Unsteady Three-Dimensional Flow in a Rotating Hybrid Nanofluid over a Stretching Sheet

Abstract

The problem of an unsteady 3D boundary layer flow induced by a stretching sheet in a rotating hybrid nanofluid is studied. A dimensionless set of variables is employed to transform the system of partial differential equations (PDEs) to a set of nonlinear ordinary differential equations (ODEs). Then, the system of ODEs is solved numerically using the MATLAB software. The impacts of different parameters, such as copper nanoparticles volume fraction, radiation, rotation, unsteadiness, and stretching parameters are graphically displayed. It is found that two solutions exist for the flow induced by the stretching sheet. Furthermore, the increasing nanoparticle volume fraction enhances the skin friction coefficient. It is noticed that the skin friction coefficient, as well as the heat transfer rate at the surface, decrease as the rotating parameter increases. Additionally, the thermal radiation as well as the unsteadiness parameter stimulate the temperature.

1. Introduction

The investigation into heat transfer is useful in various engineering applications, such as transpiration cooling, drag reduction, thrust bearing, and radial diffuser design [1]. Usually, fluids are used as heat transporters, such as in heating and cooling processes in transportation systems and industrial processes. It is also noticed that the stretching sheet has gained researchers’ attention for years. Researchers have conducted various studies on the physical phenomena and heat transmissions past a stretching plate. It has numerous important applications in industrial production, including the extrusion of plastic sheets, the process of condensation of metallic plates, and glass filer fabrication [2]. The study of flow and heat transfer is of significant importance since the quality of the final product depends on the large extent of the skin friction coefficient and the heat transfer rate at the surface [2]. Recently, the investigation of flow over a stretching sheet has been broadened to many different cases that make the study more interesting. For instance, Shahid et al. [3] studied the effects of swimming gyrotactic microorganisms using Darcy law and Vafai et al. [4] explored the effects of Dufour, Soret, and radiation on the Powell–Eyring fluid flow.

Even though the study of steady-state flows has the greatest practical significance, many scholars are now paying close attention to the study of unsteady-state flows. Steady flow can be defined as a flow in which the fluid characteristics at a given location in the system stay constant throughout time. In contrast, the unsteady flow is defined otherwise, which is time-dependent flow. Sears and Telionis [5] reported the numerical studies of steady and unsteady distinguishing boundary layer flow with Goldstein’s type singularities, and a possible comparison between the position of the singularity in relation to the time curves and the point of vanishing wall shear can be made. They discovered a significant difference between vanishing wall shear and separation. The unsteady (transient) boundary layer that is time-varying consists of mostly start-up processes, such as the movements from rest or transitions from one steady-state to another or occasional movements [6]. According to Liao [7], the unsteady flow problem may be resolved in the same manner as the steady-state similarities governed by the nonlinear ODEs. The study found that solving the problem for unsteady flow is as easy as steady flow using the homotopy analysis method (HAM). Suali et al. [8] considered both shrinking and stretching sheets with injection or suction to explore the unsteady flow towards a stagnation point on the sheet. According to this study, the spectrum of dual outcomes rises with mass suction, whereas it reduces with mass injection. The problems related to the unsteady flow can also be found in numerous literature [9,10,11,12,13,14,15,16].

The applications involved the problem of rotating flow, such as flywheels, cutting discs, rotating machinery, computer storage devices, electrical items, and many others [17]. Anuar et al. [18] stated that the fluid flow with a rotating plane was initially introduced by Kármán [19] using the momentum integral method. In the year 1988, Wang [20] explored the rotating fluid flow of the stretching plate. The solutions were determined by the rotation rate parameter, and it was found that the perturbation solutions for small and large rotation rates were comparable to other works of literature. A few years later, Rajeswari and Nath [21] broadened Wang’s problem to include the unsteady flow problem by combining the finite-difference scheme with the quasilinearization technique. Takhar et al. [22] have also extended Wang’s analysis to include the magnetic field. The application that is related to the present magnetic-rotational model was the chilling process in amalgamation reactors of liquid metal blankets. Yacob et al. [23] investigated a steady rotating flow in a nanofluid containing carbon nanotubes past a stretching/shrinking surface. Carbon nanotubes can be classified into single-walled (SWCNT) and multi-walled (MWCNT). They discovered that the heat transfer enhancement is greater in water-MWCNTs than in water-SWCNTs. Moreover, recent studies in this area may be found in references [24,25,26,27].

