Introduction

In energy-saving systems and industry, increasing the heat transfer rate in convective streams is imperative. Heat transfer is very practical in different applications such as renewable energy units, cooling systems, electronic cooling devices, etc., and those systems typically apply forced or natural convection by using typical fluids such as oil, air or water. Recently, nanofluid technology has been presented as a promising method for increasing thermophysical features of testing fluids by suspending nanopowders in the typical fluids [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]. Guestala et al. [17] scrutinized the free convection of NFs within a cylindrical tank. They found that the heat transfer grew when Re, particles volume fraction or the heated length increases. Simulation of nanomaterial treatment becomes popular in recent decade [18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35]. Selimifendigil et al. [36] numerically analyzed the mixed convection including an internal rotating cylindrical a permeable layer accumulated with covered NFs. Based on their results, heat transfer grew when the cylinder’s angular velocity rose. The evacuated tube including mini-CPC reflectors has been studied by Korrees et al. [37] who used Solidworks flow simulation software for simulating. They estimated the temperature of receiver and the thermal output of each module. Numerical approaches are extended by reviewers to find the performance of systems [38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56]. Several researchers studied the effect of nanofluid on renewable energy systems. The impact of Al2O3–H2O nanofluids on the performance of flat sheet photovoltaic thermal collectors was investigated by Bianco et al. [57]. Both microscopic [58, 59] and macroscopic [60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75] approaches can predict behavior of nanofluid. Hossain and Rees [76] scrutinized the free convective stream of a viscous liquid in an oblong cavity heated from bottom. They applied upwind differential process. In addition, they surveyed zero Darcy inversion terms in this study. Based on their results, cell shape in the tank is a feature of Grashof number and tank aspect ratio. Impact of thermophoresis and nanoflid features on thermal efficiency has been analyzed by Astanina et al. [77] who found that the intensification of stream in the cavity had inverse relationship with the heater place while it had direct relationship with Re; based on their results, the finest place of the heater was the left surface of considered tank. Also, Aham et al. [78] surveyed optimal features such as transmittance, extinction coefficient and scattering based on metal oxide, metal, graphite and grapheme for using nanofluids in solar thermal systems. Heat transfers along free convection in different fluid stream in a tank as illustrated by Drummond and Korpela [79] who found that Gr does not rely on boundary conditions because wall stability relies on the shear forces rather than gravity. Additionally, they reported that heat transfer within an adiabatic cavity was higher compared to that of in tank within convective walls.

This article discusses how magnetic force can affect the nanomaterial behavior through a permeable region, and to reduce computational cost, single-phase model has been utilized. Moreover, influence of radiation term was added in equations and several outputs were summarized in result section.

Formula explanation and simulation

In current article, 2D free convection in laminar condition was investigated. As depicted in Fig. 1, wavy wall received uniform heat flux. Geometry has 2 adiabatic walls and outer surface is cold. To evaluate temperature distribution inside the porous solid zones, new scalar was introduced. Overlooking joule heating impact and considering homogeneous carrier fluid leads to below equations [80]:

$$\frac{\partial P}{\partial x} + \frac{{\mu_{\text{nf}} }}{K}u = \left( {\sin \gamma } \right)\sigma_{\text{nf}} B_{0}^{2} \left[ { - u\left( {\sin \gamma } \right) + v\left( {\cos \gamma } \right)} \right]$$
(1)
$$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0$$
(2)
$$\begin{aligned} \frac{\partial P}{\partial y} + \frac{{\mu_{\text{nf}} }}{K}v & = \left( {T - T_{\text{c}} } \right)g\rho_{\text{nf}} \beta_{\text{nf}} \\ & \quad + {\kern 1pt} \left( {\cos \gamma } \right)\sigma_{\text{nf}} B_{0}^{2} \left[ {u\left( {\sin \gamma } \right) - v\left( {\cos \gamma } \right)} \right] \\ \end{aligned}$$
(3)
$$\frac{1}{\varepsilon }\left( {u\frac{{\partial T_{\text{nf}} }}{\partial x} + v\frac{{\partial T_{\text{nf}} }}{\partial y}} \right) + \frac{1}{{\left( {\rho C_{\text{p}} } \right)_{\text{nf}} }}\frac{{\partial q_{\text{r}} }}{\partial y} = \frac{{k_{\text{nf}} }}{{\left( {\rho C_{\text{p}} } \right)_{\text{nf}} }}\left( {\frac{{\partial^{2} T_{\text{nf}} }}{{\partial x^{2} }} + \frac{{\partial^{2} T_{\text{nf}} }}{{\partial y^{2} }}} \right) + \frac{{h_{\text{nfs}} }}{{\left( \varepsilon \right)\left( {\rho C_{\text{p}} } \right)_{\text{nf}} }}\left( { - T_{\text{nf}} + T_{\text{s}} } \right),\;\left[ {q_{\text{r}} = - \frac{{4\sigma_{\text{e}} }}{{3\beta_{\text{R}} }}\frac{{\partial T_{\text{nf}}^{4} }}{\partial y},\;T_{\text{nf}}^{4} \cong 4T_{\text{c}}^{3} T_{\text{nf}} - 3T_{\text{c}}^{4} } \right]$$
(4)
$$\frac{{k_{\text{s}} }}{{\left( {\rho C_{\text{p}} } \right)_{\text{s}} }}\left( {\frac{{\partial^{2} T_{\text{s}} }}{{\partial x^{2} }} + \frac{{\partial^{2} T_{\text{s}} }}{{\partial y^{2} }}} \right) + \frac{{h_{\text{nfs}} }}{{\left( {1 - \varepsilon } \right)\left( {\rho C_{\text{p}} } \right)_{\text{s}} }}\left( {T_{\text{nf}} - T_{\text{s}} } \right) = 0$$
(5)
Fig. 1
figure 1

