Non-similar analysis of heat generation and thermal radiation on Eyring-Powell Hybrid nanofluid flow across a stretching surface

Abstract

This paper portrays a two-dimensional mathematical model developed for the analysis of Eyring-Powell hybrid nanofluid flow, incorporating variable temperature and velocity profiles. This study investigates the flow through a porous media with a mixture of hybrid nanoparticles (copper dioxide, magnetite) integrated into ethylene glycol ( $\mathrm{EG}$ ) across a stretching sheet. The model takes magnetic effects, heat generation, and thermal radiation into account. Additionally, nonsimilar partial differential equations (PDEs) are transformed into ordinary differential equations using the local nonsimilarity approach. These equations are then numerically solved using the built-in Matlab function bvp4c. To examine the effects of adjusting parameters on heat, mass transfer, and skin friction, the results are displayed graphically. With an increase in the given values of magnetic field $M$ , the velocity profile $f\prime (\xi ,\eta )$ is seen to decay. Moreover, it is observed that the hybrid nanofluid density and dynamic viscosity increase as the volume fraction rises. It is estimated that the thermal conductivity and viscosity of the ${\mathrm{Fe}}_{3}O4+\mathrm{Cu}/\mathrm{EG}$ hybrid nanofluid are expected to increase to 4.1% and 12.35%, respectively, and to 71.55% and 78.01%, respectively, for volume concentrations ranging from 0.02% to 0.05%. It is also concluded that the local skin-friction coefficient has the opposite trend while the local Nusselt number rises monotonically with both the slip velocity parameter and the surface convection parameter. This research has broad applications In the glass and polymer industries, metallic plate cooling, plastic sheet contractions, etc.

Introduction

Nanofluid is a kind of sophisticated manufactured fluid that is made up of a base fluid (usually a liquid, such ethylene glycol, water, or oil) suspended in nanoparticles, which particles smaller than a nanometer. Frequently, metals, metal oxides, carbon nanotubes, and other nanomaterials are used to create these nanoparticles. Engineering, physics, materials science, and other disciplines are interested in nanofluids because they have special and improved features that result from the addition of nanoparticles to the basic fluid. Thermal conductivity of nanofluids is typically much higher than that of the base fluid alone. Because of this characteristic, they can be applied to thermal management and heat transfer systems, including the creation of more effective heat exchangers and electronic cooling systems. Shafiq and Sindhu ¹ conducted a statistical analysis of Williamson nanofluid on a porous stretching surface under the influence of magnetohydrodynamic (MHD) forces. Shekaramiz et al. ² investigated the MHD boundary layer flow of a nanofluid over a stretched sheet. An investigation into Maxwell nanofluid flow with convective boundary conditions was carried out by Razzaq et al. ³ Shah et al. ⁴ considered heat radiation and viscous dissipation while examining the flow of nanofluid through a stretchable Riga wall. Khan et al. ⁵ study was concerned with the Casson nanofluid flow above a nonlinear stretching plate. Farooq et al. ⁶ examined the potential of MHD bioconvective micropolar nanofluids while taking into account the soret and dufour impacts. Ali et al. ⁷ numerically investigated the transient MHD 3D rotating flow of Maxwell Cattaneo Christov heat flux model over a bidirectional stretching sheet. Khattak et al. ⁸ explored the study of nanofluid flow over a porous and deformable plate, taking into account Joule heating, thermal radiation, as well as thermophoresis and Brownian motion effects within the framework of the Cattaeno–Christov heat flux model. Razzaq et al. ⁹ examined the nanofluid flow for single-multilayer carbon nanotubes upon a vertical porous cone while considering the influences of MHD and thermal radiation.

Hybrid nanofluids represent a compelling solution for heat storage, exhibiting notably higher energy efficacious contrast to conventional nanofluids. The utilization of hybrid nanofluids, comprising manganese, zinc ferrite, copper, and silver nanocomposites on spinning discs has garnered increasing attention in both research and development due to their multifaceted applications. The MHD convection of $(A{l}_2{O}_3-\mathrm{Cu}/\mathrm{Water}$ ) with an internal heat sink/source in a porous media was investigated by Chamkha et al. ¹⁰ It is concluded that raising the number of nanoparticles, the Rayleigh numbers reduce thermal reaction and increases heat transmission in the case of high-volume percentage. Cao et al. ¹¹ investigated the effects of various nanomaterials under distinct degrees of partial slip using ternary-hybrid nanofluid models. Applying the concept of non-Fourier principles, Algehyne et al. ¹² examined the flow of a fluid over a Riga plate, considering slip impacts and activation energy, to explore the influence of MHD on the movement of $\mathrm{Ag}-\mathrm{MgO}$ / $H_{2}O$ -embedded hybrid nanofluid containing motile microorganisms. Sarada et al. ¹³ developed a heat flux model utilizing non-Fourier analysis to delve into the heat production implications within the hybridization flow of ternary nanofluids. Elnaqeeb et al. ¹⁴ developed three-dimensional flow involving dual stretching and suction of ternary hybrid nano-fluid within a closed rectangular region, with various shapes and densities. Khazayinejad and Nourazar ¹⁵ studied spatial fractional energy transmission and continuous laminar MHD boundary layer flow using a $\mathrm{Gr}-F{e}_3{O}_4-H_{2}O$ hybrid nanofluid. The flow with heat transfer containing Eyring-Powell nanoparticles from a moving stretched/shrinking sheet was analyzed by Roşca and Pop. ¹⁶ El-Zahar et al. ¹⁷ used a radioactive rolling cylinder to study the mixed convection flow of magneto-hybrid nanofluids. It was discovered that the rigid volume fraction was lessened by the force of drag.

