Numerical simulation for thermal enhancement of $H_2O$ + Ethyl Glycol base hybrid nanofluid comprising $GO + (Ag, AA7072, MoS_2)$ nano entities due to a stretched sheet

1.   Introduction

Non-Newtonian fluids have recently been revealed to be more adequate for technical and engineering implementations than Newtonian fluids. Such fluids are utilized in a vast scope of engineering and industrial mechanisms, including food manufacturing, metallurgical procedures, the extraction of petroleum goods from crude oil, drilling processes, and bio-engineering procedures. Non-Newtonian liquids encompass viscoelastic liquid, Williamson liquid, Jeffrey liquid, micro polar fluid, power-law liquid, Casson fluid, and others. In 1959, a concept of the Casson fluid ^[1] is conveyed, which is one of the most significant models for revealing the resources of yield stresses. The connection of fluid and solid stages generates the Casson fluid specimen. When yield stresses are greater than shear stresses, Casson fluid transforms into a solid. Alternatively, it begins to move whenever the yield stresses are fewer than the shear stresses. There are multiple examples like jelly, fruit drinks, beef stew, sauce, tomato, honey, and so on. Human blood can also be thought of as Casson fluid. Salahuddin et al. ^[2] investigated the stable three-dimensional momentum and inbuilt energy transform in spinning viscous Casson fluid motion with convective boundary situations using the bvp4c technique. Mittal and Patel ^[3] studied the impacts of thermophoresis, Brownian movement, non-linear heat flux, heat formation and chemical response in a two-dimensional mixed convection flow MHD instability point stream of Casson fluid just beyond a boundless sheet in permeable material numerically utilizing HAM. Likewise, Salahuddin et al. ^[4], Aneja et al. ^[5] and Abdal et al. ^[6] scrutinized the significance of Casson fluid in diverse aspects. Delhibabu et al. ^[7] discussed MHD 2-D flow flowing between two parallel plates. Ramesh et al. ^[8] discussed Joule heating and MHD effects on Casson fluid flow by using slip boundary conditions.

Although mono-nanofluids get a stronger thermal system and high rheological assets. Numerous real-time implementations necessitate a trade-off between distinct nanofluid attributes; for instance, metal oxides like $Al_2O_3$ have beneficial chemical inertia and accuracy but have lesser thermal capacitance whereas metal nanoparticles including copper, aluminum, and gold possess stronger heat conductivity but seem to be chemically reactive and inconsistent. Hybrid nanofluids possess the capability to be used in heat transfer disciplines such as naval formations, microfluidics, defense, health care, acoustic performance, transportation, and so on. There is a wealth of conceptual and experimental results examining hybrid nanofluid attitudes in diverse flowing mechanisms. Muhammad et al. ^[9] scrutinized the stream of hybrid nano liquid (MWCNTs + Cu + Water), nanofluid (MWCNTs + Water) as well as base liquid (Water) along an arched plate in the existence of viscous dispersion, mixed convection and convective boundary conditions employing the shooting technique and RK-4 methodologies (bvp4c). Souayeh et al. ^[10] enhanced the heat transfer rate by using hybrid nanoparticles in the presence of activation energy. Mondal et al. ^[11,12] used different types of hybrid nano-particles along with a sponge medium to enhance the heat transfer rate of the fluid. Biswas et al. ^[13] used a partial magnetic field along with hybrid nanoparticles for the convective system in the presence of a porous medium. Mandal et al. ^[14] used nanofluid thermal transport with a magnetic field in a split-driven porous medium. Mandal et al. ^[15] investigated the role of bioconvection and MHD for nanofluid flow with sponge medium. Chatterjee et al. ^[16] studied cooling effects on MHD hybrid nanofluid flow flowing in a cylindrical system. Manna et al. ^[17] investigated the effects of Lorentz forces on hybrid nanofluid by using entropy generation. Mandal et al. ^[18] used a non-Darcian porous medium for hybrid nanofluids to improve the heat transfer rate.

