Thermal analysis of magnetic Iron-Oxide nanoparticle with combination of Water and Ethylene Glycol passes through a partially heated permeable square enclosure

Abstract

The research explores the use of iron oxide nanoparticles to enhance the thermal properties of fluids like water and ethylene glycol. These nanoparticles find applications in engineering and medicine, including wastewater treatment and coatings. The study investigates magnetic nanoparticle movement in a permeable enclosure, considering heat, magnetic field, and electromagnetic effects. Using the MAC method, dimensionless equations were solved, and results were presented through graphs, charts, and tables. Findings show nanoparticle volume significantly impacts heat transfer in nanofluids compared to base fluids. Increasing nanoparticle volume from 0 to 0.05 improved nanofluid heat transfer by 0.278% and base fluid by 0.303%. Heat absorption and generation improved nanofluid transfer by 15.819% and base fluid by 0.596%. Nusselt number increased with higher Eckert number and heat source/sink parameters in both nanofluids and base fluids.

KEYWORDS:

• Nanofluids
• mixed convection
• porous
• viscous dissipation
• heat source/sink
• ohmic heating

Nomenclature

$x,y$	=	Dimensional Cartesian coordinates [m]
$X,Y$	=	Dimensionless Cartesian coordinate system [–]
$u,v$	=	Dimensional velocity components in x, y directions [ms−1]
$L$	=	Length of the enclosure [m]
$g$	=	Acceleration due to gravity [ms−2]
$p$	=	Dimensional pressure [Pa]
$U,V$	=	Dimensionless velocities along X and Y directions [–]
$K$	=	Permeability of the porous medium [m2]
$B_0$	=	Magnetic field strength [kg s−2A−1]
$H$	=	Height of the enclosure [m]
$U_0$	=	Constant reference velocity [ms−1]
$T_h$	=	Temperature of the hot wall [K]
$Q_0$	=	Dimensional heat generation or absorption [W m−3]
$T_c$	=	Temperature of the cold wall [K]
$P$	=	Dimensionless pressure [–]
$T$	=	Dimensional temperature of the fluid [K]
$\mathrm{Re}$	=	Reynolds number [–]
$\mathrm{Ri}$	=	Richardson number [–]
$\mathrm{Gr}$	=	Grashof number [–]
$t^{\ast }$	=	Dimensional time [s]
$Pr$	=	Prandtl number [–]
$Q$	=	Dimensionless heat generation/absorption coefficient [–]
$\mathrm{Ec}$	=	Eckert number [–]
$\mathrm{Nu}$	=	Nusselt number [–]
$\mathrm{Da}$	=	Darcy number [–]
$\mathrm{Ha}$	=	Hartmann number [–]

Greek symbols

$\rho$	=	Density [kg m−3]
$\nu$	=	Kinematic viscosity [m2 s−1]
$\mu$	=	Dynamic viscosity [kg m−1s−1]
$\alpha$	=	Thermal diffusivity [m2 s−1]
$\beta$	=	Thermal expansion coefficient [K−1]
$\theta$	=	Dimensionless temperature [−]
$\sigma$	=	Electrical conductivity [S m−1]

Subscripts

f	=	Fluid
nf	=	Nanofluid
p	=	Nanoparticle

Abbreviations

NP	=	Nanoparticle
PW	=	Paraffin Wax

1. Introduction

Nanofluids are used in numerous sectors like medical applications, electrical elements, transportation, etc. [1–5]. In general, neither metal nor metal oxide solid particles ($\mathrm{Ag},\mathrm{Cu}$, $\mathrm{CuO}$, $A{l}_2{O}_3$, $\mathrm{Ti}{O}_2$, $\mathrm{Si}{O}_2$, and $\mathrm{ZnO}$) should be used in nanoliquids. There are a number of studies [6–12] done theoretically or experimentally to examine the nanoliquid flow and its energy transmission characteristics.

In current times, $F{e}_3{O}_4$ NPs are attracting considerable attention from many researchers owing to their applications like space heating, thermal-energy storage, bone cancer treatment, removal of inorganic/organic pollutants, spin polarization, noble metals, and magnetic resonance imaging. Zhao et al. [13] investigated the $F{e}_3{O}_4$ nanoparticles in permeable lid walls. In their study, they found that higher Nusselt numbers were associated with porous materials. Hu et al. [14] studied the nanocomposites for $F{e}_3{O}_4$ -paraffin wax (PW) in a square geometry. They concluded that because the PW was doped with a greater mass fraction of $F{e}_3{O}_4$, the energy transmission efficiency within a square geometry loaded with the nanocomposites increased more rapidly. Molana et al. [15] analyzed the convective $F{e}_3{O}_4-H_{2}O$ nanoliquid hydrothermal behaviour in a novel shaped cavity. They reported that an elevated Hartmann number value enhances the rate of energy transmission. Rahman et al. [16] scrutinized the MHD $\mathrm{MWCNT}-F{e}_3{O}_4$ water conveying hybrid nanoliquid in a ⊥-shaped enclosure. They revealed that the NP volume fraction raises the average of the Bejan number. He et al. [17] reported the $F{e}_3{O}_4$ nanoparticles melting behaviour in a cubic enclosure. The researchers disclosed that the thermal storage capacity of the $F{e}_3{O}_4$ -PW phase change nanocomposites has undergone a substantial decrease. The $C_2{H}_6{O}_2$ and $H_{2}O$ conveying nanoparticles have been investigated by many other researchers [18–21].

Porous enclosures containing a nanoliquid have gained considerable attention from many other authors. Chamkha and Al-Naser [22] studied an inclined rectangular cavity filled with a porous medium. According to their findings, the permeable medium reduced fluid circulation inside the enclosure. Grosan et al. [23] deliberate the energy transmission in a square geometry loaded with a permeable medium. They revealed that the elevated values of the porosity parameter enhanced energy transmission in the interior of the enclosure. Mehmood et al. [24] evaluated the $A{l}_2{O}_3-H_{2}O$ nanoliquid flow in square permeable enclosures. They concluded that the porosity parameter augments the kinetic energy and average heat transmission. Alsabery et al. [25] analysed the $A{l}_2{O}_3-H_{2}O$ conveying of nanofluids in a wavy-walled permeable enclosure. They concluded that the isothermal lines are denser at elevated values of the Darcy number. Sumithra et al. [26] analyzed the time-dependent non-Newtonian nanofluid flow over a plate, stagnation point, and wedge. They concluded that the rate of energy transmission raise with encouraging values of the Darcy number. Sumithra and Sivaraj [27] studied the convective nanoliquid flow over a permeable square geometry. They found that the fluid flow is magnified by the amplification of Darcy number values. Sumithra and Sivaraj [28] assessed the flow of convective nanoliquid via a rectangular porous enclosure. They concluded that the transfer of heat enhances the rise of the permeability parameter.

