Unsteady Electro-Hydrodynamic Stagnating Point Flow of Hybridized Nanofluid via a Convectively Heated Enlarging (Dwindling) Surface with Velocity Slippage and Heat Generation

Abstract

In (Al₂O₃-Cu/H₂O) hybridized nanofluid (HYNF) is an unsteady electro-hydrodynamic stagnation point flow. A stretchable (shrinkable) surface that was convectively heated was studied in the past. In addition to the traditional nonslip surface, the heat generating (absorbing) and the velocity slippage constraints are deliberated in this research. An obtained nonlinear scheme is resolved by the homotopy analysis method. Governing parameters are the electric field parameters, that is, the dimensionless parameters including the magnetic parameter, Prandtl quantity, heat generating factor, Eckert quantity, and unsteady factor. We discuss in detail the effects of these variables on the movement of problems and thermal transmission characteristics. Increasing the values of the magneto and electric force parameters increased the temperature. Increasing the Prandtl number lowered the temperature. For the Eckert parameter, an increase in temperature was recognized. The symmetric form of the geometry model displayed improved the fluid flow by the same amount both above and below the stagnation streamline, while it decreased the flow pressure by the same level. The more heat source uses to increase the temperature of the HYNF over the entire area, the more heat is supplied to the plate, but with a heat sink, the opposite effect is observed.

1. Introduction

Nanofluids have gained increasing attention due to their ability to recover heat transfer efficiency in an assortment of industrial presentations, as well as the considerable upsurge in the thermal conducting of the subsequent liquid. As a continuation of nanofluids, HYNFs can be made by diffusing compound nano-powders or numerous types of nanomolecules in the solution. HYNF is an innovative nanoliquid with two separate nanoparticles immersed in the base fluid.

In recent years, many scholars have been involved in researching the heat transfer of HYNF because HYNFs exhibit a higher heat transfer coefficient than conventional nanofluids. As a result, most heat transfer applications (e.g., machined coolants and HYNF) have been explored to increase the thermal transmission coefficient of traditional nanoliquids.

Several authors used computational techniques to observe the boundary layer flow and thermal transmission of HYNFs. For example, Devi and Devi [1] used Cu-Al₂O₃ magnetically effective nanoparticles to revise the flow of HYNFs through elastic surfaces. This problem was then extended by Devi [2] to a three-dimensional flow that follows Newton’s heating conditions. They exposed that the heat transmission factor of HYNFs was greater than that of regular nanofluids in both studies. Hayat and Nadeem [3] considered the issue of (Ag-CuO/water) and HYNF that does not shake concerning rotating currents. The consequence of rapid slippage on heat transmission in an unsteady stagnating point flow of HYNF throughout a convective-heated enlarging (dwindling) plate was examined by Zainal et al. [4]. Daniel et al. [5] investigated unstable combined natural and forced convective electrical magnetohydrodynamic (MHD) flow and thermal transmission across a transparent stretched layer using the Buongiorno model. Xie et al. [6] investigated the frictional force factor and put it on a quantity of hybridized nanomolecules to evaluate their tribological assets. HYNFs have reduced wear and coefficient of friction compared to pure nanofluids, according to Devi et al. [2] examination on three-dimensional flow of (Cu-Al₂O₃/H₂O) HYNF using the RK Fehlberg integration process. Considering the Lorentz force, the movement is caused by the unidirectional linear elongation of a flat surface. Numerical results suggest that the heat exchange ratio of (Cu-Al₂O₃) HYNF is more advanced than that of the mono nanoliquid. The impact of temperature distribution and nanoparticle attention on the rheological interest of a magnetite ferrofluid silver/ethylene glycol (Fe₃O₄-Ag/EG) HYNF was studied by Afrand et al. [7].

In the existence of Lorentz forces, Ghadikolaei et al. [8] explored the physical and thermal characteristics of (TiO₂-Cu/H₂O) HYNF with shape factor. Hussian et al. [9] investigated the flow of HYNF covering (Cu-Al₂O₃/H₂O) combination via an open hollow space with an adiabatic square obstruction in the hollow space. They calculated numerical answers to the usage of the finite detail technique and explored the effect of numerous physical parameters on HYNFs.

Magnetohydrodynamics (MHD) has recently gained attention due to its extensive selection of solicitations in engineering, chemical knowledge, petroleum use, the environment, and geophysics. MHD implements a magneto force, usually orthogonal to the flow of liquid, which can generate the Lorentz force, which is a drag. This force faces the flow of the fluid and affects the velocity of the fluid, which tends to be critical [10]. We found that power-law fluid MHD combined convective flows over a nonlinearly stretched layer. Zhao et al. reported on the impact of magneto heat transmission regarding nanoliquids in micro-channels [11]. Many more major works in this field are attributed to refs. [12,13,14].

