Impact of the KKL Correlation Model on the Activation of Thermal Energy for the Hybrid Nanofluid (GO+ZnO+Water) Flow through Permeable Vertically Rotating Surface

Abstract

The thermal energy transfer characteristics during hybrid nanofluid migration are studied in the presence of a variable magnetic field, heat source, and radiation. The flow is governed by the conservation laws of mass, momentum, and energy, whereas it is modeled by the coupled set of nonlinear partial differential equations (PDEs). Suitable similarity transformations are employed to convert the developed set of PDEs to a nonlinear system of coupled ordinary differential equations (ODEs). The simplified system of ODEs is solved by using the well-established analytical procedure of homotopy analysis method (HAM). The effects of varying the strength of the physical parameters on the thermal energy transfer during hybrid nanofluid motion between two plates in which one of the plate is porous, rotating, as well as stretching are investigated through tables and two-dimensional graphs. The porosity is modeled through the Koo–Kleinstreuer model (KKL) correlation. The analysis reveals that the skin friction and Nusselt number augment with the increasing strength of the magnetic field and nanomaterials’ concentrations. The gradient in the fluid velocity has a dual dependence on the strength of the applied magnetic field and Grashof number and drops with the higher values of the unsteadiness parameter. The fluid velocity constricts with the enhancing magnetic field due to higher Lorentz forces, and it also drops with the increasing rotation rate. The enhancing buoyancy associated with higher Grashof number values augments the fluid velocity. The fluid’s temperature rises with the augmenting nanomaterial concentrations, Eckert number, nonsteadiness, heat source strength, and radiation parameter, while it drops with the higher Grashof number and Prandtl number. The applied technique of the HAM shows good convergence over a wide range of the convergent parameter. This work has potential applications in the development of efficient thermal energy transfer systems.

1. Introduction

Thermal energy transfer has great importance and utilization in almost any type of physical process. Thermal energy transfer can take place due to conduction, convection, and radiation. The conduction involves the transfer of heat due to direct contact, whereas convection occurs due to the accumulative motion of the heated substance. Radiation is the mode of thermal energy transfer due to electromagnetic radiation. The ordinary fluids that are used for convective thermal energy transport are water, ethylene, ethylene glycol, etc. Due to low thermal conductivities, these ordinary fluids have limited thermal energy transfer efficiency. The thermal energy transport ability of ordinary liquids can be increased by adding solid particles of the appropriate size and nature. If the solid particles are in the nanometer size range and are metallic in nature, then their addition to ordinary fluids can increase the thermal energy transport efficiency enormously [1,2]. The fluid obtained due to the mixing of the ordinary fluid and nanoparticles is called a nanofluid. The idea of this fascinating fluid was developed by Choi [3] for the first time. The nanofluids are prepared by uniformly dispersing the nanomaterials in the ordinary liquids. The various thermophysical properties (for example, specific heat, thermal conductivity, and viscous nature) of such a type of fluid are highly dependent on the nanomaterial’s associated characteristics, such as its nature, size, concentration, temperature, and the nature of the host fluid. Throughout the entire globe, researchers have studied the nanofluid thermal energy transfer efficiency under different constraints to model the nanofluid’s behavior [4,5]. Nanofluid has been extensively employed in the storage of solar energy [6,7], thermal energy exchangers [8,9,10], freezing processes [11], thermal energy transport through radiations [12,13,14], etc. Mebarek-Oudina in [15] investigated the nanofluid thermal energy characteristics by using different base fluids. Muhammad et al. [16] simulated the nanofluid magneto-hydrodynamic (MHD) migration by incorporating the effects of the slip condition and motile microorganisms. Anum et al. [17,18] analyzed the sensitivity of different types of fluid numerically. The Darcy–Forchheimer nanofluid motion through a curved surface was analytically investigated by Sajjad et al. [19]. The nanofluid’s mobility through an induced changing magnetic field by employing the second-grade fluid model was examined by Ambreen et al. [20]. Some recent investigations through computational and theoretical models to handle the heat exchange capabilities of nanofluids were performed in [21,22,23].