Within the field of fluid dynamics, materials’ behavior is divided into two categories: Newtonian and non-Newtonian fluids. Newton’s law of viscosity governs Newtonian fluids like water and air, but non-Newtonian fluids behave more complexly, when viscosity varying with time, stress, and shear rate, among other factors. Eyring-Powell nanofluid model is an example of non-Newtonian fluids. It is applied to explain the rheological behavior of colloidal suspensions of nanoparticles in a base fluid. Cui et al. ¹⁸ used nonsimilar modeling to study the forced convection of magnetized Eyring-Powell nanofluid flow in a porous media above a Riga sheet. Mushtaq et al. ¹⁹ examined the effects of an exponentially growing sheet under the assumption of Eyring-Powell fluid. They examined the opposite impacts of viscoelastic properties on heat-velocity boundary layer. Rahimi et al. ²⁰ utilized the collocation technique to estimate the boundary layer of the Powell-Eyring fluid. Akbar et al. ²¹ presented numerical solutions for the Powell-Eyring magnetofluid above an expanding sheet. Hassan et al. ²² conducted research focusing on examining the flow characteristics of the Powell-Eyring fluid and its associated entropy generation concerning time scale and viscosity parameters. Ghadikolaei et al. ²³ showcased the time-dependent attribute of a MHD flow of Eyring-Powell fluid constrained by a channel, influenced by Joule dissipation.

Thermal radiation is a ubiquitous phenomenon that plays a vital role in numerous natural and technological processes. It is invaluable in fields ranging from engineering and physics to astronomy and environmental science because of its distinctive qualities and applications. Determining the properties of thermal radiation is crucial for developing effective heat transfer systems, utilizing renewable energy sources, and solving cosmic mysteries. Jan et al. ²⁴ studied the influences of thermal generation-absorption and applied magnetic field on Sisko fluid model. Cui et al. ²⁵ used a model of Casson fluid with radiative nanofluids serving as the magnetized boundary layer to simulate a stretched sheet of fluid. Abrar et al. ²⁶ explored the entropy generation within cilia-driven nanofluid transport, considering the impacts of thermal radiation, internal heating, and viscous dissipation. Kumar et al. ²⁷ developed a computational model for the radiative heat transfer in the presence of magnetic field influences and viscous dissipation over an infinite vertical sheet. In addition, the author highlights the application of the under-study problem in the chemical and metallurgy sector. Agbaje et al. ²⁸ performed a computational study to determine the unstable flux in the presence of thermal radiation from Powell-Eyring nanofluid and heat obstetrics. The findings demonstrated that the fluid’s temperature and the thermal boundary layer’s thickness were elevated by the characteristics of the heat source and thermal radiation. Oreyeni et al. ²⁹ made analysis on mixed hybrid magnetite particles ( $F{e}_3{O}_4$ ) with silver (Ag) to investigate the effects of heat radiation, magnetic fields, and viscous dissipation on the thermal characterization function in solar-powered ships. The analysis methodology applied was the Galerkin weighted residual method, and thermal boundary layer’s heat flow was assessed utilizing the Cattaneo-Christov model.

The method of local non-similarity was proposed by Sparrow and Yu ³⁰ as a solution to the serious flaws and errors in the similarity method. One frequent issue with using a local similarity method is that the system is not fully converted into ordinary differential equations (ODEs). In comparison to the conventional similarity strategy, this yields a more consistent and accurate result. The local non-similarity approach addresses this constraint by considering non-similar transformations in the problem’s immediate region. In a non-Darcy porous medium, Farooq et al. ³¹ used a non-similar boundary layer model to study forced convection in the steady flow of Casson nanofluid. Similarly, non-similar solution for Maxwell hybrid nanofluid in the presence of MHD discussed by Sagheer et al. ³² The MHD convective transport of a nanofluid across a stretchy surface was investigated by Riaz et al. ³³ using the bvp4c approach, accounting for several effects. Using a non-similarity technique, Sagheer et al. ³⁴ investigated increased thermal utilization for EMHD nanofluid with variable thermal flux along the stretching surface.

This study focuses on the selection of copper oxide $(\mathrm{Cu}),$ magnetite $(F{e}_3{O}_4)$ , and ethylene glycol $(\mathrm{EG})$ based on their distinct thermal characteristics. Metallic oxide nanoparticles $F{e}_3{O}_4$ and $\mathrm{Cu}$ are easily soluble in the base fluid, while $\mathrm{EG}$ exhibits strong thermal features. Pure metallic nanoparticles $(\mathrm{Cu}$ , $F{e}_3{O}_4)\mathrm{and}\mathrm{EG}$ are recognized for their larger aspect ratio, superior heat conductivity, and low specific gravity. Although nanofluids tend to be unstable over short durations, the addition of $F{e}_3{O}_4+\mathrm{Cu}$ / $\mathrm{EG}$ ensures long-term stability and enhances thermal properties significantly. According to existing literature, it is observed that the influence of viscous dissipation on MHD heat and mass flow of hybrid-nano fluid ( $\mathrm{Cu}$ + $F{e}_3{O}_4$ ) across a stretched sheet through a permeable media using non similarity transformation remains unexplored. Utilizing the local non similarity method alongside the bvp4c algorithm up to its second truncation level enables the computation of numerical solutions to nonlinear-couple partial differential equations. Through adjustment of pertinent parameters, relevant findings on the subject matter are elucidated and visually presented using tables and diagrams.

