Abdelraheem M Aly^,1,2, Noura Alsedais³¹Department of Mathematics, King Khalid University, Abha 62529, Saudi Arabia
²Department of Mathematics, Faculty of Science, South Valley University, Qena 83523, Egypt
³Department of Mathematical Sciences, College of Science, Princess Nourah Bint Abdulrahman University, Riyadh, Saudi Arabia

Received:2021-05-24Revised:2021-08-03Accepted:2021-08-04Online:2021-09-14


Abstract
The magnetic impacts upon the transport of heat and mass of an electrically conducting nanofluid within an annulus among an inner rhombus with convex and outer cavity with periodic temperature/concentration profiles on its left wall are assessed by the ISPH method. The right wall has ${T}_{c}$ and ${C}_{c},$ flat walls are adiabatic, and the temperature and concentration of the left wall are altered sinusoidally with time. The features of the heat and mass transfer and fluid flow through an annulus are assessed across a wide scale of Hartmann number $Ha,$ Soret number $Sr,$ oscillation amplitude $A,$ Dufour number $Du,$ nanoparticles parameter $\phi ,$ oscillation frequency $f,$ Rayleigh number $Ra,$ and radius of a superellipse $a$ at Lewis number $Le=20,$ magnetic field's angle $\gamma =45^\circ ,$ Prandtl number ${\Pr }=6.2,$ a superellipse coefficient $n=3/2,$ and buoyancy parameter $N=1.$ The results reveal that the velocity's maximum reduces by $70.93 \% $ as $Ha$ boosts from 0 to 50, and by $66.24 \% $ as coefficient $a$ boosts from $0.1$ to $0.4.$ Whilst the velocity's maximum augments by $83.04 \% $ as $Sr$ increases from 0.6 to 2 plus a decrease in $Du$ from 1 to 0.03. The oscillation amplitude $A,$ and frequency $f$ are significantly affecting the nanofluid speed, and heat and mass transfer inside an annulus. Increasing the parameters $A$ and $f$ is augmenting the values of mean Nusselt number $\overline{Nu}$ and mean Sherwood number $\overline{Sh}.$ Increasing the radius of a superellipse $a$ enhances the values of $\overline{Nu}$ and $\overline{Sh}.$
Keywords： Dufour number;ISPH method;nanofluid;Soret number;rhombus;magnetic field


PDF (6681KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite
Cite this article
Abdelraheem M Aly, Noura Alsedais. Magnetic impact on heat and mass transfer utilizing nonofluid in an annulus between a superellipse obstacle and a cavity with periodic side-wall temperature and concentration. Communications in Theoretical Physics, 2021, 73(11): 115001- doi:10.1088/1572-9494/ac1a6b

Nomenclature

$a$ radius of a superellipse
${B}_{o}$ magnetic field
$C$ concentration
${D}_{1}$ mass flux
${D}_{{\rm{m}}}$ mass diffusivity
$Du$ Dufour number
${D}_{2}$ heat flux
$n$ superellipse coefficient
${\rm{g}}$ gravity, $\left({\rm{m}}\,{{\rm{s}}}^{-2}\right)$
$p$ pressure, $\left({\rm{N}}\,{{\rm{m}}}^{-2}\right)$
$L$ cavity length
$\overline{Nu}$ mean Nusselt number
$\overline{Sh}$ mean Sherwood number
$u,\,v$ velocities, $\left({\rm{m}}\,{{\rm{s}}}^{-1}\right)$
$k$ thermal conductivity, $\left({\rm{W}}\,{{\rm{m}}}^{-1}\,{{\rm{K}}}^{-1}\right)$
$T$ temperature, $\left({\rm{K}}\right)$
$t$ time, $\left({\rm{s}}\right)$
${C}_{p}$ specific heat, $\left({{\rm{Jkg}}}^{-1}\,{{\rm{K}}}^{-1}\right)$
$X,\,Y$ dimensionless Cartesian coordinates
$U,\,V$ dimensionless velocities
$x,\,y$ Cartesian coordinates, ${\rm{m}}$
Greek symbols$\gamma $ magnetic incline angle
$\phi $ nanoparticle parameter
${\rm{\Phi }}$ dimensionless concentration
$\mu $ viscosity
$\beta $ thermal expansion coefficient $\left({{\rm{K}}}^{-1}\right)$
$\theta $ dimensionless temperature
$\tau $ dimensionless time
$\rho $ density, $\left({\rm{kg}}\,{{\rm{m}}}^{-3}\right)$
$\nu $ kinematic viscosity, $\left({{\rm{m}}}^{2}\,{{\rm{s}}}^{-1}\right)$
$\sigma $ electrical conductivity
$\psi $ stream function, $\left({{\rm{m}}}^{2}\,{{\rm{s}}}^{-1}\right)$
$\alpha $ thermal diffusivity, $\left({{\rm{m}}}^{2}\,{{\rm{s}}}^{-1}\right)$
Subscripts$c$ low
$nf$ nanofluid
$f$ fluid
$h$ high
$np$ nanoparticles

1. Introduction

The research of the magnetic impacts has received a lot of consideration in engineering due to its wide range of uses for instance polymer and metallurgical industries, where hydro-magnetic practices are employed. Lo [1] studied the magnetic impacts on a buoyancy-driven flow in an enclosure. Oztop et al [2] investigated the MHD natural convection from two semi-circular heaters inside an enclosure. The magnetic field is employed in controlling heat and fluid flow. Knowledge of the magnetic influences on the heat transfer process and flow actions inside enclosures occupied by electrically conducting fluids has become increasingly important [3–10]. Recently, fluid dynamics researchers have shown a strong interest in the development of natural/mixed convection in nanofluid-filled cavities owing to their applications in various disciplines. Sherement and Pop [11] utilized the Buongiorno model to examine the natural convection in a porous cavity occupied by a nanofluid. In a heated closed rectangular enclosure, Alina and Lorenzini [12] studied the thermal behavior of ZnO-water nanofluid. Using the Lattice Boltzmann Method, Nemati et al [13] and Zhou and Yan [14] investigated natural convection through MHD flow in a cavity. The magnetic field is found to minimize cavity circulation. Mehmood et al [15] examine the magnetic impacts and thermal radiation on mixed convection of a nanofluid in a square porous cavity. More studies can be found in [16–27].

There are a considerable number of studies that consider natural convection in different cavities including the inserted bodies. Natural convection caused by a hot inner circular cylinder inside a cold outer enclosure is calculated numerically by Kim et al [28]. Sheikholeslami et al [29] investigated natural convection in a circular cavity including a sinusoidal cylinder. Jabbar et al [30] investigated natural convection in a sinusoidal enclosure having a circular cylinder. Aly [31] examined the double-diffusion in a porous enclosure contained nanofluid over two circular cylinders. Pop et al [32] studied the transmission of thermo-gravitational convection in a differentially heated chamber involving an adiabatic solid body. Sheremet et al [33] studied the thermo-gravitational of Al₂O₃–SiO₂/H₂O in a porous space holding a heat-conducting body. Bhattacharyya et al [34] examined heat and mass transport in a porous channel under the influences of Dufour, Soret, and inclined magnetic field. Kumar et al [35] explained the impacts of magnetite nanofluid over a rotating disk with considering chemical reaction and magnetic field. Shanker et al [36] checked partial velocity slip on MHD convective flow over a stretching surface. The topic of the nonuniform temperature profiles is occurring in several industrial applications, for instance, solar energy collection, building thermal isolations, energy storage, and cooling of electronic elements [37–42].

The periodic changes in the electronic components' current are providing time changes in their surface temperature. This paper treats the magnetic influences on the heat and mass transfer of an electrically conducting nanofluid inside an annulus. The regulating equations of the continuity, momentum, energy, and mass in the dimensionless form are solved by the ISPH method. The main outcomes after studying the impacts of the relevant parameters on the nanofluid flow and lineaments of the heat and mass transfer are:•The increase in the Hartmann number $Ha,$ nanoparticles parameter $\phi ,$ and radius of a superellipse $a$ is slowing down the nanofluid speed in an annulus.
•The values of $\overline{Nu}$ and $\overline{Sh}$ are augmenting as nanoparticles parameter $\phi ,$ Rayleigh number $Ra,$ amplitude $A,$ and frequency $f$ are increasing.
•Increasing $Sr$ with minimizing $Du$ is improving the strength of the concentration distributions in an annulus, and accordingly $\overline{Sh}$ is strongly decreasing.