Scientists are interested in strained surfaces for a variety of engineering applications such as blown glass, cooling of microelectronics, metal foundry quenching, wire drawing, polymer extrusion, and high-speed spraying. The theoretical boundary layer flow on an expanding surface was discovered by Crane [15]. Exponentially expanding surface area [16,17,18,19] and important industrial and technological applications have been studied by many researchers. In everyday life and industry, exponentially contracting/expanding surfaces are frequently used for liquid flow and thermal transfer. A study by Magyari and Keller [20] seems to have initially addressed the flow of the fluid boundary layer on an exponential expanding plate. The exponential comparison variable was calculated by Mushtaq et al. [21] to convert the predominant PDE to ODE. In addition, Reddy et al. [22] studied the mixed convection of nanofluids on an exponential surface and observed that the concentricity, momentum, and thickness of the thermal boundary layer increased as the viscosity ratio parameter increased. The stagnating point MHD flow model was developed by Rahman et al. [23], who generated the ODE using the exponential similarity variable.

The liquid movement at the rigid surface’s stagnating point is represented by the stagnating point flow. Hiemenz1 [24] was the main researcher to probe the problem of fluid stagnating points flowing via a rigid surface. Homann [25] extended these concerns to the case of axis symmetry for the flow of 3D stagnation spots. Meanwhile, he reported a stream of stagnation towards the contracting plate. In the work of Wang [26], the presence of stagnant flow rates could limit turbulence and maintain flow, excluding the need to apply suction to the shrinking film. Several researchers have extended the stagnation point flow problem to the effects of different flow behaviors. For example, the problem is present with or without heat transfer melting, studied by Bachok et al. [27]. They did not find a solution.

The heat transfer coefficient at the solid–liquid boundary tends to decrease only when shrinking and when increasing the melting parameters. Fang and Wang [28] also studied the flow of temporary stagnation points to the movable plate and provided an accurate result to the problem. In the fields of mathematics, fluid mechanics, and engineering, several approximation methods have been used to solve real problems. In 1992, Liao [29] learned that this method was a quick fix and better suited for solving nonlinear problems. Homotopy analysis (HAM) [30,31,32] was used for the solution.

This study aims to investigate the interactions between an electrical force, a magnetic field, a magneto force, and heat-producing (absorbing) impacts in a conductive HYNF that integrates stability analysis and rapid slippage parameters. The incorporation of hybrid nanoparticles aids in the stabilizing of a nanofluid’s motion and retains the symmetry of the moving configuration. Due to the above problems, researchers are now encouraged to perform numerical studies with unstable stagnation point flow on alumina–copper–water convection heating plates (alumina, copper-to-water) under the influence of a reduced heat transfer coefficient. Using the appropriate set of dimensionless variables to reduce the independent variables of the mathematical model equations, the analytical solution was calculated using the homotopy method.

2. Mathematical Formulation

In this research work, as illustrated in Figure 1, the unsteady 2D stagnation point flow of (Al₂O₃-Cu/H₂O) HYNF above a convective-heated stretchable (shrinkable) plate influencing the speed of slippage is deliberated.

The extending/shrinking velocity is marked by $u_w\left(x,t\right)=\frac{bx}{\left(1-ct\right)}$, where $b$ represents constant stretchable $\left(b>0\right)$ and a shrinkable $\left(b<0\right)$ cases, while $c$ indicates the time-based issue, $a>0$ signifies the asset of the flow of stagnations, and $u_e\left(x,t\right)=\frac{ax}{\left(1-ct\right)}$ is the speed of the free stream. $T_1$ and $T_0$ stand for the free temperature and the standard temperature, respectively. We let the bottom of the surface warm up by convective heat from a hot liquid at a certain temperature $T_f\left(x,t\right)=T_1-T_0\frac{a{x}^2}{2\upsilon }{\left(1-ct\right)}^{-\frac{3}{2}}$, which generated a heat transfer coefficient, denoted by $h_f$. Taking into account all of the aforementioned hypotheses, the regulating boundary layer formulas may be identified as [4]:

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

(1)

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

(2)

$$\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\frac{k_{hnf}}{{\left(\rho c_p\right)}_{hnf}}\frac{{\partial }^{2}T}{\partial y^2}+\frac{{\sigma }_{hnf}}{{\left(\rho c_p\right)}_{hnf}}{\left(B_{0}u-E_0\right)}^2+\frac{Q_0}{{\left(\rho c_p\right)}_{hnf}}\left(T-T_{\infty }\right),$$

