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, ${\rm{m}}$
$\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, $\left({{\rm{m}}}^{2}\,{{\rm{s}}}^{-1}\right)$
$c$ 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. Geometry of the problem. |
The equation of the superellipse is:
$\begin{eqnarray}{\left|\displaystyle \frac{x}{a}\right|}^{n}+{\left|\displaystyle \frac{y}{b}\right|}^{n}=1,\end{eqnarray}$
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]:
$\begin{eqnarray}\displaystyle \frac{\partial u}{\partial x}=-\displaystyle \frac{\partial v}{\partial y},\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{{\rm{d}}u}{{\rm{d}}t}+\displaystyle \frac{{\sigma }_{{\rm{nf}}}\,{B}_{0}^{2}}{{\rho }_{{\rm{nf}}}}\left(u\,{\sin }^{2}\gamma -v\,\sin \,\gamma \,\cos \,\gamma \right)\\ \,=\,-\displaystyle \frac{1}{{\rho }_{{\rm{nf}}}}\displaystyle \frac{\partial p}{\partial x}+\displaystyle \frac{{\mu }_{{\rm{nf}}}}{{\rho }_{{\rm{nf}}}}\left(\displaystyle \frac{{\partial }^{2}u}{\partial {x}^{2}}+\displaystyle \frac{{\partial }^{2}u}{\partial {y}^{2}}\right),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{{\rm{d}}v}{{\rm{d}}t}+\displaystyle \frac{{\sigma }_{{\rm{nf}}}\,{B}_{0}^{2}}{{\rho }_{{\rm{nf}}}}\left(v\,{\cos }^{2}\gamma \,-u\,\sin \,\gamma \,\cos \,\gamma \right)\\ =\,-\displaystyle \frac{1}{{\rho }_{{\rm{nf}}}}\displaystyle \frac{\partial p}{\partial y}+\displaystyle \frac{{\mu }_{{\rm{nf}}}}{{\rho }_{{\rm{nf}}}}\left(\displaystyle \frac{{\partial }^{2}v}{\partial {x}^{2}}+\displaystyle \frac{{\partial }^{2}v}{\partial {y}^{2}}\right)\\ \,+\,\displaystyle \frac{{\left(\rho {\beta }_{T}\right)}_{{\rm{nf}}}}{{\rho }_{{\rm{nf}}}}\,{g}\left(T-{T}_{c}\right)+\displaystyle \frac{{\left(\rho {\beta }_{C}\right)}_{{\rm{nf}}}}{{\rho }_{{\rm{nf}}}}\,{g}\left(C-{C}_{c}\right),\end{array}\end{eqnarray}$
$\begin{eqnarray}\displaystyle \frac{{\rm{d}}T}{{\rm{d}}t}-{\rm{\nabla }}\cdot \left({\alpha }_{{\rm{nf}}}\,{\rm{\nabla }}T\right)=\displaystyle \frac{1}{{\left(\rho {C}_{P}\right)}_{{\rm{nf}}}}{\rm{\nabla }}\cdot \left({D}_{1}\,{\rm{\nabla }}C\right),\end{eqnarray}$
$\begin{eqnarray}\displaystyle \frac{{\rm{d}}C}{{\rm{d}}t}-{\rm{\nabla }}\cdot \left({D}_{m}\,{\rm{\nabla }}C\right)={\rm{\nabla }}\cdot \left({D}_{2}\,{\rm{\nabla }}T\right).\end{eqnarray}$
The dimensionless quantities:
