Nanofluid Flow on a Shrinking Cylinder with Al₂O₃ Nanoparticles

Abstract

This study investigates the nanofluid flow towards a shrinking cylinder consisting of Al₂O₃ nanoparticles. Here, the flow is subjected to prescribed surface heat flux. The similarity variables are employed to gain the similarity equations. These equations are solved via the bvp4c solver. From the findings, a unique solution is found for the shrinking strength $\lambda \ge -1$. Meanwhile, the dual solutions are observed when ${\lambda }_c<\lambda <-1$. Furthermore, the friction factor $R{e}_x^{1/2}{C}_f$ and the heat transfer rate $R{e}_x^{-1/2}N{u}_x$ increase with the rise of Al₂O₃ nanoparticles $\phi$ and the curvature parameter $\gamma$. Quantitatively, the rates of heat transfer $R{e}_x^{-1/2}N{u}_x$ increase up to 3.87% when $\phi$ increases from 0 to 0.04, and 6.69% when $\gamma$ increases from 0.05 to 0.2. Besides, the profiles of the temperature $\theta \left(\eta \right)$ and the velocity $f’\left(\eta \right)$ on the first solution incline for larger $\gamma$, but their second solutions decline. Moreover, it is noticed that the streamlines are separated into two regions. Finally, it is found that the first solution is stable over time.

1. Introduction

The fluid flow toward a stagnation point on a fixed surface was first introduced by Hiemenz [1] in 1911. The axisymmetric flow was then studied by Homann [2]. Ariel [3] followed by examining the flow with the hydromagnetic effects. The flow on a shrinking sheet was reported by Wang [4] and Kamal et al. [5]. Different from the aforementioned studies, which considered the flow over a flat plate, Wang [6] discussed the fluid flow over a circular cylinder. This was then followed by several researchers, including Ishak et al. [7] and Awaludin et al. [8], who studied the flow over a shrinking cylinder subject to a prescribed surface heat flux. They found that the increment of the curvature parameter delayed the boundary layer separation from the surface of the cylinder. Muthtamilselvan and Prakash [9] studied the unsteady flow and heat transfer of a nanofluid over a moving surface with prescribed heat and mass fluxes, and stated that the heat flux condition is important in a microelectromechanical (MEM) condensation application. Several researchers [10,11,12,13,14,15,16] have also considered this type of surface heating condition in their studies.

Nanoparticles and structures have been used by humans in fourth century AD, by the Romans, which demonstrated one of the most interesting examples of nanotechnology in the ancient world [17]. The term nanofluid, a mixture of the base fluid and nanoparticles, was initiated by Choi and Eastman [18]. It seems that Pak and Cho [19] were the first who introduced the thermophysical correlations for the nanofluid. Several studies have considered these nanofluid correlations [20,21,22,23,24,25]. The nanofluid correlations introduced by Pak and Cho [19] were improved by Ho et al. [26]. They reported that the numerical predictions from the existing nanofluid correlations are contradicted with the experimental results. The dispersion of nanoparticles in the base fluid was observed to result in a marked reduction, instead of an enhancement. Therefore, they have introduced the new correlations of the Al₂O₃-water nanofluid through a least-square curve fitting from the experimental results. They concluded that these new correlations give more accurate predictions with the experimental data. It should be noted that the studies of the nanofluid employing these nanofluid correlations are very limited. Among them, Sheremet et al. [27] employed these correlations to study the natural convective heat transfer and fluid flow of Al₂O₃-water nanofluid in an inclined wavy-walled cavity under the effect of non-uniform heating. They found that the heat transfer rate and fluid flow rate are non-monotonic functions of the cavity inclination angle and undulation number. Similarly, these correlations have been considered by Waini et al. [28] to examine the impact of Dufour and Soret diffusions on Al₂O₃-water nanofluid flow over a moving thin needle. They reported that the skin friction coefficient and the heat transfer coefficients increase, but the mass transfer coefficient decreases in the presence of Al₂O₃ nanoparticles. This concept has been upgraded by considering two or more types of nanoparticles that dispersed simultaneously into the base fluid and is called ‘hybrid nanofluid’. Some works on such fluids can be found in references [29,30,31]. Additionally, Takabi and Salehi [32] and Devi and Devi [33] introduced the hybrid nanofluid thermophysical models, which were widely used by many researchers [34,35,36,37,38,39,40,41,42,43] in the boundary layer problems. Furthermore, Waini et al. [44,45,46,47] scrutinized the temporal stability of the hybrid nanofluid flow.

