Rotating cylinder and magnetic field on solid particles diffusion inside a porous cavity filled with a nanofluid

Abstract

This study deals with the roles of a magnetic field and circular rotation of a circular cylinder on the dissemination of solid phase within a nanofluid-filled square cavity. Two wavy layers of the non-Darcy porous media are situated on the vertical sides of a cavity. An incompressible smoothed particle hydrodynamics (ISPH) method was endorsed to carry out the blending process concerning solid phase into nanofluid and porous media layers. Initially, the solid phase is stationed in a circular cylinder containing two open gates. Implications of a buoyancy ratio (N = −2: 2), Hartmann number (Ha = 0: 100), rotational frequency $\left(\omega =1 : 10\right)$ , Darcy parameter $\left(Da={10}^{-2}:{10}^{-5}\right)$ , Rayleigh number $\left(Ra={10}^3:{10}^6\right)$ , nanoparticles parameter $\left(\phi =0:0.06\right)$ , and amplitude of wavy porous layers $\left({\rm A}=0.05:0.15\right)$ on the lineaments of heat/mass transport have been carried out. The results revealed that the diffusion of the solid phase is permanently moving toward upward except at opposing flow mode $\left(N<0\right)$ toward downward. The lower Rayleigh number reduces the solid-phase diffusions. A reduction in a Darcy parameter lessens the nanofluid speed and solid-phase diffusions in the porous layers. A reduction in $Da$ from ${10}^{-3}$ to ${10}^{-5}$ diminishes the maximum of streamlines ${\left|\mathbf{\psi }\right|}_{\mathbf{max}}$ by 13.19% at $N=-2$ , by 46.75% at $N=0$ , and by 74.75% at $N=2$ .

Introduction

Convective flows through porous structures are received several research studies and the reason for their numerous industrial and technological purposes. Applications are not restricted to the building’s insulation, thermal insulation, fibrous insulation, grain storage, packed sphere beds, fuel cells, solar collectors, and geothermal energy systems. Moreover, convective flows from double diffusion in porous media have several applications including soil pollution and nuclear waste disposal. In this context, Lewis ¹ studied multiphase flow in deformable cracked porous media. Angirasa et al. ² researched adjoined heat/mass transport on vertical surface contained porous media. Cheng ³ studied the transmission of heat/mass in an undulating surface contained a porous medium. The numerical/experimental research studies on heat transport in a vertical enclosure moderately filled by a porous medium have been introduced by Ramadhyani and Viskanta. ⁴ Beckermann et al. ⁵ examined the presence of a porous layer on natural convection in the interior of an enclosure. Singh et al. ⁶ analyzed heat/mass transport in a mixture cavity contained fluid/porous medium layers.

In recent years, many attempts were performed for the thermosolutal of nanofluids. Generally, the applications of nanofluids are not restricted to the following: solar techniques,^7-9 automotive,^10,11 heat exchangers,^12,13 and heat transfer devices ¹⁴ . Kuznetsov and Nield ¹⁵ studied thermosolutal of nanofluid inside a boundary layer flow. Dastmalchi et al. ¹⁶ examined the thermosolutal of nanofluid within a porous cavity. Aly ¹⁷ examined the impacts of embedded two circular cylinders on nanofluid flow in the interior of a porous enclosure. Aly ¹⁸ showed the magnetic inspirations on thermosolutal convection inside a nanofluid-filled cavity containing an oscillating pipe. Sarkar et al. ¹⁹ researched free convection in a semi-elliptical enclosure containing a bottom-heated. Selimefendigil et al. ²⁰ introduced numerical investigations for the turbulent flow conditions on the three-dimensional convective drying process at several porous moist blocks. Selimefendigil et al. ²¹ handled the motion of the porous moist object during heat/mass transmission in a 2D channel. Selimefendigil and Oztop ²² studied the effects of hot dry air on the transmission of heat/mass for several porous humid objects.