Nanofluid is a new amalgamation formed, as stated by Choi [28]. A nanofluid is formed by adding tiny particles in nano-dimensions to the base fluid. Nanofluids have higher thermal conductivity and are more effective in heat transport activities compared to their base fluid. Hence, it is also well acknowledged and accepted empirically and conceptually that dispersing nanoparticles in a liquid may improve the liquid’s thermo-physical properties [29]. Nanotechnology has advanced rapidly in recent years, and by combining many nanoparticle elements, stability issues and low heat conductivity can be addressed. Nanofluids will save energy, improve thermal efficiency, speed up processes, and increase the life of the equipment. Using a finite element simulation, Rana et al. [24] observed the unsteady magnetohydrodynamic (MHD) boundary layer rotating nanofluid flow on a stretching plate. Apart from that, nanofluid provides a number of advantages, including less component degradation and blockage in tiny channels than fluid containing micro-to millimeter-sized particles in suspension [30]. Similar problems in nanofluid but with different approaches were published by Ghadimi et al. [31], Noor et al. [32], Ahmad et al. [33], and Khan et al. [34].

Due to its importance in providing greater properties than nanofluid, hybrid nanofluid has recently been a topic of discussion. It’s employed in heat transfer applications using particles that are less than 100 nanometers in size. Higher energy efficiency, lower operating costs, and greater performance are among the contributions of the high thermal conductivity of a hybrid nanofluid [35]. Hybrid nanofluids have recently piqued the curiosity of many academics as a novel technological idea. Using a two-step technique, Suresh et al. [36] studied the combination of Al₂O₃-Cu/H₂O hybrid nanofluid. They found that the distinct nanoparticles improved the parameters of the hybrid nanofluid. Later, Suresh et al. [37] discussed the heat transport and the impacts of the alumina-copper/H₂O hybrid nanofluid. Numerical research of the 3D hybrid nanofluid flux with Newtonian heating and Lorentz force effects on a stretching plate was conducted by Devi and Devi [38]. Later, they continued the investigation across a stretching plate on increasing heat transmission in a copper-alumina/H₂O hybrid nanofluid [35].

Moreover, Waini et al. [39] investigated the unsteady hybrid nanofluid flow and heat transmission through a stretching, as well as a shrinking surface. For a particular unsteadiness parameter range, dual solutions exist; the results demonstrate that increasing the nanoparticle volume percentage of Cu for the first solution will improve the skin friction coefficient, while the second solutions show the opposite. Furthermore, the unsteady hybrid nanofluid flow on a porous biaxial stretching or shrinking plate, the effects of buoyancy and stagnation flow on an exponentially stretching or shrinking vertical plate, heat transfer, and MHD flow over a porous stretching/shrinking wedge, as well as the flow past a permeable non-isothermal shrinking surface were reviewed by Waini et al. [40,41,42,43]. The stability analysis was conducted by Zainal et al. [44] for the unsteady 3D magnetohydrodynamic hybrid nanofluid for Homann flow. Hayat and Nadeem [45] described how heat dissipation might be improved using Ag-CuO/H₂O hybrid nanofluid. Later, Hayat et al. [26] extended the study of rotational flow with partial slip and radiation effects. Subsequently, Anuar et al. [18] explored copper-alumina/water hybrid nanofluid with radiation on a rotating surface. They reported the stability of the solutions over time. The study of hybrid nanofluid flows has been diversified by Khan et al. [46] to various nanoparticle shape factors and for different base fluids.

Motivated by the earlier studies on rotating hybrid nanofluids, the present work intends to explore the rotation and radiation impacts on the unsteady 3D rotating flow of a hybrid nanofluid over a stretching sheet. The boundary value problem is solved numerically using the MATLAB software. The model is adopted from Rana et al. [24] and Devi and Devi [35], where the hybrid nanofluid Al₂O₃-Cu/H₂O is considered in this study. Rana et al. [24] studied the unsteady magnetohydrodynamic flow on a stretching sheet in a rotating nanofluid. The influences of the related parameters are visually depicted, and the numerical findings obtained are compared with the existing literature. The novelty of this study can also be seen in the discovery of dual solutions when the surface of the sheet is stretched. This discovery also has applications in a variety of sectors of science and technology, and it is useful for engineers as well as scientists to understand the behavior of the boundary layer flow.

