Hybrid Nanofluid Flow over a Permeable Non-Isothermal Shrinking Surface

Abstract

In this paper, we examine the influence of hybrid nanoparticles on flow and heat transfer over a permeable non-isothermal shrinking surface and we also consider the radiation and the magnetohydrodynamic (MHD) effects. A hybrid nanofluid consists of copper (Cu) and alumina (Al₂O₃) nanoparticles which are added into water to form Cu-Al₂O₃/water. The similarity equations are obtained using a similarity transformation and numerical results are obtained via bvp4c in MATLAB. The results show that dual solutions are dependent on the suction strength of the shrinking surface; in addition, the heat transfer rate is intensified with an increase in the magnetic parameter and the hybrid nanoparticles volume fractions for higher values of the radiation parameter. Furthermore, the heat transfer rate is higher for isothermal surfaces as compared with non-isothermal surfaces. Further analysis proves that the first solution is physically reliable and stable.

1. Introduction

In the history of fluid mechanics, flow development over stretching and shrinking surfaces was first described by Crane [1] and Wang [2], respectively. Meanwhile, Miklavčič and Wang [3] reported the existence of non-unique solutions for flow over a shrinking sheet. Since then, many studies have considered the effect of several physical parameters such as magnetohydrodynamic (MHD) and radiation on stretching and shrinking surfaces [4,5,6,7,8,9,10,11,12]. The effect of the MHD parameter is an important factor in many industrial and engineering applications, for example, MHD power generators, metallurgical process, crystal growth, metal casting, and cooling of nuclear reactors [13]. Thermal radiation is also important in designing innovative energy conversion systems operational at high temperatures [14].

In general, most previous studies have considered isothermal surface conditions; however, heating or cooling can occur under non-isothermal conditions for many practical applications such as in microelectromechanical (MEM) condensation applications, a thin-film solar energy collector device, the cooling of metallic plate in a cooling bath, metal spinning, paper production, and aerodynamic extrusion of plastic sheets [15,16]. In this respect, Soundalgekar and Ramana Murty [17], and Grubka and Bobba [18] considered flow over moving and stretching surfaces under non-isothermal conditions, respectively. This type of heating condition also has been reported by several researchers [19,20,21,22].

In 1995, Choi and Eastman [23] introduced nanofluids, which are a mixture of a base fluid and a single type of nanoparticle, to enhance thermal conductivity. Various studies on such fluids have been conducted [24,25,26,27,28,29]. Recently, some studies have found that advanced nanofluid consists of another type of nanoparticle that is mixed in with the regular nanofluid and improves its thermal properties, namely a ”hybrid nanofluid”. Prior experimental studies using hybrid nanoparticles have been conducted by several researchers [30,31,32] and numerical studies on the flow of hybrid nanofluids were studied by Takabi and Salehi [33]. Moreover, dual solutions of hybrid nanofluid flow were examined by Waini et al. [34,35,36,37,38,39]. Other physical aspects have been considered by several authors [40,41,42,43,44,45,46,47,48,49] and review papers are also available [50,51,52,53,54,55].

In this study, we aim at investigating the effects of Cu-Al₂O₃ hybrid nanoparticles on the radiative MHD flow over a permeable non-isothermal shrinking surface. The simultaneous effects of radiation and the hybrid nanoparticles are examined and the influence of magnetic field and variation of the temperature index is also considered. To the best of our knowledge, based on the above studies, the flow of hybrid nanofluids over non-isothermal shrinkage surfaces is not yet available in the literature, and therefore the results of this study are new. Most importantly, in this study, two solutions are discovered and the long-term stability of these solutions is investigated.

2. Mathematical Formulation

Let us consider the two-dimensional, laminar, and incompressible flow of a hybrid nanofluid over a permeable non-isothermal shrinking surface, as shown in Figure 1. The surface velocity is represented by $u_w\left(x\right)=ax$ where $a>0$ is constant and $v_0$ is the constant mass flux velocity. The flow is subjected to the combined effect of a transverse magnetic field of strength $B_0$ and the radiative heat flux $q_r$, which is assumed to be applied normal to the surface in the positive $y$-direction. Accordingly, the hybrid nanofluid equations (see Grubka and Bobba [18], Rashid et al. [20], Waini et al. [34]) are:

