1. INTRODUCTION

Improvement of heat transmission has received significant consideration over the previous few years. Hybrid nanofluids, tangent hyperbolic fluids, and magnetic fields greatly influence the heat transfer field. These phenomena have been the subject of our work.

The accessible hybrid nanomaterials are typically made of two different kinds of nanomaterials, often having a larger thermal conductivity than either pure fluids or single particle nanofluids. Heat transfer studies are one of its most important things in many applicatons. The energy consumption of thermal systems is essential in the global environment. Sundar et al. [1] provided an overview of hybrid nanofluids’ arrangement, thermal characteristics, heat transmission, and friction factor. Zainal [2] discussed moving viscous and MHD hybrid nanofluids concerning an exponentially expanding/shrinking surface. Devi and Devi [3] investigated hydromagnetic hybrid \(Cu{-}Al_{2}O_{3}\)/water nanofluid flow upon a stretchable permeable sheet with suction. Manjunatha et al. [4] enhance the boundary layer flow of hybrid nanofluids and heat transmission due to the fluid’s changing viscosity and spontaneous convection. Nurul and Nazar [5] worked on the stability investigation of MHD nanofluid hybrid flow across a stretching/shrinking sheet with quadratic velocity.

Industrial production processes and chemical engineering use non-Newtonian fluid dynamics extensively. The traditional Navier–Stokes equations fall short of explaining the typical characteristics of the non-Newtonian fluids. In light of this, several constitutive equations for non-Newtonian fluids have been presented. There is no non-Newtonian fluid model that can predict all of their properties. Among other non-Newtonian fluids, hyperbolic tangent fluid is a significant branch that applies to chemical engineering. It can describe shear thinning phenomena. Akbar [6] proposed mathematical solutions to the tangent hyperbolic fluid’s boundary layer flow, approaching a stretched sheet of magnetically conducting material. Ali and Hussain [7] worked on hydromagnetic influence on tangential hyperbolic fluid flow upon a vertically extended sheet. Besthapu Prabhakar et al. [8] measured the impression of inclined Lorentz forces upon the flow of tangent hyperbolic nanofluid with zero flux at the stretched sheet. Zakir Ullah and Zaman [9] analyzed lie group inspection of MHD tangent hyperbolic fluid flow in the direction of a stretched sheet under slip circumstances. Akbar and Hayat [10] studied heat transfer in the peristaltic flow of a Tangent hyperbolic fluid through an inclined asymmetric channel.

The study of the dynamics of electrically conducting fluids is called magnetohydrodynamics (MHD). The interaction of magnetic fields with fluid motion is formally referred to as MHD. We are only left with liquid metals, hot ionized gases, and vital electrolytes because non-magnetic and electrically conducting fluids are needed. Raptis et al. [11] studied heat radiation’s impact on the MHD fluid’s motion. Muhammad et at. [12] performed a computational investigation of MHD rotational flow of hybrid nanofluid with heat radiation across a stretched sheet. Sheikholeslam et al. [13] presented an innovative computer-based numerical technique for MHD AlO–water nanofluid transport inside a permeable media. Nawaz et al. [14] have investigated the increase in temperature caused by the presence of dust and hybrid nanoparticles in the hyperbolic tangent fluid.

An irreversible process known as viscous dissipation occurs when the stresses of the imbalanced fluids in neighbouring layers are turned into heat. Sheikholeslami et al. [16] investigate the numerical simulation of heat transfer in MHD nanofluid flow while considering viscous dissipation.

In these occurrences, we only become aware of a very small number of collaborations on these topics. In this research, we try to discover and examine what will occur inside the framework of this association.

2. GOVERNING EQUATIONS WITH PHYSICAL MODEL

\(We\) ponder the steady-state, 2D flow, boundary layer laminar flow of an incompressible hyperbolic tangent fluid with the presence of Ethylene Glycol (\(EG\)) based hybrid nanofluids (\(Fe_{3}O_{4}\) (Ferro) and \(Cu\) (Copper), another one is \(SWCNT\) (Single-walled carbon nanotubes) and \(CuO\) (Copper oxide)) past a stretching sheet. The x-axis is exposed to a transverse magnetic field of intensity B with an inclination \(\delta\). The flow is analyzed along the x-axis, parallel to the stretching surface, and the y-axis perpendicular to the sheet. \(y>0\) is conformed to the flow. The surface of the plane is fixed at a higher constant temperature \(T_{w}\) than the ambient nanofluid’s constant temperature \(T_{\infty}\).