In this study, the stagnation point flow towards a shrinking cylinder with the Al₂O₃ nanoparticle subjected to prescribed surface heat flux is investigated. Different from the previous studies, the present study examines the flow and thermal behavior of the Al₂O₃/water nanofluid by employing the correlations introduced by Ho et al. [26]. Most importantly, this is the first attempt to study the flow towards a stagnation region of a shrinking cylinder by considering these correlations. Moreover, the dual solutions and their stability are also reported in this study. The finding from this study can contribute to foresee the flow and thermal behaviors in industrial applications.

2. Mathematical Formulation

Consider the nanofluid flow on a shrinking cylinder with Al₂O₃ nanoparticles as shown in Figure 1.

Here, $u_e(x)=c_{1}x/L$ denotes the external flow velocity with $c_1>0$. The surface velocity is represented by $u_w(x)=c_{2}x/L$ where $c_2$ is a constant. Besides, $q_w(x)=T_{0}x/L$ is the prescribed heat flux where $T_0$ and $T_{\infty }$ correspond to the reference and the ambient temperatures, respectively.

Accordingly, the governing equations are [7,8]:

$$\frac{\partial (ru)}{\partial x}+\frac{\partial (rv)}{\partial r}=0$$

(1)

$$u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial r}=u_e\frac{d{u}_e}{dx}+\frac{{\mu }_{nf}}{{\rho }_{nf}}\left(\frac{{\partial }^{2}u}{\partial r^2}+\frac{1}{r}\frac{\partial u}{\partial r}\right)$$

(2)

$$u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial r}=\frac{k_{nf}}{{(\rho C_p)}_{nf}}\left(\frac{{\partial }^{2}T}{\partial r^2}+\frac{1}{r}\frac{\partial T}{\partial r}\right)$$

(3)

Subject to:

$$\begin{array}{l}u=u_w(x),v=0,k_{nf}\frac{\partial T}{\partial r}=-q_w(x) \mathrm{at} r=a\\ u\to u_e(x),T\to T_{\infty } \mathrm{as} r\to \infty \end{array}$$

(4)

where $\left(u,v\right)$ are the corresponding velocity components and T is the temperature. Further,

Table 1. Thermophysical properties.

Properties	Nanoparticle	Base Fluid
	Al2O3	water
$C_p \left(\mathrm{J}/\mathrm{kgK}\right)$	765	4179
$\rho \left(\mathrm{kg}/{\mathrm{m}}^3\right)$	3970	997.1
$k \left(\mathrm{W}/\mathrm{mK}\right)$	40	0.613

provides the properties of water and Al₂O₃ [22]. Here, Prandtl number, $\mathrm{Pr}$ is taken as $\mathrm{Pr}=6.2$. Meanwhile, the nanofluid thermophysical models are given by [19,26]:

$$\begin{array}{l}{\mu }_{nf}={\mu }_f\left(1+4.93\phi +222.4{\phi }^2\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{k}_{nf}=k_f\left(1+2.944\phi +19.672{\phi }^2\right),\\ \hspace{0.17em}{\rho }_{nf}=\left(1-\phi \right){\rho }_f+\phi {\rho }_s,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{(\rho C_p)}_{nf}=(1-\phi ){(\rho C_p)}_f+\phi {(\rho C_p)}_s\end{array}$$

(5)

where $\mu$, $k$, $\rho$, and $(\rho C_p)$ denote the dynamic viscosity, the thermal conductivity, the density, and the heat capacity, respectively with $\phi$ is the Al₂O₃ nanoparticle volume fractions and the subscript $s$ represents its solid component. Meanwhile, the subscripts $f$ and $nf$ correspond to fluid and nanofluid, respectively. Note that these thermophysical models were also considered by Sheremet et al. [27] and Waini et al. [28].