The study of a rotating cylinder is an attempt to boost heat transfer and to help flow movement within a cavity. Because of the many applications of convective flows around a rotating element like the chemical mixing devices, fuel pole in nuclear reactors, and turbomachinery, there are many studies on this problem that can be found in references.^23-34

In the current study, the influences of a turning circular cylinder on blending solid and fluid phases within a cavity filled by porous media and nanofluid have been simulated by the ISPH method. Non-Darcy porous media were in a form of two wavy layers on the vertical sides of a cavity. The circular cylinder is contained solid phase and it has a uniform circular rotation. Impacts of the key physical parameters including buoyancy ratio, rotational frequency, Darcy parameter, Rayleigh number, Hartmann number, nanoparticles parameter, and amplitude of wavy porous layers on heat and mass transport and flow speed have been discussed in detail. The performed simulations proved that the orientation of the dispersions of solid phase is reliant on the buoyancy ratio parameter and flow modes. A decrease in Ra declines buoyancy forces within a cavity, so the solid-phase dispersions are decreasing. The Darcy parameter acts as a significant element in controlling the solid-phase dispersions inside the porous layers.

Mathematical formulation

Figure 1 describes the primary model and its particle pattern. The solid phase is initially stationed in an embedded circular cylinder and carries $T_h$ and $C_h$ . Otherwise, sidewalls are cooled at $T_c$ with quiet concentration $C_c$ . The inner circular cylinder and horizontal walls are adiabatic. The two wavy partial layers are located on the vertical cavity sides with variable amplitude $a$ and saturated by the non-Darcy porous media. The nanofluid is consisting of host fluid (water) and nanoparticles (copper). The mixture density is handled by a Boussinesq approximation, and the thermal equilibrium is utilized among a nanofluid and a porous matrix.

Figure 1.

Initial model of the current problem and particles model. (i) Initial physical model and (ii) initial particles model.

The partial differential equations in a dimensionless form are expressed as^35-37

$\frac{\partial U}{\partial X}+\frac{\partial V}{\partial Y}=0$

(1)

$\frac{1}{\epsilon }\frac{dU}{d\tau }=\frac{{\mu }_{\text{nf}}}{ \epsilon {\rho }_{\text{nf}}{\alpha }_{\text{f}}}\left(\frac{{\partial }^{2}U}{\partial X^2}+\frac{{\partial }^{2}U}{\partial Y^2}\right)-\frac{\partial P}{\partial X}-\frac{{\sigma }_{\text{nf}}{\rho }_{\text{f}} }{{\sigma }_{\text{f}}{\rho }_{\text{nf}}} H{a}^2\mathrm{Pr}U-\delta \left(\frac{{\mu }_{\text{nf}}}{{\rho }_{\text{nf}}{\alpha }_{\text{f}}}\frac{1}{Da}+\frac{C}{\sqrt{Da}}\frac{\sqrt{U^2+V^2}}{{\epsilon }^{3/2}}\right)U$

(2)

$\frac{1}{\epsilon }\frac{dV}{d\tau }=\frac{{\mu }_{\text{nf}}}{\epsilon {\rho }_{\text{nf}}{\alpha }_{\text{f}}}\left(\frac{{\partial }^{2}V}{\partial X^2}+\frac{{\partial }^{2}V}{\partial Y^2}\right)-\frac{\partial P}{\partial Y}+\frac{{\left(\rho \beta \right)}_{\text{nf}}}{{\rho }_{\text{nf}} {\beta }_{\text{f}}}Ra Pr \left(\theta +N\text{Φ}\right)-\delta \left(\frac{{\mu }_{\text{nf}}}{{\rho }_{\text{nf}}{\alpha }_{\text{f}}}\frac{1}{Da}+\frac{C}{\sqrt{Da}}\frac{\sqrt{U^2+V^2}}{{\epsilon }^{3/2}}\right)V$

(3)

$\sigma \frac{d\theta }{d\tau }=\frac{\widehat{\alpha }}{{\alpha }_{\text{f}} }\left(\frac{{\partial }^2\theta }{\partial X^2}+\frac{{\partial }^2\theta }{\partial Y^2}\right)$

(4)

$\frac{d\text{Φ}}{d\tau }=\frac{1 }{Le }\left(\frac{{\partial }^2\text{Φ}}{\partial X^2}+\frac{{\partial }^2\text{Φ}}{\partial Y^2}\right)$

(5)

Darcy parameter is $Da=K/L^2$ , Prandtl number is $Pr={\nu }_{\text{f}}/{\alpha }_{\text{f}}$ , and Rayleigh number is $Ra=\text{g} {\beta }_{\text{f}}{L}^3\mathrm{\Delta }T/{\nu }_{\text{f}}{\alpha }_{\text{f}}$ . $\sigma =\left[\left(1-\epsilon \right){\left(\rho C_p\right)}_{\text{s}}+\epsilon {\left(\rho C_p\right)}_{\text{nf}}\right]/{\left(\rho C_p\right)}_{\text{nf}}$ is a capacity ratio. Hartmann number is $Ha=\sqrt{{\sigma }_{\text{f} }/{\mu }_{\text{f}}} B_{0}L$ , Lewis number is $Le={\alpha }_f/D$ , and buoyancy ratio parameter is $N={\beta }_C\left(C_h-C_c\right)/{\beta }_T\left(T_h-T_c\right)$ .