Fig. 1
figure 1

Sketch of stretching geometry of the hybrid nanofluid model.

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The governing equations for the above model are [18]:

$$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0,$$
(1)
$$u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=v_{hnf}(1-n)\frac{\partial^{2}u}{\partial y^{2}}+\sqrt{2}nv_{hnf}\Gamma\frac{\partial u}{\partial y}\frac{\partial^{2}u}{\partial y^{2}}-\frac{\sigma_{hnf}}{\rho_{hnf}}B^{2}u\sin^{2}\delta,$$
(2)
$$u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\left(\frac{k}{\rho c_{p}}\right)_{hnf}\frac{\partial^{2}T}{\partial y^{2}}+\frac{v_{hnf}}{\left(c_{p}\right)_{hnf}}(1-n)\left(\frac{\partial u}{\partial y}\right)^{2}+\frac{v_{hnf}}{\left(c_{p}\right)_{hnf}}\frac{n\Gamma}{\sqrt{2}}\frac{\partial u}{\partial y}\left(\frac{\partial u}{\partial y}\right)^{2}.$$
(3)

The respective boundary conditions,

$$y=0;\quad u=u_{w}=ax,\quad v=0,\quad T=T_{w},$$
(4)
$$y\rightarrow\infty;\quad u\rightarrow0,\quad T\rightarrow T_{\infty}.$$
(5)

In the above representation, \(u\) and \(v\) resemble the velocity components along the x and y-axes, respectively. The kinematic viscosity is denoted by \(v_{hnf}\), \(\alpha_{hnf}\) is the electrical conductivity of hybrid nanofluid and \(\alpha_{hnf}\) called the thermal diffusivity of hybrid nanofluid. \(u_{w}\), \(T_{w}\), and \(T_{\infty}\) are the stretching velocity, wall, and free stream temperatures, respectively. And \(n\) is termed for power law index.

2.2. Mathematical Formulation and Transformation

For simplicity, we employ the following transformation [19] to non-dimensional equations,

$$\eta=\sqrt{\frac{a}{v}y},$$
$$u=axf^{\prime}(\eta),$$
$$v=-\sqrt{av}f(\eta),$$

and

$$\theta(\eta)=\frac{T-T_{w}}{T_{w}-T_{\infty}},$$

where \(f\) and \(\theta\) are the dimensionless velocity function and dimensionless temperature function respectively. These following variables transform Eqs. (1)–(3) into the following form,

$$\frac{\mu_{hnf}}{\mu_{f}}\frac{\rho_{f}}{\rho_{hnf}}\left[(1-n)f^{\prime\prime\prime}+n\frac{\nu_{hnf}}{\nu_{f}}Wef^{\prime\prime}f^{\prime\prime\prime}\right]-f^{\prime2}+ff^{\prime\prime}-\frac{\sigma_{hnf}}{\sigma_{f}}\frac{\rho_{f}}{\rho_{hnf}}M^{2}f^{\prime}\sin^{2}\delta=0,$$
(6)
$$\theta^{\prime\prime}+Pr\frac{\left(\rho C_{p}\right)_{hnf}}{\left(\rho C_{p}\right)_{f}}\frac{k_{f}}{k_{hnf}}f\theta^{\prime}+Br\frac{k_{f}}{k_{hnf}}\frac{\mu_{hnf}}{\mu_{f}}\left[(1-n)f^{\prime\prime2}+\frac{nWe}{2}f^{\prime\prime3}\right]=0.$$
(7)

With corresponding boundary conditions given in Eqs. (4), (5) as follows

$$f(0)=0;\quad f^{\prime}(0)=1,\quad\theta(0)=1;\quad f^{\prime}(\infty)\rightarrow0,\quad\theta(\infty)\rightarrow0.$$
(8)

Here, the subscript “hnf” refers to Hybrid Nanofluid. The dimensionless forms of thermophysical properties of Hybrid Nanofluid and base fluid and nanoparticles following [18, 19] are represented in Tables 1 and 2.

