Dual Solutions and Stability Analysis of a Hybrid Nanofluid over a Stretching/Shrinking Sheet Executing MHD Flow

Abstract

In this paper, the unsteady magnetohydrodynamic (MHD) flow of hybrid nanofluid (HNF) composed of $Cu-A{l}_2{O}_3$/water in the presence of a thermal radiation effect over the stretching/shrinking sheet is investigated. Using similarity transformation, the governing partial differential equations (PDEs) are transformed into a system of ordinary differential equations (ODEs), which are then solved by using a shooting method. In order to validate the obtained numerical results, the comparison of the results with the published literature is made numerically as well as graphically and is found in good agreements. In addition, the effects of many emerging physical governing parameters on the profiles of velocity, temperature, skin friction coefficient, and heat transfer rate are demonstrated graphically and are elucidated theoretically. Based on the numerical results, dual solutions exist in a specific range of magnetic, suction, and unsteadiness parameters. It was also found that the values of $f^{″}\left(0\right)$ rise in the first solution and reduce in the second solution when the solid volume fraction ${\varphi }_{Cu}$ is increased. Finally, the temporal stability analysis of the solutions is conducted, and it is concluded that only the first solution is stable.

1. Introduction

Researchers are interested in studying the enhancement of heat transfer due to its significant applications in engineering and industries. The heat transfer of convectional fluids such as ethylene glycol, water, and oil can be utilized in various apparatus of engineering, for instance, devices of electrons and heat exchangers. However, these base liquids keep limited thermal conductivity or, in other words, they have low thermal conductivity. To overcome this shortcoming, engineers, mathematicians, and researchers of various fields attempt to improve thermal conductivity of previously mentioned liquids by including a solitary kind of nanosized particles to form a mixture called ‘nanofluid’, which was initially presented by Choi and Eastman [1]. It is easily shown from the previous studies that solid nanoparticles possess the ability increase thermal conductivity and the rate of heat transfer of convectional base fluids. Consequently, numerous analysts and thermal experts have carried out investigations in order to enhance the rate of heat transfer of nanofluid for various perspectives, numerically as well as experimentally. For instance, the examination of the single-phase models of nanofluids was established by Tiwari and Das [2]. This model has been constructed by considering a solid volume fraction of nanoparticles in the base fluid, and, later, the governing equations have been solved numerically by employing the finite volume method. As a result, this model has been widely considered by many scientists, engineers, and mathematicians, such as Benzema et al. [3], Dero et al. [4], Lund et al. [5,6], Dogonchi, et al. [7,8], Amini, et al. [9], Zaib et al. [10], Raza et al. [11], Rasool et al. [12,13], Dinarvand et al. [14], and Roşca, et al. [15] to investigate different types of flow. Furthermore, to keep the demand of the high heat transfer rate from industries and other sectors, researchers have introduced a new kind of the nanofluid by considering the two different types of the solid particles in the single convectional base fluid. This kind of nanofluid is known as a hybrid nanofluid [16]. It is worth mentioning that thermal conductivity of regular base fluid in hybrid nanofluid is higher than in simple nanofluid.

Due to this nice property, studies of different types of hybrid nanofluids have gained attention. Devi and Devi [17] examined the $Cu-A{l}_2{O}_3$/ water hybrid nanofluid. The obtained numerical results were compared with the experimental results of Suresh et al. [18] and were found in excellent agreement. Toghraie et al. [19] further adopted this model for the $ZnO-Ti{O}_2/EG$ hybrid nanofluid. Meanwhile, Moghadassi et al. [20] studied the $A{l}_2{O}_3-\mathrm{Cu}$/ water base hybrid nanofluid numerically and found that “for the hybrid nanofluids, the average Nusselt number increase was 4.73% and 13.46% in compared to $A{l}_2{O}_3$ water and pure water, respectively”. Moreover, this model has been widely employed by many researchers, such as Hayat et al. [21], Saba et al. [22], Acharya et al. [23], Afridi et al. [24], Shafiq et al. [25,26], and Manh et al. [27]. Furthermore, Khashi’ie et al. [28] found dual solutions of magnetohydrodynamic (MHD) flow of a hybrid nanofluid in the presence of Joule heating and noticed that higher values of Eckert number do not affect boundary layer separation. Lund et al. [29] studied hybrid nanofluids by considering copper and alumina as solid particles with water as a base fluid. Dual solutions were noticed in the presence of high suction. Dual solutions of a hybrid nanofluid over a vertical thin needle were investigated by Waini et al. [30]. They performed a stability analysis on dual solutions and discovered that only the first solution is stable. Later, Waini et al. [31] considered an unsteady flow of hybrid nanofluid over a stretching/shrinking surface and found that dual solutions appeared in certain ranges of an unsteadiness parameter.

