Unsteady Electrohydrodynamic Stagnation Point Flow of Hybrid Nanofluid Past a Convective Heated Stretch/Shrink Sheet

Department of Mathematics, University of Swabi, Swabi 23561, KPK, Pakistan Department of Mathematics, College of Sciences and Arts, ArRass, Qassim University, Saudi Arabia Laboratory of Fundamental and Applied Mathematics of Oran (LMFAO), University of Oran 1, Oran, Algeria Department of Mathematics, Abdul Wali Khan University, Mardan, 23200 Khyber Pakhtunkhwa, Pakistan College of Industrial Engineering, King Khalid University, Abha 61471, Saudi Arabia Department of Mathematics, College of Sciences, Juba University, Sudan


Introduction
The ability of nanofluids to improve heat transfer performance in a range of industrial applications due to the substantial raise in thermal conductivity of the resulting fluid has piqued interest in recent years. A liquid in which nanosized metallic or nonmetallic components are suspended is defined as this new class of fluids. By dispersing a composite nanopowder or various types of nanoparticles in the hybrid nano-and base fluids, which are a continuation of nanofluids, can be created. An enhanced nanofluid having two distinct nanoparticles scattered in the base fluid is called a hybrid nanofluid.
Many researchers have been interested in studying heat source in a hybrid nanofluid in recent years because of its ability to boost heat transfer rates when compared to ordinary nanofluid. As a result, most heat transferring uses, such as transformer cooling, electronic cooling, and coolant in machines, have used hybrid nanofluid as the heat source fluid. Nanofluid is an eminent as a high heat transferred when compared to other fluids. The hybrid nanofluid, on the other hand, is researched in this study is to strengthen the rate of heat source of the standard nanofluid. Several scientists used numerical methods to explore the heat transport and the boundary layering flow of a hybrid nanofluid. The researchers in [1] studied the flow of a hybrid nanofluid past a stretched sheet by considering Cu-Al 2 o 3 nanoparticles with magnetic impacts. Newtonian heated condition was examined by these researchers with the 3-D flow [2]. The rate of heat transferring of hybrid nanofluid was found to be greater than that of standard nanofluid. Hayat and Nadeem [3] considered the rotating flow of hybrid nanofluid which consisted of Ag-CuO/water. The authors in [4] conceptualize and investigate the flow of hybrid nanofluid over an unsteady stagnation point convective heat stretch/shrink surface including the velocity slip impact on heat transfer. Using the Buongiorno model, Daniel et al. [5] analyzed and interrogated mixed convection electrico MHD flow and heat source caused nano-fluid over a permeable stretched surface. Xie et al. [6] worked out on the friction coefficient and wear volume of hybrid nanoparticles to determine their tribological properties of the SiO 2 /Mo S 2 . When compared to pure nanofluids, they found that hybrid nanofluids have a smaller friction coefficient and wear volume. The authors in [7] scrutinized the impact of nanoparticle concentration and temperature distribution on rheological performance of ðFe 3 O 2 -Ag/EGÞ hybrid nanofluid.
Ghadikolaei et al. [8] examined the thermophysical possessions of ðTiO 2 -Cu/H 2 OÞ with a shape factor with Lorentz forces. Hussian et al. [9] considered hybrid nanofluid containing ðAl 2 o 3 -Cu/H 2 oÞ and proved excellent results in the circumstance of their assumptions. They calculated numerical solutions using the finite element scheme and analyzed the impact of various input factors on the hybrid nanofluids. Magnetohydrodynamic (MHD) has recently gotten prominent attention due to its wide range of applications in geophysics, petroleum, chemical technology, environmental, engineering, and many other branches of science and technology. The Lorentz force which is also known as the drag force produced through the magnetic field that is used by MHD. Waqas et al. [10] investigated non-Newtonian liquid MHD mixed convection flow of fluid on stretched and nonlinear surface. Furthermore, the impact of magnetic field heat transferred of nanofluids in microchannels was discussed by the researchers in their research [11]. Other researchers also worked out in this area and produced a significant work in the field [12][13][14].
Scientists are interested in the stretching surface because of its numerous uses in engineering domains such as glass blowing, polymer extrusion, fast spray, wire drawing, quenching in metal foundries, and microelectronics cooling. The theoretical boundary layer flows over stretched sheet has been scrutinized by Crane [15]. Different authors investigated the exponentially stretching sheet [16][17][18][19], the main applications in technology and industry. In everyday life and industrial challenges, the exponentially shrinked/stretched sheet is often used with fluid flow and heat transmission. The article by Magyari and Keller [20] appears to be the first to investigate at the fluid boundary layer flow on an exponentially stretched sheet. The exponential similarity variables were used by Mushtaq et al. [21] to turn the controlling PDEs into ODEs. Furthermore, the researchers in [22] interrogated the mixed convective flow of nanofluid and obtained excellent result and find out the influence of input parameter in the flow. The MHD fluid flow for the stagnation point was created by Rahman et al. [23], who employed exponential similarity variables to generate ODEs.
Fluid movement near a solid surface's stagnation area is explained by the stagnation point flow. Hiemenz [24] was the first to inspect stagnation-point flow problem in a flowing fluid over a fixed surface. Homann [25] then enlarged to a 3-D stagnation-point flow problem in the axis symmetric condition. The researcher in [26] reported a stagnation fluid flow in a diminishing surface. According to him, the presence of a stagnation fluid flow velocity can constrain the vorticity to maintain the flow, eliminating the need for suction on the shrinked sheet. Many academics have studied numerous influences on flow behavior by extending the stagnationpoint flow problem. This problem is further investigated by Bachok et al. [27] without and with heat transferred where the authors discovered the decreasing case's solutions are no unique and that increasing the melting parameter reduces the heat source rate at the solid-fluid interface. In 1992, Liao [28] discovered that this method was a faster technique to get approximated solution and that it was a better fit for solving nonlinear problems. Several researchers contributed to find the solution through a homotopy analysis scheme (HAM) [29][30][31][32]. The reader can further study about heat transfer through nanofluid flow in Refs. [33][34][35][36][37][38][39][40].
The target of this work is to look into the combined impact of magnetic field, heat generation/absorption, electric field, and velocity slip parameter for unstable stagnation-point flow of electrical conducting hybrid nanofluids with the velocity slip input factor. In addition to this, we will focus on the stability of the system. As a result of the aforementioned issues, the researchers decided to interrogate the unstable stagnation-point flow towards a convective heated stretched or shrinked surface in alumina-copper/water ðAl 2 O 3 -Cu/H 2 OÞ to see how velocity slip affects heat transfer. To decrease the independent variables in governing equations resulting from mathematical modelling, a suitable set of dimensionless variables is utilised. By applying the homotopy method, an analytical solution has been computed.