Table 1. Thermophysical properties of hybrid nanofluid
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Table 2. Experimental values of base fluid and nanoparticles
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Dimensionless parameters arise in the above equation,

Prandtl number \(Pr=\frac{\nu}{\alpha}\) 

Brinkman number \(Br=\frac{\nu u^{2}}{(\alpha(T_{w}-T_{\infty})}=Pr\cdot Ec\) 

Magnetic parameter \(M=\frac{\alpha B^{2}}{\rho a}\) 

Weissenberg number \(We=xa^{{3}/{2}}\Gamma\sqrt{\frac{2}{\nu}}\) 

Power law index \(n\) 

2.2. Skin Friction and Nusselt Number Quantities

Fig. 2
figure 2

The consequences of the parameter \(\delta\) on \(f'(\eta)\) and \(\theta(\eta)\).

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For the current model the \(C_{f}\) and \(N_{u}x\) are key engineering physical parameters that are described as

$$C_{f}=\frac{\tau_{w}}{\rho(u_{\omega})^{2}},$$
$$Nu=\frac{xq_{\omega}}{k(T_{\omega}-T_{\infty})},$$

where \(\tau_{w}\) is wall share stress and \(q_{w}\) is wall heat flux. The mathematical representation of these quantities are [14]:

$$\tau_{w}=\mu_{hnf}\left.\left[(1-n)\frac{\partial u}{\partial y}+\frac{n\Gamma}{\sqrt{2}}\left(\frac{\partial u}{\partial y}\right)^{2}\right]\right|_{y=0},$$
$$q_{w}=-\left.k_{hnf}\frac{\partial T}{\partial y}\right|_{y=0}.$$

Hence the dimensionless quantities of drag friction and Nusselt number coefficients are

$$Re^{1/2}C_{f}=\frac{\mu_{hnf}}{{\mu}}\left[(1-n)f^{\prime\prime}(0)+\frac{nWe}{2}\left(f^{\prime\prime}(0)\right)^{2}\right],$$
(9)
$$Re^{-1/2}Nu=-\frac{k_{hnf}}{k}\theta^{\prime}(0).$$
(10)

3. NUMERICAL PROCEDURE

Fig. 3
figure 3

The consequences of the parameter \(M\) on \(f'(\eta)\) and \(\theta(\eta)\).

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There are various numerical techniques to solve the linear differential equation but very few for nonlinear. Our model involves a nonlinear differential equation, so we employ the Spectral Relaxation Method (SRM). This iterative strategy breaks down the significant schemes of nonlinear equations into small schemes of linear equations [22]. The Gauss–Seidel technique is then applied to resolve linear algebraic equations. Various researchers [20–22] have used the approach to solve issues in science and engineering that are characterized by linked nonlinear systems of ODEs and PDEs. We discretize the converted Eqs. (6) and (7) using the following SRM algorithm:

  • The momentum equation is reduced for \(f(\eta)\) using the transformation \(f'(\eta)=F(\eta)\) and expressing the \(f(\eta)\) equation in \(F(\eta)\).

  • Suppose that \(f(\eta)\) is given from the preceding iteration (defined by \(f_{r}(\eta)\)), form an iteration plan for \(F(\eta)\) by considering that simply linear terms in \(F(\eta)\) are to be assessed at this iteration level (defined by \(F_{r+1}(\eta)\)) and that all linear and nonlinear terms are expected to be identified from the preceding iteration. Also, the preceding iteration evaluates nonlinear terms in \(F(\eta)\).

  • Similar iteration schemes are constructed for the remaining governing dependent variables, but now they include the revised solutions for the variables specified by the earlier equation.