The applications of MHD nanofluid flow fields are used in solar collectors, MEMS, medicine, heat exchanges, and lubrication. Chamkha [29] studied the natural convective MHD nanofluid flow via a vertical plate. They found that the local Nusselt number diminished with raising values of the Hartmann number. Oztop et al. [30] investigated the convective MHD nanoliquid flow via a wavy-shaped enclosure. The researchers concluded that increasing values of the magnetic field parameter enhanced heat transmission. MHD convective nanoliquid flow in dissimilar geometries has been investigated by several researchers [31–36].

The process by which a fluid’s viscosity absorbs heat from fluid motion and transforms it into internal fluid energy that serves as a heat source is known as ohmic heating (viscous dissipation). Viscous dissipation plays an essential role, particularly in high-velocity systems. Rahman et al. [37] numerically evaluated the effects of ohmic heating in a square geometry. They concluded that the fluid flow was significantly enhanced by an elevated value of the joule heating parameter. Ghaffarpasand [38] studied the water-conveying $F{e}_3{O}_4$ nanofluid with the effect of viscous dissipation over a lid-driven enclosure. They reveal that the entropy generation escalates with Eckert number. Kumar et al. [39] studied the time-dependent ohmic dissipation from a stretching sheet. They revealed that the thermal boundary layer thickness rise for escalating Eckert number. Reddy and Panda [40] studied the ohmic heating effects over a wavy-shaped permeable trapezoidal enclosure. They found that the temperature distribution was enhanced by raising the Eckert number.

In recent years, investigations on convective energy transmission under the effect of a heat source/sink have acquired several applications in the agricultural and engineering fields. Like, geophysics, fuel debris, geothermal systems, packed-bed reactors, etc. There are several studies [41–47] done numerically to investigate the different geometrical configurations of the nanoliquid flow and its energy transfer attributes with the impact of the heat source/sink. The renewable energy and heat transfer performances were analyzed by several researchers [48–50].

Currently, no scientific literature has mentioned this work to the best of the author’s knowledge to investigate the inclined (oblique) magnetic field effect on mixed convective nanoliquid flow within a square permeable cavity with cold and hot slits. The intention of this present analysis is to evaluate the heat transmission features of $F{e}_3{O}_4-H_{2}O$ and $F{e}_3{O}_4-C_2{H}_6{O}_2$ nanofluid flow via a square geometry with radiation, heat source or sink, porous, and a transverse magnetic field. The partially heated square geometry is used in many other engineering applications like, cooling heated electronic devices, air conditioning, thermally insulated walls, refrigerators, etc. The heat transfer features of $F{e}_3{O}_4-H_{2}O$ and $F{e}_3{O}_4-C_2{H}_6{O}_2$ are analyzed over a square geometry. The primary objective of this study is to address the following research inquiries: 1. Which combination of nanofluids (Iron-Oxide + water nanofluid and Iron-Oxide + Ethylene Glycol nanofluid) gives the better heat transfer? 2. How does the porous medium affect nanofluid flow within the enclosure? 3. What is the impact of buoyancy on fluid flow? 4. The MAC method is employed to solve the problem. The outcomes of the present results are illustrated through graphs, bar chart, and tables.

2. Mathematical model

Considering the mixed convective unsteady 2D flow and heat transmission of a permeable square geometry (shown in Figure 1) loaded with electrically conducting $F{e}_3{O}_4-H_{2}O$ and $F{e}_3{O}_4-C_2{H}_6{O}_2$ nanofluids. The Cartesian coordinates are denoted by $x$ and $y$. $L$ and $H$ represent the length and height of the cavity. The following assumptions are considered for the study,

1. The square enclosure consists of four sides, with a portion of each side being heated. The middle portion of the enclosure is designated as the cold wall.

2. The energy equation considers the impact of heat source/sink, viscous dissipation, and ohmic heating.

3. The fluid flow equations along $x$ and $y$ directions include the effects are porous, transverse magnetic field, and buoyancy term for using the Boussinesq approximation.

4. The force of gravitation acts normal to the direction.

In view of the above assumptions, the governing equations are considered as follows [28, 40, 44, 45]:

Dimensional governing equations are, (1) $\begin{array}{rl}& \frac{\mathrm{\partial }u}{\mathrm{\partial }x}+\frac{\mathrm{\partial }v}{\mathrm{\partial }y}=0,\end{array}$(1) (2) $\begin{array}{rl}& \frac{\mathrm{\partial }u}{\mathrm{\partial }{t}^{\ast }}+u\frac{\mathrm{\partial }u}{\mathrm{\partial }x}+v\frac{\mathrm{\partial }u}{\mathrm{\partial }y}=-\frac{1}{{\rho }_{\mathrm{nf}}}\frac{\mathrm{\partial }p}{\mathrm{\partial }x}+{\nu }_{\mathrm{nf}}\left[\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^2}\right]\\ & -\frac{{\nu }_{\mathrm{nf}}}{K}u+\frac{{(\mathrm{\rho \beta })}_{\mathrm{nf}}}{{\rho }_{\mathrm{nf}}}g(T-T_c)\sin \omega ,\end{array}$(2) (3) $\begin{array}{rl}& \frac{\mathrm{\partial }v}{\mathrm{\partial }{t}^{\ast }}+u\frac{\mathrm{\partial }v}{\mathrm{\partial }x}+v\frac{\mathrm{\partial }v}{\mathrm{\partial }y}=-\frac{1}{{\rho }_{\mathrm{nf}}}\frac{\mathrm{\partial }p}{\mathrm{\partial }y}+{\nu }_{\mathrm{nf}}\left[\frac{{\mathrm{\partial }}^{2}v}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^{2}v}{\mathrm{\partial }{y}^2}\right]\\ & -\frac{{\nu }_{\mathrm{nf}}}{K}v+\frac{{(\mathrm{\rho \beta })}_{\mathrm{nf}}}{{\rho }_{\mathrm{nf}}}g(T-T_c)\cos \omega -\frac{{\sigma }_{\mathrm{nf}}{B}_0^2}{{\rho }_{\mathrm{nf}}}v,\end{array}$(3) (4) $\begin{array}{rl}& \frac{\mathrm{\partial }T}{\mathrm{\partial }{t}^{\ast }}+u\frac{\mathrm{\partial }T}{\mathrm{\partial }x}+v\frac{\mathrm{\partial }T}{\mathrm{\partial }y}=\frac{k_{\mathrm{nf}}}{{(\rho c_p)}_{\mathrm{nf}}}\left(\frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{y}^2}\right)\\ & +\frac{Q_0}{{(\rho c_p)}_{\mathrm{nf}}}(T-T_c)+\frac{{\sigma }_{\mathrm{nf}}{B}_0^2{v}^2}{{(\rho c_p)}_{\mathrm{nf}}}+\frac{{\mu }_{\mathrm{nf}}}{{(\rho c_p)}_{\mathrm{nf}}K}(u^2+v^2)\\ & +\frac{{\mu }_{\mathrm{nf}}}{{(\rho c_p)}_{\mathrm{nf}}}\left(2{\left(\frac{\mathrm{\partial }u}{\mathrm{\partial }x}\right)}^2+2{\left(\frac{\mathrm{\partial }v}{\mathrm{\partial }x}\right)}^2+{\left(\frac{\mathrm{\partial }u}{\mathrm{\partial }y}+\frac{\mathrm{\partial }v}{\mathrm{\partial }x}\right)}^2\right).\end{array}$(4)