Consider the following dimensionless variables [7,8]:

$$\psi ={\left(\frac{c_1{\nu }_f}{L}\right)}^{1/2}axf(\eta ),T=T_{\infty }+\frac{q_w}{k_f}{\left(\frac{{\nu }_{f}L}{c_1}\right)}^{1/2}\theta (\eta ),\hspace{0.17em}\hspace{0.17em}\eta ={\left(\frac{c_1}{{\nu }_{f}L}\right)}^{1/2}\frac{r^2-a^2}{2a}$$

(6)

With the stream function $\psi$, the characteristic length $L$, and the fluid kinematic viscosity ${\nu }_f$. Here, $u=(\partial \psi /\partial r)/r$ and $v=-(\partial \psi /\partial x)/r$. So that:

$$u=\frac{c_{1}x}{L}f\prime (\eta ),v=-\frac{a}{r}{\left(\frac{c_1{\nu }_f}{L}\right)}^{1/2}f(\eta )$$

(7)

On using Equations (6) and (7), the continuity equation, i.e., Equation (1), is identically satisfied. Now, Equations (2) and (3) become:

$$\frac{{\mu }_{nf}/{\mu }_f}{{\rho }_{nf}/{\rho }_f}\left[2\gamma f^{″}+\left(1+2\gamma \eta \right)f^{‴}\right]+1-f{\prime }^2+f{f}^{″}=0$$

(8)

$$\frac{1}{\mathrm{Pr}}\frac{k_{nf}/k_f}{{(\rho C_p)}_{nf}/{(\rho C_p)}_f}\left[2\gamma {\theta }^{\prime }+\left(1+2\gamma \eta \right){\theta }^{″}\right]+f{\theta }^{\prime }-f^{\prime }\theta =0$$

(9)

Subject to:

$$\begin{array}{l}{f}^{\prime }(0)=\lambda ,f(0)=0,{\theta }^{\prime }(0)=-\frac{k_f}{k_{nf}},\\ f^{\prime }(\eta )\to 1,\theta (\eta )\to 0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{as}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\eta \to \infty \end{array}$$

(10)

The physical parameters appearing in Equations (8)–(10) are the stretching/shrinking parameter $\lambda$, the curvature parameter $\gamma$, and the Prandtl number Pr, given as:

$$\lambda =\frac{c_2}{c_1},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\gamma ={\left(\frac{{\nu }_{f}L}{c_1{a}^2}\right)}^{1/2},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{Pr}=\frac{{(\mu C_p)}_f}{k_f}$$

(11)

Note that, $\lambda <0$ and $\lambda >0$ signify the shrinking and stretching sheets, while $\lambda =0$ is for the static sheet. Here, by taking $\phi =\lambda =\gamma =0$, Equation (8) reduces to the Hiemenz flow, see White [48]. The local Nusselt number $N{u}_x$ and the skin friction coefficients $C_f$ are:

$$N{u}_x=-\frac{x{k}_{nf}}{k_f(T_w-T_{\infty })}{\left(\frac{\partial T}{\partial r}\right)}_{r=a},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{C}_f=\frac{{\mu }_{nf}}{{\rho }_f{u}_e^2}{\left(\frac{\partial u}{\partial r}\right)}_{r=a}$$

(12)

On using Equation (6), one obtains

$$R{e}_x^{-1/2}N{u}_x=\frac{1}{\theta (0)},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}R{e}_x^{1/2}{C}_f=\frac{{\mu }_{nf}}{{\mu }_f}{f}^{″}(0)$$

(13)

where $R{e}_x=u_{e}x/{\nu }_f$ is the local Reynolds number.

3. Stability Analysis

This temporal stability analysis was first introduced by Merkin [49] and then followed by Weidman et al. [50]. Firstly, consider the new variables as follows [8]:

$$\psi ={\left(\frac{c_1{\nu }_f}{L}\right)}^{1/2}axf(\eta ,\tau ),T=T_{\infty }+\frac{q_w}{k_f}{\left(\frac{{\nu }_{f}L}{c_1}\right)}^{1/2}\theta (\eta ,\tau ),\hspace{0.17em}\hspace{0.17em}\eta ={\left(\frac{c_1}{{\nu }_{f}L}\right)}^{1/2}\frac{r^2-a^2}{2a},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\tau =\frac{c_1}{L}t$$

