Entropy Analysis of EMHD Hybrid Nanofluid Stagnation Point Flow over a Porous Stretching Sheet with Melting Heat Transfer in the Presence of Thermal Radiation

Abstract

In this study, the entropy formation of an electromagnetohydrodynamic hybrid nanofluid at a stagnation point flow towards a stretched surface in the presence of melting heat transfer, second-order slip, porous medium, viscous dissipation and thermal radiation are investigated. Hybrid nanoparticles alumina (Al₂O₃) and copper (Cu) are considered, with the base fluid water (H₂O). Similarity transformations are used to address the governing partial differential equations (PDEs) that lead to the corresponding ordinary differential equations. The resulting ODEs are solved by employing bvp4c solver numerically in the MATLAB package. The effects of temperature, transport, production of entropy and Bejan number $Be$ are graphically exhibited. Higher radiation parameters $R$ and an electric field $E$ lead to an increase in fluid temperature. The velocity boundary layer is lowered by the magnetic field and porous media parameters. The opposite behaviour is observed in the electric field $E$. As a result, hybrid nanofluid has numerous uses in engineering cosmetics, automotive industry, home industry, for cancer treatment, food packaging, pharmaceuticals, fabrics, paper plastics, paints, ceramics, food colorants, electronics, heat exchangers, water purification, lubricants and soaps as well.

1. Introduction

Nanotechnology is gaining popularity as a result of its important function of improving the heat capacity of liquids by boosting heat and mass transfer rates. Due to its wide variety of applications in heat exchangers, biomedicine, electronic device cooling, food, double windowpane, transportation, and other fields, the idea of nanofluids has become a more expansive topic for the research community in recent years. To improve the heat capacity of common fluids like kerosene, water, and motor oil, etc., nanoparticles of different kinds must be included in the base fluids, such as graphene, silica, alumina, gold, copper, silver, carbon nanotubes, and so on. It is suggested by Choi [1] that the word nanofluid should be used to characterize the thermal conductivity of the suspension in the base fluid that is enhanced by the particle dispersion in the nanometer range. As discussed in a number of research publications that have been published in the literature, the addition of different kinds of nanoparticles may increase the thermal conductivity of base fluids. H₂O-Cu and H₂O-Ag nanofluids are used by Kaneswaran et al. [2] to investigate the influence of relevant factors on the MHD nanofluid flow across an extending sheet. The effects of viscosity dissipation and chemical reaction are taken into account, it is found that the rate of heat and mass transfers of H₂O-Cu are higher. Kumaresan and Kumar [3] explored chemically reactive 3d magneto hydrodynamic rotating flow of nanofluids across a deformable surface with joule heating through pores media. Hybrid nanofluids are a new type of nanofluid that has lately received a lot of attention. It is recognized that due to a combination of components, hybrid nanofluids have better thermal characteristics. In continuation hybrid nanofluids is a mixed nanofluids, which blend or mix two distinct kinds of nanoparticles. Jana et al. [4] conducted an experimental analysis of this fluid type in 2007. In particularly, these hybrid nanofluids are play a significant part in a wide variety of applications, including solar water heating, heat exchangers, vehicle brake fluids, defense, medical, manufacturing, microfluidics and microelectronics, amongst others.

Several studies have been conducted in order to better understand the heat transmission characteristics of hybrid nanofluids. The combined effects of melting heat transfer and viscous dissipation on MHD hybrid nanofluid flow are discussed by Kumar [5]. Suresh et al. [6,7] studied the convective heat exchange of Cu-Al₂O₃/water hybrid nanofluids via a circular pipe. For the Al₂O₃-Cu/H₂O hybrid nanofluid, the average rise in Nusselt number was 8.02%, which is much higher than that of the basic fluid (H₂O). The friction factor of Al₂O₃-Cu/H₂O hybrid nanofluids is somewhat greater than that of Al₂O₃/H₂O nanofluids, according to the findings of the experiments. As part of his research, Bhattad [8] consider a experimnental investigation of aluminium oxide–magnesium oxide hybrid nanofluid in a plate heat transfer. He looked at the exergy–energy properties of the nanofluid. The optimal temperature range for a hybrid nanofluid is 30–45° Celsius with a flow rate of 2.0–4.0 L/min. Deionized H₂O (4:1) hybrid nanofluid of alumina and magnesium oxide increased the heat exchanger performance index by 11.1%, according to the study’s findings. Some references [9,10,11] provided a few studies on nanofluids.