(3)

where $v$ is the velocity factor in $y$-axis, $u$ symbolizes the elements of rapidity in $x$-axis, ${\mu }_{hnf}$ is the (Al₂O₃-Cu/H₂O) dynamic viscosity, ${\rho }_{hnf}$ the consistency of (Al₂O₃-Cu/H₂O), $T$ is the HYNF temperature, ${\left(\rho c_p\right)}_{hnf}$ is HYNF heat capacity, and $k_{hnf}$ is the thermal conductance. The boundary conditions, composed through the restricted slippage for rapidity, are established as:

$$\begin{array}{l}u=u_w\left(x,t\right)+\upsilon H_1\frac{\partial u}{\partial y},v=0,-k_{hnf}\frac{\partial T}{\partial y}=h_f\left(T_f-T\right) at y=0,\\ u\to u_e\left(x,t\right),T\to T_{\infty } as y\to \infty .\end{array}\}$$

(4)

where $H_1=H{\left(1-ct\right)}^{\frac{1}{2}}$ is the quickness slippage variable, and $H$ denotes the original value of the speed slippage variable. Copper (Cu) physical–thermal characteristics, laterally with aluminum-oxide (Al₂O₃) and H₂O nanomolecules, are provided in

Table 1. Cu thermophysical properties laterally with Al₂O₃ and H₂O [4].

Properties	Cu	Al2O3	H2O
$\mathit{k}\left(\mathit{W}\mathbf{/}\mathit{m}\mathit{K}\right)$	400	40	0.613
$\mathit{\rho }\left(\mathit{k}\mathit{g}\mathbf{/}{\mathit{m}}^{\mathbf{3}}\right)$	8933	3970	9971
${\mathit{c}}_{\mathit{p}}\left(\mathit{J}\mathbf{/}\mathit{k}\mathit{g}\mathit{K}\right)$	385	765	4179
$\mathit{\beta }\mathbf{\times }{\mathbf{10}}^{\mathbf{5}}\left(\frac{\mathbf{1}}{\mathit{K}}\right)$	1.67	0.85	21

.

Table 2. Utilized relations for physical–thermal characteristics of HYNF [4].

Characteristic	HYNF
$\mathit{\mu }$	${\mu }_{hnf}=\frac{1}{{\left(1-{\varphi }_{hnf}\right)}^{2.5}}$
$\mathit{\rho }$	${\rho }_{hnf}=\left(1-{\varphi }_{hnf}\right){\rho }_f+{\varphi }_1{\rho }_{s1}+{\varphi }_2{\rho }_{s2}$
$\left(\mathit{\rho }{\mathit{c}}_{\mathit{p}}\right)$	${\left(\rho c_p\right)}_{hnf}=\left(1-{\varphi }_{hnf}\right){\left(\rho c_p\right)}_f+{\varphi }_1{\left(\rho c_p\right)}_{s1}+{\varphi }_2{\left(\rho c_p\right)}_{s2}$
$\mathit{k}$	$\frac{k_{hnf}}{k_f}=\left[\frac{\left(\frac{{\varphi }_1{k}_{s1}+{\varphi }_2{k}_{s2}}{{\varphi }_{hnf}}\right)+2k_f+2\left({\varphi }_1{k}_{s1}+{\varphi }_2{k}_{s2}\right)-2{\varphi }_{hnf}{k}_f}{\left(\frac{{\varphi }_1{k}_{s1}+{\varphi }_2{k}_{s2}}{{\varphi }_{hnf}}\right)+2k_f-2\left({\varphi }_1{k}_{s1}+{\varphi }_2{k}_{s2}\right)+{\varphi }_{hnf}{k}_f}\right]$

shows the physical–thermal characteristics of HYNF. The nanomolecules’ volumetric fraction is denoted by $\varphi$, ${\rho }_f$ specifies H₂O consistency, ${\rho }_s$ is the consistency of the nanosolid particles, $c_p$ is the continuous pressure of heat capacity, $k_f$ symbolizes the thermal conductivity of H₂O, and $k_s$ is the nanoparticles’ thermal conductivity.