$\begin{eqnarray}\begin{array}{l}\tau =\displaystyle \frac{t{\alpha }_{{\rm{f}}}}{{L}^{2}},X=\displaystyle \frac{x}{L},Y=\displaystyle \frac{y}{L},U=\displaystyle \frac{uL}{{\alpha }_{{\rm{f}}}},V=\displaystyle \frac{vL}{{\alpha }_{{\rm{f}}}},\\ {\alpha }_{{\rm{f}}}=\displaystyle \frac{{k}_{{\rm{f}}}}{{\left(\rho {C}_{P}\right)}_{{\rm{f}}}},\\ P=\displaystyle \frac{p{L}^{2}}{{\rho }_{{\rm{nf}}\,}{\alpha }_{{\rm{f}}}^{2}\,},\theta =\displaystyle \frac{T-{T}_{c}}{{T}_{h}-{T}_{c}},{\rm{\Phi }}=\displaystyle \frac{C-{C}_{c}}{{C}_{h}-{C}_{c}},Pr=\displaystyle \frac{{\nu }_{{\rm{f}}}}{{\alpha }_{{\rm{f}}}},\\ Ha=\sqrt{\displaystyle \frac{{\sigma }_{{\rm{f}}\,}}{{\mu }_{{\rm{f}}}}}\,{B}_{0}L,\\ Le=\displaystyle \frac{{\alpha }_{{\rm{f}}}}{{D}_{m}},Ra=\displaystyle \frac{{g}{\beta }_{T}\left({T}_{h}-{T}_{c}\right){L}^{3}}{{\nu }_{{\rm{f}}}{\alpha }_{{\rm{f}}}},N=\displaystyle \frac{{\beta }_{C}\left({C}_{h}-{C}_{c}\right)}{{\beta }_{T}\left({T}_{h}-{T}_{c}\right)}.\end{array}\end{eqnarray}$
After substitute equation (7 ) in (2 )–(6 ), the dimensionless equations are:
$\begin{eqnarray}\displaystyle \frac{\partial U}{\partial X}=-\displaystyle \frac{\partial V}{\partial Y},\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{{\rm{d}}U}{{\rm{d}}\tau }+\displaystyle \frac{{\sigma }_{{\rm{n}}{\rm{f}}}\,{\rho }_{{\rm{f}}}\,}{{\sigma }_{{\rm{f}}}\,{\rho }_{{\rm{n}}{\rm{f}}}}\,H{a}^{2}{\Pr }\left(U\,{{\rm{\sin }}}^{2}\gamma \,-V\,{\rm{\sin }}\,\gamma \,\cos \,\gamma \right)\\ \,=\,-\displaystyle \frac{\partial P}{\partial X}+\displaystyle \frac{{\mu }_{{\rm{n}}{\rm{f}}}}{\,{\rho }_{{\rm{n}}{\rm{f}}}{\alpha }_{{\rm{f}}}}\left(\displaystyle \frac{{\partial }^{2}U}{\partial {X}^{2}}+\displaystyle \frac{{\partial }^{2}U}{\partial {Y}^{2}}\right),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{{\rm{d}}V}{{\rm{d}}\tau }+\displaystyle \frac{{\sigma }_{{\rm{n}}{\rm{f}}}\,{\rho }_{{\rm{f}}}\,}{{\sigma }_{{\rm{f}}}\,{\rho }_{{\rm{n}}{\rm{f}}}}\,H{a}^{2}\Pr \left(V\,{\cos }^{2}\gamma \,-U\,\sin \,\gamma \,\cos \,\gamma \right)\\ \,=\,-\displaystyle \frac{\partial P}{\partial Y}+\displaystyle \frac{{\mu }_{{\rm{n}}{\rm{f}}}}{\,{\rho }_{{\rm{n}}{\rm{f}}}{\alpha }_{{\rm{f}}}}\left(\displaystyle \frac{{\partial }^{2}V}{\partial {X}^{2}}+\displaystyle \frac{{\partial }^{2}V}{\partial {Y}^{2}}\right)\\ \,+\,\displaystyle \frac{{\left(\rho \beta \right)}_{{\rm{n}}{\rm{f}}}}{{\rho }_{{\rm{n}}{\rm{f}}}\,{\beta }_{{\rm{f}}}}Ra\,Pr\,\left(\theta +N{\rm{\Phi }}\right),\end{array}\end{eqnarray}$
$\begin{eqnarray}\displaystyle \frac{{\rm{d}}\theta }{{\rm{d}}\tau }-\displaystyle \frac{{\alpha }_{{\rm{n}}{\rm{f}}}\,}{{\alpha }_{{\rm{f}}}\,}\left(\displaystyle \frac{{\partial }^{2}\theta }{\partial {X}^{2}}+\displaystyle \frac{{\partial }^{2}\theta }{\partial {Y}^{2}}\right)=\displaystyle \frac{1}{{\left(\rho {C}_{P}\right)}_{{\rm{n}}{\rm{f}}}}\,Du\left(\displaystyle \frac{{\partial }^{2}{\rm{\Phi }}}{\partial {X}^{2}}+\displaystyle \frac{{\partial }^{2}{\rm{\Phi }}}{\partial {Y}^{2}}\right),\end{eqnarray}$