2. Problem Formulation

The unsteady rotating flow of a hybrid nanofluid on a stretching sheet is considered as demonstrated in Figure 1, where $\left(x,y,z\right)$ are cartesian coordinates with the sheet at $z=0$. The stretching velocities in the x and y directions are denoted by $u_w\left(x,t\right)$ and $v_w\left(x,t\right)$, respectively, while $\omega$ is the uniform angular velocity of the rotation, see Figure 1. Moreover, the ambient temperature of the fluid is $T_{\infty }$ and the sheet temperature is $T_w$. The hybrid nanofluid Al₂O₃-Cu/H₂O is considered in this study. The desired hybrid nanofluid is formed by scatting copper nanoparticles in water to create Cu-H₂O nanofluid, and then aluminum oxide nanoparticles are added into that Cu-H₂O nanofluid.

The governing equations are adopted from Refs. [2,20,21,25], and may be written as

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

(1)

$$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}=2\omega v+\frac{{\mu }_{hnf}}{{\rho }_{hnf}}\frac{{\partial }^{2}u}{\partial z^2}$$

(2)

$$\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}+w\frac{\partial v}{\partial z}=-2\omega u+\frac{{\mu }_{hnf}}{{\rho }_{hnf}}\frac{{\partial }^{2}v}{\partial z^2}$$

(3)

$$\frac{\partial w}{\partial t}+u\frac{\partial w}{\partial x}+v\frac{\partial w}{\partial y}+w\frac{\partial w}{\partial z}=\frac{{\mu }_{hnf}}{{\rho }_{hnf}}\frac{{\partial }^{2}w}{\partial z^2}$$

(4)

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

(5)

The boundary conditions are

$$\begin{array}{l}u=u_w\left(x,t\right)=\frac{cx}{1-\alpha t},v=0,w=0,T=T_w \mathrm{at} z=0\\ u\to 0,v\to 0,T\to T_{\infty } \mathrm{as} z\to \infty \end{array}$$

(6)

where ($u$, v, $w$) are the velocity components along the (x, y, z) directions, t refers to time, $T$ is the fluid temperature, $c>0$ for the stretching sheet and $q_r$ is the radiative heat flux, $k_{hnf}$ is the thermal conductivity, ${\mu }_{hnf}$ is the dynamic viscosity, ${\rho }_{hnf}$ is the density, ${\left(\rho C_p\right)}_{hnf}$ is the heat capacity, and ${\sigma }_{hnf}$ is the electrical conductivities. The thermophysical properties are given in [47] as presented in

Table 1. Thermophysical properties.

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

Where ${\varphi }_{hnf}={\varphi }_{A{l}_2{O}_3}+{\varphi }_{Cu}$.

.

In Table 1, ϕ denotes the nanoparticle volume fraction where ϕ = 0 indicates the regular fluid, ${\varphi }_{A{l}_3{O}_2}$ correlates to Al₂O₃, and ${\varphi }_{Cu}$ correlates to Cu. The physical properties of the nanoparticles and the base fluid are given in

Table 2. Thermophysical properties of nanoparticles and water (base liquid).

Physical Properties	Al2O3	Cu	Water
$C_p\left(\mathrm{J}/\mathrm{KgK}\right)$	765	385	4179
$\rho \left({\mathrm{kg}/\mathrm{m}}^3\right)$	3970	8933	997.1
$k\left(\mathrm{W}/\mathrm{mK}\right)$	40	400	0.613
$\beta \times {10}^{-5}\left(1/\mathrm{K}\right)$	0.85	1.67	21

, as reported in [48].

Following Bataller [49], Ishak [50], Magyari and Pantokratoras [51], and Roşca, Roşca and Pop [52], the Rosseland approximation is applied to exhibit $q_r$ as

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

(7)

where $q_r$, $k^{*}$ and ${\sigma }^{\ast }$ respectively indicate the radiative heat flux, mean absorption coefficient, and the Stefan–Boltzmann constant. By neglecting higher-order terms and employing the Taylor series, $T^4$ may be approximated as $T^4\approx 4T_{\infty }^{3}T-3T_{\infty }^4.$ Equation (5) is now may be written as

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

(8)

The given transformation variables are according to Maqsood et al. [25],

$$\begin{array}{c}u=\frac{ax}{1-\alpha t}{f}^{\prime }\left(\eta \right),v=\frac{ax}{1-\alpha t}h\left(\eta \right),w=-\sqrt{\frac{a{\nu }_f}{1-\alpha t}}f\left(\eta \right)\\ \theta \left(\eta \right)=\frac{T-T_{\infty }}{T_w-T_{\infty }},\eta =z\sqrt{\frac{a/{\nu }_f}{1-\alpha t}}\end{array}$$

(9)

where $\left({}^{\prime }\right)$ denotes differentiation w.r.t. $\eta$, $a>0$ is the stretching constant along the x direction, $\alpha$ is a parameter indicating the flow unsteadiness, ${\nu }_f$ is the kinematic viscosity of the base fluid, and the nonlinear rotating angular velocity is $\omega ={\omega }^{\ast }/\left(1-\alpha t\right)$.