This paper is an extension of the study carried out by Waini et al. [31] by considering the unsteady magnetohydrodynamic (MHD) flow of the $Cu-A{l}_2{O}_3$/ water hybrid nanofluid in the presence of thermal radiation over the stretching/shrinking parameter. A model of Tiwari and Das [2] is used to deal with governing equations by including two solid nanoparticles, namely copper ($Cu$) and alumina ($A{l}_2{O}_3$), with water as a base fluid. To the best of authors’ knowledge and based on a survey of previous literature, this kind of study has not been investigated.

2. Problem Formulation

Let us consider an unsteady MHD incompressible flow of the $Cu-A{l}_2{O}_3$/ water nanofluid flow on a stretching/shrinking sheet in the presence of thermal radiation effect. Figure 1 shows the physical model and coordinate system of the concerned problem. The velocity of wall mass transfer is $v_w\left(x\right)=-{\left(\frac{{\vartheta }_{f}c}{\left(1-\epsilon t\right)}\right)}^{\frac{1}{2}}f\left(\eta \right)$ where $c$ is a positive constant. It is also assumed that the flow is subjected to a transverse magnetic field of strength $B=\frac{B_0}{{\left(1-\epsilon t\right)}^{1/2}}$ where $B_0$ is the constant applied magnetic field. By including the considered assumptions, the equation of mass, momentum, and energy in the form of a boundary layer are expressed in Tiwari and Das’ model [2], as below:

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

(1)

$$\frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y}=\frac{{\mu }_{hnf}}{{\rho }_{hnf}}\frac{{\partial }^{2}u}{\partial y^2}-\frac{{\sigma }^{*}{B}^{2}u}{{\rho }_{hnf}}$$

(2)

$$\frac{\partial T}{\partial t} + u\frac{\partial T}{\partial x} + v\frac{\partial T}{\partial y}=\left[\frac{k_{hnf}}{{\left(\rho c_p\right)}_{hnf}} + \frac{16{\sigma }_1{T}_{\infty }^3}{3k^{*}{\left(\rho c_p\right)}_{hnf}}\right]\frac{{\partial }^{2}T}{\partial y^2}$$

(3)

subject to the following boundary conditions:

$$\{\begin{array}{c}t<0, u=v=0,T=T_{\infty }\\ t\ge 0,v=v_w, u=\lambda u_w,T=T_w at y=0 \\ u\to 0, T\to T_{\infty } as y\to \infty \end{array}$$

(4)

where $u_w\left(x,t\right)= =\frac{cx}{\left(1-\epsilon t\right)}$ is the velocity of the surface. In the current study, thermophoresis properties of Waini et al. [31] are adopted. These properties of hybrid nanofluid are presented in Table 1 and Table 2.

In order to reduce the governing equations into a system of ordinary differential equations (ODEs), the following similarity transformation variables are employed:

$$\eta ={\left(\frac{c}{{\vartheta }_f\left(1-\epsilon t\right)}\right)}^{\frac{1}{2}}y, u=\frac{cx}{\left(1-\epsilon t\right)}f\prime \left(\eta \right), v=-{\left(\frac{{\vartheta }_{f}c}{\left(1-\epsilon t\right)}\right)}^{\frac{1}{2}}f\left(\eta \right),\theta \left(\eta \right) =\frac{ T-T_{\infty }}{T_w-T_{\infty }}$$

(5)

Substituting Equation (5) in Equations (1)–(3), Equation (1) is definitely fulfilled, and Equations (2)–(3) take the accompanying dimensionless form of ODEs:

$$f‴ + {\xi }_1\left\{ff″-f{\prime }^2-A\left(0.5\eta f″ + f\prime \right)\right\}-{\xi }_{2}M{f}^{\prime }=0$$

(6)

$$\frac{1}{{\xi }_{3}Pr} \left[\left(k_{hnf}/k_f\right) + \frac{4Rd}{3}\right]{\theta }^{″} + f{\theta }^{\prime }-0.5A\eta \theta \prime =0$$