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

(1)

$$u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\frac{{\mu }_{hnf}}{{\rho }_{hnf}}\frac{{\partial }^{2}u}{\partial y^2}-\frac{{\sigma }_{hnf}}{{\rho }_{hnf}}{B}_0^{2}u$$

(2)

$$u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\frac{k_{hnf}}{{\left(\rho C_p\right)}_{hnf}}\frac{{\partial }^{2}T}{\partial y^2}-\frac{1}{{\left(\rho C_p\right)}_{hnf}}\frac{\partial q_r}{\partial y}$$

(3)

subject to:

$$\begin{gathered} v=v_0,\hspace{1em}u=\lambda u_w\left(x\right),\hspace{1em}T=T_w\left(x\right)\hspace{1em}\mathrm{at}\hspace{1em}y=0 \\ u\to 0,\hspace{1em}T\to T_{\infty }\hspace{1em}\mathrm{as}\hspace{1em}y\to \infty \end{gathered}$$

(4)

where $u$ and $v$ represent the velocity components along the $x$- and $y$-axes and the temperature of the hybrid nanofluid is given by $T$.

The expression of the radiative heat flux is as follows [9]:

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

(5)

where ${\sigma }^{\ast }$ and $k^{\ast }$ denote the Stefan–Boltzmann constant and the mean absorption coefficient, respectively. Following Rosseland [56], after employing a Taylor series, one gets $T^4\cong 4 T_{\infty }^3 T-3T_{\infty }^4$. Then, Equation (3) becomes the following:

$$u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}= \frac{1}{{(\rho C_p)}_{hnf}}\left[k_{hnf}+\frac{16{\sigma }^{\ast }{T}_{\infty }^3}{3k^{\ast }}\right]\frac{{\partial }^{2}T}{\partial y^2}$$

(6)

Furthermore, the thermophysical properties can be referred to in

Table 1. Thermophysical properties of nanoparticles and water.

Properties	Nanoparticles		Base Fluid
	Cu	Al2O3	Water
$\rho \left(\mathrm{kg}/{\mathrm{m}}^3\right)$	8933	3970	997.1
$C_p \left(\mathrm{J}/\mathrm{kgK}\right)$	385	765	4179
$k \left(\mathrm{W}/\mathrm{mK}\right)$	400	40	0.613
$\sigma \left(\mathrm{S}/\mathrm{m}\right)$	5.96 × 107	3.69 × 107	0.05
Prandtl number, $\mathrm{Pr}$			6.2

and

Table 2. Thermophysical properties of nanofluid and hybrid nanofluid.

Thermophysical Properties	Correlations
Dynamic viscosity	${\mu }_{hnf}=\frac{{\mu }_f}{{\left(1-{\phi }_{hnf}\right)}^{2.5}}$
Density	${\rho }_{hnf}=\left(1-{\phi }_{hnf}\right){\rho }_f+{\phi }_1{\rho }_{n1}+{\phi }_2{\rho }_{n2}$
Heat capacity	${(\rho C_p)}_{hnf}=\left(1-{\phi }_{hnf}\right){(\rho C_p)}_f+{\phi }_1{(\rho C_p)}_{n1}+{\phi }_2{(\rho C_p)}_{n2}$
Thermal conductivity	$\frac{k_{hnf}}{k_f}=\frac{\frac{{\phi }_1{k}_{n1}+{\phi }_2{k}_{n2}}{{\phi }_{hnf}}+2k_f+2\left({\phi }_1{k}_{n1}+{\phi }_2{k}_{n2}\right)-2{\phi }_{hnf}{k}_f}{\frac{{\phi }_1{k}_{n1}+{\phi }_2{k}_{n2}}{{\phi }_{hnf}}+2k_f-\left({\phi }_1{k}_{n1}+{\phi }_2{k}_{n2}\right)+{\phi }_{hnf}{k}_f}$
Electrical conductivity	$\frac{{\sigma }_{hnf}}{{\sigma }_f}=\frac{\frac{{\phi }_1{\sigma }_{n1}+{\phi }_2{\sigma }_{n2}}{{\phi }_{hnf}}+2{\sigma }_f+2\left({\phi }_1{\sigma }_{n1}+{\phi }_2{\sigma }_{n2}\right)-2{\phi }_{hnf}{\sigma }_f}{\frac{{\phi }_1{\sigma }_{n1}+{\phi }_2{\sigma }_{n2}}{{\phi }_{hnf}}+2{\sigma }_f-\left({\phi }_1{\sigma }_{n1}+{\phi }_2{\sigma }_{n2}\right)+{\phi }_{hnf}{\sigma }_f}$

