Introduction

Analysis of magnetohydrodynamics \(\left( {\text{MHD}} \right)\) nanofluid flows is one of the important and noteworthy issues in heat transfer sciences. This subject has so far been studied by many scholars under different conditions [1,2,3,4,5,6,7,8,9,10,11]. This attention is due to the important role of this type of flow in the control of hydrothermal behaviors. In fact, in this type of flow, the heat transfer rates can be controlled by adding solid nanoparticles to the base fluid (nanofluid) and applying a magnetic field in the flow domain (MHD flow). There are many researches about the separate role of nanofluid [12,13,14,15,16,17,18,19,20,21,22,23,24,25,26] and magnetic field [27,28,29,30,31,32,33] on heat transfer features of fluid flow. Some of these researches have been dedicated to study the role of thermal radiation on the improvement of convection heat transfer rates [34,35,36,37,38,39,40]. Among these articles, Ghalambaz et al. [41] and Sheikholeslami et al. [42] analyzed the influences of thermal radiation and viscous dissipation on convection heat transfer of nanofluid flow in a square cavity under different conditions. Safaei et al. [43] simulated the coupling between thermal radiation and free convection heat transfer of nanofluid flow in a shallow cavity using the lattice Boltzmann method. Interacting effects of magnetic field and thermal radiation on hydrothermal behaviors of Al2O3–H2O nanofluid inside a permeable media were simulated by Sheikholeslami et al. [44]. They reported that the convection enhances as the radiation parameter increases, but it reduces by increasing the magnitudes of Hartmann number. In another research, Sheikholeslami and Rokni [45] numerically investigated the role of thermal radiation on heat transfer rates of Fe2O3–ethylene glycol nanofluid flow inside a porous enclosure considering the influences of an external electric field. It can be concluded that average Nusselt number enhances as Darcy number and radiation parameter increase. Besides, from a different point of view, the impact of radiative heat transfer on thermal behaviors of engineering problems was simulated using the discontinuous finite element method [46, 47].

The geometry of equipment used in industries and engineering applications is also one of the important factors in controlling the hydrothermal behaviors [48,49,50]. The channels with step are one of the most important geometries that are used in industries and engineering equipment. Therefore, analysis of fluid flow in these geometries is another important subject in thermal sciences. These kinds of flows are considered as a benchmark problem and have an essential role in controlling the heat transfer patterns. The hydrothermal patterns of these flows were extensively investigated in the absence of nanoparticles and magnetic field influences [51,52,53,54,55,56,57,58]. In these studies, the effects of different parameters such as baffles, obstacles, radiative heat transfer and Reynolds number on the flow and thermal characteristics were analyzed. Of course, given the industrial application of these flows, many scholars studied the impacts of solid nanoparticles on the hydrothermal features of step flows [59,60,61,62,63,64]. Among these articles, Mohammad et al. [65] and Alawi et al. [66] summarized and reviewed a large numbers of previous researches about the heat transfer characteristics of various nanofluid flows in channels having the steps. Selimefendigil and Oztop [67] numerically investigated the influences of different shaped obstacles on heat transfer rates of Cu–H2O nanofluid over a step with a corrugated bottom wall. They reported that the obstacle shapes and Reynolds number have notable impacts on the magnitudes of average Nusselt number. In other articles, two experimental and numerical studies were performed by Kherbeet et al. [68, 69] to investigate influences of various parameters on hydrothermal characteristics of nanofluid flow in the channels having the backward \(\left( {\text{BFS}} \right)\) and forward \(\left( {\text{FFS}} \right)\) facing steps. In another recent study, Atashafrooz [70] numerically simulated the impact of \({\text{Ag}}\) nanoparticles on hydrothermal behaviors of fluid flow in a 3-D channel with an inclined \({\text{BFS}}\).

It should be noted that the impacts of magnetic field on heat transfer features of step flow have been less studied compared to the nanofluids influences. Among these studies, Abbasi and Nassrallah [71] numerically analyzed the role of Lorentz force on the hydrothermal characteristics of step flow. They found that velocity field and Nusselt number distributions are dependent on the Stuart number. In other researches, Selimefendigil and Oztop [72] and Atashafrooz et al. [73, 74] examined the impacts of different parameters on thermal patterns and entropy generation of \({\text{MHD}}\) nanofluid flow in channels with step. They presented that the Hartmann number, buoyancy force, inclination angle of magnetic field and concentration of solid nanoparticles affected the flow irreversibility and heat transfer rates.

Although so far several researches have been done to analyze the hydrothermal behaviors in ducts with step, but as it is clear from the literature review, the effects of thermal radiation on forced convection heat transfer of \({\text{MHD}}\) nanofluid flow in a duct with an inclined \({\text{BFS}}\) are still not analyzed and simulated. This motivates the current research, such that the interacting impacts of radiative heat transfer, nanoparticles concentration, Brownian movement and magnetic field strength on the thermal features of \({\text{MHD}}\) nanofluid flow over an inclined \({\text{BFS}}\) are investigated here for the first time with full details.

Problem definition

The geometry and physical configuration of regarded problem for present research is depicted in Fig. 1. The relevant details and boundary conditions of considered geometry are obviously seen from this figure. This geometry includes a horizontal channel with an inclined \({\text{BFS }}\left( {\beta = 60^{ \circ } } \right)\). The Al2O3–H2O and CuO–H2O nanofluids are selected as the working fluid. Thermophysical characteristics of H2O and nanoparticles of CuO and Al2O3 are presented and tabulated in Table 1. Also, it is considered that an axial magnetic field with uniform strength of \(B_{0}\) affects the whole domain of nanofluid flow.