The following similarity transformations are presented [4] in the context of the governing Equations (1)–(3) about boundary conditions (4).

$$\psi ={\left(\frac{a\upsilon }{1-ct}\right)}^{\frac{1}{2}}xf\left(\eta \right),\theta \left(\eta \right)=\frac{T-T_{\infty }}{T_f-T_{\infty }},\eta ={\left[\frac{a}{\upsilon \left(1-ct\right)}\right]}^{\frac{1}{2}}y,$$

(5)

Here, $\psi$ is the streaming function which is obtained from relations $u=\frac{\partial \psi }{\partial y}$, $v=-\frac{\partial \psi }{\partial x}$, and $\eta$ is a similar variant. Consequently, we achieve

$$u=\frac{ax}{1-ct}{f}^{\prime }\left(\eta \right),v=-{\left(\frac{a\upsilon }{1-ct}\right)}^{\frac{1}{2}}f\left(\eta \right).$$

(6)

Concerning these relations, Equations (2) and (3) diminish the difference between the subsequent sets of nonlinear similitudes by using the similarity variables (5) and (6).

$$\frac{\frac{{\mu }_{hnf}}{{\mu }_f}}{\frac{{\rho }_{hnf}}{{\rho }_f}}{f}^{‴}+f{f}^{″}-f^{\prime 2}+1-\epsilon \left(f^{\prime }+\frac{1}{2}\eta f^{″}-1\right)+\frac{\frac{{\sigma }_{hnf}}{{\sigma }_f}}{\frac{{\rho }_{hnf}}{{\rho }_f}}M\left(E-f^{\prime }\right)=0,$$

(7)

$$\frac{1}{Pr}\frac{\frac{k_{hnf}}{k_f}}{\frac{{\left(\rho c_p\right)}_{hnf}}{{\left(\rho c_p\right)}_f}}{\theta }^{″}+f{\theta }^{\prime }-2f^{\prime }\theta +\frac{\epsilon }{2}\left(\eta {\theta }^{\prime }+3\theta \right)+\frac{\frac{{\sigma }_{hnf}}{{\sigma }_f}}{\frac{{\left(\rho c_p\right)}_{hnf}}{{\rho }_f}}MEc\left(f^{\prime }-E\right)+\frac{Q{\left(\rho c_p\right)}_f}{{\left(\rho c_p\right)}_{hnf}}\theta =0.$$

(8)

Then, the initial and limit conditions (4) are transformed in

$$\begin{array}{l}f\left(0\right)=0,f^{\prime }\left(0\right)=\lambda +\gamma f^{″}\left(0\right),-\frac{k_{hnf}}{k_f}{\theta }^{\prime }\left(0\right)=Bi\left[1-\theta \left(0\right)\right],\\ f^{\prime }(\infty )\to 1,\theta (\infty )\to 0,\end{array}\}$$

(9)

In the following, $\epsilon ,M,E,Ec,Pr,Bi,{Re}_x,\lambda ,$ and $Q$ are time-dependent parameter, magneto force, electrical force parameters, Eckert, Prandtl, Biot, Reynolds numbers in $x$-axis, the proportion of speed slippage, and heat generating (absorbing) parameter:

$$\begin{array}{l}\lambda =\frac{b}{a},\epsilon =\frac{c}{a},{Re}_x=\frac{u_{e}x}{{\nu }_f},Pr=\frac{{\nu }_f}{\alpha },M=\frac{\sigma {\beta }_0^2}{{\rho }_{f}a}, \gamma =H{\left(a{\nu }_f\right)}^{\frac{1}{2}}\\ Bi=\frac{h_f}{k_f}\sqrt{\frac{{\nu }_f\left(1-ct\right)}{a}},Ec=\frac{a{x}^2}{c_p\Delta T},Q=\frac{Q_0{\left(1-ct\right)}^2}{ax},E=\frac{E_0}{{\beta }_{0}ax}.\end{array}\}$$

(10)

For the above model, the frictional surface force $\left(C_f\right)$ and Nusselt amount $\left(N{u}_x\right)$ remain clear as follows

$$C_f=\frac{{\tau }_w}{{\rho }_f{u}_e^2},N{u}_x=\frac{x{q}_w}{k_f\left(T_f-T_{\infty }\right)}$$

(11)

where

$${\tau }_w={\mu }_{hnf}{\left(\frac{\partial u}{\partial y}\right)}_{y=0},q_w={-k_{hnf}\left(\frac{\partial T}{\partial y}\right)|}_{y=0},$$

(12)

In the dimensional form, we have from above

$$R{e}_x^{1/2}{C}_f=\frac{{\mu }_{hnf}}{{\mu }_f}{f}^{″}\left(0\right),R{e}_x^{-1/2}N{u}_x=-\frac{k_{hnf}}{k_f}{\theta }^{\prime }\left(0\right).$$

(13)

3. HAM Solution

The HAM is used to solve Equations (7) and (8) using the boundary condition (9). Mathematica software was used for this. In Figure 2, the HAM approach includes the following steps.