2. Mathematical analysis

Figure 1 describes the preliminary geometry and its particles model. In an outer cavity, the horizontal walls are presumed to be adiabatic, right wall is held at ${T}_{c}$ and ${C}_{c},$ and the left wall is altered with sinusoidal temperature/concentration in a time.

Figure 1.

New window|Download| PPT slide
Figure 1.Geometry of the problem.


The equation of the superellipse is:(1)$\begin{eqnarray}{\left|\displaystyle \frac{x}{a}\right|}^{n}+{\left|\displaystyle \frac{y}{b}\right|}^{n}=1,\end{eqnarray}$where $n,\,a$ and $b$ are positive numbers, and their values are taken as $n=3/2,$ and $a=b$ is varied throughout the computations. Hence, the superellipse shape is taken as a rhombus with convex corners. The flow assumptions are:•The Boussinesq approximation is utilized, in which density variations are ignored except via the gravity term.
•The inclined magnetic field $\left(\overline{{B}_{0}}\right)$ used with an incline angle $\gamma $ along $x-y$ axis with ignoring the viscous dissipation and Joule heating impacts.
•One phase model is employed for nanofluid modeling.
•The fluid flow is laminar, incompressible, and transitional.


The governing equations are [43, 44]:(2)$\begin{eqnarray}\displaystyle \frac{\partial u}{\partial x}=-\displaystyle \frac{\partial v}{\partial y},\end{eqnarray}$(3)$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{{\rm{d}}u}{{\rm{d}}t}+\displaystyle \frac{{\sigma }_{{\rm{nf}}}\,{B}_{0}^{2}}{{\rho }_{{\rm{nf}}}}\left(u\,{\sin }^{2}\gamma -v\,\sin \,\gamma \,\cos \,\gamma \right)\\ \,=\,-\displaystyle \frac{1}{{\rho }_{{\rm{nf}}}}\displaystyle \frac{\partial p}{\partial x}+\displaystyle \frac{{\mu }_{{\rm{nf}}}}{{\rho }_{{\rm{nf}}}}\left(\displaystyle \frac{{\partial }^{2}u}{\partial {x}^{2}}+\displaystyle \frac{{\partial }^{2}u}{\partial {y}^{2}}\right),\end{array}\end{eqnarray}$(4)$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{{\rm{d}}v}{{\rm{d}}t}+\displaystyle \frac{{\sigma }_{{\rm{nf}}}\,{B}_{0}^{2}}{{\rho }_{{\rm{nf}}}}\left(v\,{\cos }^{2}\gamma \,-u\,\sin \,\gamma \,\cos \,\gamma \right)\\ =\,-\displaystyle \frac{1}{{\rho }_{{\rm{nf}}}}\displaystyle \frac{\partial p}{\partial y}+\displaystyle \frac{{\mu }_{{\rm{nf}}}}{{\rho }_{{\rm{nf}}}}\left(\displaystyle \frac{{\partial }^{2}v}{\partial {x}^{2}}+\displaystyle \frac{{\partial }^{2}v}{\partial {y}^{2}}\right)\\ \,+\,\displaystyle \frac{{\left(\rho {\beta }_{T}\right)}_{{\rm{nf}}}}{{\rho }_{{\rm{nf}}}}\,{g}\left(T-{T}_{c}\right)+\displaystyle \frac{{\left(\rho {\beta }_{C}\right)}_{{\rm{nf}}}}{{\rho }_{{\rm{nf}}}}\,{g}\left(C-{C}_{c}\right),\end{array}\end{eqnarray}$(5)$\begin{eqnarray}\displaystyle \frac{{\rm{d}}T}{{\rm{d}}t}-{\rm{\nabla }}\cdot \left({\alpha }_{{\rm{nf}}}\,{\rm{\nabla }}T\right)=\displaystyle \frac{1}{{\left(\rho {C}_{P}\right)}_{{\rm{nf}}}}{\rm{\nabla }}\cdot \left({D}_{1}\,{\rm{\nabla }}C\right),\end{eqnarray}$(6)$\begin{eqnarray}\displaystyle \frac{{\rm{d}}C}{{\rm{d}}t}-{\rm{\nabla }}\cdot \left({D}_{m}\,{\rm{\nabla }}C\right)={\rm{\nabla }}\cdot \left({D}_{2}\,{\rm{\nabla }}T\right).\end{eqnarray}$

The dimensionless quantities:(7)$\begin{eqnarray}\begin{array}{l}\tau =\displaystyle \frac{t{\alpha }_{{\rm{f}}}}{{L}^{2}},X=\displaystyle \frac{x}{L},Y=\displaystyle \frac{y}{L},U=\displaystyle \frac{uL}{{\alpha }_{{\rm{f}}}},V=\displaystyle \frac{vL}{{\alpha }_{{\rm{f}}}},\\ {\alpha }_{{\rm{f}}}=\displaystyle \frac{{k}_{{\rm{f}}}}{{\left(\rho {C}_{P}\right)}_{{\rm{f}}}},\\ P=\displaystyle \frac{p{L}^{2}}{{\rho }_{{\rm{nf}}\,}{\alpha }_{{\rm{f}}}^{2}\,},\theta =\displaystyle \frac{T-{T}_{c}}{{T}_{h}-{T}_{c}},{\rm{\Phi }}=\displaystyle \frac{C-{C}_{c}}{{C}_{h}-{C}_{c}},Pr=\displaystyle \frac{{\nu }_{{\rm{f}}}}{{\alpha }_{{\rm{f}}}},\\ Ha=\sqrt{\displaystyle \frac{{\sigma }_{{\rm{f}}\,}}{{\mu }_{{\rm{f}}}}}\,{B}_{0}L,\\ Le=\displaystyle \frac{{\alpha }_{{\rm{f}}}}{{D}_{m}},Ra=\displaystyle \frac{{g}{\beta }_{T}\left({T}_{h}-{T}_{c}\right){L}^{3}}{{\nu }_{{\rm{f}}}{\alpha }_{{\rm{f}}}},N=\displaystyle \frac{{\beta }_{C}\left({C}_{h}-{C}_{c}\right)}{{\beta }_{T}\left({T}_{h}-{T}_{c}\right)}.\end{array}\end{eqnarray}$