(14)

where $\tau$ is the dimensionless time variable. Then, the unsteady form of Equations (2) and (3) are employed. On using Equation (14), one obtains:

$$\frac{{\mu }_{nf}/{\mu }_f}{{\rho }_{nf}/{\rho }_f}\left[2\gamma \frac{{\partial }^{2}f}{\partial {\eta }^2}+\left(1+2\gamma \eta \right)\frac{{\partial }^{3}f}{\partial {\eta }^3}\right]+1-{\left(\frac{\partial f}{\partial \eta }\right)}^2+f\frac{{\partial }^{2}f}{\partial {\eta }^2}-\frac{{\partial }^{2}f}{\partial \eta \partial \tau }=0$$

(15)

$$\frac{1}{\mathrm{Pr}}\frac{k_{nf}/k_f}{{(\rho C_p)}_{nf}/{(\rho C_p)}_f}\left[2\gamma \frac{\partial \theta }{\partial \eta }+\left(1+2\gamma \eta \right)\frac{{\partial }^2\theta }{\partial {\eta }^2}\right]+f\frac{\partial \theta }{\partial \eta }-\theta \frac{\partial f}{\partial \eta }-\frac{\partial \theta }{\partial \tau }=0$$

(16)

Subject to:

$$\begin{array}{l}\frac{\partial f}{\partial \eta }(0,\tau )=\lambda ,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} f(0,\tau )=0, \frac{\partial \theta }{\partial \eta }\left(0,\tau \right)=-\frac{k_f}{k_{nf}},\\ \frac{\partial f}{\partial \eta }\left(\infty ,\tau \right)=1,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} \theta \left(\infty ,\tau \right)=0\end{array}$$

(17)

To investigate the temporal stability, the following perturbation functions are employed [50]:

$$f(\eta ,\tau )=f_0(\eta )+e^{-\alpha \tau }F(\eta ),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\theta (\eta ,\tau )={\theta }_0(\eta )+e^{-\alpha \tau }G(\eta )$$

(18)

where $F(\eta )$ and $G(\eta )$ are comparatively small compared to $f_0(\eta )$ and ${\theta }_0(\eta )$, and $\alpha$ denotes the eigenvalue. On using Equation (18), Equations (15) and (16) respectively become:

$$\frac{{\mu }_{nf}/{\mu }_f}{{\rho }_{nf}/{\rho }_f}\left[2\gamma F^{″}+\left(1+2\gamma \eta \right)F^{‴}\right]-2f_0^{\prime }{F}^{\prime }+f_0^{″}F+f_0{F}^{″}+\alpha F^{\prime }=0$$

(19)

$$\frac{1}{\mathrm{Pr}}\frac{k_{nf}/k_f}{{(\rho C_p)}_{nf}/{(\rho C_p)}_f}\left[2\gamma G^{\prime }+\left(1+2\gamma \eta \right)G^{″}\right]+f_0{G}^{\prime }+{\theta }_0^{\prime }F-f_0^{\prime }G-{\theta }_0{F}^{\prime }+\alpha G=0$$

(20)

The boundary conditions then become:

$$\begin{array}{l}{F}^{\prime }(0)=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}F(0)=0,G^{\prime }(0)=0;\\ F^{\prime }(\infty )=0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}G(\infty )=0\end{array}$$

(21)

Without loss of generality, following Harris et al. [51], we fix the value of $F^{″}(0)$ as $F^{″}(0)=1$ to obtain the smallest eigenvalues $\alpha$ in Equations (19) and (20).

4. Results and Discussion

The solutions of Equations (8)–(10) are attained by utilizing the package bvp4c in MATLAB software [52]. The effects of various physical parameters are then examined and presented in tabular and graphical forms.

By taking $\phi =\lambda =\gamma =0$, we obtain $f^{″}(0)=1.232588$, which is in agreement with what is reported by Wang [4] and Awaludin et al. [8]. The values of $f^{″}(0)$ and $1/\theta (0)$ for several values of $\lambda$ when $\phi =\gamma =0$ are also provided in

Table 2. Values of $f^{″}(0)$ and $1/\theta (0)$ for regular fluid ($\phi =0$) under different $\lambda$ when $\gamma =0$ (flat plate).