Geometry of permeable domain with uniform heat flux

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In order to simplify the above equations, Eq. (6) has been used:

$$\begin{aligned} v & = - \frac{\partial \psi }{\partial x},\quad \theta_{\text{s}} = \left( {T_{\text{s}} - T_{\text{c}} } \right)/\Delta T, \\ u & = \frac{\partial \psi }{\partial y},\quad\theta_{\text{nf}} = \left( {T_{\text{nf}} - T_{\text{c}} } \right)/\Delta T, \\ Y & = yL^{ - 1} ,\;\Delta T = L\left( {k_{\text{f}} } \right)^{ - 1} q^{\prime\prime},\quad X = xL^{ - 1} \\ \varPsi & = \psi /\alpha_{\text{nf}} , \\ \end{aligned}$$
(6)

By including stream function definition, equations convert to below forms:

$$\begin{aligned} \frac{{\partial^{2} \varPsi }}{{\partial X^{2} }} + \frac{{\partial^{2} \varPsi }}{{\partial Y^{2} }} & = - \frac{{A_{6} }}{{A_{5} }}{\text{Ha}}\left[ {\frac{{\partial^{2} \varPsi }}{{\partial Y^{2} }}\left( {\sin^{2} \gamma } \right) + \frac{{\partial^{2} \varPsi }}{{\partial X^{2} }}\left( {\cos^{2} \gamma } \right) + 2\frac{{\partial^{2} \varPsi }}{\partial X\,\partial Y}\left( {\sin \gamma } \right)\,\left( {\cos \gamma } \right)} \right] \\ & \quad - {\kern 1pt} \frac{{A_{3} \,A_{2} }}{{A_{4} \,A_{5} }}\frac{{\partial \theta_{\text{nf}} }}{\partial X}{\text{Ra}} \\ \end{aligned}$$
(7)
$$\varepsilon \left( {\left( {1 + \frac{4}{3}\left( {\frac{{k_{\text{nf}} }}{{k_{\text{f}} }}} \right)^{ - 1} {\text{Rd}}} \right)\frac{{\partial^{2} \theta_{\text{nf}} }}{{\partial Y^{2} }} + \frac{{\partial^{2} \theta_{\text{nf}} }}{{\partial X^{2} }}} \right) + \frac{\partial \varPsi }{\partial X}\frac{{\partial \theta_{\text{nf}} }}{\partial Y} + {\text{Nhs}}\left( {\theta_{\text{s}} - \theta_{\text{nf}} } \right) = \frac{{\partial \theta_{\text{nf}} }}{\partial X}\frac{\partial \varPsi }{\partial Y}$$
(8)
$$\left( {\frac{{\partial^{2} \theta_{\text{s}} }}{{\partial Y^{2} }} + \frac{{\partial^{2} \theta_{\text{s}} }}{{\partial X^{2} }}} \right) + {\text{Nhs}}\,\delta_{\text{s}} \left( {\theta_{\text{nf}} - \theta_{\text{s}} } \right) = 0$$
(9)