$\begin{eqnarray}\displaystyle \frac{{\rm{d}}{\rm{\Phi }}}{{\rm{d}}\tau }-\displaystyle \frac{1\,}{Le\,}\left(\displaystyle \frac{{\partial }^{2}{\rm{\Phi }}}{\partial {X}^{2}}+\displaystyle \frac{{\partial }^{2}{\rm{\Phi }}}{\partial {Y}^{2}}\right)=Sr\left(\displaystyle \frac{{\partial }^{2}\theta }{\partial {X}^{2}}+\displaystyle \frac{{\partial }^{2}\theta }{\partial {Y}^{2}}\right),\end{eqnarray}$
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
$\begin{eqnarray}\begin{array}{l}{\rm{Cavity}}\mbox{'}{\rm{s}}\,\mathrm{right} \mbox{-} \mathrm{wall},{\rm{and}}\,{\rm{an}}\,{\rm{inner}}\,{\rm{blockage}}\,U=0,\\ V=0,\theta =0={\rm{\Phi }},\\ {\rm{Cavity}}\mbox{'}{\rm{s}}\,{\rm{plane}}\,{\rm{walls}}\,U=0,V=0,\displaystyle \frac{\partial \theta }{\partial Y}=0=\displaystyle \frac{\partial {\rm{\Phi }}}{\partial Y},\\ {\rm{Cavity}}\mbox{'}{\rm{s}}\,{\rm{left}}\,{\rm{wall}}\,\theta =1+A\,\sin \left(f\tau \right),\\ \,\,\,\,\,\,\,{\rm{\Phi }}=1+A\,\sin \left(f\tau \right),U=0,V=0.\end{array}\end{eqnarray}$
Mean Sherwood number:
$\begin{eqnarray}\overline{Sh}=-1{\int }_{1}\left(\frac{\partial {\rm{\Phi }}}{\partial X}\right){\rm{d}}Y,\,\end{eqnarray}$
Mean Nusselt number:
$\begin{eqnarray}\overline{Nu}=-\displaystyle {\int }_{0}^{1}\displaystyle \frac{{k}_{{\rm{nf}}}}{{k}_{{\rm{f}}}}\,\left(\displaystyle \frac{\partial \theta }{\partial X}\right){\rm{d}}{Y}.\end{eqnarray}$
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.
| $\beta \,\left(1/{\rm{K}}\right)$ | $\rho \,\left({\rm{kg}}\,{{\rm{m}}}^{-3}\right)$ | $k\,\left({\rm{W}}\,{{\rm{m}}}^{-1}\,{{\rm{K}}}^{-1}\right)$ | ${C}_{P}\left({\rm{J}}\,{\mathrm{kg}}^{-1}\,{{\rm{K}}}^{-1}\right)$ | $\sigma \,\left({\rm{S}}\,{{\rm{m}}}^{-1}\right)$ | |
|---|---|---|---|---|---|
| Copper | 1.67 $\times \,$10–5 | $8933$ | $401$ | $385$ | $5.96\,\times {10}^{7}$ |
| H2O | 21 $\times \,$10–5 | $997.1$ | $0.613$ | $4179$ | $0.05$ |
The density, specific heat, and thermal conductivity of a nanofluid [47–50], are:
$\begin{eqnarray}{\rho }_{{\rm{nf}}}=\phi {\rho }_{{\rm{np}}}+{\rho }_{{\rm{f}}}(1-\phi ),\end{eqnarray}$
$\begin{eqnarray}{\left(\rho {C}_{p}\right)}_{{\rm{nf}}}=\phi {\left(\rho {C}_{p}\right)}_{{\rm{np}}}+{\left(\rho {C}_{p}\right)}_{{\rm{f}}}-\phi {\left(\rho {C}_{p}\right)}_{{\rm{f}}},\end{eqnarray}$
$\begin{eqnarray}\begin{array}{lll}{k}_{{\rm{nf}}} & = & \left(\left({k}_{{\rm{np}}}+2{k}_{{\rm{f}}}\right){k}_{{\rm{f}}}-2\phi {k}_{{\rm{f}}}\left({k}_{{\rm{f}}}-{k}_{{\rm{np}}}\right)\right)\left(\left({k}_{{\rm{np}}}+2{k}_{{\rm{f}}}\right)\right..\\ & & {\left.+\,\phi \left({k}_{{\rm{f}}}-{k}_{{\rm{np}}}\right)\right)}^{-1}\end{array}\end{eqnarray}$