The following non-dimensional variables are acquainted: (5) $\begin{array}{rl}X& =\frac{x}{H},U=\frac{u}{U_0},Y=\frac{y}{H},t=\frac{t^{\ast }{U}_0}{L},V=\frac{v}{U_0},\\ P& =\frac{p}{{\rho }_{\mathrm{nf}}{U}_0^2},\theta =\frac{T-T_c}{T_h-T_c}.\end{array}$(5) Nanofluids thermophysical properties are essential for raising the working fluid’s thermal characteristics. According to the Tiwari-Das nanoscale model, the following describes the thermophysical properties of (Figure 1) nanoliquids:

Density: ${\rho }_{\mathrm{nf}}=(1-\phi ){\rho }_f+\phi {\rho }_p$Thermal expansion coefficient: $(\mathrm{\rho \beta }{)}_{\mathrm{nf}}=(1-\phi )(\mathrm{\rho \beta }{)}_f+\phi (\mathrm{\rho \beta }{)}_p$Heat capacitance: $(\rho C_p{)}_{\mathrm{nf}}=\phi (\rho C_p{)}_p+(1-\phi )(\rho C_p{)}_f$Thermal diffusivity: ${\alpha }_{\mathrm{nf}}=\frac{k_{\mathrm{nf}}}{{(\rho C_p)}_{\mathrm{nf}}}$Thermal conductivity: $\frac{k_{\mathrm{nf}}}{k_f}=\frac{(k_p+2k_f)-2\phi (k_f-k_p)}{(k_p+2k_f)+\phi (k_f-k_p)}$Nanofluid’s dynamic viscosity: ${\mu }_{\mathrm{nf}}=\frac{{\mu }_f}{{(1-\phi )}^{2.5}}$Electrical (Tables 1 and 2) conductivity: $\frac{{\sigma }_{\mathrm{nf}}}{{\sigma }_f}=\left[1+\frac{3(\gamma -1)\phi }{(\gamma +2)-(\gamma -1)\phi }\right]\text{ where }\gamma =\frac{{\sigma }_p}{{\sigma }_f}$Dimensionless forms of the governing equations by virtue of Equation (5)(6) $\frac{\mathrm{\partial }U}{\mathrm{\partial }X}+\frac{\mathrm{\partial }V}{\mathrm{\partial }Y}=0,$(6) are as follows, (6) $\frac{\mathrm{\partial }U}{\mathrm{\partial }X}+\frac{\mathrm{\partial }V}{\mathrm{\partial }Y}=0,$(6) (7) $\begin{array}{rl}& \frac{\mathrm{\partial }U}{\mathrm{\partial }t}+U\frac{\mathrm{\partial }U}{\mathrm{\partial }X}+V\frac{\mathrm{\partial }U}{\mathrm{\partial }Y}=-\left(\frac{{\rho }_f}{{\rho }_{\mathrm{nf}}}\right)\frac{\mathrm{\partial }P}{\mathrm{\partial }X}\\ & +\left(\frac{{\nu }_{\mathrm{nf}}}{{\nu }_f}\right)\left(\frac{{\mathrm{\partial }}^{2}U}{\mathrm{\partial }{X}^2}+\frac{{\mathrm{\partial }}^{2}U}{\mathrm{\partial }{Y}^2}\right)\frac{1}{\mathrm{Re}}-\left(\frac{{\nu }_{\mathrm{nf}}}{{\nu }_f}\right)\frac{1}{\mathrm{Re}\mathrm{Da}}U\\ & +\frac{{(\mathrm{\rho \beta })}_{\mathrm{nf}}}{{\beta }_f{\rho }_{\mathrm{nf}}}\mathrm{Risin}\mathrm{\omega \theta },\end{array}$(7) (8) $\begin{array}{rl}& \frac{\mathrm{\partial }V}{\mathrm{\partial }t}+U\frac{\mathrm{\partial }V}{\mathrm{\partial }X}+V\frac{\mathrm{\partial }V}{\mathrm{\partial }Y}=-\left(\frac{{\rho }_f}{{\rho }_{\mathrm{nf}}}\right)\frac{\mathrm{\partial }P}{\mathrm{\partial }Y}\\ & +\left(\frac{{\nu }_{\mathrm{nf}}}{{\nu }_f}\right)\left(\frac{{\mathrm{\partial }}^{2}V}{\mathrm{\partial }{X}^2}+\frac{{\mathrm{\partial }}^{2}V}{\mathrm{\partial }{Y}^2}\right)\frac{1}{\mathrm{Re}}\\ & -\left(\frac{{\nu }_{\mathrm{nf}}}{{\nu }_f}\right)\frac{1}{\mathrm{Re}\mathrm{Da}}V+\frac{{(\mathrm{\rho \beta })}_{\mathrm{nf}}}{{\beta }_f{\rho }_{\mathrm{nf}}}\mathrm{Ricos}\mathrm{\omega \theta }\\ & -\left(\frac{{\rho }_f}{{\rho }_{\mathrm{nf}}}\right)\left(\frac{{\sigma }_{\mathrm{nf}}}{{\sigma }_f}\right)H{a}^2\mathrm{Re}V,\end{array}$(8) (9) $\begin{array}{rl}& \frac{\mathrm{\partial }\theta }{\mathrm{\partial }t}+U\frac{\mathrm{\partial }\theta }{\mathrm{\partial }X}+V\frac{\mathrm{\partial }\theta }{\mathrm{\partial }Y}=\left(\frac{{\alpha }_{\mathrm{nf}}}{{\alpha }_f}\right)\frac{1}{\mathrm{Re}Pr}\left[\frac{{\mathrm{\partial }}^2\theta }{\mathrm{\partial }{X}^2}+\frac{{\mathrm{\partial }}^2\theta }{\mathrm{\partial }{Y}^2}\right]\\ & +\left(\frac{{\alpha }_{\mathrm{nf}}}{{\alpha }_f}\right)\frac{Q_1}{\mathrm{Re}Pr}\theta +\left(\frac{{\mu }_{\mathrm{nf}}}{{\mu }_f}\right)\frac{{(\rho C_p)}_f}{{(\rho C_p)}_{\mathrm{nf}}}\frac{\mathrm{Ec}}{\mathrm{Da}\mathrm{Re}}(U^2+V^2)\\ & +\left(\frac{{\sigma }_{\mathrm{nf}}}{{\sigma }_f}\right)\frac{{(\rho C_p)}_f}{{(\rho C_p)}_{\mathrm{nf}}}\frac{H{a}^2\mathrm{Ec}}{\mathrm{Re}}{V}^2\\ & +\left(\frac{{\mu }_{\mathrm{nf}}}{{\mu }_f}\right)\frac{{(\rho C_p)}_f}{{(\rho C_p)}_{\mathrm{nf}}}\frac{\mathrm{Ec}}{\mathrm{Re}}\\ & \times \left(2{\left(\frac{\mathrm{\partial }U}{\mathrm{\partial }X}\right)}^2+2{\left(\frac{\mathrm{\partial }V}{\mathrm{\partial }Y}\right)}^2+{\left(\frac{\mathrm{\partial }U}{\mathrm{\partial }Y}+\frac{\mathrm{\partial }V}{\mathrm{\partial }X}\right)}^2\right).\end{array}$(9)