In final equations, there exist some new parameters which should be defined as:

$$\begin{aligned} {\text{Ra}} & = \frac{{g\,K\,\left( {\rho \beta } \right)_{\text{f}} L\,\Delta T}}{{\mu_{\text{f}} \,\alpha_{\text{f}} }},\quad A_{3} = \frac{{\left( {\rho \beta } \right)_{\text{nf}} }}{{\left( {\rho \beta } \right)_{\text{f}} }},\quad A_{5} = \frac{{\mu_{\text{nf}} }}{{\mu_{\text{f}} }}, \\ A_{2} & = \frac{{\left( {\rho C_{\text{P}} } \right)_{\text{nf}} }}{{\left( {\rho C_{\text{P}} } \right)_{\text{f}} }},\quad A_{6} = \frac{{\sigma_{\text{nf}} }}{{\sigma_{\text{f}} }},\quad A_{4} = \frac{{k_{\text{nf}} }}{{k_{\text{f}} }}, \\ A_{1} & = \frac{{\rho_{\text{nf}} }}{{\rho_{\text{f}} }},\quad {\text{Ha}} = \frac{{\sigma_{\text{f}} K\,B_{0}^{2} }}{{\mu_{\text{f}} }},\quad {\text{Rd}} = 4\sigma_{\text{e}} T_{\text{c}}^{3} /\left( {\beta_{\text{R}} k_{\text{f}} } \right) \\ \delta_{s} & = \left[ {\left( {1 - \varepsilon } \right)k_{\text{s}} } \right]^{ - 1} k_{\text{nf}} ,\quad {\text{Nhs}} = \left( {k_{\text{nf}} } \right)^{ - 1} h_{\text{nfs}} L^{2} \\ \end{aligned}$$
(10)

The boundary conditions are summarized in Fig. 1, and we ignored to write them again, and for estimating rate of heat transfer the below factor has been defined

$${\text{Nu}}_{\text{ave}} = \frac{1}{S}\int\limits_{0}^{s} {{\text{Nu}}_{\text{loc}} } \,{\text{d}}s,\,\,\,{\text{Nu}}_{\text{loc}} = \frac{1}{\theta }\left( {1 + \frac{4}{3}\left( {\frac{{k_{\text{nf}} }}{{k_{\text{f}} }}} \right)^{ - 1} {\text{Rd}}} \right)\left( {\frac{{k_{\text{nf}} }}{{k_{\text{f}} }}} \right)$$
(11)

CuO nanopowders were dispersed into H2O, and for evaluating the properties same formulation of [80] was employed. In addition, to predict thermal conductivity, shape effect was employed [81]. To simplify the governing equations, vorticity formulations were used and CVFEM was utilized for simulation purpose. This approach was introduced first by Sheikholeslami [82] for thermal problems, and he wrote more details in his reference book. In current approach, CVFEM was employed which was utilized for various problems [83,84,85,86,87,88,89].

Results and discussion

Non-equilibrium modeling for permeable zone was involved in governing equations and simplified with using vorticity formulation and finally solved via CVFEM in this article. Validation step makes us ensure about the accuracy of used model, and Fig. 2 proves the nice agreement of our written code [90]. To reach the minimum computational costs, grid independency analysis has been analyzed and Table 1 illustrates different values of Nu for different meshes. Impact of dispersing nanopowders and imposing radiation terms are illustrated in Figs. 3 and 4. Inclusion of nanoparticles makes the flow velocity to augment and increases the temperature gradient. The position of therma plume changes with adding nanoparticles, and it shifts to right. The structure of flow is slightly affected by Rd, while according to definition of Nuave, it has direct relation with Rd.

Fig. 2
figure 2

Demonstrating deviation of present result with [90] when Gr = 104, \(\phi = 0.1\)

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Table 1 Different mesh and calculated Nuave when \({\text{Ra}} = 10,000,\;{\text{Ha}} = 20,\;\varepsilon = 0.3,\;{\text{Nhs}} = 10,\;{\text{Rd}} = 0.8\) and \(\phi = 0.04\)
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Fig. 3
figure 3

Influences of \(\phi\) on \(\theta {}_{\text{nf}}\) at \({\text{Ra}} = 10^{4} ,\;{\text{Nhs}} = 10,\;m = 5.7,\;{\text{Ha}} = 0,\;\varepsilon = 0.3,\;{\text{Rd}} = 0.8\)

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Fig. 4
figure 4

Influences of Rd on \(\theta {}_{\text{nf}}\) and \(\varPsi\) at \(\phi = 0.04,\;{\text{Nhs}} = 10,\;\varepsilon = 0.3,\;m = 5.7,\;{\text{Ra}} = 10^{4} ,\;{\text{Ha}} = 20\)

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To evaluate impact of Nhs, Ha and Ra on thermal treatment of working fluid, Figs. 5, 6, 7 and 8 were presented. Steeping of isotherms over wavy wall decreases as magnetic field is imposed and impact of Ha becomes weaker in greater Nhs. With impose of Ha, isotherms become uniform and convection weakens. This is the reason of disappearing thermal plume from with increasing Ha. In the absence of magnetic force and greatest Ra, three cells were established in domain and two smaller ones create thermal plume due to reverse direction circulation. When Ha is not zero, all cells merged and one weaker cell appears. Nhs has slight impact on nanomaterial behavior in comparison with Hartmann number. As Ha increases, the center of eddy shifted downwards. As Nhs augments, the power of eddies augments and stronger thermal plume appear when Ra = 104.