After substitute equation (7) in (2)–(6), the dimensionless equations are:(8)$\begin{eqnarray}\displaystyle \frac{\partial U}{\partial X}=-\displaystyle \frac{\partial V}{\partial Y},\end{eqnarray}$(9)$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{{\rm{d}}U}{{\rm{d}}\tau }+\displaystyle \frac{{\sigma }_{{\rm{n}}{\rm{f}}}\,{\rho }_{{\rm{f}}}\,}{{\sigma }_{{\rm{f}}}\,{\rho }_{{\rm{n}}{\rm{f}}}}\,H{a}^{2}{\Pr }\left(U\,{{\rm{\sin }}}^{2}\gamma \,-V\,{\rm{\sin }}\,\gamma \,\cos \,\gamma \right)\\ \,=\,-\displaystyle \frac{\partial P}{\partial X}+\displaystyle \frac{{\mu }_{{\rm{n}}{\rm{f}}}}{\,{\rho }_{{\rm{n}}{\rm{f}}}{\alpha }_{{\rm{f}}}}\left(\displaystyle \frac{{\partial }^{2}U}{\partial {X}^{2}}+\displaystyle \frac{{\partial }^{2}U}{\partial {Y}^{2}}\right),\end{array}\end{eqnarray}$(10)$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{{\rm{d}}V}{{\rm{d}}\tau }+\displaystyle \frac{{\sigma }_{{\rm{n}}{\rm{f}}}\,{\rho }_{{\rm{f}}}\,}{{\sigma }_{{\rm{f}}}\,{\rho }_{{\rm{n}}{\rm{f}}}}\,H{a}^{2}\Pr \left(V\,{\cos }^{2}\gamma \,-U\,\sin \,\gamma \,\cos \,\gamma \right)\\ \,=\,-\displaystyle \frac{\partial P}{\partial Y}+\displaystyle \frac{{\mu }_{{\rm{n}}{\rm{f}}}}{\,{\rho }_{{\rm{n}}{\rm{f}}}{\alpha }_{{\rm{f}}}}\left(\displaystyle \frac{{\partial }^{2}V}{\partial {X}^{2}}+\displaystyle \frac{{\partial }^{2}V}{\partial {Y}^{2}}\right)\\ \,+\,\displaystyle \frac{{\left(\rho \beta \right)}_{{\rm{n}}{\rm{f}}}}{{\rho }_{{\rm{n}}{\rm{f}}}\,{\beta }_{{\rm{f}}}}Ra\,Pr\,\left(\theta +N{\rm{\Phi }}\right),\end{array}\end{eqnarray}$(11)$\begin{eqnarray}\displaystyle \frac{{\rm{d}}\theta }{{\rm{d}}\tau }-\displaystyle \frac{{\alpha }_{{\rm{n}}{\rm{f}}}\,}{{\alpha }_{{\rm{f}}}\,}\left(\displaystyle \frac{{\partial }^{2}\theta }{\partial {X}^{2}}+\displaystyle \frac{{\partial }^{2}\theta }{\partial {Y}^{2}}\right)=\displaystyle \frac{1}{{\left(\rho {C}_{P}\right)}_{{\rm{n}}{\rm{f}}}}\,Du\left(\displaystyle \frac{{\partial }^{2}{\rm{\Phi }}}{\partial {X}^{2}}+\displaystyle \frac{{\partial }^{2}{\rm{\Phi }}}{\partial {Y}^{2}}\right),\end{eqnarray}$(12)$\begin{eqnarray}\displaystyle \frac{{\rm{d}}{\rm{\Phi }}}{{\rm{d}}\tau }-\displaystyle \frac{1\,}{Le\,}\left(\displaystyle \frac{{\partial }^{2}{\rm{\Phi }}}{\partial {X}^{2}}+\displaystyle \frac{{\partial }^{2}{\rm{\Phi }}}{\partial {Y}^{2}}\right)=Sr\left(\displaystyle \frac{{\partial }^{2}\theta }{\partial {X}^{2}}+\displaystyle \frac{{\partial }^{2}\theta }{\partial {Y}^{2}}\right),\end{eqnarray}$where $Sr=\tfrac{{D}_{2}}{{\alpha }_{{\rm{f}}}}\left(\tfrac{\left({T}_{h}-{T}_{c}\right)}{\left({C}_{h}-{C}_{c}\right)}\right)$ is a Soret number and $Du=\tfrac{{D}_{1}}{{\alpha }_{{\rm{f}}}}\left(\tfrac{\left({C}_{h}-{C}_{c}\right)}{\left({T}_{h}-{T}_{c}\right)}\right)$ is a Dufour number.

2.1. Dimensionless boundary conditions

(13)$\begin{eqnarray}\begin{array}{l}{\rm{Cavity}}\mbox{'}{\rm{s}}\,\mathrm{right} \mbox{-} \mathrm{wall},{\rm{and}}\,{\rm{an}}\,{\rm{inner}}\,{\rm{blockage}}\,U=0,\\ V=0,\theta =0={\rm{\Phi }},\\ {\rm{Cavity}}\mbox{'}{\rm{s}}\,{\rm{plane}}\,{\rm{walls}}\,U=0,V=0,\displaystyle \frac{\partial \theta }{\partial Y}=0=\displaystyle \frac{\partial {\rm{\Phi }}}{\partial Y},\\ {\rm{Cavity}}\mbox{'}{\rm{s}}\,{\rm{left}}\,{\rm{wall}}\,\theta =1+A\,\sin \left(f\tau \right),\\ \,\,\,\,\,\,\,{\rm{\Phi }}=1+A\,\sin \left(f\tau \right),U=0,V=0.\end{array}\end{eqnarray}$
Mean Sherwood number:(14)$\begin{eqnarray}\overline{Sh}=-1{\int }_{1}\left(\frac{\partial {\rm{\Phi }}}{\partial X}\right){\rm{d}}Y,\,\end{eqnarray}$

Mean Nusselt number:(15)$\begin{eqnarray}\overline{Nu}=-\displaystyle {\int }_{0}^{1}\displaystyle \frac{{k}_{{\rm{nf}}}}{{k}_{{\rm{f}}}}\,\left(\displaystyle \frac{\partial \theta }{\partial X}\right){\rm{d}}{Y}.\end{eqnarray}$

2.2. Nanofluid thermophysical properties

In this study, the water is a base fluid and copper (Cu) is the nanoparticles. The physical attributes of the copper and H₂O are shown in table 1.


Table 1.
Table 1.Physical attributes of copper (Cu) and H₂O [45, 46].

	$\beta \,\left(1/{\rm{K}}\right)$	$\rho \,\left({\rm{kg}}\,{{\rm{m}}}^{-3}\right)$	$k\,\left({\rm{W}}\,{{\rm{m}}}^{-1}\,{{\rm{K}}}^{-1}\right)$	${C}_{P}\left({\rm{J}}\,{\mathrm{kg}}^{-1}\,{{\rm{K}}}^{-1}\right)$	$\sigma \,\left({\rm{S}}\,{{\rm{m}}}^{-1}\right)$
Copper	1.67 $\times \,$10–5	$8933$	$401$	$385$	$5.96\,\times {10}^{7}$
H2O	21 $\times \,$10–5	$997.1$	$0.613$	$4179$	$0.05$

New window|CSV

The density, specific heat, and thermal conductivity of a nanofluid [47–50], are:(16)$\begin{eqnarray}{\rho }_{{\rm{nf}}}=\phi {\rho }_{{\rm{np}}}+{\rho }_{{\rm{f}}}(1-\phi ),\end{eqnarray}$(17)$\begin{eqnarray}{\left(\rho {C}_{p}\right)}_{{\rm{nf}}}=\phi {\left(\rho {C}_{p}\right)}_{{\rm{np}}}+{\left(\rho {C}_{p}\right)}_{{\rm{f}}}-\phi {\left(\rho {C}_{p}\right)}_{{\rm{f}}},\end{eqnarray}$(18)$\begin{eqnarray}\begin{array}{lll}{k}_{{\rm{nf}}} & = & \left(\left({k}_{{\rm{np}}}+2{k}_{{\rm{f}}}\right){k}_{{\rm{f}}}-2\phi {k}_{{\rm{f}}}\left({k}_{{\rm{f}}}-{k}_{{\rm{np}}}\right)\right)\left(\left({k}_{{\rm{np}}}+2{k}_{{\rm{f}}}\right)\right..\\ & & {\left.+\,\phi \left({k}_{{\rm{f}}}-{k}_{{\rm{np}}}\right)\right)}^{-1}\end{array}\end{eqnarray}$

The Brinkman model for effective dynamic viscosity of a nanofluid [51]:(19)$\begin{eqnarray}{\mu }_{{\rm{nf}}}=\displaystyle \frac{{\mu }_{{\rm{f}}}}{{\left(1-\phi \right)}^{2.5}}.\end{eqnarray}$

Electrical conductivity of a nanofluid:(20)$\begin{eqnarray}{\sigma }_{{\rm{nf}}}=\left({\sigma }_{{\rm{f}}}+\displaystyle \frac{3\phi \left({\sigma }_{{\rm{np}}}/{\sigma }_{{\rm{f}}}-1\right){\sigma }_{{\rm{f}}}}{\left({\sigma }_{{\rm{np}}}/{\sigma }_{{\rm{f}}}+2\right)-\left({\sigma }_{{\rm{np}}}/{\sigma }_{{\rm{f}}}-1\right)\phi }\right).\end{eqnarray}$