$\begin{array}{l}\epsilon =\{\begin{array}{cc}1& \text{Nanofluid layer}\\ \epsilon & \text{Porous layers}\end{array},\delta =\{\begin{array}{cc}0& \text{Nanofluid layer}\\ 1& \text{Porous layers}\end{array},\widehat{\alpha }=\{\begin{array}{cc}{\alpha }_{\text{nf}}& \text{Nanofluid layer}\\ {\alpha }_{\text{eff}}& \text{Porous layers}\end{array}\end{array}$

The following dimensionless quantities are used

$X=\frac{x}{L},Y=\frac{y}{L},\tau =\frac{t{\alpha }_{\text{f}}}{L^2},U=\frac{uL}{{\alpha }_{\text{f}}},{\alpha }_f=\frac{k_{\text{f}}}{{\left(\rho C_P\right)}_{\text{f}}}, V=\frac{vL}{{\alpha }_{\text{f}}}, P=\frac{p{L}^2}{{\rho }_{\text{nf} }{\alpha }_{\text{f}}^2 },\theta =\frac{T-T_c}{T_h-T_c},\text{Φ}=\frac{C-C_c}{C_h-C_c}$

(6)

The boundary conditions are:

$\begin{array}{l}\mathrm{Horizontal cavity walls:}U=V=0, \frac{\partial \theta }{\partial Y}=\frac{\partial \mathrm{\Phi }}{\partial Y}=0,\\ \mathrm{An embedded circular cylinder:} \frac{\partial \theta }{\partial Y}=\frac{\partial \mathrm{\Phi }}{\partial Y}=0,\\ \mathrm{Vertical cavity walls:}U= V=0, \theta =0=\mathrm{\Phi },\\ \mathrm{Solid phase: }\theta =1=\mathrm{\Phi }\end{array}$

(7)

The mean Nusselt number $\left(\overline{Nu}\right)$ and Sherwood number $\left(\overline{Sh}\right)$ are

$\overline{Nu}=-\frac{1}{L} \underset{0}{\overset{L}{{\displaystyle \int }}}\frac{k_{\text{eff}}}{k_{\text{f}}} \left(\frac{\partial \theta }{\partial X}\right)dY$

(8)

$\overline{Sh}=-\frac{1}{L} \underset{0}{\overset{L}{{\displaystyle \int }}} \left(\frac{\partial \text{Φ}}{\partial X}\right)dY$

(9)

Nanofluid properties

The nanofluid is handled by one-phase model. Table 1 signifies the nanofluid characteristics.

Table 1

Nanofluid thermophysical properties.^36,37

Property	Copper (Cu)	Water
$\rho$	8933	997.1
$C_P$	385	4179
$k$	401	0.613
$\beta$	1.67 $\times$ 10−5	21 $\times$ 10−5
$\sigma$	$5.69 \times {10}^7$	0.05

Besides, the characteristics of a nanofluid(^38-40) are

${\rho }_{\text{nf}} ={\rho }_{\text{f}}\left(1-\phi \right)+\phi {\rho }_{\text{np}}$

(10)

${\left(\rho C_p\right)}_{\text{nf}}={\left(\rho C_p\right)}_{\text{f}}\left(1-\phi \right)+\phi {\left(\rho C_p\right)}_{\text{np}}$

(11)

${\left(\rho \beta \right)}_{\text{nf}}={\left(\rho \beta \right)}_{\text{f}}\left(1-\phi \right)+\phi {\left(\rho \beta \right)}_{\text{np}}$

(12)

${\alpha }_{\text{eff}}=\frac{k_{\text{eff}}}{{\left(\rho C_p\right)}_{\text{nf}}}$

(13)

$k_{\text{eff}}=\epsilon k_{\text{nf}}+\left(1-\epsilon \right)k_{\text{s}}$

(14)