Mathematical Formulation
This research considers the unstable 2-D stagnation point flow of a hybrid system ðAl 2 o 3 -Cu/H 2 oÞ nanofluid over a convective heated stretched or shrinked surface with the impact of velocity slip. In our formulation, the term u w ðx, t Þ = bx/ð1 − ctÞ is the stretched/shrinked velocity in which b represent a fix association with shrinked ðb < 0Þ and stretched ðb > 0Þ. We have the following cases while c denotes the unsteadiness problem and the velocity of the free stream is defined by u e ðx, tÞ = ax/ð1 − ctÞ. Here, a > 0 indicates the power of the stagnation flow. T 1 and T 0 are the ambient temperature and the reference temperature, respectively. Further, heat is provided to it with some specific temperature given by T f ðx, tÞ = T 1 − T 0 ðax 2 /2υÞð1 − ctÞ −3/2 which provided a heat transferred coefficient, denoted by h f . Then, the governing equation of [4] with all our 2 Advances in Mathematical Physics assumptions is as follows: ∂T ∂t where u and v indicated the velocity components along the x-axis and along the y-axis, respectively, while the dynamic viscosity is indicated by μ hnf , T is the The above given Equation (4) show the Bcs, together with the partial slip for velocity component, the velocity slip factor is H 1 = Hð1 − ctÞ 1/2 , where the initial value of the velocity slip parameter has been denoted by H. The properties of copper ðCuÞ thermophysical along with water ðH 2 OÞ nanoparticles and with aluminum oxide ðAl 2 O 3 Þ have been presented in Table 1. In the meantime, the thermophysical properties of hybrid nanofluid were published in Table 2. All the related input factors and symbols are represented in the nomenclature with detailed description.
To simplify the abovementioned problem represented in Equations (1)-(3) along with condition (4), we take the following similarity transformations [4]: In the above, η is the similarity variable and u = ∂ψ/∂y and v = −∂ψ/∂x are satisfied through the stream function ψ ; thus, we get the following: In the case, when the similarity variables (5) and (6) are employed to (2) and (3), they are converted to the following equations: After that, the beginning and boundary constrain (4) becomes Where ε, M, E, Ec, Pr, Bi, Re x , λ, and Q unsteadiness parameter, magnetic parameter, ELECTRIC FIELD PARAMETER, Eckert number, local Reynolds number in the x-axis, Biot number, Prandtl number, ratio of velocity parameter, and heat source/sink parameter.