A Maxwell fluid is a viscoelastic solid material that may possess both properties of viscosity as well as elasticity. This Maxwell fluid has become the most dominant one in the consideration of researchers as it is the most simple rate-type fluid model. Since Maxwell fluids have less complexity so they play a significant part in the polymeric industry. Aman et al. ^[19] scrutinized the implication of second-order slip on the magneto-hydrodynamic (MHD) stream of a fractional Maxwell liquid on a moveable sheet as well as the comparative analysis of two numerical methods (Tzou and Stehfest's methodologies). Riaz and Iftikhar ^[20] utilized fractional-time derivatives to scrutinize the unsteady stream of an MHD Maxwell fluid. Na et al. ^[21] utilized the Caputo time-fractional derivative to review the free convection stream of Maxwell liquid among two parallel sheets separated by d with dulled shear and heat flux. Shehzad et al. ^[22] numerically explored the immiscible Maxwell fluid model stream through a spinning disc incorporating thermophoresis as well as the Cattaneo-Christov hypothesis by incorporating Runge-Kutta-4 and shooting technic. Correspondingly, Habib et al. ^[23], Wang et al. ^[24], Abdal et al. ^[25,26] and Habib et al. ^[27] reviewed the role of Maxwell fluid flow across the different geometries respectively.

A heat source is anything that can be used to intensify the rate of thermal transition for the desired object from different sources of energy. Heat transfer based on Nanotechnology has transformed our visualization concerning solar systems via Nano-fluids on account of the larger efficacy produced by the adjourned Nano-sized particles. With the passage of time, a method of combustion for solid fuels, which is also a heat source, went hand in hand with humanity's development from the beginning and exploding fire to adjustable, refined, and useful processes. Mahanthesh et al. ^[28] scrutinized the heat transmission assets of nano-fluid flowing across an elongating revolving disk in the appearance of an imposed gravitational flux and a convective boundary situation. Javed et al. ^[29] researched the time-dependent fluid motion of Eyring-Powell nanoparticles propelled by a bi-directionally mountable surface encoded in permeable space taking into consideration the thermal and prescribed heat modalities by using the Keller-Box technic. Similarly, Seyednezhad et al. ^[30], Nielsen et al. ^[31] and Xiao et al. ^[32] scrutinized the performance of heat sources in diverse features. Saha et al. ^[33,34] enhanced the thermal efficiency and heat transfer rate of practical fluids by using different geometries.

The word boundary layer flow relates to flow in a comparatively shallow region just across a solid substrate at which viscosity has a considerable influence. Prandtl, a German scientist, first proposed the theory of boundary layer flow in 1904. In the present era, the boundary layer is a beneficial feature of engineering owing to its innumerable applications, specifically in the prototype of discrete items to overcome drag forces all through a fluid motion for seamless execution. Ferdows et al. ^[35] used the bvp4c technic to examine the consequences of radiation, internal temperature formation, and viscous dispersion on the consistent, laminar, immiscible, convective, boundary layer stream of a viscous liquid across a moveable plate. Gangadhar et al. ^[36] demonstrated an unstable 2-dimensional nanofluid stream of thermal and mass transmission with receding nanoparticle flux. Kebede et al. ^[37] discussed a computational estimation of the thermal as well as mass transmits attributes of a 2-dimensional time-dependent stream of Williamson nanoparticles across a porous elongating plate inserted in a permeable medium while accounting for the consequences of gravitational flux, heat radiation as well as chemical response utilizing the homotopy analysis method. Berrehal et al. ^[38] assessed the innate irreversibility of a dissipating water/functionalized carbon nanostructures nanofluid, notably SWCNTs and MWCNTs, in a boundary-layer stream across a convectively warmed moveable wedge and horizontal/vertical sheets utilizing the shooting technique with the Runge-Kutta-Fehlberg integration strategy. Consequently, Shafee et al. ^[39], Khan et al. ^[40] and B. J. et al. ^[41] viewed distinct performances of boundary layer flow of fluids.