Recently, growing interest has developed in examining the thermal energy transport characteristics of hybrid nanofluids. Such types of fluids are obtained by intermixing nanomaterials of more than one type in the ordinary fluids. Hybrid nanofluids possess distinct and improved thermophysical characteristics as compared with the single-component nanofluid. There exists various kinds of hybrid nanoliquids [24,25,26]. The tremendous amount of research work displays that hybrid nanofluid are more versatile in comparison with the ordinary nanofluid. Suresh et al. [27] examined the thermal characteristics of a hybrid nanofluid consisting of copper and alumina nanoparticles using the two-step approach. Chamkha et al. [28] simulated the time-varying conjugate motion of the (Cu+Al${}_2$O${}_3$+H${}_2$O) hybrid nanofluid through a triangular container. The authors reported a substantial enhancement in the Nusselt number. Momin [29] experimentally investigated the hybrid nanofluid mixed convective laminar flow through an inclined tube. The improved thermal energy transfer characteristics and the friction parameter of the (MWCNT+Fe${}_3$O${}_4$+H${}_2$O) hybrid nanofluid were examined by Sundar et al. [30]. Ghachem et al. [31] investigated the rate of heat energy transfer through a heat exchanger employing the hybrid nanofluid. Lund et al. [32] studied the impact of viscous dissipation on the (Cu+Al${}_2$O${}_3$+H${}_2$O) hybrid nanofluid’s motion across squeezing surfaces using stability analysis. Suresh et al. [33] also investigated the possible merits of hybrid nano-powder (copper + alumina) for the efficient performance of a thermal energy system. They used water as the host fluid. Usman et al. [34] examined the hybrid nanofluid’s (Cu+Al${}_2$O${}_3$+H${}_2$O) motion from a permeable container taking into account the influences of varying thermal conduction and the source of nonlinear radiation suing LSM. The numerical examination of the impacts due to the (MWCNT+Fe${}_3$O${}_4$+H${}_2$O) hybrid nanofluid’s migration from a corrugated channel possessing separate heating/cooling sections was performed by Mohebbi et al. [35] employing the lattice Boltzmann method (LBM). Minna et al. [36] gave an overview by describing the hybrid nanofluids’ development and their advantages.

This manuscript deals with the investigation of the thermal energy transfer characteristics of the hybrid nanofluid’s (GO+ZnO+H${}_2$O) migration between two plates in the presence of a varying magnetic field, radiation, and heat source. The right plate is porous, rotating with an angular velocity $\Omega$ and stretching with velocity u. This work has potential applications in the development of efficient thermal energy transfer systems using hybrid nanofluids. The structure of the present articles is as follows:

The hybrid nanofluid flow is modeled through coupled PDEs in Section 2. The coupled system of nonlinear PDEs is transformed to a simple form by using suitable similarity transformations in Section 3. The system of coupled ODEs is analytically solved using the HAM in Section 5. The results are obtained and then graphically displayed in Section 6. The work is then concluded in Section 7.

2. Mathematical Model of the Problem

The geometry of the studied problem is shown in Figure 1. It consists of two plates, in which the right plate rotates along the y-axis with angular velocity $\Omega$ and stretches along the x-axis with velocity u. The hybrid nanofluid (GO+ZnO+H${}_2$O) is synthesized by mixing the nanoparticles of graphene oxide (Go) and zinc oxide (ZnO) with water (H${}_2$O). The hydrothermal features of the hybrid nanofluid three-dimensional (3D), incompressible, and nonsteady motion are investigated in the presence of an applied variable magnetic field. The time dependence of the magnetic field is expressed by the relation $B\left(t\right)=B_0{(1-\delta t)}^{-0.5}$ and is acting along the y-axis. The temperature at the left wall is $T_h$ and at the right wall is $T_w$. The gravitational force is acting vertically downward along the x-axis. We used the following assumptions in the current study:

• The flow is governed by the vessel wall with the stretching velocity $U_w=\chi x{(1-\delta t)}^{-1}$, where $\chi$ and $\delta$ are positive constants with $\delta t<1$.
• The effective stretching rate $\chi {(1-\delta t)}^{-1}$ enhances subject to an external force acting along the $x$-axis for $0\le \delta <1$.
• The Reynolds number (magnetic) is considered to be much less than unity to overcome the impact of the magnetic field induced.
• The magnetic field is applied in the transverse direction to the hybrid nanofluid flow.

The following equations model the hybrid nanofluid 3D flow: [37]:

$$u_x+v_y+w_z=0,$$

(1)

$${\rho }_{hnf}\left(u_t+u{u}_x+v{u}_y+2\Omega w\right)=-p_x+{\mu }_{hnf}\left(u_{xx}+u_{yy}\right)-{\sigma }_{hnf}{B}^2\left(t\right)u+g{\left(\rho \beta \right)}_{hnf}\beta (T-T_w),$$

(2)

$${\rho }_{hnf}\left(v_t+u{v}_x+v{v}_y\right)=-p_y+{\mu }_{hnf}\left(v_{xx}+v_{yy}\right),$$