$k_{\text{nf}}=\left(-2\phi k_{\text{f}}\left(k_{\text{f}}-k_{\text{np}}\right)+k_{\text{f}}\left(k_{\text{np}}+2k_{\text{f}}\right)\right){\left(\left(k_{\text{np}}+2k_{\text{f}}\right)+\phi \left(k_{\text{f}}-k_{\text{np}}\right)\right)}^{-1}$

(15)

${\mu }_{\text{nf}}=\frac{{\mu }_{\text{f}}}{{\left(1-\phi \right)}^{2.5}}$

(16)

${\sigma }_{\text{nf}}={\sigma }_{\text{f}}\left(1+\frac{3\phi \left({\sigma }_{\text{np}}/{\sigma }_{\text{f}}-1\right)}{\left({\sigma }_{\text{np}}/{\sigma }_{\text{f}}+2\right)-\phi \left({\sigma }_{\text{np}}/{\sigma }_{\text{f}}-1\right)}\right)$

(17)

ISPH method

The explaining of SPH formulation is presented in this section.

Solving steps: the predictor velocity is

$\left(1+\epsilon \mathrm{\Delta }\tau {\text{Por}}_{\text{nano}}\right) U^{\ast }=U^n+\epsilon \mathrm{\Delta }\tau \left(\frac{{\mu }_{\text{nf}}}{ {\rho }_{\text{nf}}{\alpha }_{\text{f}}}{\left(\frac{{\partial }^{2}U}{\partial X^2}+\frac{{\partial }^{2}U}{\partial Y^2}\right)}^n-\frac{{\sigma }_{\text{nf}}{\rho }_{\text{f}} }{{\sigma }_{\text{f}}{\rho }_{\text{nf}}} H{a}^2\mathrm{Pr}{U}^n\right)$

(18)

$\left(1+\epsilon \mathrm{\Delta }\tau {\text{Por}}_{\text{nano}}\right) V^{\ast }=V^n+\epsilon \mathrm{\Delta }\tau \left(\frac{{\mu }_{\text{nf}}}{ {\rho }_{\text{nf}}{\alpha }_{\text{f}}}{\left(\frac{{\partial }^{2}V}{\partial X^2}+\frac{{\partial }^{2}V}{\partial Y^2}\right)}^n+\frac{{\left(\rho \beta \right)}_{\text{nf}}}{{\rho }_{\text{nf}} {\beta }_{\text{f}}}Ra Pr \left({\theta }^n+N{\text{Φ}}^n\right)\right)$

(19)

where

${\text{Por}}_{\text{nano}}=\delta {\left(\frac{{\mu }_{\text{nf}}}{{\rho }_{\text{nf}}{\alpha }_{\text{f}}}\frac{1}{Da}+\frac{C}{\sqrt{Da}}\frac{\sqrt{U^2+V^2}}{{\epsilon }^{3/2}}\right)}^n$

Pressure Poisson equation

${\nabla }^2{P}^{n+1}=\frac{1+\epsilon \mathrm{\Delta }\tau {\text{Por}}_{\text{nano}}}{\epsilon \mathrm{\Delta }\tau }\left(\frac{\partial U^{\ast }}{\partial X}+\frac{\partial V^{\ast }}{\partial Y}\right).$

(20)

Corrected velocity is written as

$\left(1+\epsilon \mathrm{\Delta }\tau {\text{Por}}_{\text{nano}}\right)U^{n+1}=\left(1+\epsilon \mathrm{\Delta }\tau {\text{Por}}_{\text{nano}}\right)U^{\ast }-\epsilon \mathrm{\Delta }\tau {\left(\frac{\partial P}{\partial X}\right)}^{n+1}$

(21)

$\left(1+\epsilon \mathrm{\Delta }\tau {\text{Por}}_{\text{nano}}\right)V^{n+1}=\left(1+\epsilon \mathrm{\Delta }\tau {\text{Por}}_{\text{nano}}\right)V^{\ast }-\epsilon \mathrm{\Delta }\tau {\left(\frac{\partial P}{\partial Y}\right)}^{n+1}$

(22)

Thermal and concentration equations are.