Fig. 1
figure 1

Geometry and physical configuration of problem

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Table 1 Thermophysical characteristics of H2O and nanoparticles of CuO and Al2O3 [73, 74]
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Governing equations

The vector forms of the basic equations for the current study can be written as follows:

$$\nabla \cdot \vec{V} = 0$$
(1)
$$\vec{V} \cdot \nabla \vec{V} = - \frac{1}{{\rho_{\text{nf}} }}\nabla p + \frac{{\mu_{\text{nf}} }}{{\rho_{\text{nf}} }} \left( {\nabla^{2} \vec{V}} \right) + \frac{1}{{\rho_{\text{nf}} }}\overrightarrow {{F_{\text{l}} }}$$
(2)
$$\vec{V} \cdot \nabla T = \frac{{k_{\text{nf}} }}{{\rho_{\text{nf}} C_{{p_{\text{nf}} }} }} \left( {\nabla^{2} T} \right) - \frac{1}{{\rho_{\text{nf}} C_{{p_{\text{nf}} }} }}\nabla \cdot \overrightarrow {{q_{\text{r}} }}$$
(3)

The terms of \(\overrightarrow {{F_{\text{l}} }}\) and \(\nabla \cdot \overrightarrow {{q_{\text{r}} }}\) are the Lorentz force and divergence of radiative heat flux, respectively. These two terms with the effective characteristics of nanofluid would be computed in the following sections.

Computation of effective characteristics of nanofluid

To evaluate the effective viscosity \(\left( {\mu_{\text{nf}} } \right)\), density \(\left( {\rho_{\text{nf}} } \right)\), thermal conductivity \(\left( {k_{\text{nf}} } \right)\) and specific heat \(\left( {C_{{{\text{p}}_{\text{nf}} }} } \right)\) of nanofluid, the following relationships are applied [75, 76]:

$$\mu_{\text{nf}} = \frac{{\mu_{\text{f}} }}{{\left( {1 - \phi } \right)^{2.5} }} + \frac{{k_{\text{Brownain}} }}{{k_{\text{f}} }} \times \frac{{\mu_{\text{f}} }}{Pr}$$
(4)
$$\rho_{\text{nf}} = \phi \rho_{\text{s}} + \left( {1 - \phi } \right)\rho_{\text{f}}$$
(5)
$$k_{\text{nf}} = k_{\text{f}} \left( {1 + \frac{{3\left( {\frac{{k_{\text{s}} }}{{k_{\text{f}} }} - 1} \right)\phi }}{{\left( {\frac{{k_{\text{s}} }}{{k_{\text{f}} }} + 2} \right) - \left( {\frac{{k_{\text{s}} }}{{k_{\text{f}} }} - 1} \right)\phi }}} \right) + k_{\text{Brownain}}$$
(6)
$$(\rho C_{\text{p}} )_{\text{nf}} = \phi (\rho C_{\text{p}} )_{\text{s}} + \left( {1 - \phi } \right)(\rho C_{\text{p}} )_{\text{f}}$$
(7)

According to Eqs. (4) and (6), impact of Brownian movement on the parameters of \(\mu_{\text{nf}}\) and \(k_{\text{nf}}\) is considered. To obtain the term of \(k_{\text{Brownain}}\), the \({\text{KKL}}\) correlation is applied as follows [44, 75, 76]:

$$k_{\text{Brownain}} = 5 \times 10^{4} \times \left( {\phi \times \rho_{\text{f}} \times C_{{{\text{p}}_{\text{f}} }} } \right) \times \sqrt {\frac{{C_{\text{b}} T}}{{\rho_{\text{s}} d_{\text{s}} }}} \times G\left( {T,\phi ,d_{\text{s}} } \right)$$
(8)

In this equation [44, 75, 76]:

$$\begin{aligned} G\left( {T,\phi ,d_{\text{s}} } \right) & = \left( {M_{1} + M_{2 } {\text{Ln}}\left( {d_{\text{s}} } \right) + M_{3 } {\text{Ln}}\left( \phi \right) + M_{4 } {\text{Ln}}\left( {d_{\text{s}} } \right){\text{Ln}}\left( \phi \right) + M_{5 } {\text{Ln}}\left( {d_{\text{s}} } \right)^{2} } \right) {\text{Ln}}\left( T \right) \\ & \quad + M_{6} + M_{7 } {\text{Ln}}\left( {d_{\text{s}} } \right) + M_{8 } {\text{Ln}}\left( \phi \right) + M_{9 } {\text{Ln}}\left( {d_{\text{s}} } \right){\text{Ln}}\left( \phi \right) + M_{10 } {\text{Ln}}\left( {d_{\text{s}} } \right)^{2} \\ \end{aligned}$$
(9)

The coefficients of \(M_{i} \left( {i = 1, 2, \ldots , 10} \right)\) are dependent on the type of nanofluid. The values of these coefficients for CuO–H2O and Al2O3–H2O nanofluids are presented in Table 2. Besides, the subscripts of \(f\) and \(s\) in Eqs. (4)–(9) denote, respectively, the thermophysical characteristics of base fluid (H2O) and solid nanoparticles (CuO, Al2O3), where the \(\phi\) symbol is the concentration of nanoparticles. Also, the \(C_{\text{b}}\) parameter in Eq. (8) is the Boltzmann constant \(\left( {C_{\text{b}} \approx 1.38 \times 10^{ - 23} \,{\text{J}}\,{\text{K}}^{ - 1} } \right)\).