Mathematical modeling of the problem

The following assumptions are performed under the considered flow.

• • Considering a steady, two-dimensional flow of Eyring-Powell nanofluid above a linearly stretching sheet, characterized by a velocity profile of $u=\mathrm{xc}$ .

• • The Cartesian coordinate system aligns the x-axis with the extending surface, and the y-axis is normal to it.

• • The stretching surface undergoes expansion and is subjected to reheating, starting at an initial temperature $T_w$ , while $T_{\infty }$ represents the ambient temperature.

• • The flow is generated by the action of two equal and opposite forces along the x-axis and the sheet is assumed to be linearly stretching.

• • The velocity components are denoted as $v$ and $u$ in the $y$ and $x$ directions, respectively. Figure 1 illustrates the flow coordinate system and provides a detailed description of the model assumptions involved in the setup.

Figure 1.

Flow configuration.

The Cauchy stress tensor for Eyring-Powell fluid can be expressed as:

$$A=-\mathrm{PI}+\tau ,$$

(1)

$$\tau =[\mu +\frac{1}{\delta \gamma }\sin h^{-1}\left(\frac{1}{m_1}\gamma \right)]B_1,$$

(2)

In equation (1), the term “ $I$ ” represents the identity matrix, and “ $P$ ” signifies the pressure component. The symbol τ denotes the extra stress tensor, where $m_1$ and δ represent fluid parameters, μ denotes the dynamic viscosity, with $m_1$ having dimensions of $(\mathrm{time}{)}^{-1}.$ Additionally, $B_1$ represents the first Rivlin-Erickson tensor and is expressed as follows:

$$B_1=\mathrm{gradV}+\mathrm{grad}{V}^T\mathrm{and}\gamma =\sqrt{\frac{1}{2}\mathrm{tr}{B_1}^2}$$

(3)

Using the second-order approximations of the $\sinh -1$ function and ignoring the third and so on terms ² we obtain

$$\sin h^{-1}\left(\frac{1}{m_1}\gamma \right)\approx \frac{\gamma }{m_1}-\frac{1}{6}{\left(\frac{\gamma }{m_1}\right)}^3,\mathrm{with}\frac{1}{6}{\left(\frac{\gamma }{m_1}\right)}^5<<<1.$$

(4)

The governing equations related to flow problem are ³⁵ :

$$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0,$$

(5)

$$\begin{array}{c}\multicolumn{1}{c}{u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=({\nu }_{\mathrm{hnf}}+\frac{1}{\delta m_1{\rho }_{\mathrm{hnf}}})\frac{{\partial }^{2}u}{\partial y^2}-\frac{1}{2\delta {\rho }_{\mathrm{hnf}}{m_1}^3}\frac{{\partial }^{2}u}{\partial y^2}{\left(\frac{\partial u}{\partial y}\right)}^2}\\ -\frac{{\sigma }_{\mathrm{hnf}}{B}_o^2}{\mathrm{hnf}}u-\frac{{\mu }_{\mathrm{hnf}}}{{\rho }_{\mathrm{hnf}}}\frac{u}{K_1},\hfill \end{array}$$

(6)

$$\begin{array}{c}\multicolumn{1}{c}{u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\frac{k_{\mathrm{hnf}}}{{\left(\rho C_p\right)}_{\mathrm{hnf}}}\left(\frac{{\partial }^{2}T}{\partial y^2}\right)-\frac{1}{{\left(\rho C_p\right)}_{\mathrm{hnf}}}\frac{\partial q_r}{\partial y}}\\ +\frac{{\mathrm{Q}}_0}{{\left(\rho C_p\right)}_{\mathrm{hnf}}}(T-T_{\infty }),\hfill \end{array}$$

(7)

The boundary conditions for a given problem are:

$$u=\mathrm{cx},v=0,T=T_{\infty }\mathrm{at}y=0,\mathrm{as}y\to u=0,T=T_{\infty }.$$

(8)

The radiative heat flux $q_r$ is defined as :

$$q_r=-\frac{4\sigma °}{3k°}\left({\frac{\partial T}{\partial y}}^4\right).$$

(9)

Utilization of Taylor series expansion:

$$T^4=T_{\infty }^4+4T_{\infty }^3(T-T_{\infty })+6T_{\infty }^2{(T-T_{\infty })}^2+\dots$$

Substitution of equation (10) in equation (7), we obtain equation (11) as given below:

$$q_r=-\frac{16T_{\infty }^3\sigma °}{3k°}\left(\frac{\partial T}{\partial y}\right),$$

(10)

$$\begin{array}{c}\multicolumn{1}{c}{u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\frac{k_{\mathrm{hnf}}}{{\left(\rho C_p\right)}_{\mathrm{hnf}}}\left(\frac{{\partial }^{2}T}{\partial y^2}\right)-\frac{1}{{\left(\rho C_p\right)}_{\mathrm{hnf}}}}\\ \times \frac{16T_{\infty }^3\sigma °}{3K°}\frac{{\partial }^{2}T}{\partial y^2}+\frac{Q_o}{\rho C_p}(T-T_{\infty }).\hfill \end{array}$$

(11)