The study of the movement of electrically conducting fluid under the impact of magnetic and electric fields is known as electromagnetohydrodynamics. The Lorentz force is also active and interacts with the buoyancy force in regulating the flow and temperature fields when the fluid is electrically conducting and a magnetic field is applied. The Lorentz force reduces the velocities of convection currents; adding an external magnetic field may be used as a control tool in the material-building sector. Due to the many different fields that electromagnetohydrodynamic may be used in, including biomedical engineering, geophysics, magnetic drug targeting, engineering, and many more, there has been interest in the field in recent years. According to Shah et al. [12], heat radiation has a significant impact on the EMHD rotational flow of carbon nanotubes across a stretched sheet, and they discovered that the electric and magnetic parameters move in opposite directions along the momentum boundary layer. In their analysis of the 2D unstable electromagnetohydrodynamic nanofluid flow over an extending sheet with multislip and dual stratification effects, Daniel et al. [13] found that the parameter of the electric field has a positive impact on the velocity boundary. Electromagnetohydrodynamics was computationally investigated by Abbas et al. [14] in a flow of nanofluid across a porous Riga plate for gyrotactic microorganisms. Jakeer [15] examined the irreversibility analysis of an electromagnetohydrodynamic Al₂O₃-Cu/H₂O nanofluid at a stagnation point in the presence of thermal radiation and viscous dissipation. References [16,17] provide a few studies on electromagnetohydrodynamic.

Many researchers are interested in discovering EMHD in heat transport with radiation and viscous dissipation impac due to its vast variety of applications. It is used in astrophysics and geophysics to investigate stellar and solar structures as well as radio transmission via the ionosphere. It has uses in engineering, such as MHD pumps and MHD bearings engineering. Thermal radiation plays an immense role in surface heat transfer, it is used in manufacturing, the design of dependable equipment, nuclear power plants, gas turbines, aeroplanes, missiles, satellites, space vehicles, space technologies, and procedures involving high temperatures. In the medical field lukewarm radiation is in high demand. The impact of thermal radiation with double diffusion has been an important research issue for many experimenters because of its use in medical therapy. Infrared radiation is a common kind of heat treatment that is useful in many different parts of the human body it is produced by oscillations in the electromagnetic field. Radiative heat transfer allows electromagnetic waves to carry vitality. Visible brightness and microwaving are related; it helps with a variety of skin issues. The wavelength of radiation penetrating skin is determined by its radiating structure, vascularity and pigmentation. Heat treatment is aided by infrared radiation, which warms the afflicted area’s blood capillaries directly. It improves the body’s blood flow, which helps to prevent infection in superficial wounds. It also draws attention to white blood cells while removing waste products. Studies on mass and heat transmission fluid flow in porous media have a variety of applications in the field of thermal technology. A few of these applications include geothermal energy recuperation, extraction of unrefined petroleum, thermal energy storage, ground water hydrology, thermal insulation, ceramic engineering flow through filtering devices, and so on.

Electromagnetohydrodynamic (EMHD) effects on corrugated walls in microchannels through the porous medium is studied by Rashid et al. [18]. Using an expanding/shrinking surface, Zheng et al. [19] investigated the boundary layer flow and radiative transport of an incompressible micropolar liquid. Warke et al. [20] examined the effects of porous media on the mass and heat transfer of mixed convective–radiative magneto micropolar liquid flow across a vertical surface. Daniel et al. [21] analysed slip and convective conditions on MHD nanofluid flow through a porous stretching and shrinking sheet. Tripathi et al. [22] explored the effect of Coriolis body force, electromagnetohydrodynamic (EMHD) and thermal radiative heat transfer on the Casson hybrid mixed convection nanofluid propelled by an exponentially accelerated plate next to a porous medium in a rotating system. Singh et al. [23] presented the solutions of a chemically reacting and melting heat transfer on non-Newtonian flow near the stagnation point in a porous medium with nonuniform heat sources and sinks. Khatun et al. [24] conducted a computational examination of the nature of electromagnetohydrodynamic (EMHD) radiating fluid flow over an infinitely long vertical Riga plate in a rotating environment.