Substituting the similarity variables (9) into Equations (1)–(4) and (8) yields

$$\frac{{\mu }_{hnf}/{\mu }_f}{{\rho }_{hnf}/{\rho }_f}{f}^{‴}+f{f}^{″}-{f^{\prime }}^2+2\mathrm{\Omega }h-\beta \left(f^{\prime }+\frac{\eta }{2}{f}^{″}\right)=0$$

(10)

$$\frac{{\mu }_{hnf}/{\mu }_f}{{\rho }_{hnf}/{\rho }_f}{h}^{″}+f{h}^{\prime }-f^{\prime } h-2\mathrm{\Omega }{f}^{\prime }-\beta \left(h+\frac{\eta }{2}{h}^{\prime }\right)=0$$

(11)

$$\frac{1}{\mathrm{Pr}}\frac{1}{{\left(\rho C_p\right)}_{hnf}/{\left(\rho C_p\right)}_f}\left(\frac{k_{hnf}}{k_f}+\frac{4}{3}Rd\right){\theta }^{″}+f{\theta }^{\prime }-\beta \frac{\eta }{2}{\theta }^{\prime }=0$$

(12)

The boundary conditions are as follows:

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

(13)

where $\mathrm{\Omega }$ is the rotation parameter, $\beta$ the unsteadiness parameter, $\mathrm{Pr}$ indicates the Prandtl number, $Rd$ the radiation parameter, and $\lambda >0$ is the stretching parameter respectively defined as

$$\mathrm{\Omega }=\frac{{\omega }^{*}}{a},\beta =\frac{\alpha }{a},\mathrm{Pr}=\frac{{\nu }_f}{{\alpha }_f},Rd=\frac{4{\sigma }^{*}{T}_{\infty }{}^3}{k^{*}{k}_f},\lambda =\frac{c}{a}$$

(14)

It is noted that $\lambda >0$ is for stretching sheet, $\lambda <0$ for shrinking sheet, and $\lambda =0$ corresponds to static sheet.

We notice that the regular fluid $\left({\varphi }_{A{l}_2{O}_3}={\varphi }_{Cu}=0\right)$ and the absence of rotating parameter $\left(\mathrm{\Omega }=0\right),$ Equation (10) becomes Equation (15) which is consistent with Equation (6) as in Fang et al. [53].

$$f^{‴}+f{f}^{″}-{f^{\prime }}^2-\beta \left(f^{\prime }+\frac{\eta }{2}{f}^{″}\right)=0$$

(15)

The quantities of physical interest are the skin friction coefficients and the local Nusselt number which are given as follows:

$$\begin{array}{l}{C}_{fx}=\frac{{\mu }_{hnf}}{{\rho }_f{u}_e{}^2\left(x\right)}{\left(\frac{\partial u}{\partial z}\right)}_{z=0},C_{fy}=\frac{{\mu }_{hnf}}{{\rho }_f{u}_e{}^2\left(x\right)}{\left(\frac{\partial v}{\partial z}\right)}_{z=0},\\ \hspace{1em}N{u}_x=-\frac{x{k}_{hnf}}{k_f\left(T_f-T_{\infty }\right)}{\left(\frac{\partial T}{\partial z}\right)}_{z=0}+x{\left(q_r\right)}_{z=0}\end{array}$$

(16)

Using Equations (10) and (17) yields

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

(17)

where ${\mathrm{Re}}_x$ is the local Reynolds number defined as ${\mathrm{Re}}_x=u_e\left(x\right)x/{\nu }_f$.