Table 2 The coefficients values of CuO–H2O and Al2O3–H2O nanofluids in Eq. (9) [73, 74]
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Computation of Lorentz force

The Lorentz force appears in the momentum equations due to the existence of magnetic field. According to the Ohm’s low, this force can be obtained as follows [33]:

$$\overrightarrow {{F_{\text{l}} }} = \sigma_{\text{nf}} \left( {\vec{E} + \left( {\vec{V} \times \vec{B}} \right)} \right) \times \vec{B}$$
(10)

In the absence of electric field \(\left( {\vec{E} = 0} \right)\), the \(\overrightarrow {{F_{\text{l}} }}\) are computed as [33]:

$$\overrightarrow {{F_{\text{l}} }} = \sigma_{\text{nf}} \left( {\vec{V} \times \vec{B}} \right) \times \vec{B}$$
(11)

The term of \(\sigma_{\text{nf}}\) in the above equation is the effective electrical conductivity of nanofluid that can be computed as [75, 76]:

$$\sigma_{\text{nf}} = \sigma_{\text{f}} \left( {1 + \frac{{3\left( {\frac{{\sigma_{\text{s}} }}{{\sigma_{\text{f}} }} - 1} \right)\phi }}{{\left( {\frac{{\sigma_{\text{s}} }}{{\sigma_{\text{f}} }} + 2} \right) - \left( {\frac{{\sigma_{\text{s}} }}{{\sigma_{\text{f}} }} - 1} \right)\phi }}} \right)$$
(12)

Also, \(\vec{B} = B_{0} \vec{i}\) and \(\vec{V} = u\vec{i} + v\vec{j}\).

By substituting Eq. (12) and parameters of \(\vec{B}\) and \(\vec{V}\) in Eq. (11), the Lorentz force is obtained as follows:

$$\overrightarrow {{F_{\text{l}} }} = - \left( {\sigma_{\text{f}} \left( {1 + \frac{{3\left( {\frac{{\sigma_{\text{s}} }}{{\sigma_{\text{f}} }} - 1} \right)\phi }}{{\left( {\frac{{\sigma_{\text{s}} }}{{\sigma_{\text{f}} }} + 2} \right) - \left( {\frac{{\sigma_{\text{s}} }}{{\sigma_{\text{f}} }} - 1} \right)\phi }}} \right) \times v \times B_{0}^{2} } \right) \vec{j}$$
(13)

Computation of radiative heat flux

Since, the working fluid in this study is \({\text{MHD}}\) nanofluid, the radiative heat flux can be obtained using Rosseland approximation as [41]:

$$\overrightarrow {{q_{\text{r}} }} = q_{{{\text{r}}_{\text{x}} }} \vec{i} + q_{{{\text{r}}_{\text{y}} }} \vec{j}$$
(14)

and

$$q_{{{\text{r}}_{\text{x}} }} = \frac{{ - 4\sigma^{*} }}{{3\beta_{\text{R}} }}\frac{{\partial T^{4} }}{\partial x}$$
(15)
$$q_{{{\text{r}}_{\text{y}} }} = \frac{{ - 4\sigma^{*} }}{{3\beta_{\text{R}} }}\frac{{\partial T^{4} }}{\partial y}$$
(16)

where \(\sigma^{*}\) and \(\beta_{\text{R}}\) are, respectively, the Stefan–Boltzmann constant \(\left( {5.67 \times 10^{ - 8} \,{\text{W}}\,{\text{m}}^{ - 2} \,{\text{K}}^{ - 4} } \right)\) and the coefficient of mean absorption. In Eqs. (15) and (16), the \(T^{4}\) can be approximated using Taylor series as follows [39, 41, 45]:

$$T^{4} = 4\left( {T_{\text{in}} } \right)^{3} T - 3\left( {T_{\text{in}} } \right)^{4}$$
(17)

Therefore, the divergence of radiative heat flux for \({\text{MHD}}\) nanofluid flow would be calculated using the below equation [39, 41, 45]:

$$\nabla \cdot \overrightarrow {{q_{\text{r}} }} = \frac{{ - 16\left( {T_{\text{in}} } \right)^{3} \sigma^{*} }}{{3\beta_{\text{R}} }}\left( {\frac{{\partial^{2} T}}{{\partial x^{2} }} + \frac{{\partial^{2} T}}{{\partial y^{2} }}} \right)$$
(18)

Dimensionless forms of governing equations and boundary conditions

By substituting Eqs. (13) and (18) into momentum and energy equations and using the dimensionless terms, the non-dimensional forms of governing equations can be rewritten as:

$$\frac{\partial U}{\partial X} + \frac{\partial V}{\partial Y} = 0$$
(19)
$$U\frac{\partial U}{\partial X} + V\frac{\partial U}{\partial Y} = \left( {\frac{ - 1}{{\left( {1 - \phi } \right) + \phi \frac{{\rho_{\text{s}} }}{{\rho_{\text{f}} }}}}} \right)\frac{\partial P}{\partial X} + \frac{1}{Re}\left( {\frac{{\frac{1}{{\left( {1 - \phi } \right)^{2.5} }} + \frac{{k_{\text{Brownain}} }}{{k_{\text{f}} \times Pr}} }}{{\left( {1 - \phi } \right) + \phi \frac{{\rho_{\text{s}} }}{{\rho_{\text{f}} }}}}} \right) \left( {\frac{{\partial^{2} U}}{{\partial X^{2} }} + \frac{{\partial^{2} U}}{{\partial Y^{2} }}} \right)$$
(20)
$$\begin{aligned} U\frac{\partial V}{\partial X} + V\frac{\partial V}{\partial Y} & = \left( {\frac{ - 1}{{\left( {1 - \phi } \right) + \phi \frac{{\rho_{\text{s}} }}{{\rho_{\text{f}} }}}}} \right)\frac{\partial P}{\partial Y} + \frac{1}{Re}\left( {\frac{{\frac{1}{{\left( {1 - \phi } \right)^{2.5} }} + \frac{{k_{\text{Brownain}} }}{{k_{\text{f}} \times Pr}} }}{{\left( {1 - \phi } \right) + \phi \frac{{\rho_{\text{s}} }}{{\rho_{\text{f}} }}}}} \right)\left( {\frac{{\partial^{2} V}}{{\partial X^{2} }} + \frac{{\partial^{2} V}}{{\partial Y^{2} }}} \right) \\ & \quad - \left( {\frac{{1 + \frac{{3\left( {\frac{{\sigma_{\text{s}} }}{{\sigma_{\text{f}} }} - 1} \right)\phi }}{{\left( {\frac{{\sigma_{\text{s}} }}{{\sigma_{\text{f}} }} + 2} \right) - \left( {\frac{{\sigma_{\text{s}} }}{{\sigma_{\text{f}} }} - 1} \right)\phi }}}}{{\left( {1 - \phi } \right) + \phi \frac{{\rho_{\text{s}} }}{{\rho_{\text{f}} }}}}} \right)\frac{{Ha^{2} V}}{Re} \\ \end{aligned}$$
(21)
$$\begin{aligned} U\frac{\partial \varTheta }{\partial X} + V\frac{\partial \varTheta }{\partial Y} & = \frac{1}{Re Pr}\left( {\frac{{\left( {1 + \frac{{3\left( {\frac{{k_{\text{s}} }}{{k_{\text{f}} }} - 1} \right)\phi }}{{\left( {\frac{{k_{\text{s}} }}{{k_{\text{f}} }} + 2} \right) - \left( {\frac{{k_{\text{s}} }}{{k_{\text{f}} }} - 1} \right)\phi }}} \right) + \frac{{k_{\text{Brownain}} }}{{k_{\text{f}} }}}}{{\left( {1 - \phi } \right) + \phi \frac{{\left( {\rho c_{p} } \right)_{s} }}{{\left( {\rho c_{p} } \right)_{f} }}}}} \right) \\ & \times \left( {1 + \frac{4}{3}Rd\left( {\left( {1 + \frac{{3\left( {\frac{{k_{\text{s}} }}{{k_{\text{f}} }} - 1} \right)\phi }}{{\left( {\frac{{k_{\text{s}} }}{{k_{\text{f}} }} + 2} \right) - \left( {\frac{{k_{\text{s}} }}{{k_{\text{f}} }} - 1} \right)\phi }}} \right) + \frac{{k_{\text{Brownain}} }}{{k_{\text{f}} }}} \right)^{ - 1} } \right)\left( {\frac{{\partial^{2} \varTheta }}{{\partial X^{2} }} + \frac{{\partial^{2} \varTheta }}{{\partial Y^{2} }}} \right) \\ \end{aligned}$$
(22)

where

$$\begin{array}{*{20}l} {\left( {X, Y} \right) = \left( {\frac{x}{H},\frac{y}{H}} \right)} \hfill & {\left( {U, V} \right) = \left( {\frac{u}{{U_{\text{in}} }},\frac{v}{{U_{\text{in}} }}} \right)} \hfill & {P = \frac{p}{{U_{\text{in}}^{2} }}} \hfill & {Pr = \frac{{\mu_{\text{f }} C_{{{\text{p}}_{\text{f}} }} }}{{k_{\text{f}} }}} \hfill \\ {Ha = B_{0} H\sqrt {\frac{{\sigma_{\text{f}} }}{{\mu_{\text{f}} }}} } \hfill & {Re = \frac{{\rho_{\text{f}} U_{\text{in}} H}}{{\mu_{\text{f}} }}} \hfill & {\varTheta = \frac{{T - T_{\text{in}} }}{{T_{\text{w}} - T_{\text{in}} }}} \hfill & {Rd = \frac{{4\left( {T_{\text{in}} } \right)^{3} \sigma^{*} }}{{k_{\text{f}} \beta_{\text{R}} }}} \hfill \\ \end{array}$$
(23)

Besides, the non-dimensional forms of the boundary conditions can be presented as:

$$\begin{array}{*{20}l} {{\text{At}}\,{\text{ the}}\,{\text{ inlet}}\,{\text{ section}}\,{\text{ of}}\,{\text{ channel:}}} \hfill & {U = 1,\, V = 0 \,{\text{and}}\, \varTheta = 0} \hfill \\ {{\text{At}}\,{\text{ all}}\,{\text{ channel}}\,{\text{ walls:}}} \hfill & {U = V = 0 \,{\text{and }}\, \varTheta = 1} \hfill \\ {{\text{At}}\,{\text{ the}}\,{\text{ outlet}}\,{\text{ section}}\,{\text{ of }}\,{\text{channel:}}} \hfill & {\frac{\partial U}{\partial X} = \frac{\partial V}{\partial X} = 0\, {\text{and }}\, \frac{\partial \varTheta }{\partial X} = 0} \hfill \\ \end{array}$$