. Data from these tables are adapted from previous studies [26,33,34,57]. Note that ${\phi }_1$ (Al₂O₃) and ${\phi }_2$ (Cu) are nanoparticles volume fractions, and the subscripts $n1$ and $n2$ correspond to their solid components, while the subscripts $hnf$ and $f$ represent the hybrid nanofluid and the base fluid, respectively.

For the similarity solution of Equations (1), (2), and (6), the surface temperature is taken as follows (see Grubka and Bobba [18], Rashid et al. [20]):

$$T_w\left(x\right)=T_{\infty }+T_0{\left(x/L\right)}^{m }$$

(7)

where $L$ is a characteristic length of the sheet and $T_0$ is a temperature characteristic. The ambient temperature $T_{\infty }$ is assumed to be constant and $m$ represents the temperature power-law index, with $m=0$ indicating an isothermal surface and $m>0$ indicating a non-isothermal surface.

Now, using the following similarity transformation:

$$\psi =\sqrt{a{\nu }_f}xf\left(\eta \right),\hspace{1em}\theta \left(\eta \right)=\frac{T- T_{\infty }}{T_w- T_{\infty }},\hspace{1em}\eta =y\sqrt{\frac{a}{{\nu }_f}}$$

(8)

with the stream function $\psi$. Here, $u=\partial \psi /\partial y$ and $v=- \partial \psi /\partial x$, then:

$$u=ax{f}^{\prime }\left(\eta \right),\hspace{1em}v=-\sqrt{a{\nu }_f}f\left(\eta \right)$$

(9)

From Equation (9), by setting $\eta =0$, one obtains:

$$v_0=-\sqrt{a{\nu }_f} S$$

(10)

where $f\left(0\right)=S$ is the constant mass flux parameter which determines the permeability of the surface. Here, $S<0$ and $S>0$ are for injection and suction cases, respectively, while $S=0$ represents an impermeable case.

On using Equations (8) and (9), Equation (1) is identically fulfilled. Now, Equations (2) and (6) are reduced to:

$$\frac{{\mu }_{hnf}/{\mu }_f}{{\rho }_{hnf}/{\rho }_f}{f}^{‴}+f{f}^{″}-{f^{\prime }}^2-\frac{{\sigma }_{hnf}/{\sigma }_f}{{\rho }_{hnf}/{\rho }_f}M{f}^{\prime }=0$$

(11)

$$\frac{1}{\mathrm{Pr}}\frac{1}{{\left(\rho C_p\right)}_{hnf}/{\left(\rho C_p\right)}_f}\left(\frac{k_{hnf}}{k_f}+\frac{4}{3}R\right){\theta }^{″}+f{\theta }^{\prime }-m{f}^{\prime }\theta =0$$

(12)

subject to the following:

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

(13)

where primes denote differentiation with respect to $\eta$. Note that $\lambda <0$ and $\lambda >0$ represent the shrinking and stretching surfaces, while $\lambda =0$ is a rigid surface. In addition, $\mathrm{Pr}$ is the Prandtl number, while $R$ and $M$ are the radiation and the magnetic parameters, respectively, which are defined as follows:

$$\mathrm{Pr}=\frac{{\left(\mu C_p\right)}_f}{k_f},\hspace{1em}R=\frac{4 {\sigma }^{\ast }{T}_{\infty }^3}{k^{\ast }{k}_f},\hspace{1em}M=\frac{{\sigma }_f}{{\rho }_{f}a}{B}_0^2$$

(14)

The coefficient of the skin friction $C_f$ and the local Nusselt number $N{u}_x$ are given as follows [9]:

$$C_f=\frac{{\mu }_{hnf}}{{\rho }_f u_w^2}{\left(\frac{\partial u}{\partial y}\right)}_{y=0},\hspace{1em}N{u}_x=\frac{x}{k_f\left(T_w- T_{\infty }\right)}\left(- k_{hnf} {\left(\frac{\partial T}{\partial y}\right)}_{y=0}+{(q_r)}_{y=0}\right)$$

(15)

Using Equations (8) and (15), one obtains:

$${\mathrm{Re}}_x^{1/2}{C}_f=\frac{{\mu }_{hnf}}{{\mu }_f}{f}^{″}\left(0\right),\hspace{1em}{\mathrm{Re}}_x^{-1/2}N{u}_x=-\left(\frac{k_{hnf}}{k_f}+\frac{4}{3}R\right)\theta \prime \left(0\right)$$

(16)

where ${\mathrm{Re}}_x=u_w\left(x\right)x/{\nu }_f$ defines the local Reynolds number.