3. Results and Discussion

The governing non-linear ordinary differential equations (ODEs) (10)–(12) subjected to the boundary conditions (13) are solved numerically using the built-in function “bvp4c” available in the MATLAB software. The detailed settings are described in [54]. The validation for the skin friction coefficient in x and y directions $f^{″}\left(0\right)$ and $h^{\prime }\left(0\right)$, respectively, is obtained, which agrees with Wang [20], Nazar et al. [2], and Rana et al. [24]. The comparisons are for the stretching surface, $\lambda =1$ in the absence of solid volume fraction $\left({\varphi }_1={\varphi }_2=0\right)$ at a steady state for different values of $\mathrm{\Omega }$ as presented in

Table 3. Comparison of $f^{″}\left(0\right)$ and $h^{\prime }\left(0\right)$ for $\beta =0$, $\lambda =1$ and variation of $\mathrm{\Omega }$.

$\mathbf{\Omega }$	Wang [20]		Nazar et al. [2]		Rana et al. [24]		Present Study
	${\mathit{f}}^{″}\left(0\right)$	${\mathit{h}}^{\prime }\left(0\right)$	${\mathit{f}}^{″}\left(0\right)$	${\mathit{h}}^{\prime }\left(0\right)$	${\mathit{f}}^{″}\left(0\right)$	${\mathit{h}}^{\prime }\left(0\right)$	${\mathit{f}}^{″}\left(0\right)$	${\mathit{h}}^{\prime }\left(0\right)$
0	−1	0	−1	0	−1	0	−1	0
0.5	−1.1384	−0.5128	−1.1384	−0.5128	−1.1384	−0.5128	−1.1384	−0.5128
1.0	−1.3250	−0.8371	−1.3250	−0.8371	−1.3250	−0.8371	−1.3250	−0.8371
2.0	−1.6523	−1.2873	−1.6523	−1.2873	−1.6523	−1.2873	−1.6523	−1.2873
5.0	-	-	-	-	−2.3903	−2.1502	−2.3903	−2.1502

. For convenient purposes, the subscripts ‘1′ and ‘2′ indicate the alumina (Al₂O₃) and the copper (Cu), respectively.

The effects of non-dimensional parameters like Cu nanoparticle volume fraction ${\varphi }_2$, rotating parameter $\mathrm{\Omega }$, radiation parameter $Rd$, and unsteadiness parameter $\beta$, are discussed and illustrated in Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9. We can see from these diagrams that there are two solutions within the first and second solutions when $\lambda >0$. The solutions are found up to a specific critical value $\lambda ={\lambda }_c$.

Figure 2 and Figure 3 depict the changes in the skin friction $f^{″}\left(0\right)$ in the x direction and $h^{\prime }\left(0\right)$ in the y direction, respectively, for the various values of ${\varphi }_2$ when $\mathrm{Pr}=6.2$, $Rd=2.3$, $\beta =-1$, ${\varphi }_1=0.01$ and $\mathrm{\Omega }=0.01$. It is discovered that as the Cu nanoparticle volume fraction ${\varphi }_2$ increases, the values of $f^{″}\left(0\right)$ and $h^{\prime }\left(0\right)$ decrease (increase in absolute sense) for both solutions. There are two solutions for a particular range of stretching strength $\lambda >0$. It is noted that the critical value ${\lambda }_c$ for Cu/H₂O nanofluid $\left({\varphi }_1=0.01, {\varphi }_2=0.00\right)$ is 0.00087 and for the hybrid nanofluid Al₂O₃-Cu/H₂O $\left({\varphi }_1=0.01, {\varphi }_2=0.01, 0.02\right)$ are 0.00082 and 0.00071, respectively. It is found that the critical values ${\lambda }_c$ decrease when the values of ${\varphi }_2$ increase. When compared to the nanofluid, the hybrid nanofluid has a higher concentration of nanoparticles. In particular, a higher concentration of nanoparticles will lead to the boundary layer separation being delayed. In Figure 2, the magnitude of $f^{″}\left(0\right)$ rises in perfect sync with the Cu nanoparticle volume fraction ${\varphi }_2$. It is noted that the solid surface exerts a drag force on the fluid for negative values of $f^{″}\left(0\right)$. Meanwhile, if the stretching strength is less $\left(0.05<\lambda <0.4\right)$, it has the opposite behavior, in which the fluid exerts a drag force on the sheet, represented by positive values of $f^{″}\left(0\right)$. The solution exists up to the critical values of λ as shown in Figure 2, where ${\lambda }_c$ = 0.00087, 0.00082, and 0.00071 for Cu nanoparticle volume fraction parameter ${\varphi }_2=$0.00, 0.01, and 0.02, respectively. It is noted that there are positive and negative values of $f^{″}\left(0\right)$. The positive values indicate a drag force imposed by the fluid on the solid surface, while the negative sign implies a drag force imposed by the solid surface on the fluid. On the other hand, the case $f^{″}\left(0\right)=0$ indicates that the fluid-solid contact is free of friction moving at the same velocity. Overall, the higher the ratio of stretching, the higher the drag force on the surface. There are significant effects on the second solution in Figure 3, where the y-direction skin friction coefficient, $h^{\prime }\left(0\right)$ is always negative for $\lambda >0$ as the drag force is dominant on the solid surface. The effects of drag force in the y direction are less than the effects in the x direction.