${\theta }^{n+1}={\theta }^n+\frac{\mathrm{\Delta }\tau \widehat{\alpha }}{\sigma {\alpha }_{\text{f}} }{\left(\frac{{\partial }^2\theta }{\partial X^2}+\frac{{\partial }^2\theta }{\partial Y^2}\right)}^n$

(23)

${\text{Φ}}^{n+1}={\text{Φ}}^n+\frac{\mathrm{\Delta }\tau }{Le }{\left(\frac{{\partial }^2\text{Φ}}{\partial X^2}+\frac{{\partial }^2\text{Φ}}{\partial Y^2}\right)}^n$

(24)

Updated positions are

$X^{n+1}=X^n+\mathrm{\Delta }\tau U^{n+1}$

(25)

$Y^{n+1}=Y^n+\mathrm{\Delta }\tau V^{n+1}$

(26)

Shifting technique^41,42 was applied in this study

${\zeta }_{i^{\text{'}}}={\zeta }_i+{\left(\nabla \zeta \right)}_i\cdot \left(-\text{Γ} \nabla \chi \right)+\mathcal{O}\left(\delta {\mathit{r}}_{i{i}^{\text{'}}}^2\right)$

(27)

SPH formulation

The SPH concept for a function $f\left({\mathit{r}}_i\right)$ is

$f\left({\mathit{r}}_i\right)=\frac{1}{{\gamma }_i}{\displaystyle \sum _J\frac{m_j}{{\rho }_j}f\left({\mathit{r}}_j\right)W\left({\mathit{r}}_{ij},h\right)}$

(28)

Kernel renormalization parameter ${\gamma }_i$ (^{43, 44})

${\gamma }_i={\displaystyle \underset{{\text{Ω}}_i}{\int }W\left(\left|{\mathit{r}}_{ab}\right|\right)d\text{Ω}\left({\mathit{r}}_j\right)}$

(29)

$W$ is a quantic kernel function

$W=\frac{3}{16 \pi h^2}\{\begin{array}{l}\begin{array}{cc}{\left(2-r_{ij}/h\right)}^5-16{\left(1-r_{ij}/h\right)}^5& 0\le r_{ij}\le h\end{array}\\ \begin{array}{cc}{\left(2-r_{ij}/h\right)}^5& h<r_{ij}\le 2h\end{array}\\ \begin{array}{cc}0& r_{ij}>2h\end{array}\end{array}$

(30)

ISPH validation

To validate the ISPH method, the numerical test of free convection for a heater inside an air-filled cavity was introduced. The comparison between the results of the ISPH method and numerical/experimental results from Paroncini and Corvaro ⁴⁵ has been introduced in Figure 2. The results of the ISPH method agree with the given numerical/experimental data.

Figure 2.

Isothermal lines ISPH method (left), Paroncini and Corvaro ⁴⁵ numerical results (center), and experimental results (right). (a) $\chi =0.25 \& Ra=2.11\times {10}^5$ .

Results and discussion

The numerical simulations on implications of a buoyancy ratio $\left(N=-2 : 2\right)$ , Hartmann number $\left(Ha=0:100\right)$ , rotational frequency $\left(\omega =1 : 10\right)$ , Darcy parameter $\left(Da={10}^{-2}:{10}^{-5}\right)$ , Rayleigh number $\left(Ra={10}^3:{10}^6\right)$ , nanoparticles parameter $\left(\phi =0:0.06\right)$ , and amplitude of wavy porous layers ( ${\rm A}=0.05:0.15$ ) on the lineaments of heat/mass transport have been discussed. Fixed parameters are $Pr=6.2$ , $Le=20$ , porosity $\epsilon =0.6$ , speed frequency of a circular cylinder $\omega =3.15$ , and undulation number of wavy porous layers $\kappa =2$ .