Definition of Nusselt numbers

In convection heat transfer of \({\text{MHD}}\) nanofluid flow in the presence of thermal radiation, the total heat flux \(\left( {\overrightarrow {{q_{\text{t}} }} } \right)\) on the surfaces is obtained by summing the convective \(\left( {\overrightarrow {{q_{\text{c}} }} } \right)\) and radiative \(\left( {\overrightarrow {{q_{\text{r}} }} } \right)\) heat fluxes. Therefore, the total Nusselt number \(\left( {Nu_{\text{t}} } \right)\) is the sum of the convective \(\left( {Nu_{\text{c}} } \right)\) and radiative \(\left( {Nu_{\text{r}} } \right)\) Nusselt numbers [39, 45]. These Nusselt numbers are computed along the bottom wall of channel and downstream of the BFS \(\left( {0 \le X \le \frac{{L_{\text{D}} }}{H}\, {\text{and}}\, Y = 0} \right)\) as:

$$Nu_{\text{c}} \left( X \right) = - \left( {1 + \frac{{3\left( {\frac{{k_{\text{s}} }}{{k_{\text{f}} }} - 1} \right)\phi }}{{\left( {\frac{{k_{\text{s}} }}{{k_{\text{f}} }} + 2} \right) - \left( {\frac{{k_{\text{s}} }}{{k_{\text{f}} }} - 1} \right)\phi }} + \frac{{k_{\text{Brownain}} }}{{k_{\text{f}} }}} \right) \frac{1}{{\varTheta_{\text{w}} - \varTheta_{\text{M}} }} \left. {\frac{\partial \varTheta }{\partial Y}} \right|_{Y = 0}$$
(24)
$$Nu_{\text{r}} \left( X \right) = - \left( {\frac{4}{3}Rd } \right)\frac{1}{{\varTheta_{\text{w}} - \varTheta_{\text{M}} }} \left. {\frac{\partial \varTheta }{\partial Y}} \right|_{Y = 0}$$
(25)
$$Nu_{\text{t}} \left( X \right) = Nu_{\text{c}} \left( X \right) + Nu_{\text{r}} \left( X \right) = - \left( {1 + \frac{{3\left( {\frac{{k_{\text{s}} }}{{k_{\text{f}} }} - 1} \right)\phi }}{{\left( {\frac{{k_{\text{s}} }}{{k_{\text{f}} }} + 2} \right) - \left( {\frac{{k_{\text{s}} }}{{k_{\text{f}} }} - 1} \right)\phi }} + \frac{{k_{\text{Brownain}} }}{{k_{\text{f}} }} + \frac{4}{3}Rd } \right) \frac{1}{{\varTheta_{\text{w}} - \varTheta_{\text{M}} }} \left. {\frac{\partial \varTheta }{\partial Y}} \right|_{Y = 0}$$
(26)

where the mean bulk temperature \(\left( {\varTheta_{\text{M}} } \right)\) is defined as:

$$\varTheta_{\text{M}} \left( X \right) = \frac{{\mathop \smallint \nolimits_{0}^{1} U\varTheta {\text{d}}Y}}{{\mathop \smallint \nolimits_{0}^{1} U {\text{d}}Y}}$$
(27)

Numerical solution and validation results

The basic Eqs. (19)–(22) are first discretized using the finite volume method \(\left( {\text{FVM}} \right)\) and are converted to the set of algebraic equations. Also, the pressure field \(\left( P \right)\) is coupled with velocity fields \(\left( {U, V} \right)\) using the SIMPLE algorithm [77]. Then, the obtained algebraic equations are solved numerically by the line-by-line \(\left( {\text{LBL}} \right)\) method and using a FORTRAN code. The blocked-off method [78,79,80] is employed for modeling the inclined wall of \({\text{BFS}}\) in Cartesian coordinates. The convergence criterion is defined so that the sum of absolute residuals of \(U, V, P\) and \(\varTheta\) variables is less than \(10^{ - 5}\). Additionally, the following criterion is also employed to ensure the convergence of solution:

$${\text{Max}}\left| {\frac{{\varGamma^{\xi } \left( {m,n} \right) - \varGamma^{\xi - 1} \left( {m,n} \right)}}{{\varGamma^{\xi } \left( {m,n} \right)}}} \right| \le 10^{ - 6}$$

where the \(\xi\) symbol denotes the iteration level and the \(\varGamma\) term points to all variables of \(U , V, P\) and \(\varTheta\). To ensure that the obtained results are not dependent on the gird size, the grid independence analysis is carried out. Accordingly, a grid size of \(675 \times 42\)\(\left( {X \times Y } \right)\) is selected as the best mesh. Of course, the mentioned mesh is clustered near the channel and step walls to achieve more accuracy.

The numerical solution presented in the current research is validated with the results of two different studies. First, the magnitudes of maximum Nusselt number \(\left( {Nu_{\text{Max}} } \right)\) along the bottom wall of a channel with \({\text{BFS}}\) and its position \(\left( {X_{\text{Max}} } \right)\) are compared with the simulated results by Abu-Nada [59]. To reach this goal, all conditions of the problem of Ref. [59] are applied in the computer program written to analyze the problem of the current research. These comparisons are demonstrated in Fig. 2 for the various magnitudes of nanoparticles concentration and Reynolds numbers at \(ER = 2\). Moreover, a close agreement is observed between the presented results of this study and Ref. [59]. To check the correctness of the simulation results of nanofluid flow in the presence of magnetic field, the second comparison is carried out based on the findings of Aminossadati et al. [81]. The magnitudes of average Nusselt number \(\left( {\overline{Nu} } \right)\) and maximum temperature \(\left( {\varTheta_{\text{Max}} } \right)\) are compared at various values of \(Ha \left( {Ha = 0, 10, 20, 30, 40} \right)\) in Table 3. According to this table, the maximum difference between the results is about \(1.65\%\); therefore, the simulated results in this research are validated.