In addition, Figure 4 and Figure 5 elucidate the variation in the skin friction $f^{″}\left(0\right)$ in the x direction and heat transfer $-{\theta }^{\prime }\left(0\right)$ for various values of the rotating parameter $\mathrm{\Omega }$ when $\mathrm{Pr}=6.2$, $Rd=2.3$, $\beta =-1$, and ${\varphi }_1={\varphi }_2=0.01$. It is observed that the critical values ${\lambda }_c$ are getting bigger with the increasing values of the rotating parameter $\mathrm{\Omega }$. The critical values of $\lambda$ for the rotating parameter $\mathrm{\Omega }=$ 0.01, 0.02, and 0.03 are ${\lambda }_c=$0.00076, 0.00078, and 0.00079, respectively. The increment values of the rotating parameter depend on the rotation rate, as well as the stretching rate [25]. Figure 4 shows that $f^{″}\left(0\right)$ decreases as $\mathrm{\Omega }$ increases for the first solution, but it increases for the second solution, which is consistent with the results presented in Figure 2. Even though the gap is small, due to the small variation of the rotating parameter $\mathrm{\Omega }$, but the existence of the dual solutions can still be seen. However, Figure 5 shows the opposite results, where for the first solution, increasing $\mathrm{\Omega }$ leads to a decrease in the heat transfer rate $-{\theta }^{\prime }\left(0\right)$, while for the second solution, it rises.

The variations of heat transfer rate $-{\theta }^{\prime }\left(0\right)$ with $\lambda$ for various values of radiation parameter $Rd$ when $\mathrm{Pr}=6.2$, $\mathrm{\Omega }=0.01$, $\beta =-1$, and ${\varphi }_1={\varphi }_2=0.01$ are presented in Figure 6. We note that as $Rd$ increases, the absolute value of $-{\theta }^{\prime }\left(0\right)$ decreases. The critical values, ${\lambda }_c=$ 0.00914, 0.00873, and 0.00687 for $Rd=$2.0, 2.3, and 2.5, respectively, are also presented in this figure. It is noted that Rd gives no effect on the skin friction coefficients for both the x and y directions, which is expected since the velocity field is not affected by the thermal field, see Equations (10)–(13).

Figure 7, Figure 8 and Figure 9 elucidate the effects of the unsteadiness parameter $\beta$ on the fluid velocity in the x and y directions, as well as the fluid temperature. Initially, by increasing $\beta$, the velocity of the first solution decreases, while the velocity of the second solution increases. However, the opposite behavior is seen in the velocity in the y direction. When the fluid moves towards inviscid flow, the directional movement of the velocity changes in the x direction for first and second solutions, while the movement of velocity in the y direction is consistent with decreasing towards the quiescent fluid. These scenarios imply thickening of the velocity boundary layer. In addition, the temperature increases for both solutions. It shows a consistent analysis that the thermal boundary layer thickness is also rising.

4. Conclusions

The problem of the unsteady 3D rotating hybrid nanofluid flow on a stretching sheet was explored. The governing PDEs were transformed to ODEs using a suitable similarity transformation. The effects of the involved parameters on the physical quantities of interest were visually shown and analyzed. The existence of double solutions was discovered for the stretching situation. In addition, the higher concentration of the nanoparticle volume fraction slowed down the boundary layer flow separation. The function $h\left(\eta \right)$ was found to be negative, which explains that the counterclockwise fluid rotation influences the fluid flow in the negative y direction. The positive skin friction coefficient shows that the fluid imposes a drag force on the solid surface, while the negative value implies the contrary. The results showed that the radiation parameter, Rd, emits the heat energy into the boundary layer, thus leading to a temperature rise of the hybrid nanofluid and subsequently enhancing the heat transfer rate of the fluid.