Nonsimilar system

In the context of nonsimilar flow, we will introduce new parameters denoted as $\xi (x)$ for nonsimilarity variable, and $\eta (x,y)$ to represent a pseudo-similarity variable.^36,37

$$\begin{array}{c}\xi =\frac{x}{l},\eta =\sqrt{\frac{a}{\nu }}y,u=\mathrm{ax}\frac{\partial f}{\partial \eta }(\xi ,\eta ),\hfill \\ v=-\sqrt{a\nu }(\xi \frac{\partial f}{\partial \xi }+f(\xi ,\eta )),\theta (\xi ,\eta )=\frac{T-T_{\infty }}{(T_w-T_{\infty })}\hfill \end{array}$$

(12)

Putting equation (12) in equations (6)–(8), we get the following dimensionless system.

$$\begin{array}{c}\multicolumn{1}{c}{\frac{{\partial }^{3}f}{\partial {\eta }^3}(B_1\epsilon +B_2-{\xi }^2\epsilon \omega {\left(\frac{{\partial }^{2}f}{\partial {\eta }^2}\right)}^2)-M{B}_3\frac{\partial f}{\partial \eta }-B_{4}K\frac{\partial f}{\partial \eta }}\\ -{\left(\frac{\partial f}{\partial \eta }\right)}^2+f\frac{{\partial }^{2}f}{\partial {\eta }^2}=\xi (\frac{\partial f}{\partial \eta }\frac{{\partial }^{2}f}{\partial \xi \partial \eta }-\frac{{\partial }^{2}f}{\partial {\eta }^2}\frac{\partial f}{\partial \xi }),\hfill \end{array}$$

(13)

$$(B_5+\frac{4}{3}R{B}_6)\frac{{\partial }^2\theta }{\partial {\eta }^2}+\mathrm{Pr}(f\frac{\partial \theta }{\partial \eta }+B_{6}Q\theta )=\mathrm{Pr}\xi (\frac{\partial f}{\partial \eta }\frac{\partial \theta }{\partial \xi }-\frac{\partial \theta }{\partial \eta }\frac{\partial f}{\partial \xi }).$$

(14)

Subjected BCs;

$$\begin{array}{c}\multicolumn{1}{c}{\frac{\partial f(0,\xi )}{\partial \eta }=1,f(0,\xi )=0,\theta (\xi ,0)=1,\mathrm{at}y=0,\frac{\partial f(\xi ,\infty )}{\partial \eta }}\\ =0,\theta (\xi ,\infty )=0,\mathrm{at}y\to \infty \hfill \end{array}$$

$$\begin{array}{l}{B}_1=\frac{1}{\left[\left(1-{\varphi }_2\right)\left\{\left(1-{\varphi }_1\right)+{\varphi }_1{\rho }_{s1}\right\}+{\varphi }_2{\rho }_{s2}\right]},\\ B_2=\left[\frac{1}{{\left(1-{\varphi }_2\right)}^{2.5}{(1-{\varphi }_1)}^{2.5}}\right],\\ B_3=[1+\frac{3\left(S_f-1\right)\left({\sigma }_{s2}-1\right){\varphi }_2}{\left({\sigma }_{s2}+2\right)-({\sigma }_{s2}-1)\left(S_f-1\right){\varphi }_2}\\ +\frac{3\left(S_f-1\right)\left({\sigma }_{s1}-1\right){\varphi }_1}{\left({\sigma }_{s1}+2\right)-({\sigma }_{s1}-1)(S_f-1){\varphi }_1}],\\ B_4=\left[\left(1-{\varphi }_2\right)\left\{\left(1-{\varphi }_1\right)+{\varphi }_1{\rho }_{s1}\right\}+{\varphi }_2{\rho }_{s2}\right]\\ \frac{1}{{\left(1-{\varphi }_2\right)}^{2.5}{(1-{\varphi }_1)}^{2.5}},\\ B_5=\left[\left(1-{\varphi }_2\right)\left\{\left(1-{\varphi }_1\right)+{\varphi }_1{\rho }_{s1}\right\}+{\varphi }_2{\rho }_{s2}\right]\\ \times \frac{k_{s2}+\left(sf-1\right)k_{bf}-\left(sf-1\right){\varphi }_2(k_{bf}-k_{s2})}{k_{s2}+\left(sf-1\right)k_{bf}+{\varphi }_2\left(k_{bf}-k_{s1}\right)}·\\ \frac{k_{s1}+\left(sf-1\right)k_f-\left(sf-1\right){\varphi }_1\left(k_f-k_{s1}\right)}{k_{s1}+\left(sf-1\right)k_f+{\varphi }_1\left(k_f-k_{s1}\right)},\\ B_6=\left(1-{\varphi }_2\right)\left\{{\varphi }_1{\left(\rho C_p\right)}_{s1}+\left(1-{\varphi }_1\right){\left(\rho C_p\right)}_f\right\}\\ +{\varphi }_2{\left(\rho C_p\right)}_{s2},A_1=\frac{{\mu }_{hnf}}{{\mu }_f},A_2=\frac{k_{hnf}}{k_f}\end{array}$$

(15)