The Brinkman model for effective dynamic viscosity of a nanofluid [51]:
$\begin{eqnarray}{\mu }_{{\rm{nf}}}=\displaystyle \frac{{\mu }_{{\rm{f}}}}{{\left(1-\phi \right)}^{2.5}}.\end{eqnarray}$
Electrical conductivity of a nanofluid:
$\begin{eqnarray}{\sigma }_{{\rm{nf}}}=\left({\sigma }_{{\rm{f}}}+\displaystyle \frac{3\phi \left({\sigma }_{{\rm{np}}}/{\sigma }_{{\rm{f}}}-1\right){\sigma }_{{\rm{f}}}}{\left({\sigma }_{{\rm{np}}}/{\sigma }_{{\rm{f}}}+2\right)-\left({\sigma }_{{\rm{np}}}/{\sigma }_{{\rm{f}}}-1\right)\phi }\right).\end{eqnarray}$
3. ISPH formulation
The ISPH method employs a quintic kernel function $W:$
$\begin{eqnarray}W\left(q,h\right)=\displaystyle \frac{3}{16\,\pi \,{h}^{2}}\left\{\begin{array}{cc}{\left(2-q\right)}^{5}-16{\left(1-q\right)}^{5} & 0\leqslant q\leqslant 1\\ {\left(2-q\right)}^{5} & 1\lt q\leqslant 2,\\ \,0 & q\gt 2\end{array}\right.\end{eqnarray}$
where $q={{\boldsymbol{r}}}_{ij}/h.$ The description of $f\left({{\boldsymbol{r}}}_{i}\right)$ in SPH estimation: $\begin{eqnarray}f\left({{\boldsymbol{r}}}_{i}\right)=\displaystyle \frac{1}{{\xi }_{i}}\displaystyle \sum _{j}\displaystyle \frac{{m}_{j}}{{\rho }_{j}}f\left({{\boldsymbol{r}}}_{j}\right)W\left({{\boldsymbol{r}}}_{ij},h\right).\end{eqnarray}$
The first derivative is:
$\begin{eqnarray}{\rm{\nabla }}{\xi }_{ie}=-\displaystyle {\int }_{{e}_{1}}^{{e}_{2}}{\boldsymbol{n}}\left({{\boldsymbol{r}}}_{j}\right)\,W\left({{\boldsymbol{r}}}_{ij}\right){\rm{d}}{\rm{\Gamma }}\left({{\boldsymbol{r}}}_{j}\right).\end{eqnarray}$
3.1. Solving steps
The projection method [53] is employed in the ISPH method as:
The projected velocities are:
$\begin{eqnarray}\begin{array}{lcl}U* & = & {U}^{n}+{\rm{\Delta }}\tau \left(\frac{{\mu }_{{\rm{n}}{\rm{f}}}}{\,{\rho }_{{\rm{n}}{\rm{f}}}{\alpha }_{{\rm{f}}}}{\left(\frac{{\partial }^{2}U}{\partial {X}^{2}}+\frac{{\partial }^{2}U}{\partial {Y}^{2}}\right)}^{n}\right.\\ & & \left.-\,\frac{{\sigma }_{{\rm{n}}{\rm{f}}}{\rho }_{{\rm{f}}}\,}{{\sigma }_{{\rm{f}}}{\rho }_{{\rm{n}}{\rm{f}}}}\,H{a}^{2}{\Pr }\left({U}^{n}\,{\sin }^{2}\gamma \,-{V}^{n}\,\sin \,\gamma \,\cos \,\gamma \right)\right),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{lcl}{V}^{* } & = & {V}^{{n}}+{\rm{\Delta }}\tau \left(\frac{{\mu }_{{\rm{n}}{\rm{f}}}}{\,{\rho }_{{\rm{n}}{\rm{f}}}{\alpha }_{{\rm{f}}}}{\left(\frac{{\partial }^{2}V}{\partial {X}^{2}}+\frac{{\partial }^{2}V}{\partial {Y}^{2}}\right)}^{n}\right.\\ & & +\,\frac{{\left(\rho \beta \right)}_{{\rm{n}}{\rm{f}}}}{{\rho }_{{\rm{n}}{\rm{f}}}\,{\beta }_{{\rm{f}}}}Ra\,{\Pr }\,\left({\theta }^{n}+N{\Phi }^{n}\right)\\ & & \left.\,-\,\frac{{\sigma }_{{\rm{n}}{\rm{f}}}{\rho }_{{\rm{f}}}\,}{{\sigma }_{{\rm{f}}}{\rho }_{{\rm{n}}{\rm{f}}}}\,H{a}^{2}\,{\Pr }\left({V}^{n}\,{\cos }^{2}\gamma \,-{U}^{{n}}\,\sin \,\gamma \,\cos \,\gamma \right)\right).\end{array}\end{eqnarray}$