The model equations are described as fundamental derivations using HAM.

$$L_{\stackrel{⌢}{f}}\left(\stackrel{⌢}{f}\right)={\stackrel{⌢}{f}}^{‴}, L_{\stackrel{⌢}{\theta }}\left(\stackrel{⌢}{\theta }\right)={\stackrel{⌢}{\theta }}^{″}-\stackrel{⌢}{\theta }$$

(14)

Linear operators $L_{\stackrel{⌢}{f}}, and L_{\stackrel{⌢}{\theta }}$ are defined by

$$L_{\stackrel{⌢}{f}}\left({\gamma }_1+{\gamma }_2{e}^{-\eta }+{\gamma }_3{e}^{\eta }\right)=0,L_{\stackrel{⌢}{\theta }}\left({\gamma }_4{e}^{-\eta }+{\gamma }_5{e}^{\eta }\right)=0.$$

(15)

The nonlinear operators are $N_{\stackrel{⌢}{f}}$ and $N_{\stackrel{⌢}{\theta }}$ that are identified in the structure:

$$N_{\stackrel{⌢}{f}} \left[\stackrel{⌢}{f}\left(\eta ;\zeta \right)\right]=\frac{\frac{{\mu }_{hnf}}{{\mu }_f}}{\frac{{\rho }_{hnf}}{{\rho }_f}}{\stackrel{⌢}{f}}_{\eta \eta \eta }+\stackrel{⌢}{f}{\stackrel{⌢}{f}}_{\eta \eta }-{\stackrel{⌢}{f}}_{\eta }^2+1-\epsilon \left({\stackrel{⌢}{f}}_{\eta }+\frac{1}{2}\eta {\stackrel{⌢}{f}}_{\eta \eta }-1\right)+\frac{\frac{{\sigma }_{hnf}}{{\sigma }_f}}{\frac{{\rho }_{hnf}}{{\rho }_f}}M\left(E-{\stackrel{⌢}{f}}_{\eta }\right),$$

(16)

$$\begin{array}{l}{N}_{\stackrel{⌢}{\theta }}\left[\stackrel{⌢}{f}\left(\eta ;\zeta \right),\stackrel{⌢}{\theta }\left(\eta ;\zeta \right),\stackrel{⌢}{\varphi }\left(\eta ;\zeta \right)\right]\\ =\frac{1}{Pr}\frac{\frac{k_{hnf}}{k_f}}{\frac{{\left(\rho c_p\right)}_{hnf}}{{\left(\rho c_p\right)}_f}}{\stackrel{⌢}{\theta }}_{\eta \eta }+\stackrel{⌢}{f}{\stackrel{⌢}{\theta }}_{\eta }-2{\stackrel{⌢}{f}}_{\eta }\stackrel{⌢}{\theta }+\frac{\epsilon }{2}(\eta {\stackrel{⌢}{\theta }}_{\eta }+3\stackrel{⌢}{\theta })+\frac{\frac{{\sigma }_{hnf}}{{\sigma }_f}}{\frac{{\left(\rho c_p\right)}_{hnf}}{{\rho }_f}}MEc({\stackrel{⌢}{f}}_{\eta }-E)+\frac{Q{\left(\rho c_p\right)}_f}{{\left(\rho c_p\right)}_{hnf}}\theta \end{array}$$

(17)

For Equations (7) and (8) the 0th-order structure is exposed as

$$\left(1-\zeta \right)L_{\stackrel{⌢}{f}}\left[\stackrel{⌢}{f}\left(\eta ;\zeta \right)-{\stackrel{⌢}{f}}_0\left(\eta \right)\right]=p{\hslash }_{\stackrel{⌢}{f}}{N}_{\stackrel{⌢}{f}}\left[\stackrel{⌢}{f}\left(\eta ;\zeta \right)\right]$$

(18)

$$\left(1-\zeta \right) L_{\stackrel{⌢}{\theta }}\left[\stackrel{⌢}{\theta }\left(\eta ;\zeta \right)-{\stackrel{⌢}{\theta }}_0\left(\eta \right)\right]=p{\hslash }_{\stackrel{⌢}{\theta }}{N}_{\stackrel{⌢}{\theta }}\left[\stackrel{⌢}{\theta }\left(\eta ;\zeta \right),\stackrel{⌢}{f}\left(\eta ;\zeta \right)\right]$$

(19)