Now the magnetic parameter is expressed as $M=\frac{{\mathrm{\sigma }}_f{\mathrm{B}}_{\mathrm{o}}^2}{{\rho }_{\mathrm{fc}}},R=\frac{16T_{\infty }^3\sigma °}{3k_{f}K°}$ is thermal radiation parameter, $\mathrm{Pr}=\frac{{\nu }_f{\left(\rho c_p\right)}_f}{k_f}$ is Prandtl number, $w=\frac{c^3{l}^2}{2{m_1}^2{\nu }_f}\mathrm{and}\epsilon =\frac{1}{m_1{\nu }_f{\rho }_{f\delta }}$ represents the material constants of Eyring–Powell fluid. The permeability of porous medium is denoted by $K=\frac{{\nu }_f}{c{K}_1}$ and $Q=\frac{Q_o}{c{\rho }_f{C_p}_f}$ represents thermal sink/source. The skin friction and Nusselt number expressed as:

$$\begin{array}{c}R{e}^{\frac{1}{2}}{C}_f={\mathrm{A}}_1+\epsilon f″(\xi ,0)-\frac{\epsilon \omega }{3}{\xi }^2{f}^{\prime {\prime }^3}(\xi ,0),\hfill \\ R{e}^{\frac{-1}{2}}\mathrm{Nu}=({\mathrm{A}}_1+\frac{4}{3}R)\theta \prime (\xi ,0).\hfill \end{array}$$

(16)

Where $\mathrm{Re}=\frac{c{l}^2}{{\nu }_f}.$

The specific assumption for above stated physical parameters are settled as

$$M=K=0.5,\epsilon =0.1,w=0.2,{\varphi }_1=0.02,{\varphi }_1=0.04$$

When dealing with nonsimilar boundary layer problems, local nonsimilarity is an often-employed method. Before moving on to local nonsimilar solutions, it is highly beneficial to assess the local similarity technique. This method presupposes a boundary layer equation structure with associated conditions and determines that the right-hand side, denoted as term $(\partial (.))/\partial \xi )$ of equations (13)–(15) to be adequately small, thereby allowing it to be approximated as zero.

$$\begin{array}{c}\frac{{\partial }^{3}f}{\partial {\eta }^3}(B_1\epsilon +B_2-{\xi }^2\epsilon \omega {\left(\frac{{\partial }^{2}f}{\partial {\eta }^2}\right)}^2)-M{B}_3\frac{\partial f}{\partial \eta }-B_{4}K\frac{\partial f}{\partial \eta }\hfill \\ -{\left(\frac{\partial f}{\partial \eta }\right)}^2+f\frac{{\partial }^{2}f}{\partial {\eta }^2}=0,\hfill \end{array}$$

(17)

$$(B_5+\frac{4}{3}R{B}_6)\frac{{\partial }^2\theta }{\partial {\eta }^2}+\mathrm{Pr}(f\frac{\partial \theta }{\partial \eta }+B_{6}Q\theta )=0.$$

(18)

$$\begin{array}{c}\frac{\partial f(\xi ,0)}{\partial \eta }=1,f(\xi ,0)=0,\frac{\partial \theta (\xi ,0)}{\partial \eta }=0,\hfill \\ \partial f(\xi ,\infty )/\partial \eta \to 0,\theta (\xi ,\infty )\to 0\hfill \end{array}$$

(19)

The value of $\xi$ can be regarded as a constant parameter at any stream-wise point. Therefore, equations (19)–(21) mentioned above can be regarded as ordinary differential equations. It would be mentioned that some results and accuracy are questionable when terms $(\partial (.)/\partial \xi )$ are not comprehended. In order to get around these difficulties, Sparrow and Yu ³⁰ and Minkowycz and Sparrow ³⁸ suggested local non-similarity technique to determine the implications of the non-similar boundary layer equations. Now let’s introduce functions using a local non-similarity approach as.

$$\begin{array}{c}\frac{\partial f}{\partial \eta }=f\prime ,\frac{{\partial }^{2}f}{\partial {\eta }^2}=f″,\frac{{\partial }^{3}f}{\partial {\eta }^3}=f″\prime ,\frac{\partial \theta }{\partial \eta }=\theta \prime ,\frac{{\partial }^2\theta }{\partial {\eta }^2}=\theta ″,\hfill \\ g(\xi ,\eta )=\frac{\partial f(\xi ,\eta )}{\partial \xi },p(\xi ,\eta )=\frac{\partial \theta (\xi ,\eta )}{\partial \xi }.\hfill \end{array}$$

(20)

Now by using (20) in (17)–(19) the equations are transformed as

$$\begin{array}{c}f″\prime (B_1\epsilon +B_2-{\xi }^2\epsilon \omega {f″}^2)+M{B}_{3}f\prime -B_{4}Kf\prime -{f″}^2+ff″\hfill \\ =\xi ({f\prime }^{2}g-f″g).\hfill \end{array}$$

(21)

$$(B_5+\frac{4}{3}R{B}_6)\theta ″+\mathrm{Pr}(B_{6}Q\theta +\mathrm{ff}\prime )=\mathrm{Pr}\xi (f\prime p-\theta \prime g).$$

(22)

$$f\prime (\xi ,0)=1,f(\xi ,0)=0,\theta \prime (\xi ,0)=0.$$

$$f\prime (\xi ,\infty )\to 0,\theta (\xi ,\infty )\to 0.$$

(23)

We get (24)–(26) by taking partial derivatives of (21)–(23) with respect to $\xi$ :

$$\begin{array}{c}\multicolumn{1}{c}{g″\prime (B_1\epsilon +B_2-{\xi }^2\epsilon \omega {f″}^2)-2\xi \epsilon \omega f″\prime ({f″}^2+\xi g″f″)}\\ -B_{3}Mg\prime -B_{4}Kg\prime +\mathrm{g}″+2gf″+2f″g+g″f\hfill \\ -\mathrm{gf}\prime {\prime }^2=\xi g(2f\prime g\prime -g″).\hfill \end{array}$$