3. ISPH formulation

The ISPH method employs a quintic kernel function $W:$(21)$\begin{eqnarray}W\left(q,h\right)=\displaystyle \frac{3}{16\,\pi \,{h}^{2}}\left\{\begin{array}{cc}{\left(2-q\right)}^{5}-16{\left(1-q\right)}^{5} & 0\leqslant q\leqslant 1\\ {\left(2-q\right)}^{5} & 1\lt q\leqslant 2,\\ \,0 & q\gt 2\end{array}\right.\end{eqnarray}$where $q={{\boldsymbol{r}}}_{ij}/h.$ The description of $f\left({{\boldsymbol{r}}}_{i}\right)$ in SPH estimation:(22)$\begin{eqnarray}f\left({{\boldsymbol{r}}}_{i}\right)=\displaystyle \frac{1}{{\xi }_{i}}\displaystyle \sum _{j}\displaystyle \frac{{m}_{j}}{{\rho }_{j}}f\left({{\boldsymbol{r}}}_{j}\right)W\left({{\boldsymbol{r}}}_{ij},h\right).\end{eqnarray}$

The renormalization factor ${\xi }_{i}$ [43, 44, 52] is:(23)$\begin{eqnarray}{\xi }_{i}\,=\displaystyle {\int }_{{{\rm{\Omega }}}_{i}}W\left(\left|{{\boldsymbol{r}}}_{ab}\right|\right){\rm{d}}{\rm{\Omega }}\left({{\boldsymbol{r}}}_{j}\right).\end{eqnarray}$

The first derivative is:(24)$\begin{eqnarray}{\rm{\nabla }}{\xi }_{ie}=-\displaystyle {\int }_{{e}_{1}}^{{e}_{2}}{\boldsymbol{n}}\left({{\boldsymbol{r}}}_{j}\right)\,W\left({{\boldsymbol{r}}}_{ij}\right){\rm{d}}{\rm{\Gamma }}\left({{\boldsymbol{r}}}_{j}\right).\end{eqnarray}$

3.1. Solving steps

The projection method [53] is employed in the ISPH method as:

The projected velocities are:(25)$\begin{eqnarray}\begin{array}{lcl}U* & = & {U}^{n}+{\rm{\Delta }}\tau \left(\frac{{\mu }_{{\rm{n}}{\rm{f}}}}{\,{\rho }_{{\rm{n}}{\rm{f}}}{\alpha }_{{\rm{f}}}}{\left(\frac{{\partial }^{2}U}{\partial {X}^{2}}+\frac{{\partial }^{2}U}{\partial {Y}^{2}}\right)}^{n}\right.\\ & & \left.-\,\frac{{\sigma }_{{\rm{n}}{\rm{f}}}{\rho }_{{\rm{f}}}\,}{{\sigma }_{{\rm{f}}}{\rho }_{{\rm{n}}{\rm{f}}}}\,H{a}^{2}{\Pr }\left({U}^{n}\,{\sin }^{2}\gamma \,-{V}^{n}\,\sin \,\gamma \,\cos \,\gamma \right)\right),\end{array}\end{eqnarray}$(26)$\begin{eqnarray}\begin{array}{lcl}{V}^{* } & = & {V}^{{n}}+{\rm{\Delta }}\tau \left(\frac{{\mu }_{{\rm{n}}{\rm{f}}}}{\,{\rho }_{{\rm{n}}{\rm{f}}}{\alpha }_{{\rm{f}}}}{\left(\frac{{\partial }^{2}V}{\partial {X}^{2}}+\frac{{\partial }^{2}V}{\partial {Y}^{2}}\right)}^{n}\right.\\ & & +\,\frac{{\left(\rho \beta \right)}_{{\rm{n}}{\rm{f}}}}{{\rho }_{{\rm{n}}{\rm{f}}}\,{\beta }_{{\rm{f}}}}Ra\,{\Pr }\,\left({\theta }^{n}+N{\Phi }^{n}\right)\\ & & \left.\,-\,\frac{{\sigma }_{{\rm{n}}{\rm{f}}}{\rho }_{{\rm{f}}}\,}{{\sigma }_{{\rm{f}}}{\rho }_{{\rm{n}}{\rm{f}}}}\,H{a}^{2}\,{\Pr }\left({V}^{n}\,{\cos }^{2}\gamma \,-{U}^{{n}}\,\sin \,\gamma \,\cos \,\gamma \right)\right).\end{array}\end{eqnarray}$

Pressure Poisson equation:(27)$\begin{eqnarray}{{\rm{\nabla }}}^{2}{P}^{n+1}=\displaystyle \frac{1}{\,{\rm{\Delta }}\tau }\left(\displaystyle \frac{\partial {U}^{* }}{\partial X}+\displaystyle \frac{\partial {V}^{* }}{\partial Y}\right).\end{eqnarray}$

The updated velocities are:(28)$\begin{eqnarray}{U}^{n+1}={U}^{* }-{\rm{\Delta }}\tau {\left(\displaystyle \frac{\partial P}{\partial X}\right)}^{n+1},\end{eqnarray}$(29)$\begin{eqnarray}{V}^{n+1}={V}^{* }-{\rm{\Delta }}\tau {\left(\displaystyle \frac{\partial P}{\partial Y}\right)}^{n+1}.\end{eqnarray}$

The thermal and concentration equations are:(30)$\begin{eqnarray}\begin{array}{lcc}{\theta }^{n+1} & = & {\theta }^{n}+{\rm{\Delta }}\tau \left(\frac{{\alpha }_{{\rm{n}}{\rm{f}}}\,}{{\alpha }_{{\rm{f}}}\,}{\left(\frac{{\partial }^{2}\theta }{\partial {X}^{2}}+\frac{{\partial }^{2}\theta }{\partial {Y}^{2}}\right)}^{n}\right.\\ & & \left.+\,\frac{1}{{\left(\rho {C}_{P}\right)}_{{\rm{n}}{\rm{f}}}}\,Du{\left(\frac{{\partial }^{2}{\rm{\Phi }}}{\partial {X}^{2}}+\frac{{\partial }^{2}{\rm{\Phi }}}{\partial {Y}^{2}}\right)}^{n}\right),\end{array}\end{eqnarray}$(31)$\begin{eqnarray}\begin{array}{lll}{{\rm{\Phi }}}^{{n}+1} & = & {{\rm{\Phi }}}^{{n}}+{\rm{\Delta }}\tau \left(\displaystyle \frac{1\,}{Le\,}{\left(\displaystyle \frac{{\partial }^{2}{\rm{\Phi }}}{\partial {X}^{2}}+\displaystyle \frac{{\partial }^{2}{\rm{\Phi }}}{\partial {Y}^{2}}\right)}^{{n}}\right.\\ & & \left.+Sr{\left(\displaystyle \frac{{\partial }^{2}\theta }{\partial {X}^{2}}+\displaystyle \frac{{\partial }^{2}\theta }{\partial {Y}^{2}}\right)}^{{n}+1}\right).\end{array}\end{eqnarray}$

The positions are:(32)$\begin{eqnarray}{X}^{n+1}={X}^{n}+{\rm{\Delta }}\tau \,{U}^{n+1},\end{eqnarray}$(33)$\begin{eqnarray}{Y}^{n+1}={Y}^{n}+{\rm{\Delta }}\tau \,{V}^{n+1}.\end{eqnarray}$

The shifting technique is:(34)$\begin{eqnarray}{{\rm{\Upsilon }}}_{{i}^{{\prime} }}={{\rm{\Upsilon }}}_{i}+\left(-{\rm{\Gamma }}\,{\rm{\nabla }}\chi \right)\cdot {\left({\rm{\nabla }}{\rm{\Upsilon }}\right)}_{i}+{\mathscr{O}}\left(\delta {{\boldsymbol{r}}}_{i{i}^{{\prime} }}^{2}\right).\end{eqnarray}$

3.2. Validation of the ISPH method

The comparison between numerical and experimental results from Paroncini and Corvaro [54] and the ISPH results are introduced in figure 2. The comparison showed the agreement of the ISPH results compared to the experimental and numerical results [54]. Further, there are numerous validation examinations during the earlier studies of the ISPH method [43, 44, 52, 55].