It should be noted that for ${\phi }_{hnf}=S=M=R=0$, Equations (11) and (12) reduce to Equations (5) and (6) from Grubka and Bobba [18] when $\lambda =1$.

3. Stability Analysis

The temporal stability of the dual solutions as time evolves is studied. This analysis was first introduced by Merkin [58], and then followed by Weidman et al. [59]. Firstly, consider the new variables as follows:

$$\psi =\sqrt{a{\nu }_f}xf\left(\eta \right),\hspace{1em}\theta \left(\eta \right)=\frac{T- T_{\infty }}{T_w- T_{\infty }},\hspace{1em}\eta =y\sqrt{\frac{a}{{\nu }_f}},\tau =at$$

(17)

Now, the unsteady form of Equations (2) and (3) are employed, while Equation (1) remains unchanged. On using (17), one obtains:

$$\frac{{\mu }_{hnf}/{\mu }_f}{{\rho }_{hnf}/{\rho }_f}\frac{{\partial }^{3}f}{\partial {\eta }^3}+f\frac{{\partial }^{2}f}{\partial {\eta }^2}-{\left(\frac{\partial f}{\partial \eta }\right)}^2-\frac{{\sigma }_{hnf}/{\sigma }_f}{{\rho }_{hnf}/{\rho }_f}M\frac{\partial f}{\partial \eta }-\frac{{\partial }^{2}f}{\partial \eta \partial \tau }=0$$

(18)

$$\frac{1}{\mathrm{Pr}}\frac{1}{{\left(\rho C_p\right)}_{hnf}/{\left(\rho C_p\right)}_f}\left(\frac{k_{hnf}}{k_f}+\frac{4}{3}R\right)\frac{{\partial }^2\theta }{\partial {\eta }^2}+f\frac{\partial \theta }{\partial \eta }-m\frac{\partial f}{\partial \eta }\theta -\frac{\partial \theta }{\partial \tau }=0$$

(19)

subject to the following:

$$\begin{gathered} f\left(0,\tau \right)=S,\hspace{1em}\frac{\partial f}{\partial \eta }\left(0,\tau \right)=\lambda ,\hspace{1em}\theta \left(0,\tau \right)=1, \\ \frac{\partial f}{\partial \eta }\left(\infty ,\tau \right)=0,\hspace{1em}\theta \left(\infty ,\tau \right)=0 \end{gathered}$$

(20)

Then, consider the following perturbation functions [59]:

$$f\left(\eta ,\tau \right)=f_0\left(\eta \right)+e^{- \gamma \tau }F\left(\eta \right),\hspace{1em}\theta \left(\eta ,\tau \right)={\theta }_0\left(\eta \right)+e^{- \gamma \tau }G\left(\eta \right)$$

(21)

Here, Equation (21) is used to apply a small disturbance on the steady solutions $f=f_0\left(\eta \right)$ and $\theta ={\theta }_0\left(\eta \right)$ of Equations (11)–(13). The functions $F\left(\eta \right)$ and $G\left(\eta \right)$ in Equation (19) are relatively small as compared with $f_0\left(\eta \right)$ and ${\theta }_0\left(\eta \right)$. The sign (positive or negative) of the eigenvalue $\gamma$ determines the stability of the solutions. By employing Equation (21), Equations (18) to (20) become:

$$\frac{{\mu }_{hnf}/{\mu }_f}{{\rho }_{hnf}/{\rho }_f}{F}^{‴}+f_0{F}^{″}+f_0^{″}F-2f_0^{\prime }{F}^{\prime }-\frac{{\sigma }_{hnf}/{\sigma }_f}{{\rho }_{hnf}/{\rho }_f}M{F}^{\prime }+\gamma F^{\prime }= 0$$