(24)

$$\begin{array}{c}(B_5+\frac{4}{3}R{B}_6)p″+\mathrm{Pr}(B_6\mathrm{Qp}+fp\prime +g\theta \prime -f\prime p+p\prime g)\hfill \\ =\mathrm{Pr}\xi (g\prime p-p\prime g).\hfill \end{array}$$

(25)

$$g(\xi ,0)=0,g\prime (\xi ,0)=0,p\prime (\xi ,0)=0,g\prime (\xi ,\infty )\to 0,p(\xi ,\infty )\to 0.$$

(26)

Numerical solution

The precise solution to the overall problem presents significant challenges due to the highly non-linear nature of the governing equations, as addressed by the local non-similar method. To tackle this, the local non-similar equations are initially reformulated into a system of first-order differential equations by introducing new variables. These reformulated equations are then processed using the bvp4c solver, which employs the Lobatto IIIa formula-a three-stage method. This MATLAB routine ensures a solution with accuracy up to the third order over the entire interval. Consequently, the proposed problem must be translated into a system of first-order equations to facilitate the approximation of the solution, following the outlined procedure

$$\begin{array}{c}(f=\mathrm{f}\left(1\right),f\prime =\mathrm{f}\left(2\right),f″=\mathrm{f}\left(3\right)),(\theta =\mathrm{f}\left(4\right),\theta \prime =\mathrm{f}\left(5\right))\hfill \\ (g=\mathrm{f}\left(6\right),g\prime =\mathrm{f}\left(7\right),g″=\mathrm{f}(8)),(p=\mathrm{f}\left(9\right),\mathrm{P}\prime =\mathrm{f}(10))\hfill \end{array}$$

Result and discussion

This section illustrates the Cu and $F{e}_3{O}_4$ hybrid nanofluid flow embedded in ethylene glycol with expanding surfaces. Figures and tables illustrate the significance of flow limitations in relation to the energy and velocity profiles. Table 3 showed excellent consistency between the current investigation and the literature under limited settings.^39,40 Numerical solutions for the examined fluid flow issues are generated using the MATLAB-based tool bvp4c, which considers physical factors in both scenarios, particularly $F{e}_3{O}_4$ /EG (nanofluid) and $F{e}_3{O}_4+\mathrm{Cu}$ /EG used as “hybrid-nanofluid.” In Table 1 thermo-physical properties for hybrid nanofluids and nanofluids are discussed by Rashad et al. ³⁵ and Table 2 represents the thermal proprieties for Fe₃O₄, Cu, and EG.^36,37 The result is verified by contrasting it with the body of knowledge, as indicated in Table 3 below. Table 4 shows the impact of governing factors on skin friction factor and temperature gradient at boundary. Computations are performed for $F{e}_3{O}_4$  −  $\mathrm{Cu}/\mathrm{EG}$ hybrid nanofluids. At the boundary, the skin friction factor assumes an increasing trend. Furthermore, the hybrid nanofluid exhibits a thicker momentum boundary layer at a fixed value of Eyring-Powell fluid parameter in comparison to the conventional nanofluid. Figure 2 illustrate how the dimensionless magnetic parameter affects velocity distribution. With an increase in the prescribed magnetic field $M$ , $f\prime (\xi ,\eta )$ is seen to decay. The magnetic parameter $M$ gave rise to a drag force nominated the Lorentz force. The Lorentz force affects the fluid’s flow patterns by suppressing velocity in specific directions. It usually resists motion caused by velocity variations due to stretching surface, acting as a damping force. The influences of $K$ on the velocity portray is depicted in Figure 3, which also demonstrates that with higher levels of permeability, $f\prime (\xi ,\eta )$ and the momentum boundary layer thickness decreases substantially. Permeability is the property of fluid motion resistance that decreases with increasing $K$ values, which in turn causes a decrease in velocity distribution. Improved heat transfer across the porous medium is made possible by stronger fiber packing, which offsets the hybrid nanoparticles in the fluid’s inability to function physically at higher K values.

Table 1.

Thermal properties of the Fe₃ O₄, Cu, and EG are discussed by Rashad et al. ³⁵

Materials	${\mathit{C}}_{\mathit{p}}\left(\frac{\mathit{J}}{\mathit{Kkg}}\right)$	$\mathbf{\rho }\left(\frac{\mathit{Kg}}{{\mathit{m}}^3}\right)$	$\mathit{k}\left(\frac{\mathit{W}}{\mathit{mK}}\right)$	$\mathit{\sigma }\left(\frac{\mathit{S}}{\mathit{m}}\right)$
$\mathit{F}{\mathit{e}}_3{\mathit{O}}_4$	2090	5180	0.15	2500
$\mathit{Cu}$	385	8933	400	5.96 × 107
EG	2430	1115	0.253	$1.07\times {10}^{-11}$

Table 2.

Thermophysical properties for hybrid nanofluids and nanofluids are presented by Razzaq and Farooq, Razzaq et al.^36,37