Figure 2.

New window|Download| PPT slide
Figure 2.Isotherms of the numerical and experimental results of [54] and the ISPH results.

4. Results and discussion

Results are processed for a large scale of parameters. The frequency and amplitude of the temperature/concentration are varied over $\left(5\leqslant f\leqslant 100\right)$ and $\left(0.5\leqslant A\leqslant 2\right),$ respectively. Hartmann number, nanoparticles parameter, Soret number, Rayleigh number, Dufour number, and radius of a superellipse $a$ are varied as $\left(0\leqslant Ha\leqslant 50\right),$ $\left(0\leqslant \phi \leqslant 0.05\right),$ $\left(0.6\leqslant Sr\leqslant 2\right),$ $\left({10}^{3}\leqslant Ra\leqslant {10}^{5}\right),$ $\left(0.03\leqslant Du\leqslant 1\right),$ and $\left(0.03\leqslant a\leqslant 1\right),$ respectively. All over the computations, buoyancy parameter is $N=1,$ magnetic field's angle is $\gamma =45^\circ ,$ Lewis number $Le=20,$ a superellipse coefficient $n=3/2,$ and Prandtl number $Pr=6.2.$

Figure 3 shows the influences of nanoparticle's parameter $\phi $ on a nanofluid velocity, and deployments of temperature and concentration in an annulus at $\gamma =45^\circ ,N=1,$ $n=3/2,a=0.35,$ $Ra={10}^{4},$ $A=0.5,f=5,$ $Sr=1,Du=0.12,$ and $Ha=10.$ For addition of the nanoparticles, the first remark is a decline in the velocity's maximum by $17.49 \% $ as $\phi $ gets from 0 until 0.05. Physically, adding nanoparticles serves an extra effective viscosity of a nanofluid. The second remark is that an extra value of $\phi $ declines the temperature and enhances the concentration within an annulus between a cavity and an inner superellipse. Figure 4 shows the reliance of $\overline{Nu}$ and $\overline{Sh}$ on the time and nanoparticle's parameters at $\gamma =45^\circ ,N=1,$ $n=3/2,\,a=0.35,Ra={10}^{4},$ $A=0.5,f=5,$ $Sr=1,Du=0.12,$ and $Ha=10.$ It is noted that a significant enhancement is existing in the values of $\overline{Nu}$ and $\overline{Sh}$ for higher nanoparticle's parameter $\phi .$

Figure 3.

New window|Download| PPT slide
Figure 3.The influences of nanoparticle's parameter $\phi $ on nanofluid velocity, and deployments of temperature and concentration at $\gamma =45^\circ ,N=1,n=3/2,$ $a=0.35,Ra={10}^{4},$ $A=0.5,f=5,$ $Sr=1,Du=0.12,$ and $Ha=10.$

Figure 4.

New window|Download| PPT slide
Figure 4.The values of $\overline{Nu}$ and $\overline{Sh}$ below the influences of the nanoparticle's parameter at $\gamma =45^\circ ,N=1,n=3/2,$ $a=0.35,Ra={10}^{4},$ $A=0.5,f=5,$ $Sr=1,Du=0.12,$ and $Ha=10.$


The sequences of the velocity, temperature, and concentration contours are plotted at various Hartmann number $Ha$ at $\gamma =45^\circ ,N=1,n=3/2,$ $a=0.35,Ra={10}^{4},$ $A=0.5,\,f=5,$ $Sr=1,Du=0.12,$ and $\phi =0.06$ are shown in figure 5. Physically, the extra Lorentz forces of a magnetic field are produced at a higher Hartmann number. As a result, the velocity's maximum reduces by 70.93% according to an increase in $Ha$ from 0 to 50. In figures 5(b)–(c), there is a little reduction in the temperature and concentration contours within an annulus as the Hartmann number increases. Further, figure 6 presents the dependence of $\overline{Nu}$ and $\overline{Sh}$ on the Hartmann number at $\gamma =45^\circ ,N=1,n=3/2,$ $a=0.35,Ra={10}^{4},$ $A=0.5,f=5,$ $Sr=1,Du=0.12,$ and $\phi =\mathrm{0.06.}$ It is clear that an increment on the Hartmann number reduces the values of $\overline{Nu}$ and $\overline{Sh}$ which highlighting the Lorentz forces' controls on the convection flow.

Figure 5.

New window|Download| PPT slide
Figure 5.The influences of the Hartmann number on nanofluid velocity, and deployments of temperature and concentration at $\gamma =45^\circ ,N=1,$ $n=3/2,a=0.35,$ $Ra={10}^{4},$ $A=0.5,f=5,$ $Sr=1,Du=0.12,$ and $\phi =0.06.$

Figure 6.

New window|Download| PPT slide
Figure 6.The values of $\overline{Nu}$ and $\overline{Sh}$ below the influences of the Hartmann number at $\gamma =45^\circ ,N=1,n=3/2,$ $a=0.35,Ra={10}^{4},$ $A=0.5,f=5,$ $Sr=1,Du=0.12,$ and $\phi =\mathrm{0.06.}$


Figure 7 introduces the impacts of a superellipse radius $a$ on the nanofluid velocity, and deployments of temperature and concentration in an annulus at $\gamma =45^\circ ,N=1,n=3/2,$ $Ha=10,Ra={10}^{4},$ $A=0.5,f=5,$ $Sr=1,Du=0.12,$ and $\phi =\mathrm{0.06.}$ As the length $a$ controls the radius of an inner superellipse-blockage, an increment in $a$ from $0.1$ to $0.4,$ the velocity's maximum lessens by 66.24% and the temperature and concentration contours are reducing within an annulus. Physically, the inner superellipse represents a blockage for the convection flow, and consequently, as the area of a superellipse increase by an increment in $a,$ the nanofluid movement and the deployments of the temperature and concentration are shrinking within the area between a cavity and an inner blockage. The impacts of the radius of a superellipse $a$ on the values of $\overline{Nu}$ and $\overline{Sh}$ are shown in figure 8. It is noted that an expansion in the radius $a$ augments the values of $\overline{Nu}$ and $\overline{Sh}.$

Figure 7.

New window|Download| PPT slide
Figure 7.The influences of coefficient $a$ for a superellipse on nanofluid velocity, and deployments of temperature and concentration at $\gamma =45^\circ ,N=1,n=3/2,$ $Ha=10,Ra={10}^{4},$ $A=0.5,f=5,$ $Sr=1,Du=0.12,$ and $\phi =\mathrm{0.06.}$

Figure 8.

New window|Download| PPT slide
Figure 8.The values of $\overline{Nu}$ and $\overline{Sh}$ below the influences of the radius of a superellipse $a$ at $\gamma =45^\circ ,N=1,n=3/2,$ $Ha=10,Ra={10}^{4},$ $A=0.5,f=5,$ $Sr=1,Du=0.12,$ and $\phi =\mathrm{0.06.}$


Figures 9 and 10 show the impacts of combination values of the Soret–Dufour parameters (Sr and Du) on the nanofluid velocity, and deployments of temperature and concentration in an annulus as well as $\overline{Nu}$ and $\overline{Sh}$ at $\gamma =45^\circ ,N=1,\,$ $n=3/2,a=0.35,$ $Ha=10,Ra={10}^{4},$ $A=0.5,f=5,$ and $\phi =\mathrm{0.06.}$ In figure 9(a), the velocity's maximum increases by 83.04% as $Sr$ increases from 0.6 to 2 with a decrease in $Du$ from 1 to 0.03. In figures 9(b)–(c), according to an increase in $Sr$ (or a decrease in $Du$), there are slight changes in the temperature and a clear decrease in the concentration within an annulus. In figure 10, $\overline{Nu}$ is slightly enhanced and $\overline{Sh}$ is strongly decreased as $Sr$ increases with a decrease in $Du.$ Physically, Soret number is a mass alter of a temperature difference and Dufour number is a heat alter from the concentration difference. The combinations of $Sr$ and $Du$ can be found are referred in [31, 56, 57].