The mechanical and architectural implementations of a boundary layer stream across a consistent stretching surface with energy transmission are remarkable. Such a flow ordinarily comprises wire drafting, glass fiber formation, plastic sheet deformation, hot spinning, conditioning of a metal plate in a conditioning bath, and several mechanisms. Most researchers have examined flow across a sequential elongation of the surface, although there are distinguishing situations in modern and creative methodologies in which the squeezing of the substrate is not typically linear precisely, the flow triggered by a nonlinear flexing sheet performs a crucial task in the polymer extrusion procedure. Reza-E-Rabbi et al. ^[42] explored the physical implications of the Casson variable, Maxwell factor, heat flux, and viscosity dissipation as well as streamlines and isothermal assessment, on 2D higher-order MHD multiphase liquids stream owing to extending on a flat surface. Ullah et al. ^[43] utilized Lie group analysis and the shooting methodology to examine the MHD tangent hyperbolic fluid stream over a flexing plate with heat transition attributes and suction/injection influence at the boundary. Ali et al. ^[44] analyzed the performance of gravitational dipole on the thermal transition phenomena of distinct nano-materials Fe (ferromagnetic) as well as $Fe_3O_4$ (ferrimagnetic) distributed in a base fluid $(60\%$ water + $40\%$ ethylene glycol) on micro-polar liquid stream across a squeezing sheet.

The calibration of hybrid nanofluids has aroused the interest of researchers for heat transfer analysis in recent years. In the previous work ^[44], nano-particles of iron/iron oxide are utilized, these can cause clogging or rusting. Graphene oxide, $MoS_2$, and similar nano-particles are harnessed their improved thermal characteristics. Here, various aspects of hybrid nanofluid flow are taken into account which is rarely discussed in existing studies. This work addresses a hybrid base ($H_2O$ + ethyl glycol) where the enhancement in thermal transportation is attained with a homogeneous mixture of three hybrid constituents: (a) $Go$ + $AA7072$, (b) $Go$ + $MoS_2$, (c) $Go$ + $Ag$. These three cases involve vibrant thermo-physical properties as enlisted in Table 1. Two non-Newtonian models are taken into account, these are Casson fluid and Maxwell fluid. Thus a comparative study reveals some interesting results. This work surely augments our understanding of hybrid nanofluid transport phenomena. The findings herein are useful to detect the thermal efficiency of the heat exchangers of heavy industries.

Table 1.  Thermo-physical properties of water base fluid and nanoparticles ^{[52,53,54,55]}.

Physical properties	$\rm{C_2H_6O_2 - H_2O}$	$\rm{Go}$	$\rm{Ag}$	$\rm{AA}$7072	$\rm{MoS_2}$
$\rho$(kg.m$^{-3}$)	1063.80	1800	10500	2720	5060
$C_p(J(kg.°k$	3630.00	717	235	893	397
$\kappa$(W(m.°k)	0.378	5000	429	222	904.4

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2.   Materials and methods

Hybrid nanofluid heat and mass transportation is considered over a stretching sheet that lies along $x$-$axis$. The fluid is electrically conducting. The sheet is permeable and a magnetic field acts along the y-axis. A porous medium allows the fluid to permeate through. The base fluid is hybrid in the form of water plus ethyl glycol $(50\%-50\%)$. Three sets of nano-species namely Graphene oxide plus (a) AA7072, (b) Molybdenum sulfide, (c) silver are separately assumed to be dissolved homogeneously. The concentration of nano solutes is presumed to transform the mixture into non-Newtonian fluids. Two non-Newtonian fluid models are taken for comparative outputs (i) Casson fluids (ii) Maxwell fluids. It is anticipated that the flow is incompressible and velocity components are $u$, $v$. The flow passes through a permeable medium and is impacted by a magnetic field along $y$-$direction$. The flow description of the model is seen in Figure 1. Tables 1 and 2 are shown the thermo-physical properties and characteristics of hybrid base nano-particles. The fluid temperature is $T$ and volume fraction of $Go$ is $\phi_1$ and that of the other additives is $\phi_2$. Keeping in consideration, the conservation of mass, momentum, and energy, the physical problem is formulated as below ^[6,45].