Figure 1. Flow geometry.

Table 1. Dimensional initial and boundary conditions are.

$t^{\ast }\le 0,0\le x\le L,0\le y\le L.$	$u=v=0$	$T=T_c$
$\mathrm{Left}\mathrm{wall}{t}^{\ast }>0,$	$u=v=0$	$T=T_h\mathrm{for}0\le y\le 0.25L,$
$x=0,0\le y\le L.$		$T=T_c\mathrm{for}0.25L\le y\le 0.75L,T=T_h\mathrm{for}0.75L\le y\le L.$
$\mathrm{Right}\mathrm{wall}{t}^{\ast }>0,$	$u=v=0$	$T=T_h\mathrm{for}0\le y\le 0.25L,$
$x=L,0\le y\le L.$		$T=T_c\mathrm{for}0.25L\le y\le 0.75L,T=T_h\mathrm{for}0.75L\le y\le L.$
$\mathrm{Top}\mathrm{wall}{t}^{\ast }>0,$	$u=v=0$	$T=T_h\mathrm{for}0\le y\le 0.25L,$
$y=L,0\le x\le L.$		$T=T_c\mathrm{for}0.25L\le y\le 0.75L,T=T_h\mathrm{for}0.75L\le y\le L.$
$\mathrm{Bottom}\mathrm{wall}{t}^{\ast }>0,$	$u=v=0$	$T=T_h\mathrm{for}0\le y\le 0.25L,$
$y=0,0\le x\le L.$		$T=T_c\mathrm{for}0.25L\le y\le 0.75L,T=T_h\mathrm{for}0.75L\le y\le L.$

Table 2. Thermophysical properties of NPs and base fluid [15, 18, 21].

Properties	Fe3O4	H2O	C2H6O2
$\rho (\mathrm{kg}{m}^{-3})$	5200	997.1	1116.3
$C_p(\mathrm{Jk}{g}^{-1}{K}^{-1})$	670	4179	2382
$k(W{m}^{-1}{K}^{-1})$	6	0.613	0.2492
$\beta \times {10}^{-5}(K^{-1})$	$1.3\times {10}^{-5}$	$21\times {10}^{-5}$	65
$\sigma ({\mathrm{\Omega }}^{-1}{m}^{-1})$	$25\times {10}^3$	0.05	$11.07\times {10}^{-8}$
$\mu (\mathrm{Kg}{m}^{-1}{s}^{-1})$	–	$9.09\times {10}^{-4}$	0.022
$Pr$	–	7	210.3

The nondimensional parameters are, (10) $\begin{array}{rl}\mathrm{Re}& =\frac{U_{0}L}{{\nu }_f},Pr=\frac{{\nu }_f}{{\alpha }_f},\mathrm{Ha}=B_{0}L\sqrt{\frac{{\sigma }_f}{{\mu }_f}},\\ \mathrm{Da}& =\frac{K}{L^2},\mathrm{Ec}=\frac{U_0^2}{{(C_p)}_f(T_h-T_c)},\\ Q_1& =\frac{Q_0{L}^2}{{(\rho C_p)}_{\mathrm{nf}}{\alpha }_{\mathrm{nf}}},\mathrm{Gr}=\frac{g{\beta }_f(T-T_c)L^3}{{\nu }_f^2},\mathrm{Ri}=\frac{\mathrm{Gr}}{{\mathrm{Re}}^2}.\end{array}$(10) The local and average Nusselt number is, (11) $\begin{array}{rl}\mathrm{Nu}& =-\frac{k_{\mathrm{nf}}}{k_f}{\left(\frac{\mathrm{\partial }\theta }{\mathrm{\partial }Y}\right)}_{Y=0}\end{array}$(11) (12) $\begin{array}{rl}N{u}_{\mathrm{avg}}& =\frac{1}{L}\int \mathrm{Nudx}\end{array}$(12)

3. MAC solution and validation

The MAC technique was initially proposed by Harlow and Welch in 1965. The MAC technique is employed for solving Equations (6)-(9) using dimensionless initial and boundary conditions, as specified in Table 3. Reference [51] contains a comprehensive explanation of the MAC method procedure.

Table 3. Dimensionless initial and boundary conditions are.

$t\le 0,0\le X\le 1,0\le Y\le 1.$	$U=V=0$	$\theta =0$
$\mathrm{Left}\mathrm{wall}t>0,$	$U=V=0$	$\theta =1\mathrm{for}0\le Y\le 0.25,$
$X=0,0\le Y\le 1.$		$\theta =0\mathrm{for}0.25\le Y\le 0.75,$
		$\theta =1\mathrm{for}0.75\le Y\le 1.$
$\mathrm{Right}\mathrm{wall}t>0,$	$U=V=0$	$\theta =1\mathrm{for}0\le Y\le 0.25,$
$X=1,0\le Y\le 1.$		$\theta =0\mathrm{for}0.25\le Y\le 0.75,$
		$\theta =1\mathrm{for}0.75\le Y\le 1.$
$\mathrm{Bottom}\mathrm{wall}t>0,$	$U=V=0$	$\theta =1\mathrm{for}0\le Y\le 0.25,$
$Y=0,0\le X\le 1.$		$\theta =0\mathrm{for}0.25\le Y\le 0.75,$
		$\theta =1\mathrm{for}0.75\le Y\le 1.$
$\mathrm{Top}\mathrm{wall}t>0,$	$U=V=0$	$\theta =1\mathrm{for}0\le Y\le 0.25,$
$Y=1,0\le X\le 1.$		$\theta =0\mathrm{for}0.25\le Y\le 0.75,$
		$\theta =1\mathrm{for}0.75\le Y\le 1.$