(7)

subject to the following boundary conditions:

$$\{\begin{array}{c}f\left(0\right)=S,f\prime \left(0\right)=\lambda , \theta \left(0\right)=1\\ f\prime \left(\eta \right)\to 0,\theta \left(\eta \right)\to 0 \mathrm{as}\eta \to \infty \end{array}$$

(8)

The reduced quantities are expressed as $A=\frac{\epsilon }{c},M=\frac{{\sigma }^{*}{B}_0^2}{c{\rho }_f},Pr=\frac{{\vartheta }_f}{{\alpha }_f},Rd=\frac{4{\sigma }_1{T}_{\infty }^3}{k^{*}{k}_f}$.

$$\{\begin{array}{c}{\xi }_1={\left(1-{\varphi }_{A{l}_2{O}_3}\right)}^{2.5}{\left(1-{\varphi }_{Cu}\right)}^{2.5}\left\{\left(1-{\varphi }_{Cu}\right)\left[1-{\varphi }_{A{l}_2{O}_3} + {\varphi }_{A{l}_2{O}_3}\left(\raisebox{1ex}{${\rho }_{A{l}_2{O}_3}$}\!\left/ \!\raisebox{-1ex}{${\rho }_f$}\right.\right)\right] + {\varphi }_{Cu}\left(\raisebox{1ex}{${\rho }_{Cu}$}\!\left/ \!\raisebox{-1ex}{${\rho }_f$}\right.\right)\right\}\\ {\xi }_2={\left(1-{\varphi }_{Cu}\right)}^{2.5}{\left(1-{\varphi }_{A{l}_2{O}_3}\right)}^{2.5}\\ {\xi }_3=\left\{\left(1-{\varphi }_{Cu}\right)\left[1-{\varphi }_{A{l}_2{O}_3} + {\varphi }_{A{l}_2{O}_3}\frac{{\left(\rho c_p\right)}_{A{l}_2{O}_3}}{{\left(\rho c_p\right)}_f}\right] + {\varphi }_{Cu}\frac{{\left(\rho c_p\right)}_{Cu}}{{\left(\rho c_p\right)}_f}\right\}\end{array}$$

(9)

The physical quantities of attention are the skin friction coefficient $C_f$ and the local Nusselt number $N{u}_x$, which are expressed as follows:

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

(10)

Applying Equation (5) in Equation (10), leads to:

$$\sqrt{Re}{C}_f=\frac{1}{{\left(1-{\varphi }_{A{l}_2{O}_3}\right)}^{2.5}{\left(1-{\varphi }_{Cu}\right)}^{2.5}}{f}^{″}\left(0\right); \sqrt{\frac{1}{Re}}N{u}_x=-\frac{k_{hnf}}{k_f}\left[1 + \frac{4Rd}{3}\right]{\theta }^{\prime }\left(0\right)$$

(11)

where $Re=\frac{x{u}_w}{{\vartheta }_f}$ is the local Reynold number.

3. Stability Analysis

In order to perform a temporal analysis of the solutions’ stability, introducing a new dimensionless time-dependent similarity transformation variable is required, as recommended by Merkin [32], Dero et al. [33,34], and Lund et al. [35,36,37]. Letting $\tau =\frac{ct}{\left(1-\epsilon t\right)}$ yields the following new similarity transformation variables:

$$\{\begin{array}{c}u=\frac{cx}{\left(1-\epsilon t\right)}\frac{\partial f\left(\eta , \tau \right)}{\partial \eta }, v=-{\left(\frac{{\vartheta }_{f}c}{\left(1-\epsilon t\right)}\right)}^{\frac{1}{2}}f\left(\eta ,\tau \right),\theta \left(\eta ,\tau \right)=\frac{T-T_{\infty }}{T_w-T_{\infty }}\\ \eta ={\left(\frac{c}{{\vartheta }_f\left(1-\epsilon t\right)}\right)}^{\frac{1}{2}}y; \tau =\frac{ct}{\left(1-\gamma t\right)}\end{array}$$

(12)

Substituting Equation (12) in Equations (2) and (3) leads to:

$$\frac{{\partial }^{3}f}{\partial {\eta }^3} + {\xi }_1 \left\{f\frac{{\partial }^{2}f}{\partial {\eta }^2}-{\left(\frac{\partial f}{\partial \eta }\right)}^2-A\left(0.5\eta \frac{{\partial }^{2}f}{\partial {\eta }^2} + \frac{\partial f}{\partial \eta }\right)-\left(1 + A\tau \right)\frac{{\partial }^{2}f}{\partial \tau \partial \eta }\right\}-{\xi }_{2}M\frac{\partial f}{\partial \eta }=0$$