Figure 1.  Flow geometry of the problem.

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Table 2.  Thermo-physical properties of hybrid base fluid and nanoparticles ^[53,55,56].

Properties	Nanofluid	Hybrid Nanofluid
$\mu(\mbox{viscosity})$	$\mu_{nf}=\frac{\mu_f}{(1-\phi)^{2.5}}$	$\mu_{hnf}=\frac{\mu_f}{(1-\phi_1)^{2.5}(1-\phi_2)^{2.5}}$
$\rho(\mbox{density})$	$\rho_{nf}=\rho_f((1-\phi)+\phi\frac{\rho_s}{\rho_f}$	$\rho_{hnf}=\rho_f(1-\phi_2)((1-\phi_1)+\phi_1\frac{\rho_{s1}}{\rho_f})+\phi_2\rho_{s2}$
$\rho C_p(\mbox{Heat capacity})$	$(\rho C_p){nf}=(\rho C_p)_f((1-\phi)+\phi\frac{(\rho C_p)_s}{(\rho C_p)_f}$	$(\rho C_p)_{hnf}=(\rho C_p)_f(1-\phi_2)((1-\phi_1)+\phi_1\frac{(\rho C_p)_{s1}}{(\rho C_p)_f})+\phi_2(\rho C_p)_{s2}$
$\kappa (\mbox{Thermal conductivity})$	$\frac{\kappa_{nf}}{\kappa_f}=\frac{\kappa_s+(s_f-1)\kappa_f-(s_f-1)\phi(\kappa_f-\kappa_s)}{\kappa_s+(s_f-1)\kappa_f + \phi(\kappa_f-\kappa_s)}$	$\frac{\kappa_{hnf}}{\kappa_{bf}}=\frac{\kappa_{s2}+(s_f-1)\kappa_{bf}-(s_f-1)\phi_2(\kappa_{bf}-\kappa_{s2})}{\kappa_{s2}+(s_f-1)\kappa_{bf} + \phi(\kappa_{bf}-\kappa_{s2})}$ where $\frac{\kappa_{bf}}{\kappa_f}=\frac{\kappa_{s1}+(s_f-1)\kappa_f-(s_f-1)\phi_1(\kappa_f-\kappa_{s1})}{\kappa_{s1}+(s_f-1)\kappa_f + \phi(\kappa_f-\kappa_{s1})}$
$\sigma(\mbox{Electrical conductivity})$	$\frac{\sigma_{nf}}{\sigma_f}= 1+\frac{3(\sigma-1)\phi}{(\sigma+2)-(\sigma-1)\phi}$	$\frac{\sigma_{hnf}}{\sigma_{bf}}= 1+\frac{3\phi(\sigma_1\phi_1+\sigma_2\phi_2-\sigma_{bf}(\phi_1+\phi_2))}{(\sigma_1\phi_1+\sigma_2\phi_2 +2\phi\sigma_{bf})-\phi\sigma_{bf}((\sigma_1\phi_1+\sigma_2\phi_2)-\sigma_{bf}(\phi_1+\phi_2))}$

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Governing equations

The supportive boundary conditions are:

The uniform velocity at wall is $v_w$ (where, $v_w < 0$ for suction, $v_w > 0$ for injection), $h_f$ is heat transition coefficient, Employing the similarity factors:

where magnetic factor is $M$, porosity factor is $K_p$, Maxwell Deborah number is $\lambda$, $Pr$ is Prandtl number, Eckert number is $Ec$, heat source is $Q$, suction factor is $S$ and Biot number is $B_i$.