The aforementioned method is akin to the Gauss–Seidel theory for decoupling equational systems into linear algebra. By applying Chebyshev spectral collocation methods, this strategy creates a series of linear differential equations with variable coefficients that can be tackled using conventional numerical methods [23]. Here, spectral techniques are favored due to their remarkable accurateness and easiness of application in discretizing and solving linear differential equations with variable coefficients and convergent solutions over small domains. Thus, the above-described SRM iteration method, Eqs. (6) and (7) become,

$$\frac{\mu_{hnf}}{\mu}\frac{\rho}{\rho_{hnf}}\left\{ (1-n)F_{r+1}^{\prime\prime}-nWeF_{r+1}^{\prime}F_{r+1}^{\prime\prime}\right\} +f_{r}F_{r+1}^{\prime}-\left(F_{r}\right)^{2}$$
$$-\frac{\sigma_{hnf}}{\sigma}\frac{\rho}{\rho_{hnf}}MF_{r}\sin^{2}\delta=0,$$
(11)
$$f_{r+1}^{\prime}=F_{r+1},$$
(12)
$$\theta_{r+1}^{\prime\prime}+\frac{(\rho c_{p})_{hnf}}{\rho c_{p}}\frac{k}{k_{hnf}}\operatorname{Pr}f_{r+1}\theta_{r+1}{}^{\prime}+\frac{\mu_{hnf}}{\mu}\frac{k}{k_{hnf}}PrEc\Big[(1-n)(F_{r+1}{}^{\prime})^{2}$$
$$+\frac{nWe}{2}(F_{r+1}^{\prime})^{3}\Big]+\frac{\sigma_{hnf}}{\sigma}\frac{\rho}{\rho_{hnf}}PrEcMF_{r+1}^{\prime2}=0.$$
(13)

Subject to boundary conditions:

$$f_{r+1}(0)=0,\quad F_{r+1}(0)=1,\theta_{r+1}(0)=1;\quad F_{r+1}(\infty)\rightarrow0,\quad\theta_{r+1}(\infty)\rightarrow0.$$
(14)
Fig. 4
figure 4

The consequences of the parameter \(n\) on \(f'(\eta)\) and \(\theta(\eta)\).

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Fig. 5
figure 5

The consequences of the parameter \(We\) on \(f'(\eta)\) and \(\theta(\eta)\).

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Fig. 6
figure 6

The consequences of the parameter \(Br\) on \(\theta(\eta)\).

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Fig. 7
figure 7

The consequences of the parameter \(Pr\)on \(\theta(\eta)\).

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It is feasible to change the governing domain to \([-1,1]\), on which the spectral technique can be executed, before adopting the spectrum approach. We implement transformation \(\eta=L\frac{\tau+1}{2}\) to map the interval \([0,L]\) to \([-1,1]\), where \(L\) is chosen to be large enough to numerically approximate the conditions at infinity. The spectral collocation approach is predicated on the introduction of a differentiation matrix A, which is utilized to estimate the derivatives of the unknown variables at the collocation locations as the matrix vector product of the form,

$$\frac{df_{r}}{d\eta}=\sum_{k=0}^{N}D_{lk}f_{r}\left(\tau_{k}\right)=Df_{r,}\quad l=0,1,2,3,\ldots,N$$

where \(N+1\) is the number of collocation points (grid points), \(D=\frac{2A}{L}\) and

$$f=\left[f\left(\tau_{0}\right),f\left(\tau_{1}\right),f\left(\tau_{2}\right),f\left(\tau_{3}\right),\ldots,f\left(\tau_{N}\right)\right]^{T}$$

is the vector function at the locations of collocation. Higher-order derivatives are produced as powers of \(D\), or \(D\) to the power that is,

$$f_{r}^{p}=D^{p}f_{r},$$

the order of the derivative is denoted by the letter p.