Pressure Poisson equation:
$\begin{eqnarray}{{\rm{\nabla }}}^{2}{P}^{n+1}=\displaystyle \frac{1}{\,{\rm{\Delta }}\tau }\left(\displaystyle \frac{\partial {U}^{* }}{\partial X}+\displaystyle \frac{\partial {V}^{* }}{\partial Y}\right).\end{eqnarray}$
The updated velocities are:
$\begin{eqnarray}{U}^{n+1}={U}^{* }-{\rm{\Delta }}\tau {\left(\displaystyle \frac{\partial P}{\partial X}\right)}^{n+1},\end{eqnarray}$
$\begin{eqnarray}{V}^{n+1}={V}^{* }-{\rm{\Delta }}\tau {\left(\displaystyle \frac{\partial P}{\partial Y}\right)}^{n+1}.\end{eqnarray}$
The thermal and concentration equations are:
$\begin{eqnarray}\begin{array}{lcc}{\theta }^{n+1} & = & {\theta }^{n}+{\rm{\Delta }}\tau \left(\frac{{\alpha }_{{\rm{n}}{\rm{f}}}\,}{{\alpha }_{{\rm{f}}}\,}{\left(\frac{{\partial }^{2}\theta }{\partial {X}^{2}}+\frac{{\partial }^{2}\theta }{\partial {Y}^{2}}\right)}^{n}\right.\\ & & \left.+\,\frac{1}{{\left(\rho {C}_{P}\right)}_{{\rm{n}}{\rm{f}}}}\,Du{\left(\frac{{\partial }^{2}{\rm{\Phi }}}{\partial {X}^{2}}+\frac{{\partial }^{2}{\rm{\Phi }}}{\partial {Y}^{2}}\right)}^{n}\right),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{lll}{{\rm{\Phi }}}^{{n}+1} & = & {{\rm{\Phi }}}^{{n}}+{\rm{\Delta }}\tau \left(\displaystyle \frac{1\,}{Le\,}{\left(\displaystyle \frac{{\partial }^{2}{\rm{\Phi }}}{\partial {X}^{2}}+\displaystyle \frac{{\partial }^{2}{\rm{\Phi }}}{\partial {Y}^{2}}\right)}^{{n}}\right.\\ & & \left.+Sr{\left(\displaystyle \frac{{\partial }^{2}\theta }{\partial {X}^{2}}+\displaystyle \frac{{\partial }^{2}\theta }{\partial {Y}^{2}}\right)}^{{n}+1}\right).\end{array}\end{eqnarray}$
The positions are:
$\begin{eqnarray}{X}^{n+1}={X}^{n}+{\rm{\Delta }}\tau \,{U}^{n+1},\end{eqnarray}$
$\begin{eqnarray}{Y}^{n+1}={Y}^{n}+{\rm{\Delta }}\tau \,{V}^{n+1}.\end{eqnarray}$
The shifting technique is:
$\begin{eqnarray}{{\rm{\Upsilon }}}_{{i}^{{\prime} }}={{\rm{\Upsilon }}}_{i}+\left(-{\rm{\Gamma }}\,{\rm{\nabla }}\chi \right)\cdot {\left({\rm{\nabla }}{\rm{\Upsilon }}\right)}_{i}+{\mathscr{O}}\left(\delta {{\boldsymbol{r}}}_{i{i}^{{\prime} }}^{2}\right).\end{eqnarray}$
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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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.