Though BCs are

$$\begin{array}{l}{\stackrel{⌢}{f}\left(\eta ;\zeta \right)|}_{\eta =0}=0, {\frac{\partial \stackrel{⌢}{f}\left(\eta ;\zeta \right)}{\partial \eta }|}_{\eta =0}=\lambda +\gamma {\stackrel{⌢}{f}}_{\eta \eta }\left(0\right), f\left(0\right)=0,\\ {\frac{k_{nf}}{k_f}\frac{\partial \stackrel{⌢}{\theta }\left(\eta ;\zeta \right)}{\partial \eta }|}_{\eta =0}=-Bi\left(1-\stackrel{⌢}{\theta }\left(0\right)\right), {\frac{\partial \stackrel{⌢}{f}\left(\eta ;\zeta \right)}{\partial \eta }|}_{\eta =\infty }\to 1,{\stackrel{⌢}{\theta }\left(\eta ;\zeta \right)|}_{\eta =\infty }\to 0,\end{array}\}$$

(20)

The embedding constraint is $\zeta \in \left[0,1\right]$, and to regulate the result converging ${\hslash }_{\stackrel{⌢}{f}}$ and ${\hslash }_{\stackrel{⌢}{\theta }}$ are utilized. At $\zeta =0$ and $\zeta =1$ we have:

$$\stackrel{⌢}{f}\left(\eta ;1\right)=\stackrel{⌢}{f}\left(\eta \right),\stackrel{⌢}{\theta }\left(\eta ;1\right)=\stackrel{⌢}{\theta }\left(\eta \right) ,$$

(21)

Enlarging the $\stackrel{⌢}{f}\left(\eta ;\zeta \right)$ and $\stackrel{⌢}{\theta }\left(\eta ;\zeta \right)$ over Taylor’s series for $\zeta =0$, we obtain:

$$\begin{array}{l}\stackrel{⌢}{f}\left(\eta ;\zeta \right)={\stackrel{⌢}{f}}_0\left(\eta \right)+{{\displaystyle \sum }}_{n=1}^{\infty }{\stackrel{⌢}{f}}_n\left(\eta \right){\zeta }^n,\\ \stackrel{⌢}{\theta }\left(\eta ;\zeta \right)={\stackrel{⌢}{\theta }}_0\left(\eta \right)+{{\displaystyle \sum }}_{n=1}^{\infty }{\stackrel{⌢}{\theta }}_n\left(\eta \right){\zeta }^n\end{array}\}$$

(22)

$${\stackrel{⌢}{f}}_n\left(\eta \right)={\frac{1}{n!}\frac{\partial \stackrel{⌢}{f}\left(\eta ;\zeta \right)}{\partial \eta }|}_{p=0},{\stackrel{⌢}{\theta }}_n\left(\eta \right)={\frac{1}{n!}\frac{\partial \stackrel{⌢}{\theta }\left(\eta ;\zeta \right)}{\partial \eta }|}_{p=0},$$

(23)

BCs are:

$$\begin{array}{l}\stackrel{⌢}{f}\left(0\right)=0,{\stackrel{⌢}{f}}^{\prime }\left(0\right)=\lambda +\gamma {\stackrel{⌢}{f}}^{″}\left(0\right),-\frac{k_{hnf}}{k_f}{\stackrel{⌢}{\theta }}^{\prime }\left(0\right)=Bi\left[1-\stackrel{⌢}{\theta }\left(0\right)\right],\\ {\stackrel{⌢}{f}}^{\prime }(\infty )\to 1,\stackrel{⌢}{\theta }(\infty )\to 0,\end{array}\}$$

(24)

Now,

$${\Re }_n^{\stackrel{⌢}{f}}\left(\eta \right)=\frac{\frac{{\mu }_{hnf}}{{\mu }_f}}{\frac{{\rho }_{hnf}}{{\rho }_f}}{\stackrel{⌢}{f}}_{n-1}^{‴}+{{\displaystyle \sum }}_{j=0}^{w-1}{\stackrel{⌢}{f}}_{w-1-j}{f}_j^{{\stackrel{⌢}{″}}_{ }}+1-\epsilon \left({\stackrel{⌢}{f}}_{n-1}^{\prime }+\frac{1}{2}\eta {\stackrel{⌢}{f}}_{n-1}^{″}-1\right)+\frac{\frac{{\sigma }_{hnf}}{{\sigma }_f}}{\frac{{\rho }_{hnf}}{{\rho }_f}}M\left(E-{\stackrel{⌢}{f}}_{n-1}^{\prime }\right)$$