We find by using the spectral approach to equations (11) through (13):

$$A_{1}F_{r+1}=R_{1},\quad F\left(\tau_{N}\right)=\lambda,\quad F\left(\tau_{0}\right)=0,$$
$$A_{2}f_{r+1}=R_{2},\quad f\left(\tau_{N}\right)=0,$$
$$A_{3}\theta_{r+1}=R_{3},\quad\theta\left(\tau_{N}\right)=1,\quad\theta\left(\tau_{0}\right)=0,$$

where

$$A_{1}=\operatorname{diag}\left(\frac{\mu_{hnf}}{\mu}\frac{\rho}{\rho_{hnf}}(1-n)\right)D^{2}+\operatorname{diag}\left(f_{r}\right)D-\frac{\sigma_{hnf}}{\sigma}\frac{\rho}{\rho_{hnf}}M\sin^{2}\delta I,$$
(15)
$$R_{1}=(F)^{2}+\frac{\mu_{hnf}}{\mu}\frac{\rho}{\rho_{hnf}}nWeF^{\prime}{}_{r+1}F^{\prime\prime}{}_{r+1},$$
(16)
$$A_{2}=D,$$
(17)
$$R_{2}=F_{r+1},$$
(18)
$$A_{3}=D^{2}+\frac{\left(\rho c_{p}\right)_{hnf}}{\rho c_{p}}\frac{k}{k_{hnf}}\operatorname{Prdiag}\left[f_{r+1}\right]D,$$
(19)
$$R_{3}=\frac{\mu_{hnf}}{\mu}\frac{k}{k_{hnf}}Br\left\{ (1-n)\left(F_{r+1}{}^{\prime}\right)^{2}+\frac{nWe}{2}\left(F_{r+1}{}^{\prime}\right)^{3}\right\} +\frac{\sigma_{hnf}}{\sigma}\frac{\rho}{\rho_{hnf}}BrMF_{r+1}^{\prime2},$$
(20)
Table 3. Statistical comparison of \(Pr\) with previous results
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Fig. 8
figure 8

Skin friction and local Nusselt number along \(We\).

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Fig. 9
figure 9

Skin friction and local Nusselt number along \(n\).

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Fig. 10
figure 10

Local Nusselt number along \(Br\) and \(Pr\).

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 \(I\) is termed as an identity matrix. The matrix is of size \((N+1)\times1\) and diag \([~]\) is a \((N+1)\times(N+1)\) diagonal matrix, where \(N\) is the number of grid points, \(f\), \(F\), and \(\theta\) correspondingly, when evaluated at the grid points, and the subscript \(r\) represents the number of iterations. The initial estimates used to initiate the SRM method for Eqs. (11)–(13) are selected as boundary-satisfying functions. The velocity and temperature profiles for the boundary layer issue presented in this work decline exponentially at \(\eta=\infty\) based on physical considerations. The following exponential function can be used as initial predictions, as it is the most convenient choice for this purpose:

$$f_{0}=1-e^{-\eta},\quad F_{0}(\eta)=e^{-\eta},\quad\theta(\eta)=e^{-\eta}.$$

In our present investigation we take \(N=100\) collocation point. These values provided correct results for all physical quantities of interest.

3.1. Validation

Table 3 displays comparisons with previously published works made under various settings to validate the spectrum relaxation method’s correctness. Taking \(\phi_{1}=\phi_{2}=0\), \(n=0\), \(M=0\), \(\delta=\frac{\pi}{2}\), and then we compared the value of the Nusselt number for the value of \(Pr\) by altering the value of \(Pr\) with the published papers, [24], [25], and [4]. This can easily declare that confidence in the accuracy of the SRM results is justified.