(3)

$${\rho }_{hnf}\left(w_t+u{w}_x+v{w}_y-2\Omega w\right)={\mu }_{hnf}\left(w_{xx}+w_{yy}\right)-{\sigma }_{hnf}{B}^2\left(t\right)w,$$

(4)

$$\begin{array}{c}\hfill {\left(\rho C_p\right)}_{hnf}\left(T_t+u{T}_x+v{T}_y+w{T}_z\right)=k_{eff}\left(T_{xx}+T_{yy+T_{zz}}\right)+{\mu }_{hnf}(2\left[u_x^2+v_y^2+w_z^2\right]+v_x^2+v_z^2+\\ \hfill {\left(w_x+u_z\right)}^2)+Q_s(T-T_w)-q_y^r=0,\end{array}$$

(5)

here, ${\rho }_{hnf}$ (${\sigma }_{hnf}$) is the hybrid nanofluid density (electrical conductivity), ${\mu }_{hnf}$ is the fluid dynamical viscosity, $B\left(t\right)$ is the applied time-varying magnetic field, $Q_s$ is the heat generation/absorption rate, $k_{hnf}$ is the thermal conductivity, and ${\left(\rho C_p\right)}_{hnf}$ is the specific heat at constant pressure. The flux ($q_r$) of radiation is expressed as [38]:

$$\frac{\partial q_r}{\partial y}=-\frac{16{\sigma }^s{T}_{\infty }^3}{3k_1}\frac{{\partial }^{2}T}{\partial y^2};$$

(6)

here, the symbol $k_1$ (${\sigma }^s$) represents the average coefficient of absorption (Stefan–Boltzmann constant). The boundary conditions for Equations (1) to (5) are:

$$\begin{array}{c}\hfill u=U_w,v=V_w,w=0,T=T_h,\mathrm{at}y=0,\\ \hfill u\to 0,v\to 0,w\to 0,T\to T_w,\mathrm{at}y\to h.\end{array}$$

(7)

Here, $V_w=-{\left(\frac{\nu \chi }{1-\delta t}\right)}^{0.5}f\left(\eta \right)$ denotes the velocity of the water at the wall of the vessel at $\eta =0$. Moreover, the slip factors for the state variables are ignored, and $V_w$ corresponds to suction and injection subject to the cases $V_w<0$ and $V_w>0$, respectively. The hybrid nanofluid thermal expansion coefficient, thermal conductivity, electrical conductivity, specific heat, viscosity, and density are defined as follows:

$$\frac{{\left(\rho \beta \right)}_{hnf}}{{\left(\rho \beta \right)}_f}=(1-{\phi }_2)\left[(1-{\phi }_1)+{\phi }_1\left(\frac{{\left(\rho \beta \right)}_{s1}}{{\left(\rho \beta \right)}_f}\right)\right]+\frac{{\phi }_2{\left(\rho \beta \right)}_{s2}}{{\left(\rho \beta \right)}_f},$$

(8)

$$\begin{array}{c}\hfill \frac{k_{hnf}}{k_{bf}}=(1-{\phi }_2)+2{\phi }_2\left(\frac{k_{m2}}{k_{m2}-k_{bf}}\right)ln\left(\frac{k_{m2}+k_{bf}}{2k_{bf}}\right),\\ \hfill \frac{k_{bf}}{k_f}=(1-{\phi }_1)+2{\phi }_1\left(\frac{k_{m1}}{k_{m1}-k_f}\right)ln\left(\frac{k_{m1}+k_f}{2k_f}\right),\end{array}$$

(9)

$$\begin{array}{c}\hfill \frac{{\sigma }_{hnf}}{{\sigma }_{bf}}=\left[1+\frac{3\left(\frac{{\sigma }_{m2}}{{\sigma }_{bf}}-1\right){\phi }_2}{\left(\frac{{\sigma }_{m2}}{{\sigma }_{bf}}+2\right)-\left(\frac{{\sigma }_{m2}}{{\sigma }_{bf}}-1\right){\phi }_2}\right],\\ \hfill \frac{{\sigma }_{bf}}{{\sigma }_f}=\left[1+\frac{3\left(\frac{{\sigma }_{m1}}{{\sigma }_f}-1\right){\phi }_1}{\left(\frac{{\sigma }_{m1}}{{\sigma }_f}+2\right)-\left(\frac{{\sigma }_{m1}}{{\sigma }_f}-1\right){\phi }_1}\right],\end{array}$$

(10)

$$\frac{{\left(\rho C_p\right)}_{hnf}}{{\left(\rho C_p\right)}_f}=\left[(1-{\phi }_2)\left(1-\left(1-\frac{{\left(\rho C_p\right)}_{m1}}{{\left(\rho C_p\right)}_f}\right){\phi }_1\right)+{\phi }_2\frac{{\left(\rho C_p\right)}_{m2}}{{\left(\rho C_p\right)}_f}\right],$$