Fig. 2
figure 2

Comparison of \(X_{\text{Max}}\) and \(Nu_{\text{Max}}\) magnitudes at various magnitudes of \(\phi\) with the findings of Abu-Nada [57]

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Table 3 Comparison of \(\overline{Nu}\) and \(\theta_{\text{Max}}\) magnitudes at different values of \(Ha\) with the results of Ref. [79] \(\left( {\phi = 0.02, Re = 100} \right)\)
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Results and discussions

First, influences of CuO and Al2O3 nanoparticles on the magnitudes of average total Nusselt number \(\left( {\overline{{Nu_{\text{t}} }} = \mathop \smallint \limits_{0}^{{\frac{{L_{\text{D}} }}{H}}} Nu_{\text{t}} \left( X \right){\text{d}}X} \right)\) are presented in Table 4 at five various values of radiation parameter. As it is seen from this table, the magnitudes of \(\overline{{Nu_{\text{t}} }}\) are different for Al2O3–H2O and CuO–H2O nanofluids. By choosing the CuO–H2O nanofluid as the working fluid, a higher \(\overline{{Nu_{\text{t}} }}\) is obtained. This means that the impact of CuO nanoparticles on the heating and cooling procedures is greater than influence of Al2O3 nanoparticles. Therefore, the CuO–H2O nanofluid is selected to analyze the thermal behaviors at different values of radiation parameter \(\left( {Rd} \right)\), Hartmann number \(\left( {Ha} \right)\) and nanoparticles concentration \(\left( \phi \right)\).

Table 4 Influence of CuO and Al2O3 nanoparticles on the magnitudes of average total Nusselt number \(\left( {\overline{{Nu_{\text{t}} }} } \right)\) at various values of \(Rd\) parameter \(\left( {Ha = 20, \phi = 0.04} \right)\)
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To clarify the influence of radiation heat transfer on the thermal features of MHD CuO–H2O nanofluid, isotherms contours are presented in Fig. 3a–d at different magnitudes of \(Rd , \phi\) and \(Ha\) parameters. According to these figures, all three parameters have a significant impact on the isotherms contours inside the channel. To further show the details of these impacts, the dimensionless temperature distributions \(\left( \varTheta \right)\) along \(Y\) axis are also plotted in Fig. 4a–e at different locations of \(X \left( {X = 1, 2, 5, 10} \right)\). According to these figures, by augmentation of the radiation parameter, the \(\varTheta\) distribution becomes more homogeneous and approaches the temperature of the hot walls; such that the temperature gradients along the channel walls (especially along the bottom wall) reduce with increase in the \(Rd\) parameter. Besides, any increase in the nanoparticles concentration results in a decrease in magnitudes of the \(\varTheta\) distribution. Of course, this reduction is negligible near the duct walls (especially near the bottom wall), so that the difference between the temperature gradients on these walls is very small. The impact of magnetic field on the \(\varTheta\) distribution is well seen from the gradients of temperature on the bottom wall \(\left( {\left. {\frac{\partial \varTheta }{\partial Y}} \right|_{Y = 0} } \right)\). In fact, the magnitudes of \(\left. {\frac{\partial \varTheta }{\partial Y}} \right|_{Y = 0}\) decrease by augmentation of \(Ha\) number. Also, the role of recirculation zones on the \(\varTheta\) distribution is well illustrated in Figs. 3 and 4. To better understand this issue, the streamlines contours are shown in Fig. 5a and b at various magnitudes of \(\phi\) and \(Ha\) for \(Rd = 0.5\). A significant recirculation zone (lower bubble) is created on the bottom wall downstream the \({\text{BFS}}\). The length of this region on the bottom wall \(\left( {L_{\text{r}} } \right)\) enhances with augment of \(\phi\) and \(Ha\) parameters. Besides, the secondary recirculation region (upper bubble) is formed on the top wall in the absence of Lorentz force \(\left( {Ha = 0} \right)\). The extent of this region increases by enhancing the \(\phi\) parameter.

Fig. 3
figure 3

Distributions of isotherms in the duct at various magnitudes of \(Rd, \phi\) and \(Ha\) parameters

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Fig. 4
figure 4figure 4

Temperature variations along \(Y\) axis at various magnitudes of \(Rd, \phi\) and \(Ha\) parameters

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Fig. 5
figure 5

Impacts of the \(\phi\) and \(Ha\) parameters on the streamlines distributions in the duct

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The main bulk temperature \(\left( {\varTheta_{\text{M}} } \right)\) is one of the important factors in determining the thermal characteristics. The impacts of \(Rd , \phi\) and \(Ha\) parameters on the variations of \(\varTheta_{\text{M}}\) along the duct are shown in Fig. 6a–c. In all cases, the \(\varTheta_{\text{M}}\) enhances along the channel in the axial direction of flow, due to both radiative and convective heat transfer mechanisms. However, the increase in \(Rd\) parameter and consequently enhancement of thermal radiation heat transfer leads to an augmentation in the \(\varTheta_{\text{M}}\) magnitudes, while it decreases with increase in the \(\phi\) and \(Ha\) parameters.