Figure 9.

New window|Download| PPT slide
Figure 9.The influences of the Soret and Dufour parameters on nanofluid velocity, and deployments of temperature and concentration at $\gamma =45^\circ ,N=1,n=3/2,$ $a=0.35,Ha=10,$ $Ra={10}^{4},A=0.5,f=5,$ and $\phi =\mathrm{0.06.}$

Figure 10.

New window|Download| PPT slide
Figure 10.The values of $\overline{Nu}$ and $\overline{Sh}$ below the influences of Soret and Dufour numbers at $\gamma =45^\circ ,N=1,$ $n=3/2,$ $a=0.35,Ha=10,$ $Ra={10}^{4},$ $A=0.5,f=5,$ and $\phi =\mathrm{0.06.}$


Figures 11 and 12 show the influences of the Rayleigh number $Ra$ on the nanofluid velocity, and deployments of temperature and concentration in an annulus as well as $\overline{Nu}$ and $\overline{Sh}$ at $\gamma =45^\circ ,N=1,n=3/2,$ $a=0.35,Ha=10,$ $A=0.5,f=5,$ $Sr=1,Du=0.12,$ and $\phi =\mathrm{0.06.}$ In figure 11, as $Ra$ powers, the intensity of the velocity field boosts clearly and the temperature and concentration are improved from almost straight lines to the parallel lines across an annulus over a superellipse blockage. In figure 12, an increment in $Ra$ provides a clear increment in the values of $\overline{Nu}$ and $\overline{Sh}.$ Physically, increasing $Ra$ powers the buoyancy force which accelerates the nanofluid movements and enhances the heat/mass transport within an annulus.

Figure 11.

New window|Download| PPT slide
Figure 11.The influences of $Ra$ on nanofluid velocity, and deployments of temperature and concentration at $\gamma =45^\circ ,N=1,$ $n=3/2,a=0.35,$ $Ha=10,$ $A=0.5,f=5,$ $Sr=1,Du=0.12,$ and $\phi =\mathrm{0.06.}$

Figure 12.

New window|Download| PPT slide
Figure 12.The values of $\overline{Nu}$ and $\overline{Sh}$ below the influences of the $Ra$ at $\gamma =45^\circ ,N=1,$ $n=3/2,a=0.35,$ $Ha=10,$ $A=0.5,f=5,$ $Sr=1,Du=0.12,$ and $\phi =\mathrm{0.06.}$


Figures 13–15 present the influences of the amplitude $A$ and frequency $f$ of the temperature and concentration oscillation on the nanofluid velocity, temperature and concentration within an annulus at $\gamma =45^\circ ,N=1,n=3/2,$ $a=0.35,Ha=10,$ $Ra={10}^{4},$ $Sr=1,\,Du=0.12,$ and $\phi =\mathrm{0.06.}$ In figure 13, it is remarked that as an amplitude $A$ raises from 0.5 to 2, the velocity's maximum increases by 66.23% at $f=5,$ whilst it decreases by 42% at $f=50,$ and by 68.18% at $f=100.$ In figures 14 and 15, it is observed that at $f=5,$ the intensity of the temperature and concentration within an annulus is boosting extremely as $A$ increases from 0.5 to 2, whilst at $f=50$ or 100, the intensity of the temperature and concentration is decreasing as $A$ increases from 0.5 to 2. The fluctuations of the results are relevant to the definition of a sine wave for the periodic boundary condition of temperature and concentration in a left wall. Figure 16 shows a 3D-plot of $\overline{Nu}$ and $\overline{Sh}$ below the influences of the amplitude and frequency of the temperature and concentration oscillation at $\gamma =45^\circ ,N=1,n=3/2,$ $a=0.35,Ha=10,$ $Ra={10}^{4},Sr=1,$ $Du=0.12,$ and $\phi =\mathrm{0.06.}$ The values of $\overline{Nu}$ and $\overline{Sh}$ are increasing as both of amplitude $A$ and frequency $f$ are increasing and it has seen when $f=50$ and $A=2,$ the highest values of $\overline{Nu}$ and $\overline{Sh}$ are obtained.

Figure 13.

New window|Download| PPT slide
Figure 13.The influences of the amplitude and frequency of the temperature and concentration oscillation on the velocity field at $\gamma =45^\circ ,N=1,$ $n=3/2,a=0.35,$ $Ha=10,$ $Ra={10}^{4},$ $Sr=1,Du=0.12,$ and $\phi =\mathrm{0.06.}$

Figure 14.

New window|Download| PPT slide
Figure 14.The influences of the amplitude and frequency of the temperature and concentration oscillation on the temperature at $\gamma =45^\circ ,N=1,n=3/2,a=0.35,Ha=10,$ $Ra={10}^{4},Sr=1,Du=0.12,$ and $\phi =0.06.$

Figure 15.

New window|Download| PPT slide
Figure 15.The influences of the amplitude and frequency of the temperature and concentration oscillation on the concentration at $\gamma =45^\circ ,N=1,n=3/2,$ $a=0.35,Ha=10,$ $Ra={10}^{4},$ $Sr=1,Du=0.12,$ and $\phi =\mathrm{0.06.}$

Figure 16.

New window|Download| PPT slide
Figure 16.3D-plot of $\overline{Nu}$ and $\overline{Sh}$ below the influences of the amplitude and frequency of the temperature and concentration oscillation at $\gamma =45^\circ ,N=1,$ $n=3/2,a=0.35,$ $Ha=10,$ $Ra={10}^{4},$ $Sr=1,Du=0.12,$ and $\phi =\mathrm{0.06.}$

5. Conclusion

The transport of heat and mass of an oscillating concentration and temperature in the left-side of an annulus between an inner rhombus with convex corners and an outer cavity is numerically investigated. The annulus is occupied by a nanofluid and is influenced by a magnetic field, thermo-diffusion, and diffusion-thermo. The implications of the pertinent parameters like oscillation amplitude, oscillation frequency, Hartmann number, nanoparticles parameter, Soret number, Rayleigh number, Dufour number, and radius of a superellipse $a$ on the nanofluid flow and features of the heat and mass transmission have been discussed. It is remarked that the velocity's maximum reduces by $70.93 \% $ as $Ha$ raises from 0 to 50, by $66.24 \% $ as a radius of a superellipse $a$ expands from $0.1$ to $0.4.$ As $A$ raises from 0.5 to 2, the velocity's maximum declines by $42 \% $ at $f=50,$ and by $68.18 \% $ at $f=100.$ Whilst the velocity's maximum boosts by 66.23% at $f=5$ as $A$ increases from 0.5 to 2, and by $83.04 \% $ as $Sr$ boosts from 0.6 to 2 with a decrease in $Du$ from 1 to 0.03. As an oscillation amplitude $A$ increases from 0.5 to 2, the strength of the temperature and concentration is extremely boosting at an oscillation frequency $f=5,$ and decreasing at $f=50$ or $100.$ The values of $\overline{Nu}$ and $\overline{Sh}$ are increasing as amplitude $A$ and frequency $f$ are increasing. The highest values of $\overline{Nu}$ and $\overline{Sh}$ are obtained at $f=50$ and $A=2.$ Boosting $Sr$ with lower in $Du,$ leads to the followings: the temperature distributions have little changes, the strength of the concentration distributions is augmented, $\overline{Nu}$ is slightly enhanced, and $\overline{Sh}$ is strongly decreased.

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. 2/144/42). This research was funded by the Deanship of Scientific Research at Princess Nourah Bint Abdulrahman University through the Fast-track Research Funding Program.