Also

The interpretations for skin friction correlation and the local Nusselt number are as follows (see ^[46,47]):

By employing the similarity transformation equation, we acquire

Solution procedure

The mathematical formulation as finally constructed in the form of a boundary value problem (Eqs (6)–(8)) contains non-linear terms and up to third-order derivatives are involved. As usual, it is cumbersome to attain any closed-form solutions for this BVP. Numerical treatment based on the RK-4 method with shooting technique as operated by ^{[48,49,50,51]} is hired to yield a solution to the problem. The basic layout of the scheme is to convert the derivatives of high order into order first derivatives as given below:

$s_1' = s_2$

$s_2' = s_3$

$(1+ 1/\beta)s_3' = A_1(s_2^2-s_1s_3) + A_2 M s_2 + K_p s_2+\lambda A_1(S_1^2 s_3' -2s_1s_2s_3)$

$s_4' = s_5$

$s_5' = \frac{k_f}{k_{hnf}} Pr (A_3 s_1 s_5 - Ec A_4 s_3^2 -Q s_4)$

along with the boundary conditions:

$s_1 = S$, $s_2 = 1$, $\frac{k_{hnf}}{k_f} s_5 = B_i(1-s_4)$, at $\eta = 0$

$s_2 \rightarrow 0, \ s_4 \rightarrow 0 \ as \ \eta \rightarrow \infty$. This system of first-ordered differential equations makes a two-point boundary value problem. The coding of the numerical scheme is developd and rises in the Matlab environment. The commercial software Matlab R2018b is licensed at the non-linear center of studies, Northwest west university, Xian, China. The maximum value of $\eta \rightarrow \infty$ is replaced by $\eta_{max} = 6$.

3.   Results and discussions

This portion pertains to the presentation and interpretation of momentum and heat distribution for the three cases of hybrid nanofluid dynamics owing to a stretching surface. Parametric deliberation grows from the computational procedure as described in the above section. Prior to the evaluation of current results, limiting values of the parameters are used to validate the code. The values of the Nusselt number enlisted in Table 3 show reasonable agreement among themselves to ensure the validity of the current numerical procedure. Then, results for the notable physical variables are evaluated for opposite ranges of the pertinent parameters namely magnetic field ($M$), the porosity of medium ($K_p$), Maxwell fluid ($\lambda$), Casson fluid ($\beta$), mass suction ($S$) and heat source ($Q$). Table 4 indicates that magnitude of skin friction is enhanced directly with higher inputs of $M$, $\lambda$, $K_p$, and $S$ but it is reduced against parameter $\beta$. Further, the skin friction is largest for hybrid species case (a) $Go$ + $AA7072$ and it is smallest for case (c) $Go$ + $Ag$. From Table 5, the behavior of the Nusselt number convinced that the local heat transfer is enhanced with $Pr$ but it diminishes against $Ec$ and $Q$. From Table 6, it is seen that $-\theta'(0)$ is highest for case (c) $Go$ + $Ag$ and it is lowest for case (a) $Go$ + $AA7072$.

Table 3.  Comparative outputs for $Pr$ when the rest factors are zeros.

$Pr$	Gorla and Sidawi [57]	Khan and Pop [58]	Wang [59]	Devi et al. [60]	Our results
2.0	0.9114	0.9113	0.9114	0.91135	0.9113
6.13	–	–	–	1.75968	1.7596
2.0	1.8954	1.8954	1.8954	1.89540	1.8954
20.0	3.3539	3.3539	3.3539	3.35390	3.3539

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Table 4.  Findings for Skin friction factor $-f''(0)$ for Casson fluid.

