# Double-Diffusive of a Nanofluid in a Rectangle-Shape Mounted on a Cavity Saturated by Heterogeneous Porous Media

Authors: Abdelraheem M. Aly; Ehab Mohamed Mahmoud; Hijaz Ahmad; Shao-Wen Yao
Journal: Journal of Mathematics
 (2021)

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## Abstract

This study presents numerical simulations on double-diffusive flow of a nanofluid in two cavities connected with four vertical gates. Novel shape of an outer square shape mounted on a square cavity by four gates was used. Heterogeneous porous media andAl2O3-water nanofluid are filled in an inner cavity. Outer rectangle shape is filled with a nanofluid only, and its left walls carry high temperature Th and high concentration Ch. The right walls of a rectangle shape carry low temperature Tc and low concentration Cc and the other walls are adiabatic. An incompressible smoothed particle hydrodynamics (ISPH) method is applied for solving the governing equations of velocities, temperature, and concentration. Results are introduced for the effects of a buoyancy ratio −2≤N≤2, Darcy parameter 10−3≤Da≤10−5, solid volume fraction 0≤ϕ≤0.05, and porous levels. Main results are indicated in which the buoyancy ratio parameter adjusts the directions of double-diffusive convection flow in an outer shape and inner cavity. Adding more concentration of nanoparticles reduces the flow speed and maximum of the velocity field. Due to the presence of a porous medium layer in an inner cavity, the Darcy parameter has slight changes inside the rectangle shape.


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## Body

## 1. Introduction

Double-diffusive convection is a form of convection flow resulting from the variations of the diffusion rates [1]. Double-diffusive has several applications such as fuel cells, diffusion of chemical pollutants in ocean, and nuclear waste storage. In addition, heat and mass transfer in porous media are surviving in numerous engineering applications including oceanography, geophysics, chemical engineering, and thermal engineering [2]. Double-diffusive convection in an annulus has been studied intensively due to its applications in dying and cleaning operations and energy in solar ponds [3].The adjoint influences of thermal and solutal buoyancy forces generate complex flow in a porous annulus. Beji et al. [4] utilized the Darcy model to investigate double-diffusive convection in a porous annulus. Double-diffusive convection in a porous annulus subjected to mass and heat fluxes was introduced by Marcoux et al. [5]. Nithiarasu et al. [6] generalized a model for non-Darcy flow of double-diffusive convection in a porous annulus. Lee et al. [7] examined the multilayer flow in double-diffusive convection of salt water filled in a rotating annulus. Chen et al. [8] adopted lattice Boltzmann model to investigate the double-diffusive in a vertical annulus by considering opposing gradients of the temperature and concentration. Goyeau [9] studied the double-diffusive convection in a porous cavity below Darcy number and porosity impacts. Chamkha and Al-Mudhaf [10] used the finite difference method to check the effects of an inclination angle and buoyancy ratio on double-diffusive in a tilted porous cavity.In the recent years, nanofluid has many applications in solar energy systems, nuclear reactors, heat exchangers, and so on [11–19]. Bhatti et al. [20] discussed the impacts of chemical reaction and nonlinear thermal radiation on unsteady 3D boundary layer flow of a viscous nanofluid having gyrotactic microorganisms through a stretching porous cylinder. Mohebbi and Rashidi [21] used the Lattice Boltzmann Method (LBM) to study numerically the natural convection flow of nanofluid in L−shaped enclosure containing a heating obstacle. Chowdhury et al. [22] studied the double-diffusive convection in an enclosure filled by porous media and nanofluid with considering heat generation effects. The ISPH method was adopted by Aly and Raizah [23] to simulate double-diffusive convection in an enclosure filled with a nanofluid.The topic of the fluid flow and heat transfer in heterogeneous porous media has many applications in remediation, enhanced oil recovery, and geological CO2 reservation. The porous media are heterogeneous porous media when the permeability varies from point to point in a medium [24–26].Since the last decade, mesh-free methods became a well-alternative tool for overcoming disadvantage of mesh methods in simulating free-surface flows and high deformation problems. Lucy [27] and Gingold and Monaghan [28] firstly introduced smoothed particle hydrodynamics to simulate astrophysical problems. The smoothed particle hydrodynamics (SPH) method is employed in several fields [29–35]. An incompressible SPH (ISPH) method has the ability for solving several complex problems by showing a good efficiency [36–42].The SPH method is considered an excellent numerical method for the problems of a highly deforming surface/interface such as free-surface flow, dam break, wave dynamics, impact flows, and bubble dynamics. In addition, the SPH method deals with interfaces naturally without any special requirements. The incompressible version of the SPH method entitled the ISPH method and has mesh-free advantages. The ISPH method can handle the double-diffusive convection flow in a complex cavity by an easy way.In the current research, the ISPH method is used for the simulation of thermo-solutal convection in an outer square shape connected with a center square cavity. The outer square shape and inner square cavity are filled withAl2O3-water nanofluid, and the inner cavity is saturated by a heterogeneous porous medium. It is found that an augmentation in buoyancy ratio parameter increases the thermo-solutal convection in both of outer square shape and inner cavity. Moreover, the directions of thermo-solutal convection are strongly depending on the values of buoyancy ratio parameter. Presence of a porous medium in an inner square cavity only makes slight effects of Darcy parameter and porous medium levels in maximum velocity field. An incremental in solid volume fraction augments viscosity of water, and consequently, the flow speed decreases.