The discretized X and Y velocity equations are, (13) $\begin{array}{rl}{U}^{n+1}& =U^n+\mathrm{dt}(-\left[U\frac{\mathrm{\partial }U}{\mathrm{\partial }X}+V\frac{\mathrm{\partial }U}{\mathrm{\partial }Y}\right]\\ & +{\alpha }_1\left[\frac{{\mathrm{\partial }}^{2}U}{\mathrm{\partial }{X}^2}+\frac{{\mathrm{\partial }}^{2}U}{\mathrm{\partial }{Y}^2}\right]-{\alpha }_{2}U+{\alpha }_3\sin \mathrm{\omega \theta }).\end{array}$(13) (14) $\begin{array}{rl}{V}^{n+1}& =V^n+\mathrm{dt}(-\left[U\frac{\mathrm{\partial }V}{\mathrm{\partial }X}+V\frac{\mathrm{\partial }V}{\mathrm{\partial }Y}\right]\\ & +{\alpha }_1\left[\frac{{\mathrm{\partial }}^{2}V}{\mathrm{\partial }{X}^2}+\frac{{\mathrm{\partial }}^{2}V}{\mathrm{\partial }{Y}^2}\right]-{\alpha }_{2}V+{\alpha }_3\sin \mathrm{\omega \theta }-{\alpha }_{4}V).\end{array}$(14) Here, ${\alpha }_1=\left(\frac{{\nu }_{\mathrm{nf}}}{{\nu }_f}\right)\frac{1}{\mathrm{Re}},{\alpha }_2=\left(\frac{{\nu }_{\mathrm{nf}}}{{\nu }_f}\right)\frac{1}{\mathrm{Re}\mathrm{Da}},{\alpha }_3=\frac{{(\mathrm{\rho \beta })}_{\mathrm{nf}}}{{\rho }_{\mathrm{nf}}{\beta }_{\mathrm{nf}}}\mathrm{Ri},{\alpha }_4=\left(\frac{{\rho }_f}{{\rho }_{\mathrm{nf}}}\right)\left(\frac{{\sigma }_{\mathrm{nf}}}{{\sigma }_f}\right)H{a}^2\mathrm{Re}.$

The discretized energy equation is, (15) $\begin{array}{rl}{\theta }^{n+1}& ={\theta }^n+\mathrm{dt}\left(\begin{array}{l}-\left[U\frac{\mathrm{\partial }\theta }{\mathrm{\partial }X}+V\frac{\mathrm{\partial }\theta }{\mathrm{\partial }Y}\right]+{\chi }_1\left[\frac{{\mathrm{\partial }}^2\theta }{\mathrm{\partial }{X}^2}+\frac{{\mathrm{\partial }}^2\theta }{\mathrm{\partial }{Y}^2}\right]\\ +{\chi }_2\theta +{\chi }_3(U^2+V^2)\\ +{\chi }_4{V}^2+{\chi }_5(2{\left(\frac{\mathrm{\partial }U}{\mathrm{\partial }X}\right)}^2+2{\left(\frac{\mathrm{\partial }V}{\mathrm{\partial }Y}\right)}^2\\ +{\left(\frac{\mathrm{\partial }U}{\mathrm{\partial }X}+\frac{\mathrm{\partial }V}{\mathrm{\partial }Y}\right)}^2)\end{array}\right).\end{array}$(15) Here, $\begin{array}{rl}{\chi }_1& =\left(\frac{{\alpha }_{\mathrm{nf}}}{{\alpha }_f}\right)\frac{1}{\mathrm{Re}Pr},{\chi }_1=\left(\frac{{\alpha }_{\mathrm{nf}}}{{\alpha }_f}\right)\frac{Q_1}{\mathrm{Re}Pr},\\ {\chi }_3& =\left(\frac{{\mu }_{\mathrm{nf}}}{{\mu }_f}\right)\frac{{(\rho C_p)}_f}{{(\rho C_p)}_{\mathrm{nf}}}\frac{\mathrm{Ec}}{\mathrm{Da}\mathrm{Re}},\\ {\chi }_4& =\left(\frac{{\sigma }_{\mathrm{nf}}}{{\sigma }_f}\right)\frac{{(\rho C_p)}_f}{{(\rho C_p)}_{\mathrm{nf}}}\frac{H{a}^2\mathrm{Ec}}{\mathrm{Re}},{\chi }_5=\left(\frac{{\mu }_{\mathrm{nf}}}{{\mu }_f}\right)\frac{{(\rho C_p)}_f}{{(\rho C_p)}_{\mathrm{nf}}}\frac{\mathrm{Ec}}{\mathrm{Re}}.\end{array}$

4. Result and discussion

The behaviour of mixed convective MHD nanoliquid flow in a permeable square geometry with heat source/sink, viscous dissipation, and ohmic heating was analyzed. The outcomes of the present results are shown in tables and graphs. The range of the Reynolds number $(\mathrm{Re})$, heat source/sink $(Q_1)$, magnetic parameter $(\mathrm{Ha})$, Joule heating and Darcy number $(\mathrm{Da})$, Eckert number $(\mathrm{Ec})$, nanoparticle volume fraction ( $\varphi$ ), Richardson number $(\mathrm{Ri})$, and Prandtl number $(Pr)$, used in this study are: $-6\le Q_1\le 6,0\le \mathrm{Ha}\le 100,0\le \mathrm{Ec}\le 0.05,0.1\le \mathrm{Da}\le 0.001,0.01\le \mathrm{Ri}\le 10,0\le \varphi \le 0.05,$ Prandtl number $H_{2}O=7,C_2{H}_6{O}_2=210.3$ respectively. Throughout the study, the pertinent parameters values are frozen with $\mathrm{Da}=\left(\frac{1}{{10}^2}\right),\mathrm{Re}=10,\mathrm{Ri}=1,\omega =\frac{\pi }{4},\mathrm{Ha}=100,\mathrm{Ec}=0.05,Q_1=-6,$ and ${\varphi }_1=0.05$ unless otherwise specified.