(13)

$$\frac{1}{{\xi }_{3}Pr} \left[\left(k_{hnf}/k_f\right) + \frac{4Rd}{3}\right]\frac{{\partial }^2\theta }{\partial {\eta }^2} + f\frac{\partial \theta }{\partial \eta }-0.5A\eta \frac{\partial \theta }{\partial \eta }-\left(1 + A\tau \right)\frac{\partial \theta }{\partial \tau }=0$$

(14)

subject to the following boundary conditions:

$$\{\begin{array}{c}f\left(0,\tau \right)=S,\frac{\partial f\left(0, \tau \right)}{\partial \eta }=\lambda ,\theta \left(0,\tau \right)=1\\ \frac{\partial f\eta \left(, \tau \right)}{\partial \eta }= \theta \left(\eta ,\tau \right)=0 as \eta \to \infty \end{array}$$

(15)

To check the stability of the steady flow solutions, where $f\left(\eta \right)=f_0\left(\eta \right),\theta \left(\eta \right)={\theta }_0\left(\eta \right),$ and $\varnothing \left(\eta \right)={\varnothing }_0\left(\eta \right)$, the following boundary value problem (8)-(11) must be satisfied:

$$\{\begin{array}{c}f\left(\eta ,\tau \right)=f_0\left(\eta \right) + e^{-\gamma \tau }F\left(\eta ,\tau \right)\\ \theta \left(\eta ,\tau \right)={\theta }_0\left(\eta \right) + e^{-\gamma \tau }G\left(\eta ,\tau \right)\end{array}$$

(16)

where $F\left(\eta \right)$ and $G\left(\eta \right)$ are small relatives of $f_0\left(\eta \right)$ and ${\theta }_0\left(\eta \right),$ respectively, and $\gamma$ is an unknown eigenvalue parameter, which need to be determined. Substituting Equation (16) in Equations (13)–(14) by considering $\tau =0$, gives the following linearized eigenvalue problem:

$$F_0^{‴} + {\xi }_1\left\{f_0{F}_0^{″}-2f_0^{\prime }{F}_0^{\prime } + F_0{f}_0^{″}-A\left(0.5\eta F_0^{″} + F_0^{\prime }\right) + F_0^{\prime }\right\}-{\xi }_{2}M{F}_0^{\prime }=0$$

(17)

$$\frac{1}{{\xi }_{3}Pr} \left[\left(k_{hnf}/k_f\right) + \frac{4Rd}{3}\right]G_0^{″} + f_0{G}_0^{\prime } + F_0{\theta }_0^{\prime }-0.5\eta A{G}_0^{\prime } + G_0=0$$

(18)

subject to the following boundary conditions:

$$\{\begin{array}{c}{F}_0\left(0\right)=0, F_0^{\prime }\left(0\right)=0, G_0\left(0\right)=0\\ F_0^{\prime }\left(\eta \right)\to 0, G_0\left(\eta \right)\to 0 as \eta \to \infty \end{array}$$

(19)

According to Haris et al. [38] and Khashi’ie et al. [39], one boundary condition has to be relaxed in order to obtain the value(s) of the smallest eigenvalue. It should be noted that this relaxation in boundary condition does not affect the results [40]. In this particular problem, $F_0^{\prime }\left(\eta \right)\to 0 \mathrm{as} \eta \to \infty$ is relaxed into $F_0^{″}\left(0\right)=1$.

4. Result and Discussion

In this section, Equations (6) and (7) with boundary conditions (8) have been solved numerically by employing the shooting method in Maple software. The shooting method has been employed by many researchers [41,42,43,44]. As suggested by Devi and Deci [17], the solid volume fraction ${\varphi }_{A{l}_2{O}_3}=0.1$ is kept as a constant in many graphs. Furthermore, the solid volume fraction ${\varphi }_{Cu}$ is added in order to make the $Cu-A{l}_2{O}_3$/ water hybrid nanofluid. As recommended by Dero et al. [4] and Iqbal et al. [45], the Prandtl number $Pr=6.2$ is kept as constant and employed to produce the results of the current study. Table 3 shows the comparison of the numerical values of $f^{″}\left(0\right)$ and $-{\theta }^{\prime }\left(0\right)$ with the published results of waini et al. [31] for Cu–water nanofluid. It is worth mentioning that Waini et al. [31] used the 3-stage Labatto IIIa formula in a BVP4C solver to solve the resultant similarity equations. It is found that the current results show outstanding agreement with the results of Waini et al. [31]. Therefore, it can be concluded that the accuracy of the present method is fine, and it can be used in this study. Figure 2 is demonstrated for the comparison purpose in order to check the accuracy of the used method. It is observed that the critical values of the current study have a good agreement with the critical values of published paper (allude Figure 6 in Waini et al. [31]). After comparison, this method can be used confidently in this problem.