In heat transport analysis, viscous dissipation is important, especially in boundary layer flow. As a result of higher velocity gradients (V-G) in boundary layer flow, the lower temperature functions as a heat source, resulting in a considerable rise in fluid temperature. The kinetic energy of the liquid is converted to heat energy when it passes through the boundary layer flow due to the larger gradations in velocity inside the liquid flow. Several manufacturing and technical applications necessitate the study of fluid flow in a porous medium, including irrigation and drainage, extraction of gas or fuel, petrochemicals, flow of blood, cosmetic products, extraction of particulates from soils and specific separation mechanisms such as filtering and absorption in industrial engineering. Using viscous dissipation, Mallikarjuna et al. [25] investigated the impact of melting heat transfer and viscous dissipation on the two-phase flow of a dusty hybrid nanofluid using the Forchheimer–Darcy model. The movements of blood conveying Au nanosized particles are valuable to the field of medication during the testing process. Aanalysis of Koriko et al. [26] states, the space experts found that the impact of conveying dust particles over rockets was helpful in air dynamics. First time, Gebhart et al. [27] noticed an increase in the temperature of fluid with a convective heat transmission rate. Awais et. al. [28] Investigate the impact of organisms on the exploration of heat, stratification and viscous dissipation in bio-convectional nanomaterials. Industrial operations of a higher level and viscosity dissipation on nanofluids with a continuous stretchy medium of porous are reported by Gopal et al. [29]. Aziz et al. [30] presented hybrid Powell–Eyring nanofluid entropy production.

To the best of authors knowledge, the idea of electromagnetohydrodynamics hybrid nanofluid flow across a stretching surface with melting heat transfer, thermal radiation, entropy generation and stagnation point not yet been addressed. For this hybrid nanofluid flow, both electric and magnetic field are considered. In addition, the effects of viscous dissipation and thermal radiation are studied in heat transfer in order to assess their relevance in the heat transfer process. The Runge–Kutta fourth order, along with the shooting technique, are applied to execute the results of modelled energy and momentum equations. Finally, with the use of tables and graphs, the physical characteristics of the necessary parameters are investigated. In addition, profiles of entropy generation have been drafted. This model might be good for cancer therapy because cancer cells make a large amount of heat; so, the entropy of cancer cells is very high. This means that analyzing tumor cells can help in healing their surrounding healthy cells.

2. Mathematical Formulation

Consider a stable, incompressible, two-dimensional, electrically conductive hybrid nanofluid and stagnation point flow over a stretched sheet coated in a permeable medium in the presence of thermal radiation and viscous dissipation, as shown in Figure 1.

• The velocity of the stretching sheet is $u_w(x)=cx$ and the outer flow velocity of is $u_e(x)=\mathsf{\omega }x$, where $\mathsf{\omega }$ and $c$ both are positive constants.
• The boundary layer equations of the fluid flow are composed from the continuity equation, the momentum equation and the energy equation, which are formulated based on Maxwell’s equation and Ohm’s law.
• The effects of a magnetic field $B$ and an electric field $E$ on the incompressible flow of a viscous fluid are considered.
• The magnetic and electric fields obey Ohm’s law defining magnetic induction as $\overrightarrow{J}=\sigma \left(\overrightarrow{E}+\overrightarrow{v}\times \overrightarrow{B}\right)$, where σ is the electric conductivity, $\overrightarrow{v}$ represents the fluid velocity and $\overrightarrow{J}$ flui the current.
• The melting surface and ambient temperature are represented by $T_m$ and $T_{\infty }$, where $T_m< T_{\infty }$.
• A magnetic field and electric field are applied normal to the flow, such that the Reynolds number is assumed to be smaller. The induced magnetic field is smaller than applied magnetic field. As a result, for low magnetic Reynolds numbers, the induced magnetic field is ignored.
• Entropy analysis is also taken into account.
• As a result of the above analysis, the flow is governed by [31,32,33].

Figure 1. Configuration of the flow model.

Figure 1. Configuration of the flow model.