(22)

$$\frac{1}{\mathrm{Pr}}\frac{1}{{\left(\rho C_p\right)}_{hnf}/{\left(\rho C_p\right)}_f}\left(\frac{k_{hnf}}{k_f}+\frac{4}{3}R\right)G^{″}+f_0{G}^{\prime }+{\theta }_0^{\prime }F-m\left(f_0^{\prime }G+{\theta }_0{F}^{\prime }\right)+\gamma G=0$$

(23)

subject to the following:

$$\begin{gathered} F\left(0\right)=0,\hspace{1em}{F}^{\prime }\left(0\right)= 0,\hspace{1em}G\left(0\right)=0, \\ {F}^{\prime }\left(\infty \right)=0,\hspace{1em}G\left(\infty \right)=0 \end{gathered}$$

(24)

Without loss of generality, we set $F^{″}\left(0\right)=1$ [60] to get the eigenvalues $\gamma$ in Equations (22) and (23).

4. Results and Discussion

By utilising the package bvp4c in MATLAB software, Equations (11)–(13) were solved numerically. This solver employs the three-stage Lobatto IIIa formula [61]. The effect of several physical parameters on the flow behaviour is examined. The total composition of Al₂O₃ and Cu volume fractions are applied in a one-to-one ratio. For instance, $1\%$ of Al₂O₃ $\left({\phi }_1=1\%\right)$ and $1\%$ of Cu $\left({\phi }_2=1\%\right)$ are mixed to produce $2\%$ of Al₂O₃-Cu hybrid nanoparticles volume fractions, i.e., ${\phi }_{hnf}=2\%$. Meanwhile, ${\phi }_{hnf}=0$ indicates a regular viscous fluid.

The values of $-{\theta }^{\prime } \left(0\right)$ for various values of $m$ and Pr when ${\phi }_{hnf}=S=M=R=0$ and $\lambda =1$ (stretching sheet) are compared with Grubka and Bobba [18], and Ishak et al. [15] and the results for each m and Pr considered are comparable, as shown in

Table 3. Values of $-{\theta }^{\prime }\left(0\right)$ under different values of $m$ and Pr when ${\phi }_{hnf}=S=M=R=0$ and $\lambda =1$ (stretching sheet).

$\mathit{m}$	Pr	Grubka and Bobba [18]	Ishak et al. [15]	Present Results
0	1	0.5820	-	0.5820
1	-	1.0000	-	1.0000
2	-	1.3333	-	1.3333
3	-	1.6154	-	1.6154
1	0.72	0.8086	0.8086	0.8086
-	1	1.0000	1.0000	1.0000
-	3	1.9237	1.9237	1.9237
-	10	3.7207	3.7207	3.7207

. In addition, it should be noted that the values of $-{\theta }^{\prime } \left(0\right)$ increase for higher values of $m$ and Pr. Furthermore,

Table 4. Values of ${\mathrm{Re}}_x^{1/2}{C}_f$ and ${\mathrm{Re}}_x^{-1/2}N{u}_x$ when ${\phi }_{hnf}=2\%$, $S=2$ and $\lambda =-1$ (shrinking sheet) for different physical parameters.

$\mathit{m}$	$\mathit{R}$	$\mathit{M}$	First Solution		Second Solution
			$\mathbf{R}{\mathbf{e}}_{\mathit{x}}^{1/2}{\mathit{C}}_{\mathit{f}}$	$\mathbf{R}{\mathbf{e}}_{\mathit{x}}^{-1/2}\mathit{N}{\mathit{u}}_{\mathit{x}}$	$\mathbf{R}{\mathbf{e}}_{\mathit{x}}^{1/2}{\mathit{C}}_{\mathit{f}}$	$\mathbf{R}{\mathbf{e}}_{\mathit{x}}^{-1/2}\mathit{N}{\mathit{u}}_{\mathit{x}}$
0	0	0	1.3622	11.8319	0.8566	11.8066
0.5	-	-	1.3622	11.5596	0.8566	11.5177
1	-	-	1.3622	11.2748	0.8566	11.2126
1	1	-	1.3622	9.9890	0.8566	9.5366
-	2	-	1.3622	8.8910	0.8566	7.6301
-	3	-	1.3622	8.0105	0.8566	5.7594
-	3	0.01	1.3834	8.0575	0.8354	5.5222
-	-	0.05	1.4554	8.2064	0.7634	4.2952
-	-	0.1	1.5284	8.3426	0.6904	0.4505