$M$	$K_p$	$\beta$	$S$	$Go + AA7072$	$Go + MoS_2$	$Go + Ag$
0.0	0.5	1.0	0.5	2.2987	2.2712	2.2108
0.5				2.6317	2.6008	2.5328
1.0				2.9232	2.8893	2.8146
0.5	0.1			2.3962	2.3677	2.3050
	0.3			2.5170	2.4873	2.4219
	0.5			2.6317	2.6008	2.5328
	0.1	0.5		3.1498	3.1136	3.0339
		1.0		2.6317	2.6008	2.5328
		1.5		2.4315	2.4027	2.3392
		1.0	0.1	2.3774	2.3519	2.2958
			0.3	2.5016	2.4735	2.4116
			0.5	2.6317	2.6008	2.5328

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Table 5.  Findings for Skin friction factor $-f''(0)$ for Maxwell fluid.

$M$	$K_p$	$\lambda$	$S$	$Go + AA7072$	$Go + MoS_2$	$Go + Ag$
0.0	0.5	0.5	0.5	2.4738	2.4659	2.4474
0.5				2.7993	2.7913	2.7728
1.0				3.0823	3.0743	3.0556
0.5	0.1			2.5695	2.5615	2.5430
	0.3			2.6876	2.6796	2.6611
	0.5			2.7993	2.7913	2.7728
	0.1	0.5		2.7993	2.7919	2.7728
		1.0		3.6939	3.6747	3.6308
		1.5		4.5719	4.5393	4.4651
		0.5	0.1	2.5265	2.5247	2.5206
			0.3	2.6945	2.6902	2.6802
			0.5	2.7993	2.7913	2.7728

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Table 6.  Findings for Nusselt number $-\theta'(0)$.

$Pr$	$Ec$	$Q$	$Go + AA7072$	$Go + MoS_2$	$Go + Ag$
6.135	0.1	0.01	0.5447	0.5460	0.5488
7.0			0.5653	0.5666	0.5696
8.0			0.5854	0.5868	0.5899
6.135	0.01		0.6255	0.6257	0.6262
	0.05		0.5896	0.5903	0.5918
	0.1		0.5447	0.5460	0.5488
	0.1	0.01	0.5447	0.5460	0.5488
		0.02	0.5420	0.5434	0.5462
		0.03	0.5393	0.5406	0.5435

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Figures 2–7 are plotted for velocity and temperature distribution for three cases of hybrid nanofluid namely (a) $Go$ + $AA7072$ / hybrid base, (b) $Go$ + $MoS_2$ / hybrid base, (c) $Go$ + $Ag$ / hybrid base. For the plots of Figure 2(a), it readily comes to know that the flow speed $f'(\eta)$ declines against magnetic field strength $M$ (see ^[56]). Actually, greater values of $M$ cross pond to reduce larger resistive force (Lorentz force) because magnetic and electric fields intervene. Similarly in Figure 2(b), the velocity curves slope down against larger values of $K_p$ (porosity parameter). The larger $K_p$ means lesser permeability of the medium that offers increased resistance to the motion and hence the speed reduction (see ^[53]). Figure 3(a) depicts the impact of Casson factor $\beta$ on $f'(\eta)$. It is revealed that the speed of the fluid decelerates when the nature of fluid transform from Casson fluid to Newtonian fluid ($\beta$ enhances). The larger values of the Maxwell parameter enhance flow speed as depicted in Figure 3(b). The implications of suction/injection factor $S$ on the flow velocity $f'(\eta)$ are mapped in Figure 3(c). The flow becomes faster for injection $(S < 0)$ and it is slowed down against suction $(S > 0)$. Figure 4(a), (b) respectively show the impacts of volume fraction $\phi_2$ on the fluid velocity for the Casson model and Maxwell model. It is seen that the velocity $f'(\eta)$ goes down with increments in $\phi_2$. Further, Figures 2 to 4 indicate that the velocity of the hybrid nanofluid is faster for case (a) $Go$ + $AA7072$ and it is slower for case (c) $Go$ + $Ag$. Figure 5(a), (b), graphs are drawn respectively for fluids temperature $\theta(\eta)$ for the Casson fluid and Maxwell fluid. The fluid temperature $\theta(\eta)$ is incremented with a rise in the volume fraction $\phi_2$ of nano-species. As seen from Figure 6(a), the hybrid nanofluids temperature is decreased against the higher valuation of Prandtl number $Pr$. The higher Prandtl number means the lesser heat diffusivity. The fluid's temperature rises with increasing strengths of heat source factor $Q$ as well as Eckert number $Ec$ as noticed respectively in Figures 6(b) and 7(a) (see ^[55]). The growing values of Biot number $B_i$ also enhance the temperature of the fluid as seen in Figure 7(b). Moreover, fluids temperature for hybrid nanofluid case (a) $Go$ + $AA7072$ remains the highest whereas it is smallest for the case (c) $Go$ + $Ag$.