Density $(\mathit{Kg}(\mathit{m}{)}^{-3})$	${\mathbf{\rho }}_{\mathit{hnf}}=[{\varphi }_2{\mathbf{\rho }}_{\mathit{s}2}+(1-{\varphi }_2)\{{\varphi }_1{\mathbf{\rho }}_{\mathit{s}1}+(1-{\varphi }_1){\mathbf{\rho }}_{\mathit{f}}\}]$
Dynamic viscosity $(\mathit{kg}(\mathit{ms}{)}^{-1})$	${\mu }_{\mathrm{hnf}}=\frac{{\mu }_f}{{(1-{\varphi }_2)}^{2.5}{(1-{\varphi }_1)}^{2.5}}$
Heat capacity $(\mathit{J}(\mathit{kg}.\mathit{K}{)}^{-1})$	${\left(\rho C_p\right)}_{\mathrm{hnf}}=(1-{\varphi }_2)\{{\varphi }_1{\left(\rho C_p\right)}_{s1}+(1-{\varphi }_1){\left(\rho C_p\right)}_f\}+{\varphi }_{21}{\left(\rho C_p\right)}_{s2}$
Thermal conductivity $(\mathit{W}(\mathit{mK}{)}^{-1})$	$\frac{k_{\mathrm{hnf}}}{k_{\mathrm{bf}}}\mathrm{L}=\frac{k_{s2}+(\mathrm{sf}-1)k_{\mathrm{bf}}-(\mathrm{sf}-1){\varphi }_{2}l(k_{\mathrm{bf}}-k_{s2})}{k_{s1}+(\mathrm{sf}-1)k_{\mathrm{bf}}+\varphi l(k_{\mathrm{bf}}-k_{s1})},$ $\frac{k_{\mathrm{bf}}}{k_f}=\frac{k_{s1}+(\mathrm{sf}-1)k_f-L(\mathrm{sf}-1){\varphi }_1(k_f-k_{s1})}{k_{s1}+(\mathrm{sf}-1)k_f+L\varphi (k_{\mathrm{bf}}-k_{s1})},$
Electrical conductivity $(\mathit{S}{\mathit{m}}^{-1})$	$\frac{{\sigma }_{\mathrm{hnf}}}{{\sigma }_{\mathrm{bf}}}=\frac{3\varphi ({\sigma }_1{\varphi }_1+{\sigma }_2{\varphi }_{2-}{\sigma }_{\mathrm{bf}}({\varphi }_1+{\varphi }_2))}{({\sigma }_1{\varphi }_1+{\sigma }_2{\varphi }_2+2\varphi {\sigma }_{\mathrm{bf}})-\varphi {\sigma }_{\mathrm{bf}}(({\sigma }_1{\varphi }_1+{\sigma }_2{\varphi }_2)-{\sigma }_{\mathrm{bf}}({\varphi }_1+{\varphi }_2))}+1,$ where $\frac{{\sigma }_{\mathrm{nf}}}{{\sigma }_f}=1+\frac{3(\frac{{\sigma }_s}{{\sigma }_f}-1)\varphi }{(\frac{{\sigma }_s}{{\sigma }_f}+2)-(\frac{{\sigma }_s}{{\sigma }_f}-1)\varphi }$ ,

Table 3.

$-\theta \prime (0)$ values are compared for various $\mathrm{Pr}$ inputs when the remaining parameters are zero.

$\mathit{Pr}$	Ishak et al. 39	Alsaedi et al. 40	Present results
2	1.3349	1.3333	1.4236
5	3.2927	3.3165	3.8953
10	4.7742	4.7969	4.9523
15	5.9097	5.9320	5.9377

Table 4.

Results for skin friction for some values of $M,K,\epsilon ,\omega \mathrm{and}R$ .

$\mathit{M}$	$\mathit{K}$	$\mathbf{\epsilon }$	$\mathit{\omega }$	$\mathit{R}$	$\varphi$	$-{\mathbf{C}}_{\mathbf{fx}}{\left(\mathbf{R}{\mathbf{e}}_{\mathbf{x}}\right)}^{\frac{1}{2}}$	$-\mathbf{Nu}{\left(\mathbf{R}{\mathbf{e}}_{\mathbf{x}}\right)}^{\frac{1}{2}}$
0.2	0.5	0.2	0.2	0.5	0.04	1.8543	0.1736
0.5						2.1017	0.1766
1.0						2.3240	0.1798
0.5	0.1					1.9425	0.1766
	0.3					1.9838	0.1777
	0.5					2.0240	0.1792
	0.1	0.1				1.5029	0.17363
		0.2				1.4672	0.1688
		0.3				1.3341	0.1624
			0.1			1.6749	0.1762
			0.3			1.6812	0.1769
			0.5			1.6929	0.1798
				0.2		1.5016	0.1793
				0.5		1.5763	0.1889
				0.8		1.5765	0.1926
					0.02	0.1362	0.1453
					0.03	0.1455	0.1597
					0.05	0.1673	0.1761

Figure 2.

Variation of $\mathit{M}$ versus $\mathit{f}\prime (\mathit{\xi },\mathit{\eta })$ .

Figure 3.

Variation of $\mathit{K}$ versus $\mathit{f}\prime (\mathit{\xi },\mathit{\eta })$ .