Figure 3 and Figure 4 show the variation of $f^{″}\left(0\right)$ and $-{\theta }^{\prime }\left(0\right)$ against $A$ for ${\varphi }_{Cu}=0.001, 0.01$, and $0.1$. By keeping $A$ fixed, an expansion of ${\varphi }_{Cu}$ creates a reduction in the heat transfer rate, which implies that the heat transfer coefficient is decreased on the sheet, while an opposite nature is observed for the $f^{″}\left(0\right)$ in the second solution. On the other hand, the coefficient of skin friction decreases in the first solution. Moreover, it is noticed that multiple solutions exist for $A >A_c$, whereas no solution exists for $A<A_c$. It is worth mentioning here that $A_c$ is the critical value where multiple solutions exist. Variations of $f^{″}\left(0\right)$ and $-{\theta }^{\prime }\left(0\right)$ along $M$ for numerous values of ${\varphi }_{Cu}$ were drawn in Figure 5 and Figure 6, respectively. It is observed from them that when the solid volume fraction ${\varphi }_{Cu}$ increases the values of the critical point of $M$ are also increased. Furthermore, $M_{c1}=0.4492, M_{c2}=0.3912$ are the respective critical values of $M$ for ${\varphi }_{Cu}=0.001$ and $0.01$. It is examined that the critical values of $M$ get smaller for higher values of the solid volume fraction ${\varphi }_{Cu}$, which is the physical reason of the extension of the thickness of boundary layer separation. Moreover, it can be seen that the values of $f^{″}\left(0\right)$ rises in the first solution and reduces in the second solution when the solid volume fraction ${\varphi }_{Cu}$ is increased. In the meantime, the rate of heat transfer decreases for the unstable solution and increases for the stable solution with the increasing of ${\varphi }_{Cu}$.

In detail, Figure 7 and Figure 8 show the variation of $f^{″}\left(0\right)$ and $-{\theta }^{\prime }\left(0\right)$ with $A$ for the numerous values of the mass suction parameter $S$. It is observed that the domain of the solutions increases along the critical value of $A,$ moving to one side when the mass suction parameter $S$ is decreased. In light of our calculations, $A_{c1}=-10.1708, A_{c2}=-7.2207,$ and $A_{c3}=-5.2042$ are the respective critical values of $A$ for $S=1.8, 1.75$ and $1.7$. At the same time, it is examined that the smaller critical value is for $S=1.8$ as compared to $S=1.75$ and $S=1.7$. Henceforth, boundary layer separation is delayed for the higher values of the mass suction $S$. In addition, the coefficient of skin friction increases in the first solution with the developing of $S,$ whereas it declines in the second solution. The expanding of $S$ prompts to rise the effect of the heat transfer rate $-{\theta }^{\prime }\left(0\right)$ in both solutions. Figure 9 and Figure 10 demonstrate the variation of $f^{″}\left(0\right)$ and $-{\theta }^{\prime }\left(0\right)$ with $S$ for the numerous values of ${\varphi }_{Cu}$. It can be found that the increments in ${\varphi }_{Cu}$ lead to smaller critical values of $S_c$. According to our computation, $S_{c1}=1.7886, S_{c2}=1.78234,$ and $S_{c3}=1.7496$ are the respective critical values of $S$ for ${\varphi }_{Cu}=0.001, 0.01,$ and $0.1$. Furthermore, the heat transfer rate rises in both solutions for the higher values of the mass suction $S.$ This is due to fact that mass suction $S$ helps the molecules of the hybrid nanofluid to transfer the heat effectively. On the other hand, $f^{″}\left(0\right)$ increases (decreases) in the first (second) solution when ${\varphi }_{Cu}$ is enhanced.