(11)

$$\frac{{\mu }_{hnf}}{{\mu }_f}=\frac{1}{{(1-{\phi }_1)}^{2.5}{(1-{\phi }_2)}^{2.5}},$$

(12)

$$\frac{{\rho }_{hnf}}{{\rho }_f}=\left[(1-{\phi }_2)\left(1-\left(1-\frac{{\rho }_{m1}}{{\rho }_f}\right){\phi }_1\right)+{\phi }_2\frac{{\rho }_{m2}}{{\rho }_f}\right].$$

(13)

The effective thermal conductivity is defined as follows:

$$k_{ef}=k_{sta}+k_{Brow},$$

(14)

where, $k_{sta}=\left[(1-{\phi }_2)+2{\phi }_2\left(\frac{k_{m2}}{k_{m2}-k_{bf}}\right)ln\left(\frac{k_{m2}+k_{bf}}{2k_{bf}}\right)\right]\left[(1-{\phi }_1)+2{\phi }_1\left(\frac{k_{m1}}{k_{m1}-k_f}\right)ln\left(\frac{k_{m1}+k_f}{2k_f}\right)\right]$. Using the KKL correlation [39], $k_{Brow}$ is given by:

$$k_{Brow}=5\times {10}^4\phi {\left(\rho C_p\right)}_f\sqrt{\frac{k_{b}T}{{\rho }_p{d}_p}}f(\phi ,T),$$

(15)

where $k_{f,ef}=\frac{1}{R_f{d}_p+\frac{d_p^2}{K_f}}$ is widely used for $k_f$. Furthermore,

$$\begin{array}{c}\hfill g^{\prime }(\phi ,T,d_p)=\left(a_1+a_{2}ln\left(d_p\right)+a_{3}ln\left(\phi \right)+a_{4}ln\left(\phi \right)ln\left(d_p\right)+a_{5}ln{\left(d_p\right)}^2\right)ln\left(T\right)\\ \hfill +\left(a_6+a_{7}ln\left(d_p\right)+a_{8}ln\left(\phi \right)+a_{9}ln\left(\phi \right)ln\left(d_p\right)+a_{10}ln{\left(d_p\right)}^2\right),\end{array}$$

(16)

where the coefficients $a_i$ for $i=0,1,2,\dots ,10$ are used for the nanoparticles, and we have from Equation (15),

$$k_{Brow}=5\times {10}^4\phi {\left(\rho C_p\right)}_f\sqrt{\frac{k_{b}T}{{\rho }_p{d}_p}}{g}^{\prime }(\phi ,d_p,T).$$

(17)

On the other hand, the dynamic viscosity in functional form can be defined as:

$${\mu }_{ef}={\mu }_{sta}+{\mu }_{Brow}=\frac{k_{Brow}}{k_f}\times \frac{{\mu }_f}{P{r}_f}+{\mu }_{sta},$$

(18)

where ${\mu }_{sta}=\frac{{\mu }_f}{{(1-{\phi }_1)}^{2.5}{(1-{\phi }_2)}^{2.5}}$.

3. Similarity Transformations

The similarity transformations that are used to transform Equations (1)–(6) are expressed as [37]:

$$\begin{array}{cc}\hfill & u=\frac{\chi x}{(1-\delta t)}{f}^{\prime }\left(\eta \right),v=-{\left(\frac{\chi \nu }{1-\delta t}\right)}^{0.5}f\left(\eta \right),\theta =\frac{T-T_w}{T_h-T_w},\hfill \\ \hfill & w=-{\left(\frac{\chi x}{1-\delta t}\right)}^{0.5}g\left(\eta \right),\eta =y{\left(\frac{\chi }{\nu (1-\delta t)}\right)}^{0.5}.\hfill \end{array}$$

(19)

After applying the similarity transformations (19), we obtain:

$$f^{⁗}-R\frac{{ϵ}_1}{{ϵ}_2}(f^{\prime }{f}^{‴}-f{f}^{⁗})-2Kr\frac{{ϵ}_1}{{ϵ}_2}{g}^{\prime }-{ϵ}_5\frac{M}{A_2}{f}^{″}-Re\frac{{ϵ}_1}{{ϵ}_2}G{r}_m{\beta }^{*}{\theta }^{\prime }-A_0(\frac{\eta }{2}{f}^{″}+f^{\prime })=0,$$