The impact of $\mathrm{Da}$ on streamline is displayed in Figure 2. In general, the porosity and permeability are typically related. It describes the capacity of a porous medium to permit fluid passage, and an increase in porosity improves permeability, which results in an enhancement in velocity. The characteristics of $\mathrm{Da}$ on the streamline contour of water conveying iron oxide nanofluid are represented in Figure 2(A). In $\mathrm{Da}=0.001$, the three circulatory cells appear, and the secondary and primary circulatory cell sizes are enlarged compared with the single-layer tertiary cell. Whenever increasing the $\mathrm{Da}$ values from 0.01–0.1, the small circulatory cell vanished, and we have only two cells that cover the whole space of the enclosure. The characteristics of $\mathrm{Da}$ on the streamline contour of ethylene glycol conveying iron oxide nanofluid are revealed in Figure 2(B). At $\mathrm{Da}=0.001$, the double convective cells appear, and they are stretched at both horizontal walls. The secondary circulatory cell has three layers. At $\mathrm{Da}=0.01$, the secondary circulatory cell’s third layer merges with the primary circulatory cell, and a new tertiary cell is observed on the enclosure’s bottom right. At $\mathrm{Da}=0.1$, a single-layer tertiary cell eventually moves to left middle enclosure. In the overall case, Figure 2 concludes that the porous media does not affect the influence of the fluid flow, but the streamline circulation changes significantly within the enclosure. Increases in $\mathrm{Da}$ do not cause any changes in the magnitude of the streamlines.

Figure 2. Streamlines of (A) $F{e}_3{O}_4-H_{2}O$ and (B) $F{e}_3{O}_4-C_2{H}_6{O}_2$ for distinct values of $\mathrm{Da}=0.001,0.01,0.1$.

The streamline contour for various $\mathrm{Ha}$ is revealed in Figure 3. Figure 3(A) depicts the influence of $\mathrm{Ha}$ on streamlines contour for $F{e}_3{O}_4-H_{2}O$ nanofluid. At $\mathrm{Ha}=0$, the two convective circulation zones are established, and they are located in the opposite diagonal direction. The secondary circulation zone has a smaller streamline magnitude value compared with the primary circulation zone. The primary circulation streamline magnitude is 1.001. An increase in $\mathrm{Ha}$ ( $\mathrm{Ha}=50$ and 100), streamlines circulation inside the cavity, which is markedly changed. The interior of the streamlines core and magnitude values are abundantly shrinking. Uplifting values of $\mathrm{Ha}$ fluid flow decrease from 1.001–1.0003 and 1.0001. The fluid flow contour presents a declining tendency for elevated magnetic parameter values $\mathrm{Ha}$. Figure 3(B) demonstrates the effect of $\mathrm{Ha}$ on streamlines contour for $F{e}_3{O}_4-C_2{H}_6{O}_2$ nanofluid. The streamline magnitude value decreases, the shape of the circulation cells shrinks, and new cells are appeared for raising values of $\mathrm{Ha}$. Uplifting values of $\mathrm{Ha}$ fluid flow decrease from 1.0003–1.0001. The Lorentz force is a combination of the magnetic and electric forces on a moving point charge due to electric fields. With an augmentation in Hartmann number, Lorentz force enhances, which opposes the velocity. It is evident from these figures that the fluid flow diminishes in magnitude when the magnetic field parameter $(\mathrm{Ha})$ increases.

Figure 3. Streamlines of (A) $F{e}_3{O}_4-H_{2}O$ and (B) $F{e}_3{O}_4-C_2{H}_6{O}_2$ for distinct values of $\mathrm{Ha}=0,50,100$.

Figure 4 describes the features of the $Q_1$. Figure 4(A) demonstrates the influence of $Q_1$ on isotherms contour for $F{e}_3{O}_4-H_{2}O$ nanofluid. Here, $Q_1<0,$ represented the heat absorption, $Q_1=0,$ represented the non-existence of the heat absorption/generation case, and $Q_1>0,$ represented the heat generation case. Energy transfer in the enclosure depends on $Q_1$. In general, both $F{e}_3{O}_4-H_{2}O$ and $F{e}_3{O}_4-C_2{H}_6{O}_2$ cases, inside the square geometry, the yellow colour represented high thermal conductivity, and the blue colour represented low thermal conductivity. A low heat transmission was visible on the side of the cold walls, while a large heat transmission was seen on the side of the hot walls. At $Q_1=-6,$ the maximum of the isotherm lines evolves from the cold wall side, and the minimum of the isotherm lines appeared on the hot wall side. In the middle of the portion, there is a square-shaped cell with a single layer. In the absence of $Q_1$ case the heat transfer increases from 0.80078–0.90041. At $Q_1=6,$ the heat generation case the internal heat transmission increases from 0.90041–0.97195. An uplifting value of the $Q_1$ enhances the isotherm magnitude value from 0.80078–0.90041 and 0.97195. Figure 4(B) demonstrates the influence of $Q_1$ on isotherms contour for $F{e}_3{O}_4-C_2{H}_6{O}_2$ nanofluid. Ethylene glycol has more sites for hydrogen bonding than $H_{2}O.$ As a consequence of this, it is substantially more viscous than water and has strong intermolecular forces of attraction. The raising value of the $Q_1$ parameter enhances the heat transmission and the isotherm magnitude value from 0.9–0.90025 and 0.90289. Physically, the increasing values of $Q_1$ enhance more heat in the interior of the enclosure. This implies that the isotherm profiles exaggerate with the amassed estimation of $Q_1$. Therefore, the influence of $Q_1$ parameter is more dominant in the $F{e}_3{O}_4-H_{2}O$ nanofluid compared to the $F{e}_3{O}_4-C_2{H}_6{O}_2$ nanofluid. It can be noted from the contours that the heat transmission enhances in magnitude when $Q_1$ increases on both $F{e}_3{O}_4-H_{2}O$ and $F{e}_3{O}_4-C_2{H}_6{O}_2$ nanofluid. By replacing the heat absorption $Q_1=-6,$ with heat generation $Q_1=6,$ the transfer of heat can be improved up to 15.819% for $F{e}_3{O}_4-H_{2}O$ nanofluid and 0.596% for $F{e}_3{O}_4-C_2{H}_6{O}_2$ nanoliquid. So, increasing $Q_1$ from −6 to 6 enhances energy transfer in enclosure for both $F{e}_3{O}_4-H_{2}O$ and $F{e}_3{O}_4-C_2{H}_6{O}_2$ nanofluids.

Figure 4. Isotherms of (A) $F{e}_3{O}_4-H_{2}O$ and (B) $F{e}_3{O}_4-C_2{H}_6{O}_2$ for distinct values of $Q_1=-6,0,6$.

The fluid flow of the nanoliquids is not remarkably influenced by the volume fraction’s variation. Figure 5(A-B) reveals that the streamline contour of $F{e}_3{O}_4-H_{2}O$ and $F{e}_3{O}_4-C_2{H}_6{O}_2$ nanoliquids are unchanged as the volume fraction enhances. It is distinguished that in the $F{e}_3{O}_4-H_{2}O$ nanofluid flow, circulation of the vortices forms in the whole portion of the cavity. In $F{e}_3{O}_4-C_2{H}_6{O}_2$ case, ${\varphi }_1=0$, the enclosure has three convection cells: top, bottom, and left middle. At ${\varphi }_1=0.03$ the third cell is merged with primary circulatory cell. At ${\varphi }_1=0.05$ a new cell appeared at the bottom right of the enclosure.