Figure 11 and Figure 12 illustrate the profiles of velocity $f^{\prime }\left(\eta \right)$ and the temperature $\theta \left(\eta \right)$ for S. It was demonstrated that the rise in $S$ decreases the velocity of the fluid in the first solution but it increases the velocity of the fluid in the second solution. On the other hand, it is perceived that the temperature of fluid decreases with the expansion of the mass suction $S$. The effect of the thermal radiation parameter on $\theta \left(\eta \right)$ is shown in Figure 13. It can be noticed that the thickness of the thermal boundary layer, as well as temperature of the fluid, increases with the increasing values of $Rd.$

Finally, the linearized eigenvalue Equations (17) and (18) have been solved using a 3-stage Labatto IIIa formula in a BVP4C solver in MATLAB 2017b. The results of the stability analysis of the solutions are demonstrated graphically in Figure 14. It can be easily observed from the figure that the initial decay of disturbance is denoted by the positive value of $\gamma$ and a stable flow, while the initial growth of disturbance is indicated by the negative values of $\gamma$ and an unstable flow. Additionally, it is noticed that $\gamma$ tends to zero at the critical values of $A$ from both solutions.

Table 1. Thermophysical properties of the hybrid nanofluid [46].

Properties	Hybrid Nanofluid
Dynamic viscosity	${\mu }_{hnf}=\frac{{\mu }_f}{{\left(1-{\varphi }_{A{l}_2{O}_3}\right)}^{2.5}{\left(1-{\varphi }_{Cu}\right)}^{2.5}}$
Density	${\rho }_{hnf}=\left(1-{\varphi }_{Cu}\right)\left[\left(1-{\varphi }_{A{l}_2{O}_3}\right){\rho }_f + {\varphi }_{A{l}_2{O}_3}{\rho }_{A{l}_2{O}_3}\right] + {\varphi }_{Cu}{\rho }_{Cu}$
Thermal conductivity	$k_{hnf}=\frac{k_{Cu} + 2k_{nf}-2{\varphi }_{Cu}\left(k_{nf}-k_{Cu}\right)}{k_{Cu} + 2k_{nf} + {\varphi }_{Cu}\left(k_{nf}-k_{Cu}\right)}\times \left(k_{nf}\right)$ where $k_{nf}=\frac{k_{A{l}_2{O}_3} + 2k_f-2{\varphi }_{A{l}_2{O}_3}\left(k_f-k_{A{l}_2{O}_3}\right)}{k_{A{l}_2{O}_3} + 2k_f + {\varphi }_{A{l}_2{O}_3}\left(k_f-k_{A{l}_2{O}_3}\right)}\times \left(k_f\right)$
Heat capacity	${\left(\rho c_p\right)}_{hnf}=\left(1-{\varphi }_{Cu}\right)\left[\left(1-{\varphi }_{A{l}_2{O}_3}\right){\left(\rho c_p\right)}_f + {\varphi }_{A{l}_2{O}_3}{\left(\rho c_p\right)}_{A{l}_2{O}_3}\right] + {\varphi }_{Cu}{\left(\rho c_p\right)}_{Cu}$

Table 1. Thermophysical properties of the hybrid nanofluid [46].

Properties	Hybrid Nanofluid
Dynamic viscosity	${\mu }_{hnf}=\frac{{\mu }_f}{{\left(1-{\varphi }_{A{l}_2{O}_3}\right)}^{2.5}{\left(1-{\varphi }_{Cu}\right)}^{2.5}}$
Density	${\rho }_{hnf}=\left(1-{\varphi }_{Cu}\right)\left[\left(1-{\varphi }_{A{l}_2{O}_3}\right){\rho }_f + {\varphi }_{A{l}_2{O}_3}{\rho }_{A{l}_2{O}_3}\right] + {\varphi }_{Cu}{\rho }_{Cu}$
Thermal conductivity	$k_{hnf}=\frac{k_{Cu} + 2k_{nf}-2{\varphi }_{Cu}\left(k_{nf}-k_{Cu}\right)}{k_{Cu} + 2k_{nf} + {\varphi }_{Cu}\left(k_{nf}-k_{Cu}\right)}\times \left(k_{nf}\right)$ where $k_{nf}=\frac{k_{A{l}_2{O}_3} + 2k_f-2{\varphi }_{A{l}_2{O}_3}\left(k_f-k_{A{l}_2{O}_3}\right)}{k_{A{l}_2{O}_3} + 2k_f + {\varphi }_{A{l}_2{O}_3}\left(k_f-k_{A{l}_2{O}_3}\right)}\times \left(k_f\right)$
Heat capacity	${\left(\rho c_p\right)}_{hnf}=\left(1-{\varphi }_{Cu}\right)\left[\left(1-{\varphi }_{A{l}_2{O}_3}\right){\left(\rho c_p\right)}_f + {\varphi }_{A{l}_2{O}_3}{\left(\rho c_p\right)}_{A{l}_2{O}_3}\right] + {\varphi }_{Cu}{\left(\rho c_p\right)}_{Cu}$

Table 2. The thermo physical properties of the base fluid (water) and the nanoparticles [47,48].