(25)

$$\begin{array}{l}{\Re }_n^{\stackrel{⌢}{\theta }}(\eta )=\frac{1}{\mathrm{Pr}}\frac{\frac{k_{hnf}}{k_f}}{{\left(\rho c_p\right)}_{hnf}/{\left(\rho c_p\right)}_f}\left({\stackrel{⌢}{\theta }}_{n-1}^{″}\right)+{\displaystyle \sum _{j=0}^{w-1}{\stackrel{⌢}{\theta }}_{w-1-j}^{\prime }}{\widehat{f}}_j-2{\displaystyle \sum _{j=0}^{w-1}{\stackrel{⌢}{\theta }}_{w-1-j}}{\stackrel{⌢}{f}}_j^{\prime }+\frac{\epsilon }{2}\left(\eta {\stackrel{⌢}{\theta }}_{n-1}^{\prime }+3{\stackrel{⌢}{\theta }}_{n-1}\right)+\\ \frac{{\sigma }_{hnf}/{\sigma }_f}{{\left(\rho c_p\right)}_{hnf}/{\rho }_f}MEc\left({\stackrel{⌢}{f}}_{n-1}^{\prime }-E\right)+\frac{Q{\left(\rho c_p\right)}_f}{{\left(\rho c_p\right)}_{hnf}}{\stackrel{⌢}{\theta }}_{n-1},\end{array}$$

(26)

where

$${\chi }_n=\{\begin{array}{c}0, \mathrm{if} n\le 1\\ 1, \mathrm{if} n>1.\end{array}$$

(27)

4. Results and Discussion

This study solved a homotopy analysis system for the transformed calculations of momentum, energy, and concentration ((7), (8)). Computational analysis was performed using various parameters such as Prandtl quantity, magneto force, electric field, time-dependent parameter, Eckert number, and parameter of heat source/sink. Figure 3, Figure 4 and Figure 5 show various implanted parameters: electric field, magnetic field, and discontinuous velocity profile $f^{\prime }\left(\eta \right)$.

Figure 3 specifies that as the electric parameter values rise, the rate of HYNF will increase. Electrical parameters act as accelerating forces. The better the Lorentz pressure and the smaller the Lorentz pressure, the better the electric confinement. A more potent Lorentz pressure will increase and release the stick impact with liquid nanoparticles that imply improved convection warmth switch and the width of the boundary momentum layer. The electric force increases the stored internal energy of the particles, which leads to an increase in the movement within the liquid, as a result of the increased collisions between the particles and each other and as a result of the internal acceleration among them. Figure 4 indicates the impact of the magneto force constraint $M$ on the rate profile of HYNF. The better the magneto parameter value the more the width of the rate-momentum boundary layer is diminished. The magnetic area creates a drag referred to as the Lorenz wave that opposes the flow. This creates resistance to fluid flow, slows down, and decreases the width of the boundary layer. Thus, it works to slow the movement of particles within the fluid, which in turn reduces the general velocity of the fluid as a result of this increased resistance. Figure 5 indicates the impact of the unsteadiness parameter at the nanoliquid pace profile. This conduct is because of the deceleration case, which boosts the momentum boundary layer thickness and decreases HYNF flow. As the acceleration increases, the flow rate profile decreases. Over time, the internal energy of the molecules is lost, and thus the overall movement of the fluid is reduced.