Figure 3 presents the variations of the buoyancy ratio $N$ and Darcy parameter $Da$ on tracking solid phase, streamlines, isotherms, and isoconcentration for different buoyancy ratio and Darcy parameter at $Ra={10}^4$ , $Ha=1, Le=20,\hspace{0.17em}\text{Α}=0.05$ , $\omega =3.15,\hspace{0.17em}\phi =0.01, \hspace{0.17em}\kappa =2$ , and $\tau =0.25$ . In this figure, it is seen that $N$ regulates the blending of solid phase within a nanofluid. At $N<0$ (opposing flow), the solid phase is shifting downward, and when $N\ge 0$ (aiding flow), the solid phase is shifting upward. As $N$ increases from 0 to 2, the solid-phase dispersions are increasing. Physically, the increase in $N$ powers the buoyancy forces in a cavity and accordingly the fluid movements are powered. On other hand, a decrease in $Da$ shrinks the distribution of the solid phase in the interior of the nanofluid flow. Physical reason returns the higher permeability which grows the porous resistance at a lower $Da$ . Further, the highest values of a maximum of an absolute stream function ${\left|\text{ψ}\right|}_{\text{max}}$ are obtained at $N=0$ . A decrease in $Da$ powers the porous resistance, accordingly it slows down the fluid speed and then reduces ${\left|\text{ψ}\right|}_{\text{max}}$ . A reduction in $Da$ from ${10}^{-3}$ to ${10}^{-5}$ , reduces ${\left|\text{ψ}\right|}_{\text{max}}$ by 13.19% at $N=-2$ , by 46.75% at $N=0$ , and by 74.75% at $N=2$ , respectively. Mainly, the streamlines contours are affected by the buoyancy ratio and Darcy factors. The isotherms and isoconcentration below the variations on the buoyancy ratio and Darcy parameters are conducted. The contours of the isotherms and isoconcentration have received clear variations below the variations of $N$ . It is remarked that the isotherms and isoconcentration are following the solid-phase positions.

Figure 3.

Tracking solid phase, streamlines, isotherms, and isoconcentration for different buoyancy ratio and Darcy parameter at $Ra={10}^4$ , $Ha=1, Le=20, \text{Α}=0.05$ , $\omega =3.15, \phi =0.01, \kappa =2$ , and $\tau =0.25$ .

The solid-phase positions below the impacts of different angular frequencies at two various values of $Da={10}^{-2}$ and $Da={10}^{-4}$ are introduced in Figure 4. It is remarked that an increase in an angular frequency increases the rotational velocity of an open circular cylinder and it changes the blending of solid phase within nanofluid. The solid-phase dispersions are decreasing as the Darcy parameter reduces. Here, the contours of the streamlines under the variations in an angular frequency at two different values of a Darcy parameter are introduced. It is observed that an angular frequency augments the absolute value of the stream function’s maximum ${\left|\text{ψ}\right|}_{\text{max}}$ and a decrease in $Da$ reduces ${\left|\text{ψ}\right|}_{\text{max}}$ . An angular frequency changes the mixing processes of the solid phase within the mixture of the nanofluid. As a result, the contours of the isotherms and isoconcentration were varied under the variations on angular frequency.

Figure 4.

Tracking solid phase, streamlines, isotherms, and isoconcentration for the different angular frequency at $Le=20$ , $Da={10}^{-4}, N=2, Ha=1, \phi =0.01$ , $\text{Α}=0.05, \kappa =2$ , and $\tau =0.25$ .

The effects of a Hartmann number $Ha$ on the solid-phase tracking within the mixture of the nanofluid have been shown in Figure 5. A growth in $Ha$ reduces the solid-phase blending in a mixture of the nanofluid. The results are returning to an extra $Ha$ augments the Lorentz force which decelerates down the velocity speed. Therefore, an augmentation in $Ha$ reduces the values of ${\left|\text{ψ}\right|}_{\text{max}}$ . Because of the Lorentz force which declines the convection flow, an increase in $Ha$ declines the intensity of the contours of the isotherms and isoconcentrations inside a cavity.

Figure 5.

Tracking solid phase, streamlines, isotherms, and isoconcentration for different Hartmann number at $\phi =0.01, Ra={10}^4$ , $Da={10}^{-4}, N=2, Le=20, \text{Α}=0.05$ , $\omega =3.15, \kappa =2$ and $\tau =0.25$ .

Effects of nanoparticles parameter $\phi$ on the solid-phase diffusion, streamlines, isotherms, and isoconcentration have been shown in Figure 6. It is found that adding nanoparticles concentration until 6% has slight changes on the solid-phase dispersions in a cavity. Therefore, the ${\left|\text{ψ}\right|}_{\text{max}}$ is slightly varied following the variations on $\phi$ . Besides, the isotherms and isoconcentration are receiving slight changes under the changes of $\phi$ .

Figure 6.

Tracking solid phase, streamlines, isotherms, and isoconcentration for different nanoparticles parameter at $Le=20, Ra={10}^4$ , $Da={10}^{-4}, N=2, Ha=1, \text{Α}=0.05$ , $\omega =3.15, \kappa =2$ , and $\tau =0.25$ .