(20)

$$g^{″}-R\frac{{ϵ}_1}{{ϵ}_2}(f^{\prime }g-f{g}^{\prime })+2Kr\frac{{ϵ}_1}{{ϵ}_2}{f}^{\prime }-{ϵ}_5\frac{M}{{ϵ}_2}g+A_0(\frac{\eta }{2}{g}^{\prime }+g)=0,$$

(21)

$${\theta }^{″}+Pr\frac{{ϵ}_2{ϵ}_3}{{ϵ}_1{ϵ}_4}\left[Re\frac{{ϵ}_1}{{ϵ}_2}f{\theta }^{\prime }+Ec\frac{{ϵ}_1}{{ϵ}_3}(4f^{\prime }{}^2+g^2)+\frac{{ϵ}_1}{{ϵ}_2{ϵ}_3}ReQ\theta \right]/(1+\frac{4Rd}{3{ϵ}_4})-A_0\frac{\eta }{2}{\theta }^{\prime }=0,$$

(22)

and the boundary conditions become

$$f\left(\eta \right)=S,f^{\prime }\left(\eta \right)=1,\theta \left(\eta \right)=1,g\left(\eta \right)=0:\eta =0,f^{\prime }\left(\eta \right)\to 0,f\left(\eta \right)\to 0,g\left(\eta \right)\to 0,//\theta \left(\eta \right)\to 0:\eta \to 1.$$

(23)

Here, ${ϵ}_1=\frac{{\rho }_{hnf}}{{\rho }_f}$, ${ϵ}_2=\frac{{\mu }_{eff}}{{\mu }_f}$, ${ϵ}_3=\frac{{\left(\rho C_P\right)}_{hnf}}{{\left(\rho C_p\right)}_f}$, ${ϵ}_4=\frac{k_{eff}}{K_f}$, ${ϵ}_5=\frac{{\sigma }_{hnf}}{{\sigma }_f}$, and $A_0=\frac{\delta }{\chi }$ are constants, $G{r}_n=\frac{g{\beta }_f(T_h-T_w)}{{\chi }^{2}x}$ is the Grashof number, $\beta =\frac{{\beta }_{hnf}}{{\beta }_f}$ is the thermal expansion coefficient, $Q=\frac{Q_0}{\chi {\left(\rho C_p\right)}_f}$ is the heat source/sink parameter, $M=\frac{{\sigma }_f{B}_0^2{h}^2}{{\rho }_f{\nu }_f}$ is the magnetic parameter, $kr=\frac{\Omega h^2}{{\nu }_f}$ is the rotation parameter, $Pr=\frac{\mu {\left(\rho C_p\right)}_f}{{\rho }_f{k}_f}$ is the Prandtl number, $Ec=\frac{{\rho }_f{a}^2{h}^2}{{\left(\rho C_p\right)}_f({\theta }_w-{\theta }_h)}$ is the Eckert number, $Rd=\frac{16{\sigma }^s{T}_h^3}{3k{k}^s}$ is the radiation parameter, and $Re=\frac{\chi h^2}{{\nu }_f}$ is the Reynolds number.

4. Engineering Quantities of Interest and Their Calculation

The quantities of engineering interest are defined as:

$$C_{fx}=\frac{{\tau }_w}{0.5{\rho }_{hnf}\left(U_w\right)},N{u}_x=\frac{x{q}_x}{k_{hnf}(T_w-T_{\infty })},$$

(24)

and

$$C{f}_{x}R{e}^{1/2}=\frac{2{(1-{\phi }_2)}^{-2.5}}{{(1-{\phi }_1)}^{2.5}}{f}^{″}\left(0\right),\frac{N{u}_x}{R{e}_x^{1/2}}=-k_{hnf}{k}_f^{-1}{\theta }^{\prime }\left(0\right).$$

(25)

The thermo-physical characteristics of of hybrid nanomaterials, the physical model for the computation of the hybrid nanofluid’s different properties, and the calculation of the skin friction and Nusselt number are given in

Table 1. Thermo-physical properties of GO, ZnO, and H${}_2$O [40].

Properties (Thermo-Physical)	$\mathbf{k}\left(\frac{\mathbf{W}}{\mathbf{m}.\mathbf{K}}\right)$	$\mathbf{\ae }\left(\frac{\mathbf{Kg}}{{\mathbf{m}}^3}\right)$	${\mathbf{c}}_{\mathbf{p}}\left(\frac{\mathbf{J}}{\mathbf{Kg}.\mathbf{K}}\right)$
GO	5000	3600	765
H${}_2$O	0.613	997.1	4179
ZnO	25	5700	523

,

Table 2. Variations in the $C{f}_{x}R{e}_x^{0.5}$ values under the impact of different parameters.