Fluids	$\mathit{\rho }$	${\mathit{c}}_{\mathit{p}}(\mathbf{J}/\mathbf{kg}\mathbf{K})$	k (W/m K)
Alumina ($A{l}_2{O}_3$)	3970	765	40
Copper (Cu)	8933	385	400
Water ($H_{2}O$)	997.1	4179	0.613

Table 2. The thermo physical properties of the base fluid (water) and the nanoparticles [47,48].

Fluids	$\mathit{\rho }$	${\mathit{c}}_{\mathit{p}}(\mathbf{J}/\mathbf{kg}\mathbf{K})$	k (W/m K)
Alumina ($A{l}_2{O}_3$)	3970	765	40
Copper (Cu)	8933	385	400
Water ($H_{2}O$)	997.1	4179	0.613

Table 3. Values of $f^{″}\left(0\right)$ and ${\theta }^{\prime }\left(0\right)$ for the Cu–water nanofluid (${\varphi }_{Cu}=0.2$) with various values of $A$ when $M=Rd=0,S=2.1$, $Pr=6.2,$ and $\lambda =-1$.

$\mathit{A}$	${\mathit{\varphi }}_{\mathit{A}{\mathit{l}}_2{\mathit{O}}_3}$	${\mathit{f}}^{″}\left(0\right)$				$-{\mathit{\theta }}^{\prime }\left(0\right)$
		Waini	[31]	Present	Results	Waini	[31]	Present	Results
		1st Soln	2nd Soln	1st Soln	2nd Soln	1st Soln	2nd Soln	1st Soln	2nd Soln
−1	0	2.194247	−1.491281	2.19424658	−1.49128119	7.073680	6.884548	7.0736798	6.88454793
−1	0.1	--	--	1.60888878	−0.69818494	--	--	5.2238150	5.0737595
−3	0	1.521197	−4.144746	1.52119229	−4.14474572	7.497151	7.296176	7.4971508	7.2961762
−3	0.1	--	--	0.83697819	−2.53135605	--	--	5.6316794	5.4925121
−5	0	0.844435	−6.431507	0.84443506	−6.43150738	7.858446	7.657801	7.8584457	7.6578012
−5	0.01	--	--	0.77597819	−6.20435575	--	--	7.6375695	7.4426329
−9	0	−0.517287	−10.58983	−0.51728551	−10.5898303	8.473316	8.277676	8.4733163	8.27767616
−9	0.01	--	--	−0.60021181	−10.2417356	--	--	8.24709639	8.05819003

Table 3. Values of $f^{″}\left(0\right)$ and ${\theta }^{\prime }\left(0\right)$ for the Cu–water nanofluid (${\varphi }_{Cu}=0.2$) with various values of $A$ when $M=Rd=0,S=2.1$, $Pr=6.2,$ and $\lambda =-1$.

$\mathit{A}$	${\mathit{\varphi }}_{\mathit{A}{\mathit{l}}_2{\mathit{O}}_3}$	${\mathit{f}}^{″}\left(0\right)$				$-{\mathit{\theta }}^{\prime }\left(0\right)$
		Waini	[31]	Present	Results	Waini	[31]	Present	Results
		1st Soln	2nd Soln	1st Soln	2nd Soln	1st Soln	2nd Soln	1st Soln	2nd Soln
−1	0	2.194247	−1.491281	2.19424658	−1.49128119	7.073680	6.884548	7.0736798	6.88454793
−1	0.1	--	--	1.60888878	−0.69818494	--	--	5.2238150	5.0737595
−3	0	1.521197	−4.144746	1.52119229	−4.14474572	7.497151	7.296176	7.4971508	7.2961762
−3	0.1	--	--	0.83697819	−2.53135605	--	--	5.6316794	5.4925121
−5	0	0.844435	−6.431507	0.84443506	−6.43150738	7.858446	7.657801	7.8584457	7.6578012
−5	0.01	--	--	0.77597819	−6.20435575	--	--	7.6375695	7.4426329
−9	0	−0.517287	−10.58983	−0.51728551	−10.5898303	8.473316	8.277676	8.4733163	8.27767616
−9	0.01	--	--	−0.60021181	−10.2417356	--	--	8.24709639	8.05819003