## 2. Mathematical Formulations

Figure1 presents the initial physical model and initial particle distributions. In the current physical model, the outer square shape has a height 2L and the width of a square shape equals 0.1. The inner center cavity is a square cavity with a height L, and there are four normal gates between an outer square shape and an inner square cavity. Only the left walls of an outer square shape are carrying Th and Ch and right walls have Tc and Cc, and the other walls are adiabatic. The nanofluid occupied an outer square shape and inner square cavity, and the nanofluid is modeled by a one-phase model. Heterogeneous porous media are saturated inner square cavity only and the porous media are in thermal equilibrium with nanofluid. Thermophysical properties of water (base fluid) and Al2O3 at T=310K are introduced in Table 1.Figure 1
Initial schematic diagram and initial particle distributions.
(a)(b)Table 1
Nanofluid thermo-physical properties atT=310K [45–47].
ρKgm−3KW/mKCpJ/kgKβ×10−5K−1dnmH2O9930.628417836.20.385Al2O33970407650.8533Time dependent of double-diffusive convection for nanofluid flow according to [43, 44] is(1)∂U∂X+∂V∂Y=0,ρnfρf1ε∂U∂τ+Uε2∂U∂X+Vε2∂U∂Y=−ρnfρf∂P∂X+1ε∂∂XPrμnfμf∂U∂X+1ε∂∂YPrμnfμf∂U∂Y−δPrμeffμnfK∗X,YDaU,ρnfρf1ε∂V∂τ+Uε2∂V∂X+Vε2∂V∂Y=−ρnfρf∂P∂X+1ε∂∂XPrμnfμf∂V∂X+1ε∂∂YPrμnfμf∂V∂Y−δPrμeffμnfK∗X,YDaV+ρβnfρβfPrRaθ+NΦ,γ∂θ∂τ+U∂θ∂X+V∂θ∂Y=ρcpfρcpnf∂∂XζX,Y∂θ∂X+∂∂YζX,Y∂θ∂Y,∂Φ∂τ+Uε∂Φ∂X+Vε∂Φ∂Y=1Le∂∂XΠX,Y∂Φ∂X+∂∂YΠX,Y∂Φ∂Y.The porous matrix expressions are(2)K∗X,Y=eη1HX+η2HY,ζX,Y=1−εeη1HX+η2HY+εknfkf,ΠX,Y=δDeffX,YD0=δDoeη1HX+η2HYD0=δeη1HX+η2HY,γ=1−ερcpp+ερcpnfρcpnf,where(3)δ=0,ε=1.0,nanofluidonly,1,ε=0.6,nanofluidandporousmedium.The expressions of a nanofluid according to [45–47] are(4)ρnf=ϕρp+1−ϕρf,αnf=knfρcpnf,ρβnf=ϕρβp+1−ϕρβf,ρcpnf=ϕρcpp+1−ϕρcpf,μnf=μf1−34.87dp/df−0.3φ1.03,knf=kf1+4.4ReB0.4Pr0.66TTrf10kpkf0.03φ0.66,ReB=ρfuBdpμf,uB=2KBTπμfdp2,where uB is Brownian velocity, Trf is a freezing point of a base fluid, and KB=1.380648×10−23J/K is Boltzmann’s coefficient.


### 2.1. Dimensionless Boundary Conditions

The dimensionless boundary conditions are(5)At a left wall of an outer square shape,0≤Y≤2L;θ=1,Φ=1,U=V=0,At a right wall of an outer square shape,0≤Y≤2L;θ=0,Φ=0,U=V=0,At top/bottom walls of an outer square shape ,0≤X≤2L;∂θ∂Y=0,∂Φ∂Y=0,U=V=0,At the walls of an inner cavity,∂θ∂n=0,∂Φ∂n=0,U=V=0,where n is a normal vector.