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

(1)

$$u\frac{\partial u}{\partial x}+v\frac{\partial v}{\partial y}=u_e\frac{d{u}_e}{dx}+\frac{{\mu }_{hnf}}{{\rho }_{hnf}}\frac{{\partial }^{2}u}{\partial y^2}+\frac{{\sigma }_{hnf}}{{\rho }_{hnf}}\left(E_0{B}_0-B_0^2\left(u_e-u\right)\right)-\frac{{\mu }_{hnf}}{{\rho }_{hnf K^{*}}}(u_e-u)$$

(2)

$$\begin{array}{l}u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\frac{k_{hnf}}{{\left(\rho c_p\right)}_{hnf}}\frac{{\partial }^{2}T}{\partial y^2}+\frac{{\mu }_{hnf}}{{\left(\rho c_p\right)}_{hnf}}{\left(\frac{\partial u}{\partial y}\right)}^2-\frac{1}{{\left(\rho c_p\right)}_{hnf}}\frac{\partial q_r}{\partial y}+\frac{{\sigma }_{hnf}}{{\left(\rho c_p\right)}_{hnf}}{\left(u{B}_0-E_0\right)}^2\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\frac{Q_0}{{\left(\rho c_p\right)}_{hnf}}\left(T-T_{\infty }\right)\end{array}$$

(3)

The adequate corresponding initial and boundary conditions are given by [31].

$$u=u_{slip}+u_w\left(x\right),T_m=T$$

$$T\to T_{\infty }u\to u_e\left(x\right)y\to \infty$$

(4)

$$k_{hnf}{\left(\frac{\partial T}{\partial y}\right)}_{y=0}={\rho }_{hnf}\left[c_s \left(T_m-T_0\right)+L\right]v\left(x,0\right)$$

(5)

The thermophysical characteristics of [34] hybrid nanofluids are specified as follows:

$$\frac{k_{hnf}}{k_f}=\frac{2\left(1-\varphi \right)k_{f+}\left(1+2{\varphi }_{1s}\right)k_{1s}+\left(1+2{\varphi }_{2s}\right)k_{2s}}{\left(2-\varphi \right)k_{f+}\left(1-{\varphi }_1\right)k_{1s}+\left(1-{\varphi }_2\right)k_{2s}}, \frac{{\mu }_{hnf}}{{\mu }_f}={\left(1-\varphi \right)}^{-2.5}$$

$$\frac{{\rho }_{hnf}}{{\rho }_f}=\left(1-\varphi \right)+\frac{{\varphi }_{1s}{\rho }_{1s}}{{\rho }_f}+\frac{{\varphi }_{2s}{\rho }_{2s}}{{\rho }_f}, \frac{{\left(\rho c_p\right)}_{hnf}}{{\left(\rho c_p\right)}_f}=\left(1-\varphi \right)+\frac{{\varphi }_{1s}{\left(\rho c_p\right)}_{1s}+{\varphi }_{2s}{\left(\rho c_p\right)}_{2s}}{{\left(\rho c_p\right)}_f}$$

(6)

$$\frac{{\sigma }_{hnf}}{{\sigma }_f}=\left[1+\frac{3{\sigma }_{1s}{\varphi }_{1s}+{\varphi }_{2s}{\sigma }_{2s}-3\varphi {\sigma }_f}{{\sigma }_{1s}\left(1-{\varphi }_{1s}\right)+{\sigma }_{2s}\left(1-{\varphi }_{2s}\right)+\left(2+\varphi \right){\sigma }_f}\right], \varphi ={\varphi }_{1s}+{\varphi }_{2s}$$

Here, $u$ and $v$ are the velocity components of x and y axes T and L represent the hybrid nanofluid’s temperature and latent heat. Roberts [35] established the melting surface condition in Equation (5) $c_s$ denotes heat capacity. Furthermore, Lin Wu [36] develops and defines the velocity slip as follows.

$$u_{slip}=A\frac{\partial u}{\partial y}+B\frac{{\partial }^{2}u}{\partial y^2}$$

(7)

Famous astrophysicist Svein Roseland provided an approximation of radiation heat flux for a boundary layer that is optically thick as the following:

$$q_r^{\ast }=-\frac{4{\sigma }^{\ast }}{3k^{\ast }}\frac{\partial T^4}{\partial z}$$

(8)

where the Stefan–Boltzmann constant and Rosseland mean absorption coefficient are represented by ${\sigma }^{*}$ and $k^{*}$, respectively, expanding Taylor’s series on $T^4$ about the ambient temperature $T_{\infty }$ as follows:

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

(9)

Since the temperature differences are very small in the natural convection process, the terms from second order onwards may be neglected, and then it takes the form of:

$$T^4\cong (4T-3T_{\infty })T_{\infty }{}^3$$

(10)