Reference View Option By original order
By published year
By cited within times
By Impact factor

[1]Lo D C 2010 High-resolution simulations of magnetohydrodynamic free convection in an enclosure with a transverse magnetic field using a velocity–vorticity formulation
Int. Commun. Heat Mass Transfer 37 514 523
DOI:10.1016/j.icheatmasstransfer.2009.12.013 [Cited within: 1]

[2]Öztop H F Rahman M M Ahsan A Hasanuzzaman M Saidur R Al-Salem K Rahim N A 2012 MHD natural convection in an enclosure from two semi-circular heaters on the bottom wall
Int. J. Heat Mass Transfer 55 1844 1854
DOI:10.1016/j.ijheatmasstransfer.2011.11.037 [Cited within: 1]

[3]Garandet J P Alboussiere T Moreau R 1992 Buoyancy driven convection in a rectangular enclosure with a transverse magnetic field
Int. J. Heat Mass Transfer 35 741 748
DOI:10.1016/0017-9310(92)90242-K [Cited within: 1]

[4]Alchaar S Vasseur P Bilgen E 1995 Natural convection heat transfer in a rectangular enclosure with a transverse magnetic field
Trans.—Am. Soc. Mech. Eng. J. Heat Transfer 117 668
DOI:10.1115/1.2822628

[5]Mehmet Cem E Elif B 2006 Natural-convection flow under a magnetic field in an inclined rectangular enclosure heated and cooled on adjacent walls
Fluid Dyn. Res. 38 564 590
DOI:10.1016/j.fluiddyn.2006.04.002

[6]Teamah M A 2008 Numerical simulation of double diffusive natural convection in rectangular enclosure in the presences of magnetic field and heat source
Int. J. Therm. Sci. 47 237 248
DOI:10.1016/j.ijthermalsci.2007.02.003

[7]Sathiyamoorthy M Chamkha A 2010 Effect of magnetic field on natural convection flow in a liquid gallium filled square cavity for linearly heated side wall(s)
Int. J. Therm. Sci. 49 1856 1865
DOI:10.1016/j.ijthermalsci.2010.04.014

[8]Doostani A Ghalambaz M Chamkha A J 2017 MHD natural convection phase-change heat transfer in a cavity: analysis of the magnetic field effect
J. Braz. Soc. Mech. Sci. Eng. 39 2831 2846
DOI:10.1007/s40430-017-0722-z

[9]Aly A M Ahmed S E Raizah Z 2021 Impacts of variable magnetic field on a ferrofluid flow inside a cavity including a helix using ISPH method
Int. J. Numer. Methods Heat Fluid Flow 31 2150 2171
DOI:10.1108/HFF-08-2020-0501

[10]Alsabery A I Sheremet M A Chamkha A J Hashim I 2018 MHD convective heat transfer in a discretely heated square cavity with conductive inner block using two-phase nanofluid model
Sci. Rep. 8 7410
DOI:10.1038/s41598-018-25749-2 [Cited within: 1]

[11]Sheremet M A Pop I 2014 Natural convection in a square porous cavity with sinusoidal temperature distributions on both side walls filled with a nanofluid: Buongiorno's mathematical model
Transp. Porous Media 105 411 429
DOI:10.1007/s11242-014-0375-7 [Cited within: 1]

[12]Minea A A Lorenzini G 2017 A numerical study on ZnO based nanofluids behavior on natural convection
Int. J. Heat Mass Transfer 114 286 296
DOI:10.1016/j.ijheatmasstransfer.2017.06.069 [Cited within: 1]

[13]Nemati H Farhadi M Sedighi K Ashorynejad H R Fattahi E 2012 Magnetic field effects on natural convection flow of nanofluid in a rectangular cavity using the Lattice Boltzmann model
Scientia Iranica 19 303 310
DOI:10.1016/j.scient.2012.02.016 [Cited within: 1]

[14]Zhou W N Yan Y Y 2015 Numerical investigation of the effects of a magnetic field on nanofluid flow and heat transfer by the lattice boltzmann method
Numer. Heat Transfer A 68 1 16
DOI:10.1080/10407782.2014.965017 [Cited within: 1]

[15]Mehmood K Hussain S Sagheer M 2017 Numerical simulation of MHD mixed convection in alumina–water nanofluid filled square porous cavity using KKL model: effects of non-linear thermal radiation and inclined magnetic field
J. Mol. Liq. 238 485 498
DOI:10.1016/j.molliq.2017.05.019 [Cited within: 1]

[16]Kefayati G R 2013 Effect of a magnetic field on natural convection in an open cavity subjugated to water/alumina nanofluid using Lattice Boltzmann method
Int. Commun. Heat Mass Transfer 40 67 77
DOI:10.1016/j.icheatmasstransfer.2012.10.024 [Cited within: 1]

[17]Rashad A M Rashidi M M Lorenzini G Ahmed S E Aly A M 2017 Magnetic field and internal heat generation effects on the free convection in a rectangular cavity filled with a porous medium saturated with Cu–water nanofluid
Int. J. Heat Mass Transfer 104 878 889
DOI:10.1016/j.ijheatmasstransfer.2016.08.025

[18]Sheikholeslami M Rokni H B 2017 Simulation of nanofluid heat transfer in presence of magnetic field: a review
Int. J. Heat Mass Transfer 115 1203 1233
DOI:10.1016/j.ijheatmasstransfer.2017.08.108

[19]Hussam W K Khanafer K Salem H J Hussam W K Sheard G J 2019 Natural convection heat transfer utilizing nanofluid in a cavity with a periodic side-wall temperature in the presence of a magnetic field
Int. Commun. Heat Mass Transfer 104 127 135
DOI:10.1016/j.icheatmasstransfer.2019.02.018

[20]Ahmed S E Rashed Z Z 2019 MHD natural convection in a heat generating porous medium-filled wavy enclosures using Buongiorno's nanofluid model
Case Stud. Therm. Eng. 14 100430
DOI:10.1016/j.csite.2019.100430

[21]Shehzad S A Sheikholeslami M Ambreen T Shafee A 2020 Convective MHD flow of hybrid-nanofluid within an elliptic porous enclosure
Phys. Lett. A 384 126727
DOI:10.1016/j.physleta.2020.126727

[22]Abdulkadhim A Hamzah H K Ali F H Yildiz C Abed A M Abed E M Arici M 2021 Effect of heat generation and heat absorption on natural convection of Cu-water nanofluid in a wavy enclosure under magnetic field
Int. Commun. Heat Mass Transfer 120 105024
DOI:10.1016/j.icheatmasstransfer.2020.105024

[23]Dogonchi A S Sheremet M A Ganji D D Pop I 2019 Free convection of copper-water nanofluid in a porous gap between hot rectangular cylinder and cold circular cylinder under the effect of inclined magnetic field
J. Therm. Anal. Calorimetry 135 1171 1184
DOI:10.1007/s10973-018-7396-3

[24]Dogonchi A S Tayebi T Chamkha A J Ganji D D 2020 Natural convection analysis in a square enclosure with a wavy circular heater under magnetic field and nanoparticles
J. Therm. Anal. Calorim. 139 661 671
DOI:10.1007/s10973-019-08408-0

[25]Selimefendigil F Öztop H F 2020 Combined effects of double rotating cones and magnetic field on the mixed convection of nanofluid in a porous 3D U-bend
Int. Commun. Heat Mass Transfer 116 104703
DOI:10.1016/j.icheatmasstransfer.2020.104703

[26]Chamkha A J Mansour M A Rashad A M Kargarsharifabad H Armaghani T 2020 Magnetohydrodynamic mixed convection and entropy analysis of nanofluid in gamma-shaped porous cavity
J. Thermophys. Heat Transfer 34 836 847
DOI:10.2514/1.T5983

[27]Aly A M Aldosary A Mohamed E M 2021 Double diffusion in a nanofluid cavity with a wavy hot source subjected to a magnetic field using ISPH method
Alexandria Eng. J. 60 1647 1664
DOI:10.1016/j.aej.2020.11.016 [Cited within: 1]

[28]Kim B S Lee D S Ha M Y Yoon H S 2008 A numerical study of natural convection in a square enclosure with a circular cylinder at different vertical locations
Int. J. Heat Mass Transfer 51 1888 1906
DOI:10.1016/j.ijheatmasstransfer.2007.06.033 [Cited within: 1]