Figure 5. Streamlines of (A) $F{e}_3{O}_4-H_{2}O$ and (B) $F{e}_3{O}_4-C_2{H}_6{O}_2$ for distinct values of ${\varphi }_1=0,0.03,0.05$.

Figure 6 illustrates the volume fraction factor ${\varphi }_1$ on isotherm contours for $F{e}_3{O}_4-H_{2}O$ and $F{e}_3{O}_4-C_2{H}_6{O}_2$ nanofluids. The outcomes of this study exhibit that the heat transmission performance of the considered nanoliquids rises significantly with the volume fraction of the nanoparticles. As the volume fraction increases, energy transfer rates increase due to irregular and unpredictable particle movements, and nanoliquid thermal distribution improves. Figure 6(A) reveals the $F{e}_3{O}_4-H_{2}O$ nanofluid for distinct values of ${\varphi }_1$. Mounting values of ${\varphi }_1$ parameter enhance isotherm contour magnitude values and energy transmission in the hot side wall directions. The maximum of the isotherm magnitude values rises from 0.90083–0.90086 and 0.90087. Figure 6(B) reveals the $F{e}_3{O}_4-C_2{H}_6{O}_2$ nanoliquid for distinct values of ${\varphi }_1$. In this case, the maximum of the isotherm lines occurs on the cold wall side, and the heated wall has a high isotherm magnitude value. The rising value of ${\varphi }_1$ from 0 to 0.05 transfer of heat can be improved up to 0.278% for $F{e}_3{O}_4-H_{2}O$ nanofluid and 0.303% for $F{e}_3{O}_4-C_2{H}_6{O}_2$ nanoliquid. So, mounting the value of ${\varphi }_1$ from 0 to 0.05 has the tendency to improve the energy transfer within the enclosure for both $F{e}_3{O}_4-H_{2}O$ and $F{e}_3{O}_4-C_2{H}_6{O}_2$ nanofluids.

Figure 6. Isotherms of (A) $F{e}_3{O}_4-H_{2}O$ and (B) $F{e}_3{O}_4-C_2{H}_6{O}_2$ for distinct values of ${\varphi }_1=0,0.03,0.05$.

Figure 7 shows the streamline contour for various Richardson numbers. The Richardson number represents the mode of convection. Here $\mathrm{Ri}=0.01,$ is the forced convection. $\mathrm{Ri}=1,$ is the mixed convection, $\mathrm{Ri}=10$ is the natural convection. Figure 7(A and B) illustrates the $F{e}_3{O}_4-H_{2}O,$ and $F{e}_3{O}_4-C_2{H}_6{O}_2$ nanofluid streamline contour. In both combinations of water conveying iron oxide and ethylene glycol conveying nanofluid, the streamline contour does not change. Water conveying iron oxide nanofluid interior values of the streamline’s magnitude increases from 1 to 1.0001 and 1.0012. Ethylene glycol conveying iron oxide nanofluid interior values of the streamlines magnitude increases from 1 to 1.0001 and 1.0009. Therefore, the influence of $\mathrm{Ri}$ parameter is more dominant in the $F{e}_3{O}_4-H_{2}O$ nanofluid compared to the $F{e}_3{O}_4-C_2{H}_6{O}_2$ nanofluid.

Figure 7. Streamlines of (A) $F{e}_3{O}_4-H_{2}O$ and (B) $F{e}_3{O}_4-C_2{H}_6{O}_2$ for distinct values of $\mathrm{Ri}=0.01,1,10$.

The transport properties of viscous fluids are significantly influenced by the Reynolds number. The Reynolds number is used in weather forecasting systems. Figure 8(A) illustrates the $F{e}_3{O}_4-H_{2}O$ nanofluid in a streamline contour. At $\mathrm{Re}=1,$ the four convective circulation zones are formed, and they are grouped close to the cavity’s right side. At $\mathrm{Re}=5,$ the circulation cells count decreases, and it appears on both horizontal sides of the enclosure. At $\mathrm{Re}=10,$ the third circulation cell merged with the primary circulation cell. This implies that the streamline magnitude values rise from 1 to 1.0001. Figure 8(B) illustrates the $F{e}_3{O}_4-C_2{H}_6{O}_2$ nanofluid in a streamline contour. At $\mathrm{Re}=1,$ in this case, three circulation zones occurred. At $\mathrm{Re}=5,$ the third circulation cell, a new cell form in the enclosure’s right midsection., and at the primary circulation cell, the magnitude of the streamlines increases from 1 to 1.0001. At $\mathrm{Re}=10,$ the number of circulation cells decreases.

Figure 8. Streamlines of (A) $F{e}_3{O}_4-H_{2}O$ and (B) $F{e}_3{O}_4-C_2{H}_6{O}_2$ for distinct values of $\mathrm{Re}=1,5,10$.

Figure 9. Local Nusselt number of (a) $F{e}_3{O}_4-H_{2}O$ and (b) $F{e}_3{O}_4-C_2{H}_6{O}_2$ for distinct values of $\mathrm{Ec}=0,0.03,0.05$.

Figure 9(A) demonstrates the local Nusselt number profile of $F{e}_3{O}_4-H_{2}O$ nanofluid, and Figure 9(B) reveals the local Nusselt number profile of $F{e}_3{O}_4-C_2{H}_6{O}_2$ nanofluid for various values of $\mathrm{Ec}$. Enlarging the $\mathrm{Ec}$ value from 0 to 0.05, the local rate of heat transmission profiles are enhanced on both $F{e}_3{O}_4-H_{2}O$ and $F{e}_3{O}_4-C_2{H}_6{O}_2$ nanofluids.

Figure 10(A) demonstrates the $\mathrm{Nu}$ profile of $F{e}_3{O}_4-H_{2}O$ nanoliquid, and Figure 10(B) reveals the local Nusselt number profile of $F{e}_3{O}_4-C_2{H}_6{O}_2$ nanofluid for various values of $Q_1$. Enlarging the $Q_1$ value from −6 to 6, the local rate of heat transmission profiles is enhanced on both $F{e}_3{O}_4-H_{2}O$ and $F{e}_3{O}_4-C_2{H}_6{O}_2$ cases. An uplifting value of $Q_1$ produced more energy in the interior of the square geometry.

Figure 10. Local Nusselt number ( $\mathrm{Nu}$ ) of (A) $F{e}_3{O}_4-H_{2}O$ and (B) $F{e}_3{O}_4-C_2{H}_6{O}_2$ for distinct values of $Q_1=-6,0,6$.