Figure 2.  Fluctuation of $f'(\eta)$ along with (a) $M$ and (b) $K_p$.

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Figure 3.  Fluctuation of $f'(\eta)$ along with (a) $\beta$, (b) $\lambda$ and (c) $S$.

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Figure 4.  Fluctuation of $f'(\eta)$ along with $\phi_2$ (a) Casson fluid $\beta$ and (b) Maxwell fluid $\lambda$.

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Figure 5.  Fluctuation of $\theta(\eta)$ along with $\phi_2$ (a) Casson fluid $\beta$ and (b) Maxwell fluid $\lambda$.

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Figure 6.  Fluctuation of $\theta(\eta)$ along with (a) $Pr$ and (b) $Q$.

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Figure 7.  Fluctuation of $\theta(\eta)$ along with (a) $Ec$ and (b) $B_i$.

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4.   Conclusions

The central point of this work is the consideration of hybrid base hybrid nanofluid transportation across a stretching sheet. The bulk fluid is a homogeneous mixture of $(50\%-50 \%)$ water + ethyl glycol. Three hybrid nano-species: Silver, Molybdenum sulfide, and AA7072 are taken separately with Graphene oxide. These blends of nano-materials are mixed in hybrid base fluids. A detailed analysis of the practical designs are presented to elucidate the flow and heat transportation pertaining to the enhanced thermal efficiency of heat exchangers. The fluid flow models namely Casson fluid and Maxwell fluid are theoretically considered for non-Newtonian behavior of the nanofluids with concentration greater than 0.03 $\%$. Numerical outcomes are briefly defined below:

● Velocity $f'(\eta)$ goes down against the upsurge values of magnetic factor $M$, $K_p$, $\beta$ as well as $S$.

● Elevated Maxwell fluid factor $\lambda$ inputs explicitly upsurge velocity $f'(\eta)$.

● With the up surged values of volume fraction $\phi_2$, the fluid velocity for the Casson model and Maxwell model shows decreasing behavior.

● It is noted that when $Pr$ is uplifted, temperature $\theta(\eta)$ goes down.

● Temperature $\theta(\eta)$ rises rapidly when $Q$, $Ec$ and $B_i$ takes larger values.

● It is interesting to note that temperature $\theta(\eta)$ increases quickly for both the fluid models when $\phi_2$ takes higher inputs.

● For Casson fluid model, skin friction coefficient boosted with $M$, $S$ and $K_p$ but with the developing values of $\beta$ induce for reduction of $-f''(0)$.

● With the higher inputs of Prandtl number, $-\theta'(0)$ uplifted while opposite behavior observed for $Q$, $Bi$ and Eckert number $Ec$.

Acknowledgments

This research work was funded by institutional fund projects under no. (IFP-A- 2022-2-5-24). Therefore, the authors gratefully acknowledge technical and financial support from the ministry of education and the University of Hafr Al Batin, Saudi Arabia.

Conflict of interest

The authors declare no conflict of interest.