Fig. 5
figure 5

Contours of \(\left( \varPsi \right)\), \(\left( {\theta_{\text{nf}} } \right)\) and \(\left( {\theta_{s} } \right)\) when \(\varepsilon = 0.3,\;\phi = 0.04,\;m = 5.7,\;{\text{Ra}} = 1000,\;{\text{Rd}} = 0.8,\;{\text{Nhs}} = 10\)

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Fig. 6
figure 6

Contours of \(\left( \varPsi \right)\), \(\left( {\theta_{\text{nf}} } \right)\) and \(\left( {\theta_{\text{s}} } \right)\) when \({\text{Nhs}} = 1000,\;\varepsilon = 0.3,\;\phi = 0.04,\;{\text{Ra}} = 1000,\;{\text{Rd}} = 0.8,\;{\text{m}} = 5.7\)

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Fig. 7
figure 7

Contours of \(\left( \varPsi \right)\), \(\left( {\theta_{\text{nf}} } \right)\) and \(\left( {\theta_{\text{s}} } \right)\) when \(\varepsilon = 0.3,\;\phi = 0.04,\;m = 5.7,\;{\text{Rd}} = 0.8,\;{\text{Ra}} = 10^{4} ,\;{\text{Nhs}} = 10\)

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Fig. 8
figure 8

Contours of \(\left( \varPsi \right)\), \(\left( {\theta_{\text{nf}} } \right)\) and \(\left( {\theta_{\text{s}} } \right)\) when \({\text{Ra}} = 10^{4} ,\;\varepsilon = 0.3,\;\phi = 0.04,\;{\text{Nhs}} = 1000,\;m = 5.7,\;{\text{Rd}} = 0.8\)

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Impacts of changing scrutinized parameters on Nuave are illustrated in Fig. 9. As Nhs augments, temperature gradient deteriorates which indicates lower Nuave. Magnetic force reduces the nanoparticle velocity and generates thicker boundary layer which results in lower Nuave. Direct relation exists between Ra and Nuave which is attributed to thinner boundary layer with augment of Ra. With impose of radiation impact, Nusselt number enhances, but it is no sensible in greatest Nhs. The below equation can present influences of parameters as good mathematic formula.

$$\begin{aligned} {\text{Nu}}_{\text{ave}} & = 3.02 + 1.62\;{\text{Ra}}^{*} - 0.95\;{\text{Nhs}}^{*} + 1.35\;{\text{Rd}} - 1.42\;{\text{Ha}}^{*} \\ & \quad - {\kern 1pt} 1.07\;{\text{Ra}}^{*} {\text{Nhs}}^{*} - 1.19\;{\text{Ra}}^{*} {\text{Ha}}^{*} \\ & \quad - {\kern 1pt} 0.6\;{\text{Nhs}}^{*} {\text{Rd}} + 1.42\;{\text{Nhs}}^{*} \;{\text{Ha}}^{*} - 0.77\;{\text{Ha}}^{*} \;{\text{Rd}} \\ {\text{Ra}}^{*} & = 10^{ - 3} \;{\text{Ra}},\quad {\text{Ha}}^{*} = 0.1\;{\text{Ha,}}\quad {\text{Nhs}}^{*} = 10^{ - 3}\;{\text{Nhs}} \\ \end{aligned}$$
(12)
Fig. 9
figure 9figure 9

Influences of \({\text{Ra}},\;{\text{Ha}},\;{\text{Rd}},\;{\text{Nhs}}\) on Nuave at \(\varepsilon = 0.3,\phi = 0.04\)

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As demonstrated in Table 2, shape of powder can influence the thermal behavior due to its impact on thermal conductivity. As shape factor augments, Nuave enhances and its effect reduces with rise of Hartmann number. Spherical shape has minimum heat transfer rate.

Table 2 Impact of shape of nanoparticles on Nusselt number when \({\text{Rd}} = 0.8,\;{\text{Ra}} = 10,000,\;\varepsilon = 0.3,\;{\text{Nhs}} = 10,\;\phi = 0.04\)
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Conclusions

In this article, not only the Lorenz force impact but also the radiation impact were analyzed and to simplify the governing equations, Joule heating effect was overlooked and homogeneous model has been employed. As the Nhs is increased, sinusoidal wall temperature augments while increasing Ra has reverse effect. So, Nuave reduces with augment of Nhs while it rises with rise of Ra. Influence of Ha on temperature profile is converting convection to conduction which provide lower Nuave. According to formula of calculating Nuave, this function has direct relation with Rd. Three eddies are formed inside the domain in absence of magnetic force which intensify the convective flow.