In addition, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11 show the properties of Eckert amount $Ec$, time-dependent variant $\epsilon$, heat source/sink $Q$, magneto force constraint $M$, electrical force $E$, and Prandtl amount $Pr$ on the temperature outlines $\theta \left(\eta \right)$. In Figure 6, when the kinetic energy to enthalpy ratio rises with a rise in the $Ec$, the temperatures of the nanoliquid increase, as does the width of the thermal boundary layer. This leads to an increase in heat dissipation within the HYNF. The effect of the unsteadiness parameter on $\theta \left(\eta \right)$ outlined in Figure 7 is contrary to the speed behavior in Figure 5. The unsteadiness parameter works overtime to make the molecules dissipate the stored energy internally, which boosts the heat transfer process, which in turn raises the temperature of the hybrid nanoliquid. It is obvious that the temperature is more affected by unsteadiness and increment values of it. The consequence of heat generation (absorption) on $\theta \left(\eta \right)$ is shown in Figure 8 and Figure 9. It is shown that the heat source $\left(Q>0\right)$ rises the fluid’s temperature and the width of the temperature boundary layer, although the heat absorption $\left(Q<0\right)$ delivers a diminution in $\theta \left(\eta \right)$ and slimmer temperature boundary layer width. $Q=0$ implies the lack of heat generation (absorption). Figure 10 launches the effect of magneto force variant $M$ on the nanoliquid energy contours. The width of the thermal boundary layer has risen owing to the transverse magnetic field. As a result, the thickness of the thermal boundary layer and $\theta \left(\eta \right)$ outlines are improved. The magneto force performs as a sturdy Lorentz power, growing $\theta \left(\eta \right)$ of the nanoliquid inside a border area. Lorentz forces work to boost the internally stored energy inside the nanomolecules, which leads to an excess in the temperature of the nanofluid and thus restores its movement, which reverses the effect between the rapidity of movement and the temperature as a result of raising the value of the magnetic field applied to the fluid. The influence of the electrical force variant at the temperature outline is found in Figure 11. The electric power performs as a speed-up force, boosting the nanoliquid temperature and developing the thickness of the temperature boundary layer. A wider, higher capacity $\theta \left(\eta \right)$ outline inside the boundary layer in proximity to the HYNF is hooked up with a larger amount of an electrical power variant. Higher Prandtl’s quantities range is very satisfactory and differs from one fluid to the next. Figure 11 demonstrates that as it improves, the nanoliquid energy drops. The diffusivity impetus is more than the thermal diffusion for big values. As a result, the thickener of the energy boundary layer diminishes. The excess in the Prandtl quantity diminishes the thermal diffusivity factor as well as the boost in the viscidness of the hybrid nanoliquid, which in turn reduces the temperature of the hybrid nanoliquid as a whole, and this is the effect that is observed in the figure. In most warmness switch problems, the comparative thickness of each thermal boundary layer is reduced.

Table 3. Effect of various physical parameters on skin friction $R{e}_x^{1/2}{C}_f=\frac{{\mu }_{hnf}}{{\mu }_f}{f}^{″}\left(0\right)$.

ε	E	M	$\frac{{\mathit{\mu }}_{\mathit{h}\mathit{n}\mathit{f}}}{{\mathit{\mu }}_{\mathit{f}}}{\mathit{f}}^{\mathbf{″}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}$
0.3	0.1	0.4	0.72059328
0.5			0.83542092
0.7			1.03614135
	0.1		1.86313569
	0.2		1.64385204
	0.3		1.76103193
		0.4	1.03873708
		0.8	1.30863981
		1.0	1.58376213

shows that $C_f$ is enlarged after the standards of $M,\epsilon ,E$ are enlarged.

Table 4. Consequence of various physical parameters on Nusselt number $\left[-\frac{k_{hnf}}{k_f}{\theta }^{\prime }\left(0\right)\right]$.

Ec	Q	Pr	M	E	$\mathbf{-}\frac{{\mathit{k}}_{\mathit{h}\mathit{n}\mathit{f}}}{{\mathit{k}}_{\mathit{f}}}{\mathit{\theta }}^{\mathbf{\prime }}\mathbf{\left(}\mathbf{0}\mathbf{\right)}$
0.3	0.5	4.5	0.4	0.1	1.07386504
0.5					1.17290347
0.7					1.23893104
	0.5				2.30319769
	1.0				2.15912307
	1.5				2.02463073
		4.5			0.54354079
		5.5			0.73865302
		6.5			0.93865321
			0.4		1.13159603
			0.8		1.09764384
			1.0		1.05346068
				0.1	1.12183304
				0.2	1.23583931
				0.3	1.30346893

displays that $N{u}_x$ is enlarged once the standards of $M,Q$ are enlarged. $N{u}_x$ is diminished as the standards of $Pr,E,Ec$ are enlarged.

5. Conclusions

An evaluation of the unstable electro-hydrodynamic stagnation factor float of HYNF via a convective heated stretchable (shrinkable) plate including the speed slippage effect on the heat switch is investigated in this study. Electrical and magneto forces and heat generating (absorbing) parameters were measured. The principal equations of the system were transmuted to nonlinear regular differential equalities with the use of similar variants and then resolved via the homotopy analysis method. The effects of numerous variables on pace and temperature were examined. Mathematical consequences of pace gradient and warmth switch quotes in opposition to numerous parameters were deliberated. The following are the most important observations of the study:

• Speed and temperature rise with a rise in an electrical constraint.
• Magnetic constraint has an inverted influence on rapidity and energy parameters.
• Heat generation increases the temperature, while the converse happens with a heat sink.
• Higher values of the unsteadiness constraint decrease the velocity and temperature.
• An augmentation in temperature is observed for Eckert amount, while it decreases for the Prandtl number.

The current technique could be implemented in other fields of science and engineering, especially when related to the simulation of fluid dynamics in the future [33,34,35,36].