## 2.1. Dimensionless Boundary Conditions

The dimensionless boundary conditions are(5)At a left wall of an outer square shape,0≤Y≤2L;θ=1,Φ=1,U=V=0,At a right wall of an outer square shape,0≤Y≤2L;θ=0,Φ=0,U=V=0,At top/bottom walls of an outer square shape ,0≤X≤2L;∂θ∂Y=0,∂Φ∂Y=0,U=V=0,At the walls of an inner cavity,∂θ∂n=0,∂Φ∂n=0,U=V=0,where n is a normal vector.


## 3. ISPH Method

This section summarizes the following steps of the ISPH method.Predicted velocities:(6)U∗=Un+ερfρnfΔτ1ε∂∂XPrμnfμf∂U∂X+1ε∂∂YPrμnfμf∂U∂Y−δPrμeffμnfK∗X,YDaUn,V∗=Vn+ερfρnfΔτ1ε∂∂XPrμnfμf∂V∂X+1ε∂∂YPrμnfμf∂V∂Y−δPrμeffμnfK∗X,YDaV+ρβnfρβfPrRaθ+NΦn.Pressure Poisson equation (PPE):(7)∇2Pn+1=ρΔτ∂U∗∂X+∂V∗∂Y+γρ−ρnumΔτ2.Corrected velocities:(8)Un+1=U∗−Δτρ∇Pn+1,Vn+1=V∗−Δτρ∇Pn+1.Thermal energy:(9)θn+1=θn+Δτρcpfρcpnf∂∂XζX,Y∂θ∂X+∂∂YζX,Y∂θ∂Yn.Concentration equation:(10)Φn+1=Φn+Δτ1Le∂∂XΠX,Y∂Φ∂X+∂∂YΠX,Y∂Φ∂Yn.Positions are updated as(11)Xn+1=Xn+ΔτUn+1,Yn+1=Yn+ΔτVn+1.Shifting technique is(12)ϒi′=ϒi+∇ϒiδRii′+oδRii′2.


### 3.1. SPH Formulation

Any hydrodynamic functionΩ can be defined as(13)Ωxi,t=∫Wrij,hΩxj,tdv,where h is a smoothing length and W is a kernel function. Equation (13) is defined by SPH approximation as(14)Ωxi,t=∑jmjρjWrij,hΩjxj,t.Quintic spline kernel function is(15)Wrij,h=αd3−rijh5−62−rijh5+151−rijh5,0≤rij<h,3−rijh5−62−rijh5,h≤rij<2h,3−rijh5,2h≤rij<3h,0,rij≥3h,where αd=7/478πh2. First derivative is approximated by SPH:(16)∇Ωri=1ρi∑jmjΩj−Ωi∇Wrij,h,∇ϕri=ρi∑jmjΩjρj2+Ωiρi2∇Wrij,h.First derivatives of velocity∇ui and pressure ∇Pi are defined as(17)∇ui=1ρi∑jmjuj−ui⋅∇Wrij,h,∇Pi=ρi∑jmjPjρj2+Piρi2∇Wrij,h.Here, first derivative of a velocity is corrected by a kernel gradient normalizationL^rij as(18)∇^Wij=L^rij∇Wij,L^rij=∑j=1nmjρjxj−xi∂Wij∂xi∑j=1nmjρjxj−xi∂Wij∂yi∑j=1nmjρjyj−yi∂Wij∂xi∑j=1nmjρjyj−yi∂Wij∂yi−1.SPH approximations of second derivatives for the velocity and pressure are(19)∇⋅ν∇⋅u=∑jmjρiνi+ρjνjρiρjrij⋅∇Wijrij2+η2uij,∇2Pi=2ρi∑jmjrij⋅∇Wijrij2+η2Pij.The calculations of FORTRAN-90 code of the ISPH method were performed by Shaheen II supercomputer owned by King Abdullah University of Science and Technology, Thuwal, Saudi Arabia (project number K1467) (https://www.hpc.kaust.edu.sa/content/shaheen-ii).