Fig. 6
figure 6

Impacts of different parameters on the \(\varTheta_{\text{M}}\) variations along the duct

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Considering the significant effects of \(Rd , \phi\) and \(Ha\) parameters on the temperature distributions and mean bulk temperature; it is expected that the heat transfer rates are affected by these parameters. In the next figures, an attempt is performed to analyze this issue. Impacts of \(Rd , \phi\) and \(Ha\) parameters on the convective Nusselt number \(\left( {Nu_{\text{c}} } \right)\) along the bottom \(\left( {0 \le X \le \frac{{L_{\text{D}} }}{H} , Y = 0} \right)\) are displayed in Fig. 7a–e. As it is seen from these figures, both the trends and magnitudes of \(Nu_{\text{c}}\) are dependent on the \(Ha\) number, whereas the \(Rd\) and \(\phi\) parameters only affect its values. In fact, the difference between the trends of \(Nu_{\text{c}}\) distributions at various magnitudes of \(Ha\) is seen at the lower recirculation zone and near to the reattachment point \(\left( {X_{\text{r}} } \right)\). In the case of \(Ha = 0\), the \(Nu_{\text{c}}\) has a local maximum value \(\left( {Nu_{{{\text{c}}_{\text{Max}} }} } \right)\) near the reattachment point, while the magnitude of \(Nu_{{{\text{c}}_{\text{Max}} }}\) decreases with augmentation of \(Ha\) number, such that the \(Nu_{\text{c}}\) distribution lacks a peak in the highest value of \(Ha\) number \(\left( {Ha = 60} \right)\). However, the \(Nu_{\text{c}}\) decreases with enhancing the radiation parameter and Hartmann number, but it enhances with increase in the concentration of \({\text{CuO}}\) nanoparticles. These treatments can be also seen from the magnitudes of average convective Nusselt number \(\left( {\overline{{Nu_{\text{c}} }} = \mathop \smallint \nolimits_{0}^{{\frac{{L_{\text{D}} }}{H}}} Nu_{\text{c}} \left( X \right){\text{d}}X} \right)\) against the various values of \(Rd , \phi\) and \(Ha\) parameters, which are reported in Fig. 8. Based on Fig. 8, the maximum rate of convective heat transfer occurs in the absence of thermal radiation and magnetic field and for the nanofluid with highest value of nanoparticles concentration \(\left( {Rd = 0, Ha = 0, \phi = 0.04} \right)\). With respect to Eq. (24) and the presented explanations about the \(\frac{\partial \varTheta }{\partial Y}\) and \(\varTheta_{\text{M}}\), it can be concluded that the reduction of \(Nu_{\text{c}}\) in terms of \(Rd\) parameter is due to the decrease of \(\left. {\frac{\partial \varTheta }{\partial Y}} \right|_{Y = 0}\), whereas its enhancement against the \(\phi\) parameter is related to the increase in \(\frac{{k_{\text{nf}} }}{{k_{\text{f}} }}\) term. Also, the decrease of \(\left. {\frac{\partial \varTheta }{\partial Y}} \right|_{Y = 0}\) and \(\varTheta_{\text{M}}\) terms against \(Ha\) number is the main reason for reducing the \(Nu_{\text{c}}\) in terms of this number.

Fig. 7
figure 7

Variations of \(Nu_{\text{c}}\) along the bottom wall at various magnitudes of \(Rd, \phi\) and \(Ha\) parameters

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Fig. 8
figure 8

Impacts of the \(Rd, \phi\) and \(Ha\) parameters on the values of \(\overline{{Nu_{\text{c}} }}\)

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Influences of \(Rd , \phi\) and \(Ha\) parameters on the radiative Nusselt number \(\left( {Nu_{\text{r}} } \right)\) distributions along the bottom wall are shown in Fig. 9a–e. Comparing the presented results in Figs. 7 and 9 clearly shows that trends of \(Nu_{\text{r}}\) distribution are similar to the trends of \(Nu_{\text{c}}\) distribution. Therefore, additional explanations are not presented to avoid repetition. According to Figs. 9 and 10, the \(Nu_{\text{r}}\) and its average \(\left( {\overline{{Nu_{\text{r}} }} = \mathop \smallint \nolimits_{0}^{{\frac{{L_{\text{D}} }}{H}}} Nu_{\text{r}} \left( X \right){\text{d}}X} \right)\) enhance significantly with augmentation of radiation parameter. This enhancement is due to an increase in the magnitude of \(\varTheta_{\text{M}}\) parameter and the direct presence of \(Rd\) parameter in calculating the radiative Nusselt number. But, the \(Nu_{\text{r}}\) and \(\overline{{Nu_{\text{r}} }}\) reduce with the increase in the \(Ha\) number because of the decrease of \(\varTheta_{\text{M}}\) and \(\left. {\frac{\partial \varTheta }{\partial Y}} \right|_{Y = 0}\) terms against this number. A noteworthy point in Fig. 9 is that the impact of \(\phi\) parameter on the magnitudes of \(Nu_{\text{r}}\) is dependent on the values of \(Ha\) number. In the case of \(Ha = 60\), this impact is generally decreasing, while in the absence of magnetic field \(\left( {Ha = 0} \right)\), this influence is different in different zones. However, Fig. 10 clearly presents that the overall effect of concentration of nanoparticles on the \(\overline{{Nu_{\text{r}} }}\) is small and decreasing. In fact, this reduction is due to a decrease in the values of \(\varTheta_{\text{M}}\) against the \(\phi\) variable.