4. RESULT AND DISCUSSION

This section evaluates the physical trend and mechanism behind the graphical representation of nonlinear hybrid nanofluid flow upon a stretching surface with various novel effects. Dimension-free equations are recovered for fluid flow problems using similar variables and solved via SRM. During the process, many parameters have emerged, which are discussed in detail. The effects of non-dimensional parameters are measured for various types of hybrid nanofluid particles, which are Ferro-Copper, (\(Fe_{3}O_{4}{-}Cu\)) and Single-walled carbon nanotubes-Copper Oxide (\(SWCNT{-}CuO\)) with the base fluid ethylene-glycol (\(EG\)). At first, the effects of the inclination angle of magnetic field \(\delta\) and magnetic parameter \(M\) are illustrated in Figs. 2 and 3 for base fluid, nanofluid, hybrid nanofluid, \(SWCNT{-}CuO/EG\), and \(Fe_{3}O_{4}/EG\). In both cases, the flow velocity decreases, and the temperature increases. The impacts created by varying the strength of the magnetic field can also roughly be obtained by manipulating the magnetic field’s inclination angle in practical applications linked to managing the velocity and heat transmission of fluid flow. Here \(SWCNT{-}CuO/EG\) gives a better result as its flow velocity’s diminishing is less than \(Fe_{3}O_{4}/EG\). Thermal efficiency is higher for higher \(\delta\). So working with \(SWCNT{-}CuO/EG\) instead of \(Fe_{3}O_{4}/EG\) for thermal efficiency is also preferred since \(SWCNT{-}CuO/EG\) has a greater heat transmission rate. Also, enhancement in the magnetic parameter \(M\) causes the occurrence of the Lorentz force. The Lorentz force provides resistance to fluid motion. As a result, increasing Lorentz force causes the fluid velocity close to the central region to slow down. In fact, larger magnetic fields can apparently disturb the fluid’s temperature. This is primarily caused by the associated changes in fluid frictional heat generation and joule heating that occur with a high magnetic field. These graphs ensure that \(SWCNT{-}CuO/EG\) is more preferred than \(Fe_{3}O_{4}-Cu/EG\). So, it is more convenient to function with \(SWCNT{-}CuO/EG\).

Figure 4 deals with predicting the effects of the parameter \(n\) on velocity and temperature profiles. As shown in Fig. 4, with increasing n the fluid’s velocity slows down. It is happening because as n grows, the character of the fluid shifts from shear-thinning to shear-thickening, resulting in a decrease in the velocity profile as the power-law index rises. Both fluids have a propensity to decrease in quantity. Figure 4 also displays the variation of temperature with the changes of \(n\). It is quite evident from the graph that temperature has increasing phenomena with power law index \(n\). However, \(SWCNT{-}CuO/EG\) gives an improved approximation in both cases.

Weissenberg number \(We\) has an impact, as can be shown in Fig. 5, respectively, upon the velocity and temperature boundary layers. These profiles are obtained to drop by growing the parameter \(We\). Because the growing values of the parameter \(We\) escalate, the fluid particle’s relaxation time. As a result, viscosity increases, creating resistance to the fluid flow and a consequent drop in the fluid velocity. While working with two different hybrid nanofluids, our primary focus is on determining which of the two is superior. It is clear from Fig. 4 that \(SWCNT{-}CuO/EG\) is finer than \(Fe_{3}O_{4}{-}Cu/EG\).

Fig. 6 presents the changes in temperature profile along with Brinkman number \(Br\). The figure shows that the temperature is increasing with increasing values of \(Br\). The Brinkman number Br measures the heat generated by viscous dissipation to heat transferred by molecular diffusivity, i.e., the proportion of internal to external heating. Thus, the higher values of the parameter \(Br\) slowers the heat conduction created by viscous dissipation and hence the greater temperature increase. Nanofluid gives a better approximation than base fluid, and of course, Hybrid nanofluid gives the best result among the three of them. We ignore the impact that \(Br\) has on flow velocity is relatively insignificant. Figure 6 further shows that Ferro Copper/Ethylene Glycol is superior to Single-Walled Carbon nano-tubes-Copper oxide/Ethylene Glycol.

Figure 7 demonstrates the consequences of the Prandtl number \(Pr\) upon temperature. It is reckoned that a deduction in fluid temperature because of an escalation in Pr reduces the thermal boundary layer thickness. In this circumstance also stated, \(SWCNT{-}CuO/EG\) heat transfer rate reducing less than \(Fe_{3}O_{4}{-}Cu/EG\).