[29]Sheikholeslami M Gorji-Bandpy M Pop I Soleimani S 2013 Numerical study of natural convection between a circular enclosure and a sinusoidal cylinder using control volume based finite element method
Int. J. Therm. Sci. 72 147 158
DOI:10.1016/j.ijthermalsci.2013.05.004 [Cited within: 1]

[30]Jabbar M Ahmed S Hamzah H Ali F 2019 Heat and entropy lines visualization of natural convection between hot inner circular cylinder and cold outer sinusoidal cylinder
Int. J. Heat Technol. 37 1151 1162
DOI:10.18280/ijht.370425 [Cited within: 1]

[31]Aly A M 2017 Natural convection over circular cylinders in a porous enclosure filled with a nanofluid under thermo-diffusion effects
J. Taiwan Inst. Chem. Eng. 70 88 103
DOI:10.1016/j.jtice.2016.10.050 [Cited within: 2]

[32]Pop I Sheremet M A Groşan T 2020 Thermal convection of nanoliquid in a double-connected chamber
Nanomaterials 10 588
DOI:10.3390/nano10030588 [Cited within: 1]

[33]Sheremet M A Cimpean D S Pop I 2020 Thermogravitational convection of hybrid nanofluid in a porous chamber with a central heat-conducting body
Symmetry 12 593
DOI:10.3390/sym12040593 [Cited within: 1]

[34]Bhattacharyya A Kumar R Seth G S 2021 Capturing the features of peristaltic transport of a chemically reacting couple stress fluid through an inclined asymmetric channel with Dufour and Soret effects in presence of inclined magnetic field
Indian J. Phys.
DOI:10.1007/s12648-020-01936-8 [Cited within: 1]

[35]Kumar R Bhattacharyya A Seth G S Chamkha A J 2021 Transportation of magnetite nanofluid flow and heat transfer over a rotating porous disk with Arrhenius activation energy: fourth order Noumerov's method
Chin. J. Phys. 69 172 185
DOI:10.1016/j.cjph.2020.11.018 [Cited within: 1]

[36]Seth G S Bhattacharyya A Mishra M K 2019 Study of partial slip mechanism on free convection flow of viscoelastic fluid past a nonlinearly stretching surface
Comput. Therm. Sci.: Int. J. 11 105 117
DOI:10.1615/ComputThermalScien.2018024728 [Cited within: 1]

[37]Kazmierczak M Chinoda Z 1992 Buoyancy-driven flow in an enclosure with time periodic boundary conditions
Int. J. Heat Mass Transfer 35 1507 1518
DOI:10.1016/0017-9310(92)90040-Y [Cited within: 1]

[38]Burgess I W El-Rimawi J A Plank R J 1990 Analysis of beams with non-uniform temperature profile due to fire exposure
J. Constr. Steel Res. 16 169 192
DOI:10.1016/0143-974X(90)90008-5

[39]Wang G Meng X Zeng M Ozoe H Wang Q W 2014 Natural convection heat transfer of Copper–Water nanofluid in a square cavity with time-periodic boundary temperature
Heat Transfer Eng. 35 630 640
DOI:10.1080/01457632.2013.837684

[40]Aly A M Chamkha A J Lee S-W Al-Mudhaf A F 2016 On mixed convection in an inclined lid-driven cavity with sinusoidal heated walls using the ISPH method
Comput. Therm. Sci.: Int. J. 8 337 354
DOI:10.1615/ComputThermalScien.2016016527

[41]Revnic C Ghalambaz M Groşan T Sheremet M Pop I 2019 Impacts of non-uniform border temperature variations on time-dependent nanofluid free convection within a trapezium: Buongiorno's nanofluid model
Energies 12
DOI:10.3390/en12081461

[42]Hussam W K Khanafer K Salem H J Sheard G J 2019 Natural convection heat transfer utilizing nanofluid in a cavity with a periodic side-wall temperature in the presence of a magnetic field
Int. Commun. Heat Mass Transfer 104 127 135
DOI:10.1016/j.icheatmasstransfer.2019.02.018 [Cited within: 1]

[43]Nguyen M T Aly A M Lee S-W 2017 Effect of a wavy interface on the natural convection of a nanofluid in a cavity with a partially layered porous medium using the ISPH method
Numer. Heat Transfer A 72 68 88
DOI:10.1080/10407782.2017.1353385 [Cited within: 3]

[44]Nguyen M T Aly A M Lee S-W 2018 ISPH modeling of natural convection heat transfer with an analytical kernel renormalization factor
Meccanica 53 2299 2318
DOI:10.1007/s11012-018-0825-3 [Cited within: 3]

[45]Ahmed S E Elshehabey H M 2018 Buoyancy-driven flow of nanofluids in an inclined enclosure containing an adiabatic obstacle with heat generation/absorption: Effects of periodic thermal conditions
Int. J. Heat Mass Transfer 124 58 73
DOI:10.1016/j.ijheatmasstransfer.2018.03.044 [Cited within: 1]

[46]Aly A M Raizah Z Ahmed S E 2018 Mixed convection in a cavity saturated with wavy layer porous medium: entropy generation
J. Thermophys. Heat Transfer 32 764 780
DOI:10.2514/1.T5369 [Cited within: 1]

[47]Kefayati G R 2015 FDLBM simulation of mixed convection in a lid-driven cavity filled with non-Newtonian nanofluid in the presence of magnetic field
Int. J. Therm. Sci. 95 29 46
DOI:10.1016/j.ijthermalsci.2015.03.018 [Cited within: 1]

[48]Mahapatra T Saha B C Pal D 2018 Magnetohydrodynamic double-diffusive natural convection for nanofluid within a trapezoidal enclosure
Comput. Appl. Math. 37 6132 6151
DOI:10.1007/s40314-018-0676-5

[49]Xuan Y Roetzel W 2000 Conceptions for heat transfer correlation of nanofluids
Int. J. Heat Mass Transfer 43 3701 3707
DOI:10.1016/S0017-9310(99)00369-5

[50]Aly A M Raizah Z Ahmed S E 2018 Mixed convection in a cavity saturated with wavy layer porous medium: entropy generation
J. Porous Media 32 764 780
[Cited within: 1]

[51]Brinkman H C 1952 The viscosity of concentrated suspensions and solutions
J. Chem. Phys. 20 571 571
DOI:10.1063/1.1700493 [Cited within: 1]

[52]Nguyen Tuan M Aly Abdelraheem M Lee S-W 2018 Improved wall boundary conditions in the incompressible smoothed particle hydrodynamics method
Int. J. Numer. Methods Heat Fluid Flow 28 704 725
DOI:10.1108/HFF-02-2017-0056 [Cited within: 2]

[53]Cummins S J Rudman M 1999 An SPH projection method
J. Comput. Phys. 152 584 607
DOI:10.1006/jcph.1999.6246 [Cited within: 1]

[54]Paroncini M Corvaro F 2009 Natural convection in a square enclosure with a hot source
Int. J. Therm. Sci. 48 1683 1695
DOI:10.1016/j.ijthermalsci.2009.02.005 [Cited within: 3]

[55]Nguyen M T Aly A M Lee S-W 2018 A numerical study on unsteady natural/mixed convection in a cavity with fixed and moving rigid bodies using the ISPH method
Int. J. Numer. Methods Heat Fluid Flow 28 684 703
DOI:10.1108/HFF-02-2017-0058 [Cited within: 1]

[56]Mansour M A El-Anssary N F Aly A M 2008 Effects of chemical reaction and thermal stratification on MHD free convective heat and mass transfer over a vertical stretching surface embedded in a porous media considering Soret and Dufour numbers
Chem. Eng. J. 145 340 345
DOI:10.1016/j.cej.2008.08.016 [Cited within: 1]

[57]Chamkha A J Aly A M 2010 Heat and mass transfer in stagnation-point flow of a polar fluid towards a stretching surface in porous media in the presence of soret, dufour and chemical reaction effects
Chem. Eng. Commun. 198 214 234
DOI:10.1080/00986445.2010.500161 [Cited within: 1]