5. Conclusions

The problem of the unsteady MHD flow of the $Cu-A{l}_2{O}_3$/ water hybrid nanofluid in the presence of the thermal radiation effect over the stretching/shrinking sheet is examined. Two methods are adopted to carry out the numerical computations of the current problem, a shooting method is used to get the dual solutions of ODEs (6–8) in MAPLE software 2018, and the computation of the stability analysis is carried out by employing a 3-stage Labatto IIIa formula in a BVP4C solver in MATLAB 2017b. The impacts of ${\varphi }_{Cu}$ unsteadiness parameter S and suction parameter S on the $f″\left(0\right), -\theta \prime \left(0\right), f\prime \left(\eta \right)$, and $\theta \left(\eta \right)$ were exhibited graphically and examined. The results show that the development (reduction) of $f″\left(0\right)$ and the decline (reduction) of $-\theta \prime \left(0\right)$ past a shrinking sheet is found with the expansion of ${\varphi }_{Cu}$ in the first (second) solution. The temperature of fluid in both solutions increase (decrease) for the higher values of $Rd$ and $S$. The fluid velocity decreases when the effect of suction parameter $S$ enhanced in the first solution, whereas it increases in the second solution. Furthermore, it is found that the skin friction coefficient increases with an increase in solid volume friction of ${\varphi }_{Cu}$ and suction parameter $S$ in the first solution, while it decreases in the second solution. On the other hand, an improvement in the heat transfer rate is noticed when suction parameter $S$ is increased for both solutions, whereas the totally opposite trend is noticed for the case of solid volume friction of ${\varphi }_{Cu}$. For the certain ranges of the suction parameter $S$, unsteadiness parameter $A$, and magnetic parameter $M$, dual solutions are noticed. Furthermore, the results of the temporal stability analysis confirm that the first solution is stable, whereas the second solution is unstable. In future, this work can be continued to assess the impact of viscous dissipation, joule heating, boundary slip conditions, and convective boundary conditions. Other than these impacts, it can be extended for three-dimensional boundary layer flows by conserving various possible effects.

Abbreviation

	Nomenclature
$T_{ 0}$	a constant	${\varphi }_{A{l}_2{O}_3}$	nanoparticle volume fraction of the iron oxide.
$T_{\infty }$	ambient temperature	M	magnetic parameter
$\prime$	differentiation with respect to $\eta$	$K_1$	Porous parameter
$Rd$	Thermal radiation	c	constant
${\rho }_{hnf}$	effective density of hybrid nanofluid	$Pr$	Prandtl number
${\rho }_{nf}$	effective density of nanofluid	$C_f$	skin friction coefficient
${\mu }_{hnf}$	effective dynamic viscosity of hybrid nanofluid	${\gamma }_1$	smallest eigen value
${\mu }_{nf}$	effective dynamic viscosity of nanofluid	$\tau$	Stability transformed variable
${\sigma }^{*}$	electrical conductivity	$v_w$	suction/injection velocity
$f$	fluid fraction	T	Temperature
M	Hartmann/magnetic number	$k_{hnf}$	thermal conductivity of the hybrid nanofluid
${\left(\rho c_p\right)}_{hnf}$	heat capacitance of the hybrid nanofluid	$k_{nf}$	thermal conductivity of the nanofluid
${\left(\rho c_p\right)}_{nf}$	heat capacitance of the nanofluid	t	time
hnf	Hybrid nanofluid	$\eta$	transformed variable
$N_u$	local Nusselt number	$A$	Unsteadiness parameter
$Re$	local Reynolds number	$T_{ w}$	variable temperature at the sheet
B	Magnetic field	u, v	velocity components
$nf$	nanofluid fraction	$\lambda$	shrinking/stretching parameter
$k^{*}$	mean absorption coefficient	${\sigma }_1$	Stefan–Boltzmann constant
${\varphi }_{Cu}$	nanoparticle volume fraction of the copper	$S$	$S<0$ for suction parameter and $S>0$ for blowing parameter