Figure 8 illustrates the Weissenberg number’s influence on skin friction and the Nusselt number. The Weissenberg number directly correlates with an escalation in the skin friction coefficient. Also, from Fig. 8, it can be analyzed that \(Fe_{3}O_{4}{-}Cu/EG\) has a better skin friction value than \(SWCNT{-}CuO/EG\). And the consequence of the Nusselt number is decaying, while the Weissenberg number is improving, as shown in Fig. 8. The above figure convinces that \(SWCNT{-}CuO/EG\) has a greater value related to \(Fe_{3}O_{4}Cu/EG\).

Figure 9 represents the diversity of skin friction and Nusselt number along with changes of \(n\). Firstly, in Fig. 9, we observe that with higher estimations of \(n\), the skin friction has an increment, and it also shows that \(Fe_{3}O_{4}{-}Cu/EG\) has an improved skin friction coefficient than \(SWCNT{-}CuO/EG\). Then from the evaluation of Fig. 9, we conclude that the Nusselt number has a growing phenomenon with increasing values of \(n\). It is also assured that \(Fe_{3}O_{4}{-}Cu/EG\) has enhanced the importance of the Nusselt number more than the \(SWCNT{-}CuO/EG\).

Figure 10 indicates the variation of the Nusselt number, in addition to the Prandtl and Brinkman numbers. Here, the Nusselt number goes down when the Prandtl number rises. The fluid \(SWCNT{-}CuO/EG\) has a higher Nusselt number than the fluid \(Fe_{3}O_{4}{-}Cu/EG\). However, the Nusselt number goes higher with an enhancement in Brinkman. The figure demonstrates that \(SWCNT{-}CuO/EG\) has a higher local Nusselt number than \(Fe_{3}O_{4}{-}Cu/EG\). The Prandtl number and the Brinkman number have relatively little of an influence on the skin friction coefficient. Therefore, we can choose to disregard this consequence.

5. CONCLUSIONS

The current research investigates magnetohydrodynamic boundary layer flow and heat transfer phenomena when a hybrid nanofluid film is presented upon a steady stretched sheet. The effects of an angled magnetic field, tangent hyperbolic flow, and viscous dissipation on the momentum and thermal boundary layer are also investigated. The similarity transformation yields a nonlinear, ordinary differential equations system, which is analyzed using an iterative numerical approach termed SRM. MATLAB is utilized in an attempt to perform this simulation. One of this research focuses on comparing basic fluids, nanofluids, and hybrid nanofluids to determine which is best for various applications. The primary focus is to evaluate the efficiency of two hybrid nanofluids in different circumstances. The outcomes of this experiment are summarized as follows:

  • Compared with the base fluid and the nanofluid, the hybrid nanofluid provides more accurate predictions regarding velocity and temperature predictions.

  • This study indicates that \(SWCNT{-}CuO/EG\) is more advantageous than \(Fe_{3}O_{4}{-}Cu/EG\) in terms of Prandtl number \(Pr\), Magnetic parameter \(M\), Weissenberg number \(We\), Power law index \(n\), and \(\delta\).

  • Another instance of the Brinkman number \(Br\) in which the combination \(Fe_{3}O_{4}{-}Cu/EG\) is favored over the combination \(SWCNT{-}CuO/EG\).

  • When dimensionless characteristics such as the Weissenberg number \(We\), the power law index \(n\), the skin friction coefficient seems to have a better prediction for \(Fe_{3}O_{4}{-}Cu/EG\) than it does for \(SWCNT{-}CuO/EG\).

  • When local approximations of the Nusselt number are performed, \(SWCNT{-}CuO/EG\) outperforms \(Fe_{3}O_{4}{-}Cu/EG\) for the same dimensionless parameters mentioned above.

In industrial applications, such as polymer processing flows and aerodynamic heating, the influence of viscous dissipation in hybrid nanofluids is significant. According to one of the project’s findings, the hybrid nanofluid has a higher thermal conductivity than normal fluids. One further discovery is about the question of who is more prominent in between \(SWCNT{-}CuO/EG\) and \(Fe_{3}O_{4}{-}Cu/EG\) to utilize in such industries.

FUNDING

This work was supported by ongoing institutional funding. No additional grants to carry out or direct this particular research were obtained.

CONFLICT OF INTEREST

The authors of this work declare that they have no conflicts of interest.