From Equations (8)–(10), Equation (3) reduces to:

$$\begin{array}{cc}\hfill & u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\frac{k_{hnf}}{{\left(\rho c_p\right)}_{hnf}}\frac{{\partial }^{2}T}{\partial y^2}+\frac{{\mu }_{hnf}}{{\left(\rho c_p\right)}_{hnf}}{\left(\frac{\partial u}{\partial y}\right)}^2+\frac{16{\sigma }^{*}{T_{\infty }}^3}{{\left(\rho c_p\right)}_{hnf}3K^{*}}\frac{{\partial }^{2}T}{\partial y^2}+\frac{{\sigma }_{hnf}}{{\left(\rho c_p\right)}_{hnf}}{\left(u{B}_0-E_0\right)}^2\hfill \\ \hfill & \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\frac{Q_0}{{\left(\rho c_p\right)}_{hnf}}\left(T-T_{\infty }\right)\hfill \end{array}$$

(11)

To obtain ordinary differential equations, we exploited the following similarity transformations:

$$u=\omega x{f}^{\prime }\left(\eta \right), v=-f\left(\eta \right)\sqrt{\left(\omega v_f\right)},\frac{T-T_m}{T_{\infty }-T_m}=\theta \left(\eta \right), \eta =y{\left(\frac{\omega }{v_f}\right)}^{1/2}$$

(12)

By using similarity variables (12), Equations (2) and (11) transformed as:

$$\frac{A_1}{A_2}{f}^{‴}+f{f}^{″}-{f^{\prime }}^2+1+\frac{A_3}{A_2}M\left(E-\left(1-f^{\prime }\right)\right)+\frac{A_1}{A_2}K(1-f^{\prime })=0$$

(13)

$$\begin{array}{l}\frac{A_4}{Pr}\left(A_5+\frac{4}{3}R\right){\theta }^{″}+\frac{A_1}{A_4}Ec{f^{″}}^2+f{\theta }^{\prime }+\frac{A_3}{A_4}MEc{\left(f^{\prime }-E\right)}^2+\frac{1}{A_4}Q\theta =0\end{array}$$

(14)

The nondimensional boundary conditions are taken into consideration:

$$\left.\begin{array}{l}f\left(0\right)Pr\frac{{\rho }_{hnf}}{{\rho }_f}+{\theta }^{\prime }\left(0\right)M_1\frac{k_{hnf}}{k_f}=0,\\ f\left(0\right)=f^{‴}\left(0\right)\beta +f^{″}\left(0\right)\alpha +\lambda , \theta \left(0\right)=0 as \eta \to 0\\ f^{\prime }\left(\eta \right)\to 1,\theta \left(\eta \right)\to 1, as \eta \to \infty \end{array}\right\}$$

(15)

Together with the corresponding nondimensional parameters, are given

$$\begin{array}{l}Pr=\frac{{\left(\mu C_P\right)}_f}{k_f} , \alpha =A\sqrt{\frac{\omega }{v_f}} , \beta =\frac{\omega B}{v_f}, M_1=\frac{\left(T_{\infty }-T_m\right){\left(C_P\right)}_f}{L+\left(T_m-T_0\right)c_s} ,E=\frac{E_0}{{\beta }_0{u}_e},\\ M=\frac{{\sigma }_f{\beta }_0^2}{{\rho }_f\omega }, Ec=\frac{\omega x^2}{C_P\Delta T}, K=\frac{{\gamma }_f}{w{k}^{*}} , R=\frac{4{\sigma }^{*}{T}_{\infty }{}^{*}}{K^{*}{k}_f} ,\lambda =\frac{\gamma }{\omega }, Q=\frac{Q_0}{w} .\end{array}$$

(16)

The mathematical kinds of constants A₁, A₂, A₃, A₄ and A₅ may now be expressed as follows:

$$\left.\begin{array}{l}{A}_1={\mu }_{hnf}/{\mu }_f , A_2={\rho }_{hnf}/{\rho }_f, A_3={\sigma }_{hnf}/{\sigma }_f \\ A_4={(\rho c_p)}_{hnf}/{(\rho c_p)}_f, A_5=k_{hnf}/k_f\end{array}\right\}$$

(17)