provides the values of ${\mathrm{Re}}_x^{1/2}{C}_f$ and ${\mathrm{Re}}_x^{-1/2}N{u}_x$ when ${\phi }_{hnf}=2\%$, $S=2,$ and $\lambda =-1$ (shrinking sheet) for different physical parameters. The consequence of increasing $m$ and $R$ values is to reduce the local Nusselt number ${\mathrm{Re}}_x^{-1/2}N{u}_x$ for both branch solutions. However, the skin friction coefficient ${\mathrm{Re}}_x^{1/2}{C}_f$ is not affected by these parameters. Moreover, the values of ${\mathrm{Re}}_x^{1/2}{C}_f$ and ${\mathrm{Re}}_x^{-1/2}N{u}_x$ for the first solution increase, but they decrease for the second solution as $M$ increases.

The variations of ${\mathrm{Re}}_x^{-1/2}N{u}_x$ against $R$ when $\lambda =-1, S=2, M=0.1, {\phi }_{hnf}= 2\%, \mathrm{and} \mathrm{Pr}=6.2$ for various values of $m$ are presented in Figure 2. Reductions in the values of ${\mathrm{Re}}_x^{-1/2}N{u}_x$ on both solutions are observed with an increase in $R$ and $m$. Moreover, the simultaneous effect of $R$ and ${\phi }_{hnf}$ on ${\mathrm{Re}}_x^{-1/2}N{u}_x$ when $\lambda = -1, S = 2, M = 0.1, m = 1, \mathrm{and} \mathrm{Pr}=6.2$ can be observed in Figure 3. The values of ${\mathrm{Re}}_x^{-1/2}N{u}_x$ on the first solution decrease with a high percentage of ${\phi }_{hnf}$ for smaller values of $R$. This finding seems to contradict the fact that the added hybrid nanoparticles improve the heat transfer rate due to synergistic effects as discussed by Sarkar et al. [50]. However, it is interesting to note that this behaviour is opposite when higher values of $R$ are applied to the system where the enhancement in the values of ${\mathrm{Re}}_x^{-1/2}N{u}_x$ are observed with a high percentage of ${\phi }_{hnf}$. From these observations, we conclude that the rate of heat transfer could be controlled by manipulating the values of $R$ and ${\phi }_{hnf}$.

Next, the variations of ${\mathrm{Re}}_x^{1/2}{C}_f$ and ${\mathrm{Re}}_x^{-1/2}N{u}_x$ against $S$ for various values of ${\phi }_{hnf}$ and $M$ are presented in Figure 4, Figure 5, Figure 6 and Figure 7, respectively. The enhancement in the values of ${\mathrm{Re}}_x^{1/2}{C}_f$ and ${\mathrm{Re}}_x^{-1/2}N{u}_x$ on the first solution are observed with an increase in S, ${\phi }_{hnf}$ and $M$ values. The dual solutions are also obtained when a suitable suction strength is imposed on the shrinking surface. The flow is unlikely to exist since the vorticity could not be confined in the boundary layer. These figures reveal that a sufficient suction strength is needed to preserve the flow over a shrinking sheet. The similarity solutions are terminated at $S=S_c$ (critical value) and this point is known as the bifurcation point of the solutions. The boundary layer separation is also delayed with an increase in ${\phi }_{hnf}$ and $M$ by expanding the domain of $S$. Here, the critical values are $S_{c1}= 1.8974, S_{c2}= 1.8733, \mathrm{and} S_{c3}= 1.8519$ for ${\phi }_{hnf}= 0\%, 1\%, \mathrm{and} 2\%$, respectively. Meanwhile, for $M= 0, 0.05, \mathrm{and}, 0.1$, the critical values are $S_{c1}=1.9474, S_{c2}= 1.9003, \mathrm{and} S_{c3}= 1.8519$, respectively. It can be seen that the presence of those parameters suppressed the vorticity generation due to the shrinking of the sheet and the steady boundary layer flow is maintained.