## 3.1. SPH Formulation

Any hydrodynamic functionΩ can be defined as(13)Ωxi,t=∫Wrij,hΩxj,tdv,where h is a smoothing length and W is a kernel function. Equation (13) is defined by SPH approximation as(14)Ωxi,t=∑jmjρjWrij,hΩjxj,t.Quintic spline kernel function is(15)Wrij,h=αd3−rijh5−62−rijh5+151−rijh5,0≤rij<h,3−rijh5−62−rijh5,h≤rij<2h,3−rijh5,2h≤rij<3h,0,rij≥3h,where αd=7/478πh2. First derivative is approximated by SPH:(16)∇Ωri=1ρi∑jmjΩj−Ωi∇Wrij,h,∇ϕri=ρi∑jmjΩjρj2+Ωiρi2∇Wrij,h.First derivatives of velocity∇ui and pressure ∇Pi are defined as(17)∇ui=1ρi∑jmjuj−ui⋅∇Wrij,h,∇Pi=ρi∑jmjPjρj2+Piρi2∇Wrij,h.Here, first derivative of a velocity is corrected by a kernel gradient normalizationL^rij as(18)∇^Wij=L^rij∇Wij,L^rij=∑j=1nmjρjxj−xi∂Wij∂xi∑j=1nmjρjxj−xi∂Wij∂yi∑j=1nmjρjyj−yi∂Wij∂xi∑j=1nmjρjyj−yi∂Wij∂yi−1.SPH approximations of second derivatives for the velocity and pressure are(19)∇⋅ν∇⋅u=∑jmjρiνi+ρjνjρiρjrij⋅∇Wijrij2+η2uij,∇2Pi=2ρi∑jmjrij⋅∇Wijrij2+η2Pij.The calculations of FORTRAN-90 code of the ISPH method were performed by Shaheen II supercomputer owned by King Abdullah University of Science and Technology, Thuwal, Saudi Arabia (project number K1467) (https://www.hpc.kaust.edu.sa/content/shaheen-ii).


## 4. Validation Test

In this section, the validation test of the ISPH method for the problem of natural convection in a partitioned porous cavity has been conducted. Figure2 shows the temperature profiles in a partitioned porous cavity among the results from the current results of the ISPH method and Beckermann et al. [48]. The temperature profiles were evaluated at three different locations in a cavity: Y/L=0.055, Y/L=0.496, and Y/L=0.874. It is clear that the results of the ISPH method have a well agreement with the experimental and numerical results of Beckermann et al. [48].Figure 2
Comparison of the temperature profiles in a partitioned porous cavity between Beckermann et al. [48] and ISPH method.