$\mathit{\lambda }$	Wang [4]	Awaludin et al. [8]	Present Results
	${\mathit{f}}^{″}(0)$	${\mathit{f}}^{″}(0)$	${\mathit{f}}^{″}(0)$	$1/\mathit{\theta }(0)$
−1	1.32882		1.328817	−2.359393
−0.5	1.49567		1.495670	0.314542
0	1.232588	1.232588	1.232588	1.573433
0.1	1.14656	1.146561	1.146561	1.767533
0.2	1.051130	1.051130	1.051130	1.949500
0.5	0.7133	0.713295	0.713295	2.438276
1	0	0	0	3.120727
2	−1.88731	−1.887307	−1.887307	4.203068
5	−10.26475	−10.264749	−10.264749	6.491300

for future reference. Further, the values of $R{e}_x^{-1/2}N{u}_x$ and $R{e}_x^{1/2}{C}_f$ when $\mathrm{Pr}=6.2$ with various values of $\phi$, $\gamma$, and $\lambda$ are given in

Table 3. Values of $R{e}_x^{-1/2}N{u}_x$ and $R{e}_x^{1/2}{C}_f$ for $\phi$, $\gamma$, and $\lambda$ when $\mathrm{Pr}=6.2$.

$\mathit{\phi }$	$\mathit{\gamma }$	$\mathit{\lambda }$	$\mathit{R}{\mathit{e}}_{\mathit{x}}^{-1/2}\mathit{N}{\mathit{u}}_{\mathit{x}}$	$\mathit{R}{\mathit{e}}_{\mathit{x}}^{1/2}{\mathit{C}}_{\mathit{f}}$
0	0	0	1.573433	1.232588
0.02			1.610281	1.382684
0.04			1.634333	1.625081
0.04	0.05		1.673416	1.667025
	0.1		1.711566	1.708036
	0.2		1.785416	1.787623
	0.1	−0.5	0.354240	2.110589
		−0.2	1.242946	1.914480
		0.5	2.645346	0.979397

. The values of $R{e}_x^{-1/2}N{u}_x$ and $R{e}_x^{1/2}{C}_f$ are intensified with the rise of $\gamma$ and $\phi$. Quantitatively, a 3.87% increment of $R{e}_x^{-1/2}N{u}_x$ is observed when $\phi$ increases from 0 to 0.04. Moreover, it is noticeable that the values of $R{e}_x^{-1/2}N{u}_x$ increase up to 6.69% when $\gamma$ increases from 0.05 to 0.2. Meanwhile, the values of $R{e}_x^{1/2}{C}_f$ reduce, but the values of $R{e}_x^{-1/2}N{u}_x$ increase when $\lambda$ increases from −0.5 to 0.5. It is seen that the nanoparticle volume fractions, the curvature, and the stretching/shrinking parameters can be utilized to control the heat transfer rate.

Next, the results in graphical forms are provided to have a better insight into the effect of the physical parameters. The variations of the local Nusselt number $R{e}_x^{-1/2}N{u}_x$ and the skin friction coefficient $R{e}_x^{1/2}{C}_f$ against $\lambda$ when $\phi =0.02$ and $\mathrm{Pr}=6.2$ for several values of $\gamma$ are shown in Figure 2 and Figure 3. Larger $\gamma$ gives higher values of $R{e}_x^{1/2}{C}_f$ and $R{e}_x^{-1/2}N{u}_x$ on the first solution compared to the flat plate case $(\gamma =0)$. Besides, a unique solution is found when $\lambda \ge -1$. Meanwhile, two solutions are observed for the limited range of $\lambda$ when the sheet is shrunk $({\lambda }_c<\lambda <-1)$. The similarity solutions also terminate in this region at $\lambda ={\lambda }_c$ (critical value). Here, the critical values are respectively ${\lambda }_c=-1.24657,$ $-1.32099$, and $-1.38801$ for $\gamma =0,$ 0.1 and $0.2$. The velocity $f^{\prime }(\eta )$ and temperature $\theta (\eta )$ profiles for $\phi =0,0.02,$ and $0.04$ when $\mathrm{Pr}=6.2$, $\lambda =-1.24$, and $\gamma =0.1$ are given in Figure 4 and Figure 5. The reduction of $f^{\prime }(\eta )$ and $\theta (\eta )$ are observed for both branches with the rising of $\phi$. Physically, the addition of the nanoparticles makes the fluid more viscous and thus, slows down the flow. Consequently, the fluid velocity decreases. Also, the added nanoparticles dissipate energy in the form of heat and consequently exert more energy, which enhances the temperature. However, in this study, we discover that the temperature decreases as $\phi$ increases. This behavior is due to the prescribed heat flux on the shrinking surface of the cylinder.