The influence of $m$ and $R$ on the variations ${\mathrm{Re}}_x^{-1/2}N{u}_x$ against $S$ are given in Figure 8 and Figure 9, respectively. The heat transfer rate is higher for the isothermal surface $\left(m=0\right)$ as compared with the non-isothermal surface $(m>0)$. An increase in $R$ leads to a reduction in the values of ${\mathrm{Re}}_x^{-1/2}N{u}_x$. In addition, the boundary layer separation occurs at the same point where the critical value is $S_c=1.8519$ for all values of $m$ and $R$ considered.

The profiles of the velocity $f^{\prime }\left(\eta \right)$ and the temperature $\theta \left(\eta \right)$ for several pertinent parameters are presented in Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16 and Figure 17. There are dual solutions for $f^{\prime }\left(\eta \right)$ and $\theta \left(\eta \right)$ which satisfy the infinity boundary conditions (13) asymptotically. For more detail, the profiles of $f^{\prime }\left(\eta \right)$ and $\theta \left(\eta \right)$ for several values of $S$ when $\lambda =-1, M=0.1,{\phi }_{hnf}=2\%,m=1,R=3, \mathrm{and} \mathrm{Pr}=6.2$ are given in Figure 10 and Figure 11. Note that the profiles of the first and the second solutions are merged towards some values of $S$. This behaviour can also be seen in Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9 where the similarity solutions ended at $S=S_c$.

Next, an increase in ${\phi }_{hnf}$ and $M$ values lead to an upsurge in the velocity $f^{\prime }\left(\eta \right)$ but reduces the temperature $\theta \left(\eta \right)$ on the first solution, as shown in Figure 12, Figure 13, Figure 14 and Figure 15, respectively. Physically, the addition of the nanoparticles makes the fluid more viscous, and thus slows down the flow; the added nanoparticles also dissipate energy in the form of heat and consequently exert more energy which enhances the temperature. However, in this study, we discover that the velocity increases, but the temperature decreases, as ${\phi }_{hnf}$ increases. Furthermore, an increase in magnetic strength enhances the magnitude of Lorentz force and results in an increment in the velocity and a reduction in the temperature for the shrinking sheet case.

Moreover, Figure 16 and Figure 17 show the consequence effects of $m$ and $R$ on the temperature $\theta \left(\eta \right)$. It is seen that both branch solutions of $\theta \left(\eta \right)$ show an increasing pattern for larger values of $m$ and $R$; in addition, the boundary layer thickness of the first and the second solutions expand as $m$ and $R$ increase. For $m>0$, the temperature in the flow field increases due to direct variation of the wall temperature along the shrinking surface. Moreover, the radiation is dominant over conduction with an increase in $R$. Therefore, the temperature $\theta \left(\eta \right)$ increases due to the high radiation energy presence in the flow field.

The variations of $\gamma$ against $S$ when $\lambda =-1,{\phi }_{hnf}=2\%, \mathrm{and} M=0.1$ are described in Figure 18. For the positive value of $\gamma$, it is noted that $e^{- \gamma \tau }\to 0$ as time evolves $\left(\tau \to \infty \right)$. In the meantime, for the negative value of $\gamma$, $e^{- \gamma \tau }\to \infty$. These behaviours show that the first solution is stable and physically reliable, while the second solution becomes unstable over time.

5. Conclusions

The flow and heat transfer over a permeable non-isothermal shrinking surface with radiation and magnetohydrodynamic (MHD) effects were examined in this paper. The findings revealed that dual solutions appeared when satisfactory suction strength was applied on the shrinking surface. Moreover, the heat transfer rate was enhanced with a high percentage of ${\phi }_{hnf}$ when higher values of the radiation parameter, $R$, were applied to the system; additionally, the heat transfer rate was higher for the isothermal surface $\left(m=0\right)$ as compared with the non-isothermal surface $(m>0)$. Increased ${\phi }_{hnf}$ and $M$ values also enhanced the skin friction coefficient ${\mathrm{Re}}_x^{1/2}{C}_f$ and the local Nusselt number ${\mathrm{Re}}_{\mathrm{x}}^{-1/2}N{u}_x$. The effect of $m$, as well as $R$, was to increase the temperature $\theta \left(\eta \right)$ inside the boundary layer. Lastly, it was discovered that the first solution was stable, and thus physically reliable in the long run.