The $C_f$ is the friction drag coefficient and $N{u}_x$ is the local rate of heat transmission, as follows:

$$C_f=\frac{{\mu }_{hnf}}{{\rho }_f{u}_w^2}{\left.\left(\frac{\partial u}{\partial y}\right)\right|}_{y=0} , N{u}_x=- \frac{k_{hnf}x{q}_w}{k_f\left(T_f-T_{\infty }\right)} {\left.\left(\frac{\partial T}{\partial y}\right)\right|}_{y=0}$$

(18)

Using Equations (7) and (14), we obtained

$$c_f{\left({\mathrm{Re}}_x\right)}^{1/2}=\frac{1}{{\left(1-{\varphi }_1\right)}^{2.5}{(1-{\varphi }_2)}^{2.5}}{f}^{″}(0)$$

(19)

$$N{u}_x{\left({\mathrm{Re}}_x\right)}^{1/2}=-\left(\frac{k_{hnf}}{k_f}+\frac{4}{3}R\right){\theta }^{\prime }(0)$$

(20)

Here, ${\mathrm{Re}}_x=\raisebox{1ex}{$u_{e}x$}\!\left/ \!\raisebox{-1ex}{${\gamma }_f$}\right.$ is the local Reynolds number.

3. Entropy Analysis or Analysis of Energy Loss

Entropy generation in a hybrid nanofluid with the effects of electric field, magnetic field and thermal radiative heat flow is characterized as follows [37,38,39]:

$$S_G=\underset{\begin{array}{l}energy loss via \\ heat transfer\end{array}}{\underbrace{\frac{1}{T_{\infty }{}^2}\left(K_{hnf}+\frac{16{\sigma }^{*}{T}_{\infty }{}^3}{3k^{*}}\right){\left(\frac{\partial T}{\partial y}\right)}^2}}+\underset{\begin{array}{l}energy loss via \\ fluid friction\end{array}}{\underbrace{\frac{{\mu }_{hnf}}{T_{\infty }}{\left(\frac{\partial u}{\partial y}\right)}^2}}+\underset{\begin{array}{l}energy loss via\\ Joule dissipation and \\ electric field\end{array}}{\underbrace{\frac{\sigma }{T_{\infty }}{\left(u{B}_0-E_0\right)}^2}}$$

(21)

Using Equation (7), Equation (16) yields the regenerated dimensionless form as follows:

$$N_G=\left(\frac{K_{hnf}}{K_f}+\frac{4}{3}R\right){\alpha }_1{{\theta }^{\prime }}^2+Br {(1-{\varphi }_1)}^{-2.5}{(1-{\varphi }_2)}^{-2.5}{f}^{″}+M Br {\left(f^{\prime }-E\right)}^2$$

(22)

where, the local entropy is $N_G=\frac{S_G {\gamma }_f T_{\infty }}{K_f \omega \Delta T}$, ${\alpha }_1=\frac{\Delta T}{T_{\infty }}$ is the dimensionless ratio variable, $Br=\frac{{\mu }_f u_{\omega }{}^2}{K_f \Delta T}$ is the Brinkman number.

The Bejan number is demonstrated as,

$$Be=\frac{\left(\frac{K_{hnf}}{K_f}+\frac{4}{3}R\right){\alpha }_1 {{\mathsf{\theta }}^{\prime }}^2}{\left(\frac{K_{hnf}}{K_f}+\frac{4}{3}R\right){\alpha }_1 {{\mathsf{\theta }}^{\prime }}^2+\frac{Br}{{(1-{\mathsf{\varphi }}_1)}^{2.5}{(1-{\mathsf{\varphi }}_2)}^{2.5}}{f}^{″}+M Br{(f^{\prime }-E)}^2}$$

(23)

4. Solution for Numerical Technique

The nondimensional system of Equations (13) and (14) as well as the boundary conditions (15) are numerically solved by using the Runge–Kutta fourth order along with shooting technique. Using similarity variables, the governing PDEs are converted into a set of ordinary differential equations and then transformed into a collection of first-order ODEs for this scheme. The detailed procedure is as follows

$$\begin{array}{l}f=S_1\\ f^{\prime }=S_2\\ f^{″}=S_3\\ f^{‴}=-\frac{A_2}{A_1}\left(S_1{S}_3-{\left(S_2\right)}^2+1+\frac{A_3}{A_2}M\left(E-\left(1-S_2\right)\right)+\frac{A_1}{A_2}K\left(1-S_2\right)\right)\\ \theta =S_4\\ {\theta }^{\prime }=S_5\\ {\theta }^{″}=-\frac{pr}{\left(A_5+\frac{4}{3}Rd\right)A_4} \left(\frac{A_1}{A_4}Ec{\left(S_3\right)}^2+S_1{S}_5+\frac{A_3}{A_4}M Ec{\left(S_2-E\right)}^2+\frac{1}{A_4}Q{S}_4\right)\end{array}$$