M	${\mathit{\phi }}_1-{\mathit{\phi }}_2$	$\mathit{kr}$	${\mathit{Gr}}_{\mathit{n}}$	${\mathit{Cf}}_{\mathit{x}}$
0.2	0.01	0.3	0.1	1.24321
0.4				1.49334
0.6	0.03			1.58438
	0.06			1.67854
		0.5		1.79825
		0.7		1.98252
			0.3	1.82112
			0.5	1.64678

and

Table 3. Variations in the $N{u}_{x}R{e}_x^{-0.5}$ values under the impact of different parameters.

${\mathit{\phi }}_1-{\mathit{\phi }}_2$	M	Q	$\mathit{Rd}$	$\mathit{Ec}$	$\mathit{Pr}$	${\mathit{Nu}}_{\mathit{x}}$
0.01	0.2	0.1	0.1	1	6.0	0.112530
0.03						0.223450
0.06	0.4					0.238790
	0.6					0.339988
		0.3				0.441230
		0.6				0.541537
			0.3			0.651980
			0.6			0.767098
				2		0.889043
				3		0.892081
					6.3	0.742980
					6.6	0.705678

.

Table 2 shows the computation of the skin friction $C{f}_x$ with changing $M,{\phi }_1-{\phi }_2,kr$, and $G{r}_n$. It is clear that the skin friction augments with the rising values of all these parameters, except $G{r}_n$. The skin friction first augments and then drops with the rising $G{r}_n$.

The computation of the Nusselt number ($N{u}_x$) with changing values of the parameters ${\phi }_1-{\phi }_2,M,Q,Rd,Ec$, and $Pr$ is displayed in Table 3. The table shows that the Nusselt number augments with the increasing values of all these parameters, except $Pr$. The Nusselt number first rises and then drops with the rising $Pr$ values.

5. Solution by the HAM

For a solution of the system of Equations (20)–(22) together with the boundary conditions (23), we used the HAM. The basic idea and use of the HAM are explained in [41,42,43]. For the solution purpose, we used the following conditions:

$$\widehat{f}=1-exp\left(\eta \right),\widehat{\theta }=exp(-\eta ),\widehat{\varphi }=exp\left(\eta \right),$$

(26)

satisfying the operators:

$$L_{\widehat{f}}\left(\widehat{f}\right)=f^{‴},L_{\widehat{\theta }}\left(\widehat{\theta }\right)={\theta }^{\prime },L_{\widehat{\varphi }}\left(\widehat{\varphi }\right)={\varphi }^{\prime }.$$

(27)

6. Results and Discussion

This section explains the hydrothermal features of the assumed hybrid nanofluid 3D motion through rotatory plates in the presence of a varying applied magnetic field. The variations of the state variables with the parameters of practical importance are graphically depicted.

Figure 2 displays the variation of $f^{\prime }\left(\eta \right)$ with increasing strength of unsteadiness parameter $A_0$. The values of $A_0$ were taken as $1.0,1.5,2.0,2.5$. At fixed $A_0$, the velocity field gradient drops with enhancing $\eta$. The augmenting $A_0$ causes the profile of $f^{\prime }\left(\eta \right)$ to drop. The drop rate enhances with the increasing $\eta$. The different curves become almost parallel at the highest $\eta$. Thus, the increasing unsteadiness during the hybrid nanofluid motion mitigates the velocity field gradient due to the increasing turbulence associated with higher values of $A_0$.

The dependence of $f^{\prime }\left(\eta \right)$ on the magnetic field strength through M is shown in Figure 3. Dual behavior is observed in the $f^{\prime }\left(\eta \right)$ profile with increasing M. It is observed that $f^{\prime }\left(\eta \right)$ drops with increasing M up to about $\eta =0.5$ (mid the boundary layer) and then enhances. Both rates (drop and rise) augment with the increasing M as is clear from the increasing separation between different curves in both regions. It is therefore concluded that the increasing strength of the magnetic parameter results in a uniform velocity distribution for smaller $\eta$ (near the hot plate), whereas this makes the velocity distribution non-uniform at higher $\eta$ values due to higher dynamic viscosity of the fluid.

The impact of enhancing magnetic field strength through the magnetic parameter (M) on the velocity $f\left(\eta \right)$ is displayed in Figure 4. The values of M were taken as $5.0,10.0,15.0,20.0$. The graph shows that at fixed M, the velocity first enhances, attains the maximum value, and then, drops with the rising values of $\eta$. The augmenting M results in a drop in the $f\left(\eta \right)$ profile. The dropping rate enhances with the rising values of M as is clear from the increasing separation between different curves and is more drastic at the intermediate values of $\eta$. The various curves overlap at about $\eta =0.9$. This drop in the fluid velocity is due to higher Lorentz forces, corresponding to higher values of M.