Figure 7 introduces the tracking of the solid phase into a mixture of the nanofluid, streamlines, isotherms, and isoconcentration under the effects of an amplitude parameter ${\rm A}$ of the wavy porous layers at $Le=20$ , $Da={10}^{-4}, N=2,\hspace{0.17em}Ha=1,\hspace{0.17em} \phi =0.01$ , $\omega =3.15, \kappa =2$ , and $\tau =0.25$ . An augmentation on a wavy porous amplitude is trying to reduce the diffusion of the solid phase within the porous layers. Therefore, the streamlines vertical cells are shrinking when an undulating porous amplitude equals $\text{Α}=0.2$ . The values of ${\left|\text{ψ}\right|}_{\text{max}}$ are increasing as ${\rm A}$ increases from $0.15$ to $0.2$ . The intensity of the isotherms and isoconcentrations contours is slightly decreasing according to an increase in amplitude parameter of the wavy porous layers.

Figure 7.

Tracking solid phase, streamlines, isotherms, and isoconcentration for different amplitude parameter at $Le=20$ , $Da={10}^{-4}, N=2, Ha=1, \phi =0.01$ , $\omega =3.15, \kappa =2$ , and $\tau =0.25$ .

The impacts of a Rayleigh number $Ra$ on the solid-phase tracking have been indicated in Figure 8. At a similar time instant $\tau =0.25$ , the solid-phase diffusion is greatly dependent on the Rayleigh number. When $Ra={10}^3$ , little solid phase is spreading from the embedded circular cylinder, while as $Ra$ increases to ${10}^4$ and ${10}^5$ , the solid phase is completely spreading upward. Due to the existence of the solid phase in the top area of a cavity, ${\left|\text{ψ}\right|}_{\text{max}}$ is fluctuating. The isotherms and isoconcentrations contours are concentrating nearby the top wall of a cavity. The higher values of $Ra$ are generating higher buoyancy forces that are causing the above observations.

Figure 8.

Tracking solid phase, streamlines, isotherms, and isoconcentration for different Rayleigh number at $Ha=1$ , $Da={10}^{-4}, N=2, Le=20, \text{Α}=0.05$ , $\omega =3.15, \kappa =2, \phi =0.01,$ and $\tau =0.25$ .

Figure 9 represents the values of $\overline{Nu}$ and $\overline{Sh}$ below the influences of buoyancy ratio with Darcy parameter. Lower in a Darcy parameter creates the smallest values of $\overline{Nu}$ and $\overline{Sh}$ . An incrementation in a buoyancy ratio parameter augments $\overline{Nu}$ and $\overline{Sh}$ at higher Darcy parameter $Da\ge {10}^{-4}$ , while there are roughly no alterations on $\overline{Nu}$ and $\overline{Sh}$ below the impacts of a buoyancy ratio at a lower Darcy parameter $Da={10}^{-5}$ .

Figure 9.

Values of $\overline{Nu}$ and $\overline{Sh}$ below the influences of a buoyancy ratio with Darcy parameter at $Ra={10}^4, Ha=1$ , $Le=20, \text{Α}=0.05$ , $\omega =3.15, \phi =0.01, \kappa =2$ , and $\tau =0.25$ . (a) $\overline{Nu}$ and (b). $\overline{Sh}$

Conclusion

The ISPH method is utilized for simulating the diffusion of the solid phase within a nanofluid flow and undulating porous layers in a cavity. Moreover, the influences of a rotating circular cylinder with variable speed frequencies in the solid particle’s diffusion were conducted. The main results concluded that the solid-phase diffusion is always upwards when the mode of flow is aiding. The solid-phase diffusion is moving in downward at opposing mode. A decline in $Ra$ is reducing the buoyancy forces and it reduces the solid-phase diffusion. The lower Darcy parameter diminishes the diffusion of the solid phase within the porous layers. Hartmann number augments Lorentz force that declines nanofluid speed and consequently the solid-phase diffusion decreases. A reduction in a Darcy parameter diminishes $\overline{Nu}$ and $\overline{Sh}$ . The variations of a buoyancy ratio on $\overline{Nu}$ and $\overline{Sh}$ are appearing at a higher Darcy parameter $Da\ge {10}^{-4}$ , and when the buoyancy ratio parameter boosts, the values of $\overline{Nu}$ and $\overline{Sh}$ are improving.