Along with the boundary conditions are:

$$pr A_2 S_1\left(0\right)+M_1 A_5 S_5\left(0\right)=0$$

$$S_1\left(0\right)=\lambda +\alpha \psi \left(0\right)+\beta S_2\left(0\right), S_4(0)=0, as\eta \to 0$$

$$S_2\left(\eta \right)\to 1,\hspace{1em}{S}_4\left(\eta \right)\to 1 \mathrm{as} \eta \to \infty$$

$$\begin{array}{c}\mathrm{Where},\\ \psi =f^{‴}=-\frac{A_2}{A_1}\left(S_1{S}_3-{\left(S_2\right)}^2+1+\frac{A_3}{A_2}M\left(E-\left(1-S_2\right)\right)+\frac{A_1}{A_2}K\left(1-S_2\right)\right)\end{array}$$

Equations (13) to (14) are converted into first-order differential equations with the help of above expressions and also the boundary conditions we get

$$\begin{array}{l}\left[\begin{array}{l}{s}_1{}^{\prime }\\ s_2{}^{\prime }\\ s_3{}^{\prime }\\ s_4{}^{\prime }\\ s_5{}^{\prime }\end{array}\right]=\left[\begin{array}{l}{s}_2\\ s_3\\ -\frac{A_2}{A_1}\left(S_1{S}_3-{\left(S_2\right)}^2+1+\frac{A_3}{A_2}M\left(E-\left(1-S_2\right)\right)+\frac{A_1}{A_2}K\left(1-S_2\right)\right)\\ s_5\\ {\theta }^{″}=-\frac{pr}{\left(A_5+\frac{4}{3}Rd\right)A_4} \left(\frac{A_1}{A_4}Ec{\left(S_3\right)}^2+S_1{S}_5+\frac{A_3}{A_4}M Ec{\left(S_2-E\right)}^2+\frac{1}{A_4}Q{S}_4\right)\end{array}\right]\\ \end{array}$$

5. Results and Discussion

This analysis has been accomplished to understand the physical mechanism of the present problems. On the governing model, computed numerical solutions and the computed values are plotted in terms of graphs and tables below. By using the Runge–Kutta 4th order along with the shooting technique approach, a set of ordinary differential Equations (13) and (14) with regard to boundary conditions (15), are solved numerically. Not only that, the Bejan number is used to investigate entropy creation. The impacts of non dimensional factors, such as magnetic field $M$, porous medium $K$, electric field $E$, radiation parameter $R$, heat source/sink $Q$, Eckert number $Ec$, Bejan number $Be$ and Brinkman number $Br$, on flow transport $f^{\prime }(\eta )$ and fluid temperature $\theta (\eta )$ are discussed using graphs. Hybrid nanoparticles and base fluid physical parameters are listed in

Table 1. Thermophoresis properties of water and nanoparticles [40].

Thermophysical Property	$\mathit{\rho }(\mathit{k}\mathit{g}/{\mathit{m}}^3)$	${\mathit{C}}_{\mathit{p}}(\mathit{J}/\mathit{k}\mathit{g}\mathit{K})$	$\mathit{k}(\mathit{w}/\mathit{m}\mathit{K})$	$\mathit{\sigma }{\left(\mathit{\Omega }\mathit{m}\right)}^{-1}$
$H_{2}O$	997.1	4180	0.613	0.05
$Cu$	8933	385	401	59.5 × 106
$A{l}_2{O}_3$	3970	765	40	35 × 106

.

Table 2. Comparison of results of $K$ with MATLAB and HPM for $\frac{1}{{\left(1-{\mathsf{\varphi }}_1\right)}^{2.5}{\left(1-{\mathsf{\varphi }}_2\right)}^{2.5}}\hspace{0.33em}{f}^{″}$.