Fig. 9
figure 9

Variations of \(Nu_{\text{r}}\) along the bottom wall at various magnitudes of \(Rd, \phi\) and \(Ha\) parameters

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Fig. 10
figure 10

Impacts of the \(Rd, \phi\) and \(Ha\) parameters on the values of \(\overline{{Nu_{\text{r}} }}\)

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Originally, overall heat transfer characteristics of the thermal systems can be specified by calculating the total Nusselt number \(\left( {Nu_{\text{t}} } \right)\). Distributions of \(Nu_{\text{t}}\) along the bottom wall are displayed in Fig. 11a–e at various values of \(Rd , \phi\) and \(Ha\) parameters. These figures clearly show that the trends of \(Nu_{\text{t}}\) distribution are similar to the variations of the \(Nu_{\text{c}}\) and \(Nu_{\text{r}}\). As it is shown in Fig. 11a–e, the \(Nu_{\text{t}}\) enhances with augmentation of the \(Rd\) and \(\phi\) parameters, but it reduces with increase in the \(Ha\) number. These results are also confirmed by calculating the magnitudes of average total Nusselt number \(\left( {\overline{{Nu_{\text{t}} }} } \right)\) in Fig. 12. Based on this figure, the highest magnitude of total heat transfer rate occurs in the absence of Lorentz force and for the highest values of radiation parameter and nanoparticles concentration \(\left( {Ha = 0, Rd = 1, \phi = 0.04} \right)\).

Fig. 11
figure 11

Variations of \(Nu_{\text{t}}\) along the bottom wall at various magnitudes of \(Rd, \phi\) and \(Ha\) parameters

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Fig. 12
figure 12

Impacts of the \(Rd, \phi\) and \(Ha\) parameters on the values of \(\overline{{Nu_{\text{t}} }}\)

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Conclusions

Interacting impacts of radiative heart transfer, Lorentz force and Brownian movement on the thermal features of \({\text{MHD}}\) nanofluid flow over an inclined \({\text{BFS}}\) are analyzed with details. The main results of this research can be summarized as follows:

  • The CuO nanoparticles have a greater impact on heat transfer rates in comparison to the Al2O3 nanoparticles.

  • The trends and magnitudes of \(Nu_{\text{c}}\), \(Nu_{\text{r}}\) and \(Nu_{\text{t}}\) along the bottom wall are dependent on the \(Ha\) number, while the \(Rd\) and \(\phi\) parameters only affect their values.

  • An enhancement in the radiation parameter leads to an increase in the magnitudes of radiative and total heat transfer rates, while the values of convective heat transfer rate reduce with increase in this parameter. As for the case of \(Ha = 20\) and \(\phi = 0.04\), the \(\overline{{Nu_{\text{r}} }}\) and \(\overline{{Nu_{\text{t}} }}\) enhance about \(261.92\%\) and \(39.96\%\), respectively, when the \(Rd\) parameter rises from \(0.25\) to \(1\), while the reduction rate of the \(\overline{{Nu_{\text{c}} }}\) is about \(8.28\%\)\(\left( {\left| {\frac{{\left. {\overline{{Nu_{\text{t}} }} } \right|_{Rd = 1} - \left. {\overline{{Nu_{\text{t}} }} } \right|_{Rd = 0.25} }}{{\left. {\overline{{Nu_{\text{t}} }} } \right|_{Rd = 0.25} }}} \right| \times 100 = 39.96\% } \right).\)

  • All heat transfer rates decrease with augmentation of the magnetic field strength, such that for the case of \(Rd = 0.5\) and \(\phi = 0.04\), the \(\overline{{Nu_{\text{c}} }}\), \(\overline{{Nu_{\text{r}} }}\) and \(\overline{{Nu_{\text{t}} }}\) variables reduce, respectively, about \(19.54\%\), \(19.55\%\) and \(19.57\%\) when the \(Ha\) number changes from \(0\) to \(60\)\(\left( {\left| {\frac{{\left. {\overline{{Nu_{\text{t}} }} } \right|_{Ha = 60} - \left. {\overline{{Nu_{\text{t}} }} } \right|_{Ha = 0} }}{{\left. {\overline{{Nu_{\text{t}} }} } \right|_{Ha = 0} }}} \right| \times 100 = 19.57\% } \right).\)

  • By adding the CuO nanoparticle to the water-based flow, convective and total heat transfer rates enhance. As for the case of \(Rd = 0.5\) and \(Ha = 20\), the \(\overline{{Nu_{\text{c}} }}\) and \(\overline{{Nu_{\text{t}} }}\) increase, respectively, about \(13.96\%\) and \(8.27\%\) when the nanoparticles concentration rises to \(4\%\)\(\left( {\left| {\frac{{\left. {\overline{{Nu_{\text{t}} }} } \right|_{\phi = 0.04} - \left. {\overline{{Nu_{\text{t}} }} } \right|_{\phi = 0} }}{{\left. {\overline{{Nu_{\text{t}} }} } \right|_{\phi = 0} }}} \right| \times 100 = 8.27\% } \right) .\) Also, the impact of \(\phi\) parameter on the \(\overline{{Nu_{\text{r}} }}\) variable decreases about \(3.07\%\).