Figure 11 displays the average heat transfer for $F{e}_3{O}_4-H_{2}O$ and $F{e}_3{O}_4-C_2{H}_6{O}_2$ on the bar diagram for a different value of $\mathrm{Ec}.$ In $F{e}_3{O}_4-H_{2}O$ case the raising value of $\mathrm{Ec}$ enhances the mean heat transfer from 1.80372–1.80374 and finally 1.80375. In $F{e}_3{O}_4-C_2{H}_6{O}_2$ case the raising values of $\mathrm{Ec}$ enhance the mean heat transfer from 12.94453–12.94454 and finally 2.94455.

Figure 11. Average Nusselt number of $F{e}_3{O}_4-H_{2}O$ and $F{e}_3{O}_4-C_2{H}_6{O}_2$ for distinct values of $\mathrm{Ec}=0,0.03,0.05$.

Figure 12 demonstrates the average Nusselt number of $F{e}_3{O}_4-H_{2}O$ and $F{e}_3{O}_4-C_2{H}_6{O}_2$ on the bar diagram for different values of $(Q_1=-6,0,6).$ In $F{e}_3{O}_4-H_{2}O$ case the raising values of $Q_1$ enhances the mean heat transfer from 1.80375–2.33965 and finally 2.92114. In $F{e}_3{O}_4-C_2{H}_6{O}_2$ case the raising values of $Q_1$ enhances the mean heat transfer from 12.94455–13.04112 and finally 13.13802.

Figure 12. Average Nusselt number of $F{e}_3{O}_4-H_{2}O$ and $F{e}_3{O}_4-C_2{H}_6{O}_2$ for distinct values of $Q_1=-6,0,6$.

Figure 13 portrays the effect of nanoparticle ($F{e}_3{O}_4-C_2{H}_6{O}_2$ and $F{e}_3{O}_4-H_{2}O$) volume fractions on a bar diagram for distinct values of ${\varphi }_1.$ In $F{e}_3{O}_4-H_{2}O$ case the raising values of ${\varphi }_1$ enhance the mean heat transmission from 1.7988–1.8036 and finally 1.8038. In $F{e}_3{O}_4-C_2{H}_6{O}_2$ case the raising values of ${\varphi }_1$ enhance the mean heat transmission from 11.88777–12.51868 and finally 12.94455. In general, the enhancement of ${\varphi }_1$ leads to an improvement in overall energy transport; this can be expected owing to the fact that the change in ${\varphi }_1$ percentage primarily enhances the thermal conductivity of nanofluid and for this reason, elevates flow strength in both $F{e}_3{O}_4-H_{2}O$ and $F{e}_3{O}_4-C_2{H}_6{O}_2$ cases.

Figure 13. Average Nusselt number of $F{e}_3{O}_4-H_{2}O$ and $F{e}_3{O}_4-C_2{H}_6{O}_2$ for distinct values of ${\varphi }_1=0,0.03,0.05$.

Tables 4 and 5 illustrate the rate of heat transmission of $F{e}_3{O}_4-H_{2}O$ and $F{e}_3{O}_4-C_2{H}_6{O}_2$ with $Q_1$ and $\mathrm{Ec}$. The ethylene glycol conveying iron oxide has a higher percentage value compared with the ethylene glycol conveying iron oxide nanofluid. This means that the base fluid ethylene glycol increases, the $N{u}_{\mathrm{avg}}$ on $F{e}_3{O}_4$ nanoparticles, raising the values of $Q_1$ and $\mathrm{Ec}$. The Eckert number improves fluid temperature by improving thermal conduction.

Table 4. Average heat transfer for rising values of $Q_1$.

$Q_1$	$F{e}_3{O}_4-H_{2}O$	$F{e}_3{O}_4-C_2{H}_6{O}_2$	Increase
−6	0.656246	0.920045	40.198
0	0.705743	0.922782	30.753
6	0.760016	0.925527	21.777

Table 5. Average heat transfer for rising values of $\mathrm{Ec}.$.

$\mathrm{Ec}$	$F{e}_3{O}_4-H_{2}O$	$F{e}_3{O}_4-C_2{H}_6{O}_2$	Increase
0	0.656245	0.920045	40.198401511631
0.3	0.656252	0.920047	40.197210827548
0.5	0.656256	0.920049	40.196661059099

5. Conclusions

The present study investigates the influence of the heat transferred MHD mixed convective nanofluid flow inside a square cavity with the effects of heat source/sink, viscous dissipation, and ohmic heating. The thermal analyses of $F{e}_3{O}_4-H_{2}O$ and $F{e}_3{O}_4-C_2{H}_6{O}_2$ are investigated. The enclosure’s four walls are partially heated. The MAC technique is used to resolve the dimensionless governing equations. The outcome of the present results is illustrated via graphs, tables, and a bar chart. This analysis gives the following findings:

• The permeability of the porous medium does not affect the velocity on both $F{e}_3{O}_4-H_{2}O$ and $F{e}_3{O}_4-C_2{H}_6{O}_2$ nanofluids.

• Fluid flow diminishes for mounting values of the magnetic field parameter $(\mathrm{Ha})$ on both $F{e}_3{O}_4-H_{2}O$ and $F{e}_3{O}_4-C_2{H}_6{O}_2$ nanofluids.

• The influence of $\mathrm{Ri}$ parameter is more dominant in the $F{e}_3{O}_4-H_{2}O$ nanofluid compared to the $F{e}_3{O}_4-C_2{H}_6{O}_2$ nanofluid flow.

• The rising value of ${\varphi }_1$ from 0 to 0.05 transfer of heat can be improved up to 0.278% for $F{e}_3{O}_4-H_{2}O$ nanofluid and 0.303% for $F{e}_3{O}_4-C_2{H}_6{O}_2$ nanofluid.

• The heat absorption $Q_1=-6,$ with heat generation $Q_1=6,$ the transfer of heat can be improved up to 15.819% for $F{e}_3{O}_4-H_{2}O$ nanofluid and 0.596% for $F{e}_3{O}_4-C_2{H}_6{O}_2$ nanofluid.

• The $N{u}_{\mathrm{avg}}$ increases for rising values of $\mathrm{Ec}$, $Q_1$, and ${\varphi }_1$ parameter in both $F{e}_3{O}_4-H_{2}O$ and $F{e}_3{O}_4-C_2{H}_6{O}_2$ nanofluids.

• Magnifying $Q_1$ and $\mathrm{Ec}$ values, enhances the $\mathrm{Nu}$.

In the current analysis, the thermal analysis of MHD-mixed convective nanofluids is evaluated. Forthcoming investigations may extend the present analyses to include hybrid nanofluids in a different geometrical configuration.

Disclosure statement

No potential conflict of interest was reported by the author(s).