S. No	$\mathit{K}$	$\frac{1}{{\left(1-{\mathsf{\varphi }}_1\right)}^{2.5}{\left(1-{\mathsf{\varphi }}_2\right)}^{2.5}}\hspace{0.33em}{\mathit{f}}^{″}$
		MATLAB	HPM
1	0.0	1.000003	1.000003
2	0.1	1.000536	1.000195
3	0.2	1.000459	1.000402
4	0.3	1.000098	1.000086
5	0.4	1.000005	1.000019

shows a good agreement between the numerical results bvp4c (MATLAB 2021b) compared with the HPM.

The impact of the magnetic field parameter is seen in Figure 2. It is shown that hybrid nanofluid velocity has a negative impact on the function of $M$. According to an actual perspective, when the magnetic field M grows, it slows down the fluid by producing resistance in the flow direction known as the Lorentz force. Because the Lorentz force is shown in the radial direction, the flow velocity decreases due to the additional resistance, resulting in a decrease in the radial velocity field velocity. Figure 3 displays the velocity profiles for various values of porous medium $K$ other parameters are constant. The thickness of the momentum boundary layer grows as $K$ values increase, as indicated by the graph. The physical reason for this is that increasing K makes a porous medium less resistant, which improves the regime’s momentum development and makes the fluid move faster.

As shown in Figure 4, the electric field $E$ generated by the Lorentz force greatly improves the velocity profile of hybrid nanofluids. Figure 5 illustrates the influence of various values of $M$ on dimensionless temperature. It is found that the temperature gradient increases as $M$ increases. Physically, this results from the retarding characteristic of the external magnetic field that produces an opposing Lorentz force that prevents momentum transmission and increases the viscosity of the thermal boundary layer. According to Figure 6, raising the radiation parameter $R$ causes an increase in fluid temperature and thermal radiation improves the fluid temperature throughout the boundary area. Because the thermal radiation parameter increases the amount of diffuse energy, the system remains stable.

Figure 7 depicts the effect of heat source/sink on the hybrid nanofluid’s temperature. Specifically, the temperature rises as the heat source parameter increases. Figure 8 demonstrates that Ec values are increased, temperature profiles rise in the same way. Scientifically speaking, the Eckert number $Ec$ is the ratio of the kinetic energy to the difference in specific enthalpy between the sheet and the fluid. Therefore, when the Eckert number increases, work is done against the stresses of the viscous fluid to turn kinetic energy into internal energy. Because of this, increasing Ec makes the fluid temperature increase.

Figure 9, Figure 10, Figure 11 and Figure 12 demonstrate the entropy profiles for the Brinkmann, radiation and magnetic parameters. The effect of the magnetic field $M$ on the results of the entropy generation $N_G$ is shown in Figure 9. Physically, there is greater friction when the magnetic parameter $M$ causes the Lorentz force to grow. As a result, the rate of entropy generation ($N_G$) increases. Entropy generation rates are exemplified in Figure 10 for radiation parameter $R$. When $R$ is increased to higher magnitudes, the thermal irreversibility also rises, which causes the system to become more disordered and causes $N_G$ to rise. Figure 11 shows that the Bejan number decreases when the magnetic parameter is increased. Because of an increase in physical entropy production, the Bejan number drops. In Figure 12 displays the Bejan number is increased by varying the Brinkman number $Br$.

6. Conclusions

The rate of entropy generation on EMHD $Cu-A{l}_2{O}_3/H_{2}O$ hybrid nanofluid at a stagnation point flow across a stretching surface is investigated in this paper. Second-order slip, porous medium, radiative heat flux, viscous dissipation and melting heat transmission are considered. A similarity solution method is employed to convert the dimensionless partial differential equations into a set of first - order ordinary differential equations. The resultant equations are solved numerically with the use of popular R-K method with shooting technique. The impacts of active factors on fluid transport, fluid temperature distribution, Bejan number and entropy generation are shown graphically.

The results of this latest investigation are summarized below.

• The velocity profile decreases when the magnetic parameter and porous medium are increased.
• As increasing the value of electric field, the momentum boundary layer is increases
• The thermal boundary layer expands as the magnetic parameter, radiation parameter, heat source and Eckert number are raised.
• Increases in radiation parameter and magnetic field cause an increase in the rate of entropy generation.
• The Bejan number rises as the Brinkman number rises, but the magnetic field shows the opposite trend.