The variation of the fluid velocity $f\left(\eta \right)$ with enhancing Grashof number ($G{r}_n$) is depicted in Figure 5. It is observed that for a given value of $G{r}_n$, the fluid velocity first enhances, reaches the maximum, and then, drops to smaller values with the increasing $\eta$. The velocity profile rises with the augmenting $G{r}_n$. The increase in the velocity profile is more prominent at the intermediate values of $\eta$ as is clear from the wider separation between the different curves. The different curves overlap at about $\eta =0.7$. Thus, the fluid presents a smaller opposing force to the flow due to lower viscosity at higher $G{r}_n$ values and, hence, augments the fluid velocity. This increase in velocity is appreciable only in the middle region of the boundary layer.

The impact of the enhancing Grashof number $G{r}_n$ on the velocity gradient $f^{\prime }\left(\eta \right)$ is depicted in Figure 6. The values of $G{r}_n$ chosen were $10,30,50,70$. The velocity gradient displays dual behavior with the augmenting $G{r}_n$. At smaller $\eta$, the velocity gradient profile enhances, whereas at larger $\eta$, it drops with the increasing $G{r}_n$. The figure also displays that the rate of drop is higher as compared to the rate of enhancement with the increasing $G{r}_n$. Thus, the enhancing buoyancy associated with the higher values of $G{r}_n$ reduces the gradient in the fluid velocity beyond about the mid-point of the boundary layer. Therefore, it is concluded that the increasing buoyancy (decreasing viscosity) associated with higher $G{r}_n$ does not affect the velocity distribution of the hybrid nanofluid drastically.

The effect of enhancing values of the rotation parameter $kr$ on the hybrid nanofluid velocity $f\left(\eta \right)$ is plotted in Figure 7. The values of $kr$ were chosen as $1,3,5,7$. Figure shows that at a given value of $kr$, the fluid velocity first enhances, reaches the maximum, and then, drops with the rising $\eta$. The figure also displays that the augmenting $kr$ causes the fluid velocity profile to drop. The rate of velocity drop decreases with the increasing $kr$ as displayed by the decreasing separation between the velocity curves at higher values of $kr$. The different curves overlap at about $\eta =0.82$. Thus, the enhancing rotation rate offers a higher constriction to the fluid velocity, which causes the fluid velocity to drop. Thus, the enhancing rotation rate associated with the higher values of $kr$ saturates the velocity drop during the hybrid nanofluid migration.

Figure 8 depicts the dependence of $g\left(\eta \right)$ on the Grashof number $G{r}_n$. It is evident that at a fixed value of $G{r}_n$, the velocity component first increases, attains the maximum value, and then, drops to smaller values with augmenting $\eta$. The increasing Grashof number values result in an enhancement in the velocity of the hybrid nanofluid. The increase in the velocity is more prominent at the intermediate values of $\eta$. The different curves overlap at about $\eta =0.78$. Thus, the increasing buoyancy augments the hybrid nanofluid velocity.

The impact of magnetic field strength through magnetic parameter M on the velocity component $g\left(\eta \right)$ is exhibited in Figure 9. The values of M were chosen as $5.0,10.0,15.0,20.0$. The graph exhibits that the augmenting magnetic parameter strength drops the fluid velocity due to enhancing Lorentz forces acting on the fluid motion. Thus, the increasing magnetic parameter strength affects $g\left(\eta \right)$ drastically in the far region of the boundary layer due to the higher constriction of the flow in the comparatively colder region of the fluid.

The variation of fluid’s temperature $\theta \left(\eta \right)$ with a changing nanomaterials’ concentration is depicted in Figure 10. The concentration of both types of nanomaterials was taken as equal. The figure displays that the enhancing fluid concentration causes the fluid’s temperature to augment. The increase in fluid’s temperature is more prominent with the increasing values of $\eta$, and this increase has almost the same rate for different values of the nanomaterials’ concentrations as is clear from the equal spacing between different curves. It is also clear from the figure that at a fixed concentration, the fluid’s temperature drops as we move away form the hot plate of the configuration.

The variation of the fluid’s temperature $\theta$ with enhancing values of the unsteadiness parameter $A_0$ is displayed in Figure 11. The figure shows that the increasing $A_0$ results in an enhancement in the fluid’s temperature. The rate of enhancement with the rising $A_0$ remains almost constant as is clear from the equal separation between different curves. The different curves becomes almost parallel at the highest $\eta$. Furthermore, at fixed $A_0$, the fluid’s temperature drops as we move away from the hot plate.

