , ϕ = 0.01 \phi =0.01 , and Le = 20 {\rm{Le}}=20 .$

Figure 15

Variation of wall-boundary conditions on temperature at N = 1 , η 1 = η 2 = 1.5 , L h = 0.25 , R c = 0.1 , Da = 1 0 − 3 , Ra = 1 0 4 , ϕ = 0.01 , and Le = 20 .

$Figure 16 Variation of wall-boundary conditions on concentration at N = 1 N=1 , L h = 0.25 {L}_{h}=0.25 , η 1 = η 2 = 1.5 {\eta }_{1}={\eta }_{2}=1.5 , R c = 0.1 {R}_{c}=0.1 , Da = 1 0 − 3 {\rm{Da}}=1{0}^{-3} , Ra = 1 0 4 {\rm{Ra}}=1{0}^{4} , ϕ = 0.01 \phi =0.01 , ε = 0.6 \varepsilon =0.6 , and Le = 20 {\rm{Le}}=20 .$

Figure 16

Variation of wall-boundary conditions on concentration at N = 1 , L h = 0.25 , η 1 = η 2 = 1.5 , R c = 0.1 , Da = 1 0 − 3 , Ra = 1 0 4 , ϕ = 0.01 , ε = 0.6 , and Le = 20 .

$Figure 17 Variation of wall-boundary conditions on velocity fields at N = 1 N=1 , L h = 0.25 {L}_{h}=0.25 , η 1 = η 2 = 1.5 {\eta }_{1}={\eta }_{2}=1.5 , R c = 0.1 {R}_{c}=0.1 , Da = 1 0 − 3 {\rm{Da}}=1{0}^{-3} , Ra = 1 0 4 {\rm{Ra}}=1{0}^{4} , ϕ = 0.01 \phi =0.01 , and Le = 20 {\rm{Le}}=20 .$

Figure 17

Variation of wall-boundary conditions on velocity fields at N = 1 , L h = 0.25 , η 1 = η 2 = 1.5 , R c = 0.1 , Da = 1 0 − 3 , Ra = 1 0 4 , ϕ = 0.01 , and Le = 20 .

Figure 18 shows Nu ¯ and Sh ¯ along a circular cylinder wall below the effects of Da with two levels of porous media. Nu ¯ has lower values at a homogeneous porous medium compared to a heterogeneous porous medium. However, Sh ¯ has lower values at a heterogeneous porous medium. Moreover, a decrease in Da boosts the values of Nu ¯ .

$Figure 18 Average Nusselt Nu ¯ \overline{{\rm{Nu}}} & Sherwood Sh ¯ \overline{{\rm{Sh}}} numbers a long a circular cylinder wall below the effects of Da with two levels of porous media at N = 2 N=2 , Ra = 1 0 4 {\rm{Ra}}=1{0}^{4} , ϕ = 0.01 \phi =0.01 , ε = 0.6 \varepsilon =0.6 and Le = 20 {\rm{Le}}=20 .$

Figure 18

Average Nusselt Nu ¯ & Sherwood Sh ¯ numbers a long a circular cylinder wall below the effects of Da with two levels of porous media at N = 2 , Ra = 1 0 4 , ϕ = 0.01 , ε = 0.6 and Le = 20 .

5 Conclusion

In this study, the double diffusion within a combined cavity of a circular cylinder over rectangular shape filled with Al 2 O 3 -water and a heterogeneous/homogeneous porous medium was examined numerically by the ISPH method. The variations of buoyancy ratio with Rayleigh number, various boundary conditions, solid volume fraction, Lewis number, and two levels of the porous media with Darcy parameter on the features of heat/mass transport and nanofluid velocity are conducted.

The accompanying points are the principal discoveries of the current study:

A homogeneous porous medium reduces the temperature and concentration distributions as well as the velocity field compared to a heterogeneous porous medium.
As a Darcy parameter signifies a porous resistance for nanofluid flow, the velocity field is decreasing as Darcy parameter decreases.
An increase in Rayleigh number with higher value of a buoyancy ratio are augmenting heat/mass transport as well as a nanofluid velocity.
An increase in solid volume fraction decelerates the nanofluid motion due to extra viscosity a nanofluid.
Variations in wall boundary conditions change the transmission of heat/mass within a combined cavity.
A homogeneous porous medium supports stable nanofluid movements inside a combined cavity compared to a heterogeneous porous medium.

Nomenclature

C: mass concentration
Da: Darcy parameter
Pr: Prandtl number
U , V: dimensionless velocities
K B: Boltamann’s coefficient
C p: heat capacity
x , y: dimensional Cartesian coordinates
P: dimensionless pressure
k: thermal conductivity
Ra: Rayleigh number
W: kernel function
t: dimensional time
K 0: permeability
X , Y: dimensionless Cartesian coordinates
u B: Brownian velocity
T: dimensional temperature
L: height of a rectangular cavity
u , v: dimensional velocities
: Greek symbols
ε: porosity
η 1: change rate of ln ( K ) in X
η 2: change rate of ln ( K ) in Y
Φ: dimensionless mass
μ: dynamic viscosity
ϕ: solid volume fraction
τ: dimensionless time
ρ: density
θ: dimensionless temperature
: Subscripts
f: fluid
nf: Nanofluid
p: porous medium
c: cold
h: hot

Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University, Abha, Saudi Arabia, for funding this work through the Research Group Project under Grant Number (RGP. 2/36/43). This research was funded by the Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2022R102), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

Funding information: This research was funded by the Deanship of Scientific Research at King Khalid University, Abha, Saudi Arabia, for funding this work through the Research Group Project under Grant Number (RGP. 2/36/43). This research was funded by the Princess Nourah Bint Abdulrahman University Researchers Supporting Project number (PNURSP2022R102), 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.
Data availability statement: The data that support the findings of this study are available upon request from the first author (Abdelraheem M. Aly; E-mail: abdelreheam.abdallah@sci.svu.edu.eg).

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Received: 2022-02-01

Revised: 2022-06-25

Accepted: 2022-07-14

Published Online: 2022-09-16

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

Double diffusion in a combined cavity occupied by a nanofluid and heterogeneous porous media

Abstract

1 Introduction

2 Mathematical formulations

3 Proposed methodology

3.1 Boundary treatment

3.2 Validation tests

4 Results and discussions

5 Conclusion

Nomenclature

Acknowledgments

References

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