Further, Figure 6 and Figure 7 show the effect of $\gamma$ on $f^{\prime }(\eta )$ and $\theta (\eta )$ when $\phi =0.02,\lambda =-1.24$ and $\mathrm{Pr}=6.2$. The profiles of $f^{\prime }(\eta )$ and $\theta (\eta )$ on the first solution incline for larger $\gamma$. However, the profiles on the second solution decline. Besides, the negative values of $\theta (\eta )$ are noticed in Figure 5 and Figure 7. The definition of the curvature parameter $\gamma$ is inversely proportional to the radius of the cylinder, see Equation (11). Thus, the radius of the cylinder decreases as $\gamma$ increases. Hence, the fluid velocity amplifies due to less resistance occurring between the surface of the cylinder and the fluid. Consequently, the fluid temperature increases for cumulative $\gamma$. Since the Kelvin temperature of substances is defined as the average kinetic energy of the particles of substances, as velocity enhances with $\gamma$, the kinetic energy increases, and consequently intensifies the temperature [16].

Figure 8 and Figure 9 display the streamlines when $\lambda =-1.24$ (shrinking sheet), $\phi =0.02$, and $\gamma =0.1$ for the first and the second solutions, respectively. Here, the streamlines are plotted for several values of $\overline{\psi }=\psi /a{(c_1{\nu }_f/L)}^{1/2}$. The streamlines are separated into two regions by the horizontal line for both solutions. It is notable that the horizontal line that separates the flow is nearer to the shrinking sheet for the first solution. Besides, the reverse rotating flow occurs in the lower region. Meanwhile, the flow pattern on the upper region behaves as the normal stagnation point.

The variation of $\alpha$ against $\lambda$ when $\phi =0.02$ and $\gamma =0.1$ is described in Figure 10. For positive values of $\alpha$, it is noted that $e^{-\alpha \tau }\to 0$ as time evolves $(\tau \to \infty )$. In contrast, negative values of $\alpha$, $e^{-\alpha \tau }\to \infty$ as $\tau \to \infty$ show a growth of disturbance as time evolves. These behaviors show that the first solution is stable, while the second solution is unstable in the long run.

5. Conclusions

This study examined the stagnation point flow on a shrinking cylinder filled with Al₂O₃ nanoparticles. The surface of the cylinder is subjected to prescribed surface heat flux. The correlations of Al₂O₃/water nanofluid introduced by Ho et al. [25] were employed. Findings revealed two solutions to be observed for the limited range of $\lambda$ when the sheet is shrunk $({\lambda }_c<\lambda <-1)$. The similarity solutions terminated in this region at $\lambda ={\lambda }_c$. Meanwhile, a unique solution was found when $\lambda \ge -1$. The skin friction coefficient $R{e}_x^{1/2}{C}_f$ and the local Nusselt number $R{e}_x^{-1/2}N{u}_x$ were intensified with the rising of the nanoparticle volume fraction $\phi$ and the curvature parameter $\gamma$. Quantitatively, the values of $R{e}_x^{-1/2}N{u}_x$ increased up to 3.87% when $\phi$ is increased from 0 to 0.04, and 6.69% when $\gamma$ is increased from 0.05 to 0.2. Furthermore, Al₂O₃/water nanofluid produced higher values of $R{e}_x^{1/2}{C}_f$ and $R{e}_x^{-1/2}N{u}_x$ compared to water. Moreover, the rising of $\phi$ tended to reduce the velocity $f^{\prime }(\eta )$ and the temperature $\theta (\eta )$ for both branches. Besides, the profiles on the first solution incline when larger values of $\gamma$ are applied. Finally, the temporal stability analysis showed that the first solution is stable while the second solution is unstable over time.