Physical Quantities of Interest
For the above model, we take the coefficient of skin friction ðc f Þ and local Nusselt number ðNu x Þ as follows: In which The dimensional form of the above is

Solution by HAM
In this section of the paper, HAM will be used to solve Equations (7) and (8) with boundary conditions (9). Mathematica software is utilized for this purpose. The basic HAM derivation is given as follows: Linear operators L f _ and L θ _ are specified by The nonlinear operators Ν f _ and Ν θ _ are as follows: For Equations (7) and (8), we have shown the 0 th -order system as  The BCs are: While the embedded constraint isζ ∈ ½0, 1, ℏ f _ and ℏ θ _ are utilized to regulate the convergence. When ζ = 0 and ζ = 1, we have After that, f _ ðη ; ζÞ and θ _ ðη ; ζÞ are expended through Taylor's expansion for ζ = 0: The BCs are Now, with the condition

Results and Discussion
The transformed equations of concentration, energy, and momentum (7) and (8) with the boundary condition (9) are solved using the homotopy analysis technique in this work. Pr, M, E, ε, Ec, and Q are some of the parameters used in the computational analysis. f ′ðηÞ for the different embedded parameters, such as E, M , and ε, are shown in Figures 1-3 to highlight the influence of these input factors on the system. Higher values of E enhance the hybrid nanofluid velocity, as seen in Figure 1. An accelerating force is provided by an electric parameter. The stronger Lorentz force is connected with the higher value of E, whereas the weaker Lorentz force is related with the lower value of E . The larger Lorentz force enhances and resolves the sticky effect caused by nanoparticles in the fluid, indicating an enhancement in convective heating transferred and the thickness of the momentum barrier layer. In Figure 2, we have shown the effect of M on the hybrid nanofluid f ′ðηÞ. In which, we observed that the thickness of the velocity and momentum boundary layers decreased as the magnetic parameter was raised. Lorentz flow is a resistive type force that occurs when a magnetic field is applied in the opposite flow direction. As a result, the fluid flow behavior becomes more resistive, resulting in a decline in f ′ðηÞ and a thickness boundary layer. In addition to this, we highlighted the impact of ε on the nanofluid f ′ ðηÞ in Figure 3.
Because of the acceleration situation ðε > 0Þ, the hybrid nanofluid flow rate is slower, and the thickness of boundary layer of momentum is thinner. For enhanced acceleration, the f ′ ðηÞ decreases. In Figures 4-9, we illustrated the effect of Ec, E, Q, M, ε, and Pr on θðηÞ. Figure 4 shows that the thickness of the thermal boundary layer and the nanofluid temperature are increased as the kinetic energy to enthalpy ratio rises with an enhance in the Ec values. In Figures 5 and 6, the influence of heat absorption or generation on θðηÞ is highlighted numerically. The parameter θðηÞ of the fluid and the thickness of the thermal boundary layer are both increased when ðQ > 0Þ is used.    Advances in Mathematical Physics On the other hand, ðQ < 0Þ outcomes in a lower fluid temperature and a smaller thickness of the thermal boundary layer. Q = 0 denotes the lack of heat generation or absorption. Figure 7 reveals the impact of M on θðηÞ of the nanofluids. We noticed that the transverse magnetic field increases the thickness of the thermal boundary layer. As a result, the temperature of the fluid as well as the thickness of the thermal boundary layer increased. The magnetic field acts as a strong Lorentz force, increasing the temperature of the nanofluid in the boundary region. Figure 8 shows the depicts the impact of E on θðηÞ.
The impact of E on θðηÞ is shown in Figure 8, which showed that the electric field acts as an accelerating force, raising the fluid flow temperature and increasing the thickness of the thermal boundary layer. A thicker, higher quantity temperature distribution within the boundary layer region of the hybrid nanofluid's vicinity is connected with a greater value of an electric field. The Prandtl number is a material property that varies depending on the fluid. From Figure 9, it reveals that as it raises in Pr, θðη Þ declines. Moreover, the higher value of Pr implies smaller thermal diffusivity than the momentum diffusivity. As a result, the energy boundary layer thickness declines. Table 3 represents that C f is raised when the values of M, ε, and E are increased. Table 4 shows that Nu x is increased when the values of M, Q are increased. The Nu x is decreased, when the values of Pr, E, and Ec is increased.

Concluding Remarks
This study investigates the unsteady electrohydrodynamic stagnation-point fluid flow of a hybrid nanofluid over a convective heat stretch or shrink sheet, taking into account the velocity slip effect on heat transferred. The effects of heat generation and absorption, as well as the electric and magnetic fields, are all taken into account. The following are the main observations of the present study: (i) f ′ ðηÞ and θðηÞ raises with an enhance in the value of E (ii) The magnetic factor has an inverse impact for velocity and temperature gradient (iii) A heat source raises the temperature, whereas a heat sink does the opposite (iv) The velocity and temperature are reduced at higher levels of the unsteadiness parameter