Figure 12 shows the variation of the temperature distribution with the enhancing Eckert number $Ec$. The values of $Ec$ were taken as $1.0,1.5,2.0,2.5$. It is evident from the figure that the rising $Ec$ values augment the fluid’s temperature. The rate of increase rises with the higher $Ec$ values as evident from the wider separation between different curves. The different curves overlap at about $0.9$. Thus, the enhancing fluid velocity associated with higher values of $Ec$ results in an increase in the heat dissipation, which augments the fluid’s temperature distribution.

The dependence of the fluid’s temperature on the enhancing values of the Grashof number $G{r}_n$ is plotted in Figure 13. The chosen values of $G{r}_n$ were $10,30,50,70$. It is clear that the at given values of the Grashof number, the fluid’s temperature first enhances, reaches the maximum, and then, drops with the rising values of the independent variable $\eta$. The augmenting strength of $G{r}_n$ reduces the fluid’s temperature. The rate of the reduction of the fluid’s temperature increases with the rising values of $G{r}_n$ as is clear from the greater separation between different curves. Furthermore, the dropping rate of the fluid’s temperature is more drastic for the intermediate values of $\eta$. Thus, the decreasing viscous dissipation associated with the higher values of $G{r}_n$ reduces the fluid’s temperature distribution.

The impact of the enhancing Prandtl number $Pr$ on the fluid’s temperature distribution is displayed in Figure 14. The figure shows that the rising values of the Prandtl number result in a reduction in the fluid’s temperature distribution. The rate of reduction of $\theta \left(\eta \right)$ is more prominent at the intermediate values of $\eta$. The different curves overlap at about $\eta =0.9$. Thus, the increasing thermal diffusivity associated with the higher values of $Pr$ reduces the fluid’s temperature distribution, which is more prominent in the middle region of the system.

The impact of enhancing heat source strength Q on the fluid’s temperature is displayed in Figure 15. The values of Q were chosen as $0.1,0.2,0.3,0.4$. It is clear that the augmenting values of Q enhance the fluid’s temperature distribution. The enhancement rate augments with the rising values of the source strength as evident from the larger separation between different curves. Furthermore, the enhancement rate is more obvious at the intermediate values of $\eta$.

The variation of the fluid’s temperature with the rising strength of the radiation parameter $Rd$ is exhibited in Figure 16. The graph shows that at a given value of $Rd$, the fluid’s temperature drops with the rising $\eta$. The increasing values of $Rd$ result in an enhancement in the fluid’s temperature. The enhancement rate rises with the augmenting $Rd$ values as evident from the increasing separation between different curves.

Convergence Analysis

The validity of the implemented technique for the transformed system of equations is based on the convergence of the series solution at $q=1$, which is directly related to the h-curves. The suitable and educated guess of this parameter bounds the series to converge. Based on the assumptions and choice of certain parameters, Figure 17 displays the convergence analysis of the applied analytical procedure, the HAM. The figure shows that the HAM is convergent over a wide range of the convergent parameter h.

7. Conclusions

The research work is concluded in this section. The heat transfer characteristics during the hybrid nanofluid’s (GO+ZnO+H${}_2$O) 3D incompressible unsteady motion between two parallel plates in the presence of an applied varying magnetic field were investigated. The right plate was porous, rotating with angular velocity $\Omega$ and stretching with velocity u. The flow was modeled using continuity, momentum, and energy equations. The system of developed coupled PDEs was transformed to a simple set of coupled ODEs through similarity transformations and then solved by employing the standard technique of the HAM. The impacts of the parameters of interest on the fluid motion were investigated through tables and graphs. The main findings of the current study are presented as follows:

• The skin friction augments with the rising magnetic parameter, the difference in nanomaterials’ concentrations, and the rotation parameter.
• The Nusselt number rises with the enhancing magnetic field, heat source strength, radiation parameter, and Prandtl number.
• The gradient in the fluid velocity’s horizontal component increases with the enhancing nonsteadiness parameter and has dual dependence on the augmenting magnetic field intensity.
• The horizontal component of the fluid velocity drops with the rising magnetic field and rotation parameter, whereas it rises with the increasing Grashof number.
• The vertical component of the fluid velocity rises with the increasing Grashof number and falls with the enhancing magnetic field strength.
• The hybrid nanofluid’s temperature augments with the enhancing radiation parameter and the heat source strength, while it drops with the increasing Prandtl number, Grashof number, nonsteady parameter, and nanomaterials’ concentrations.