As seen in Figures 4 and 5, the function with $R$ is in opposition to $f\prime (\xi ,\eta ),\theta (\xi ,\eta ),$ and the thickness of the significant boundary layer. This is because increasing $R$ causes thermal diffusivity to lag, which in turn causes the area of the thermal boundary layer’s thickness to decrease. In Figures 6 and 7 the behavior of heat generation parameter $(Q)$ is represented with respect to velocity distribution $f\prime (\xi ,\eta )$ and thermal distribution $\theta (\xi ,\eta )$ . The interpretation of Figures 8 and 9 clearly points to an increase in $Q$ values, $f\prime (\xi ,\eta )$ and $\theta (\xi ,\eta )$ as a result of an improvement in thermal boundary layer thickness due to internal heat energy growth in the ascending heat source parameter circumstance. Figures 8 and 9 illustrate the velocity diagram for varying volume fractions of $F{e}_3{O}_4$ and $\mathrm{Cu},$ when $0\le \xi \le 1$ . The hybrid nanofluid density and dynamic viscosity increase as the volume fraction rises. As shown in these figures, this subsequently modifies $f\prime (\xi ,\eta )$ and results in decreased momentum generation. As can be shown in Figure 10, the temperature directly depends on the $\mathrm{Cu}$ nanoparticles. This pattern is consistent with the fact that copper nanoparticles have higher thermal conductivity than basic fluids. Figure 11 presents the temperature account influenced by the momentum to mass diffusivity ratio; a decreasing trend is noted toward increased values of the parameter that is employed, namely $\mathrm{Pr}$ . The main reason for the temperature drop is that when $\mathrm{Pr}$ increases, thermal diffusivity falls. Because fluids with higher Prandtl numbers have relatively poorer thermal conductivity, conduction, and the thickness of the thermal boundary layer decreases, lowering temperature. Figures 12 and 13 display the computed estimates of skin friction and Nusselt number for various values of $M$ versus nonsimilarity variable $(\xi )$ . The figures indicate that an increase in $M$ leads to an improvement in both heat transmission and skin friction coefficient because of the Lorentz force’s resistance; nevertheless, an opposite pattern is observed as $\xi$ increases. With respect to the mixed convection parameter, and for various permeability values, Figures 14 and 15 illustrate that the permeability parameter exhibits the same behavior as the magnetic factor on the Nusselt number and friction coefficient. The Nusselt number and skin friction profiles against $\xi$ with different inputs of $R$ are shown in Figures 16 and 17. When the $\xi$ and R parameters rise, the drag force and Nusselt factor also rise. The reason for this is that when the flux improves and absorbs more heat, the momentum boundary layer thickness increases.

Figure 4.

Variation of $\mathit{R}$ versus $\mathit{f}\prime (\mathit{\xi },\mathit{\eta })$ .

Figure 5.

Variation of $\mathit{R}$ versus $\mathit{\theta }(\mathit{\xi },\mathit{\eta })$ .

Figure 6.

Variation of $\mathit{Q}$ versus $\mathit{f}\prime (\mathit{\xi },\mathit{\eta })$ .

Figure 7.

Variation of $\mathit{Q}$ versus $\mathit{\theta }(\mathit{\xi },\mathit{\eta })$ .

Figure 8.

Variation of ${\mathit{\varphi }}_1$ versus $\mathit{\theta }(\mathit{\xi },\mathit{\eta })$ .

Figure 9.

Variation of ${\varphi }_2$ versus $\mathit{f}\prime (\mathit{\xi },\mathit{\eta })$ .

Figure 10.

Variation of ${\mathit{\varphi }}_2$ versus $\mathit{\theta }(\mathit{\xi },\mathit{\eta })$ .

Figure 11.

Variation of $\mathit{Pr}$ versus $\mathit{\theta }(\mathit{\xi },\mathit{\eta })$ .

Figure 12.

Variation of $\mathit{M}$ versus $\mathit{R}{\mathit{e}}^{\mathit{1}/2}\mathit{Cf}$ .

Figure 13.

Variation of $\mathit{M}$ versus $\mathit{R}{\mathit{e}}^{-\mathit{1}/2}\mathit{Nu}$ .

Figure 14.

Variation of $\mathit{K}$ versus $\mathit{R}{\mathit{e}}^{\mathit{1}/2}\mathit{Cf}$ .

Figure 15.

Variation of $\mathit{K}$ versus $\mathit{R}{\mathit{e}}^{-\mathit{1}/2}\mathit{Nu}$ .

Figure 16.

Variation of $\mathit{R}$ versus $\mathit{R}{\mathit{e}}^{\mathit{1}/2}\mathit{Cf}$ .

Figure 17.

Variation of $\mathit{R}$ versus $\mathit{R}{\mathit{e}}^{-1/2}\mathit{Nu}$ .

Conclusion

This article examines how heat generation and radiative heat affect the mixed convection flux in a porous medium containing a magnetic Eyring–Powell hybrid nanofluid. With the aid of non similarity method which solved by using the built-in Matlab algorithm bvp4c, the results have been obtained. The main conclusions are.

• • With an increase in the parameter values of $M,K,R,{\varphi }_1$ , and ${\varphi }_2$ , $f\prime (\xi ,\eta )$ is dropped. However, it is the heat sink/source that exhibits the contrary behavior.

• • Raising the values of $Q$ and ${\varphi }_2$ parameters cause an increase in $\theta (\xi ,\eta ).$

• • Thermal enhancement is caused by an increase in thermal radiation and $F{e}_3{O}_4$ nanoparticle concentrations.

• • Drag force decreases with increasing magnetic field, porous material permeability, and heat radiation.

• • As the magnetic field, the permeability of porous media, and the heat source/sink increase, heat transfer decreases.

• • It is estimated that the thermal conductivity and viscosity of the ${\mathrm{Fe}}_3{O}_4+\mathrm{Cu}/\mathrm{EG}$ hybrid nanofluid are expected to increase to 4.1% and 12.35%, respectively, and to 71.55% and 78.01%, respectively, for volume concentrations ranging from 0.02% to 0.05 and temperatures between 20°C and 60°C.

The methodologies and findings of this study have practical implications for various engineering applications, including aerospace, automotive, and environmental engineering. Researchers in these fields can leverage this investigation insight to improve the design and performance of systems involving boundary layer flows. By contributing these advancements, the study serves as a valuable resource for researchers looking to solve complex boundary layer problems more effectively and efficiently.