## 5. Results and Discussion

The numerical results are presented in terms of concentration, temperature, and velocity field characteristics in order to get a clear insight on the problem of double-diffusive of a nanofluid in an outer square shape mounted on the four vertical gates in a porous cavity. Figures3–5 show the impacts of a buoyancy ratio parameter on the characteristic of the temperature, concentration, and velocity field. It is revealed that an increase in a buoyancy ratio parameter rises the temperature and concentration distributions inside an outer square shape and inner square cavity. Moreover, at opposing flow mode N<0, the double-diffusive convection occurs in the bottom area of an outer square shape and in an inner cavity. While at aiding flow mode N>0, the double-diffusive convection occurs in the top area of an outer square shape and in a cavity. The maximum of the velocity field increases by 175.75% as the buoyancy ratio parameter increases from N=−2 to N=2. Moreover, the maximum of the velocity field appears in the left-top area of an outer square shape at opposing flow mode N<0, and it appears in the bottom-left part of an outer square shape at aiding flow mode N>0.Figure 3
Features of temperature below the variations of a buoyancy ratio parameter atRa=104,ϕ=0.01,Da=10−3, η1=η2=1.5, ε=0.6,Le=20, and τ=1.Figure 4
Features of concentration below the variations in a buoyancy ratio parameter atRa=104,ϕ=0.01,Da=10−3, η1=η2=1.5, ε=0.6,Le=20, and τ=1.Figure 5
Velocity field below the variations of a buoyancy ratio parameter atRa=104,ϕ=0.01,Da=10−3, η1=η2=1.5, ε=0.6,Le=20, and τ=1.The effects of the Darcy parameter on the characteristic of the temperature, concentration, and velocity field have been shown in Figures6–8. As a Darcy parameter represents a useful factor for explaining the fluid flow in the porous materials, a little decrease in the temperature and concentration distributions is occurring when the Darcy parameter decreases from 10−3 to 10−5. Maximum of the velocity field is reducing as a Darcy parameter decreases from 10−3 to 10−5. Moreover, a lower velocity field in the center of a square cavity appears at a lower Darcy parameter Da≤10−4. From these figures, it is noted there are slight changes in the characteristics of the temperature, concentration, and velocity field below the effects of a Darcy parameter. The physical reason returns to the presence of a porous medium layer in the inner cavity only, while an outer square shape contains nanofluid only.Figure 6
Features of temperature below the variations of a Darcy parameter atN=1,Ra=104,ϕ=0.01, η1=η2=1.5, ε=0.6,Le=20, and τ=1.Figure 7
Features of concentration below the variations of a Darcy parameter atN=1,Ra=104,ϕ=0.01, η1=η2=1.5, ε=0.6,Le=20, and τ=1.Figure 8
Velocity field below the variations of a Darcy parameter atN=1,Ra=104,ϕ=0.01, η1=η2=1.5, ε=0.6,Le=20, and τ=1.Figures9–11 introduce the influences of the solid volume fraction on the characteristics of temperature, concentration, and velocity field. It is seen that the temperature and concentration distributions have slight decrease according to adding more concentration of nanoparticles until 5%. The concentration of the nanoparticles is limited to 5% to avoid solidification between the nanofluid and porous media. Adding more concentration of the nanoparticles increases the nanofluid viscosity. As a result, the maximum of the velocity field decreases by 44.77% as the solid volume fraction increases from 0 to 0.05.Figure 9
Features of temperature below variations of a solid volume fraction parameter atN=1,Ra=104,Da=10−3, η1=η2=1.5, ε=0.6,Le=20, and τ=1.Figure 10
Features of concentration below the variations of a solid volume fraction parameter atN=1,Ra=104,Da=10−3, η1=η2=1.5, ε=0.6,Le=20, and τ=1.Figure 11
Velocity field below the variations of a solid volume fraction parameter atN=1,Ra=104,Da=10−3, η1=η2=1.5, ε=0.6,Le=20, and τ=1.The variations of the porous levels according to differentη1 and η2 on the temperature, concentration, and velocity field profiles have been shown in Figures 12–14. In Figure 12, the highest distributions of the temperature are obtained at a heterogeneous porous medium η1=η2=1.5 and lowest temperature distributions are obtained at a homogeneous porous medium η1=η2=0. In Figure 13, it is seen that the concentration distributions are affected strongly by changing the porous medium levels. A heterogeneous porous medium η1=η2=1.5 gives the highest concentration distributions in an inner cavity and lowest concentration distributions in an outer square shape. In contrast, a homogeneous porous medium η1=η2=0 gives the lowest concentration distributions in an inner cavity and the highest concentration distributions in an outer square shape. In Figure 12, there are slight changes in the velocity field below different levels of porous media. The physical reason returns to the presence of porous media in an inner cavity only.Figure 12
Features of temperature below the variations of a porous level,η1 and η2, at N=1,ϕ=0.01,Ra=104,Da=10−4, ε=0.6, Le=20, and τ=1.Figure 13
Features of concentration below the variations of a porous level,η1 and η2, at N=1,ϕ=0.01,Ra=104,Da=10−4, ε=0.6, Le=20, and τ=1.Figure 14
Features of concentration below the variations of a porous level,η1 and η2, at N=1,ϕ=0.01,Ra=104,Da=10−4, ε=0.6, Le=20, and τ=1.


## 6. Conclusion

The novelty of this work is to simulate double-diffusive convection of nanofluid inside a new novel shape of two square cavities connected by four vertical gates. Three different levels of porous media related to the values ofη1 and η2 were conducted. An inner cavity is filled by a heterogeneous porous medium and Al2O3-water nanofluid. The outer square shape with four gates is filled by a nanofluid only. An incompressible version of the SPH method was utilized to solve the nondimensional governing equations of the current problem. The main finding of the current results can be summarized as follows:(i)
An augmentation in a buoyancy ratio parameter increases the temperature and concentration distributions in an outer square shape and in an inner cavity(ii)
At opposing flow mode,N<0, and the direction of buoyancy force from upwards to downwards and the maximum of velocity field occur at a top-left part of an outer square shape(iii)
At aiding flow mode,N>0, and the direction of buoyancy force from downwards into upwards and the maximum of velocity field occur at a bottom-left part of an outer square shape(iv)
A decrease in the Darcy parameter reduces the flow speed in an inner porous cavity, and it has slight effects in an outer square shape(v)
Adding more concentration of nanoparticles increases the viscosity of a host fluid, and consequently, the flow speed inside an outer square shape and in an inner cavity decreases(vi)
Variation in porous medium levels changes the concentration distributions inside an outer square shape and in an inner cavity



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