Hybrid nanofluid flow over a slippery surface for thermal exploration

Abstract

The potential of hybrid nanofluid (HNF) to maximize heat transportation has captured the attention of many researchers, inspiring them to further investigate the performance of the common base fluid. Conventional flow in Cu-Al₂O₃/H₂O hybrid nanofluid (HNF) toward a high permeability horizontal flat plate incorporated in Darcy porous medium has been explored in this research to determine how Cu-Al₂O₃/H₂O hybrid fluid will respond thermodynamically when physical factors like suction/injection and slip boundary conditions are present. In addition, the effects of radiation, dissipations, energy engagement, and inclined magnetized field associated with the fluid flow were studied. The governing system is transformed by similarity transformations to a solvable ordinary differential equation by employing HAM (Homotopy Analysis Method) scheme. The main results show that Cu-Al₂O₃/H₂O has high thermal conductivity compared to Cu/H₂O. As a result, hybrid fluids are essential for the development of thermal phenomena.

Keywords

suction-injection exponential stretchable surface MHD slip condition

Introduction

Heat transfer intensiﬁcation is essential in many industrial, engineering, and medical fields with achievable cost and less energy. The demand for tiny gadgets with remarkable attributes including excellent performance, precise operation, and prolonged lifetime is nowadays being encouraged by breakthroughs in science and technology. Viewed in this way, the attention of many researchers and scientists has gathered in a concerted effort to analyze heat and mass transfer.^1–3 Scientists and scholars introduced a novel class of liquid they declared “nanofluids” (NF) as a result of solids’ superior heat transmission characteristics to those of ordinary fluids.⁴ Gowda et al.⁵ scrutinized the impact of activation heat with chemical reactions on energy transmission of the Marangoni flow of a non-Newtonian nanofluid. KKL (Koo-Kleinstreuer and Li) correlations and a refined model of Fourier heat flux are used in Punith et al.⁶ to investigate the idea of computational simulation of nanocomposite flow across a curved stretchable surface. With H₂O as a common fluid and Cu as the nanoparticle, Sheikholeslami and Ganji⁷ calculated the heat transfer rate and demonstrated the influence of volume fraction and Nusselt number on identical surfaces.⁸ looked into the Casson nanofluid flow between parallel discs that are convectively heated.⁹ investigated the radiative-reactive magnetized flow of viscoelastic nanofluid, which contains gyrotactic microorganisms due to a vertical stretching surface while being subject to various convective and stratification limitations.¹⁰ According to the research on the mathematical replication of nanofluids flow classifying as root canal, the root canal’s wall shear stress rises as nanofluids concentration and irritating volumetric flow rate increase. In an intelligent computing model, Shoaib et al.,¹¹ examined the Ohmic thermal effect and entropy creation for a microfluidics scheme of Ree-Eyring fluid.

A novel and advanced nanotechnological fluid is known as a hybrid nanofluid (HNF) is developed by distributing two nanostructures into an established heat transfer fluid. Hybrid nanofluid (HNF) is the combination of two different nanoparticles in the same base fluid. HNFs have played an extensive role to enhance the HT rates, scientists and researchers have recently examined ternary hybrid nanofluids (THNFs). THNFs have a higher TC and dynamic viscosity than convectional fluids, HNFs, and NFs. THNF is made up of three different types of NP, water being the basic fluid. These liquids are widely used in solar power and photovoltaic thermal collectors. Recent studies suggest that hybrid nanofluids, particularly those operating at very high temperatures, can successfully replace convectional cooling. Hybrid nanofluids were shown to have excellent thermal consistency and characteristics when compared to simple nanofluids, which makes them ideal for heating purposes such as solar thermal systems, automotive cooling systems, heat sinks, or thermal energy storage. The idea of a hybrid nanofluid was developed by Suresh et al.¹² and Baghbanzadeh et al.¹³ and investigated by various studies and numerical outcomes. In contrast to nanofluid and simple fluid, the hybrid nanofluid, in his opinion, increases energy transmission. Many several researchers were inspired to work in the area of hybrid nanofluids by this concept. This concept inspired a wide spectrum of researchers to work in the hybrid nanofluid sector. Mishra and Upreti and Upreti¹⁴ used the Buongiorno model to compare the Ag-MgO/water0and Fe3O4-CoFe2O4/EG-water hybrid0nanofluid flows along curved surfaces with chemical reactions. They examined the Ag–MgO/H2O hybrid nanofluid flow for the enhancement of heat transfer (HT) under the inspiration of viscous dissipation.

Tlili et al.¹⁵ developed a novel framework for hybrid nanofluidic (methanol) three-dimension motion with the effect of MHD on an irregular, thick surface with slip factors. According to the results, the transmission of the heat of the flow of hybrid fluid is much higher than that of the corresponding nanofluid with the unique nanoparticle. The resistive type force caused due to the magnetic force on the fluid is less pronounced in the hybridized fluid that contains the nanofluid. Kumar et al.¹⁶ proposed exploring the effects of activation heating on hybrid nanocomposite flow on a curved stretchy surface. Darcy-Forchheimer SWCNT-MWCNT’s thermal management was examined by Sahu et al.¹⁷ due to a vertically stretched cylinder-induced cross-hybrid nanofluid flow with and without inertia effects. The characteristics of the flow of hybrid nanofluids are studied by Hussain,¹⁸ who reveals that the mixture with the highest energy transfer coefficient, graphene, and molybdenum disulfide (MoS₂), utilizes a marginally greater particle pressure than water. The impact of the non-Fourier heat flux models of interior energy production on the tri-hybrid nanofluid flow, in which water takes as a base fluid was studied.¹⁹ Diamond-Co3O4/ethylene glycol hybrid nanofluid flow with nonlinear radiative MHD flow was examined by²⁰ for entropy effects. Waini et al.²¹ reveal that there are two anti-flow solutions when he discusses the motion of hybrid nanostructure of different sizes of complex nanoparticles with the extension impact of mixed convective through a permeable medium. Shaw et al.²² investigated for each Prandtl number of different impacts of thermal radiations (linear, nonlinear, and quadratic) on MHD flow. Naveen et al.²³ conduct a comparative study on heat transfer analysis in tri-hybrid nanofluid that is transporting various nanocomposite shapes in three dimensions of unstable magnetic fluid flow. Nayak et al.²⁴ highlighted the importance of non-Newtonian fluids with hybrid nanostructures in terms of thermodynamics. The studied literature specifies that the hybridized fluid with different nanoparticles is an essential replacement for conventional thermal systems. Nadeem et al.²⁵ investigated the flow of hybrid nanofluid through a curled medium and during this analysis, they predicted that when suction or injection are related, the hybrid nanofluid allocates energy faster than the simple nanofluid.

Due to its applications across many technical fields, boundary film moves through a stretchable medium and has attracted a lot of attention. This kind of flow occurs in many different contexts, such as the boundary film along a conveyer of material handling, liquid film, and condensing mechanism, sheet, and fabric cooling, or drying processes, manufacturing of fiberglass, and many other scientific, and technical sectors. In fluid mechanics, the fluid flow over a stretchy surface is crucial for biomedical sciences and industrial operations. Crane²⁶ was the first, who discovers a way of getting the same outcomes for the nanofluid stream through an extending sheet or surface. However, compared to the stretching scenario, the flow passing through the stretch film received little attention. The use of the KKL relationship in the computation of nanofluid flow over a stretched sheet, taking magnetic dipole, and chemical reaction into account was emphasized in.²⁷ Vjravelu et al..²⁸ explored the viscous nanocomposite motion over a linearly expanding medium. In the presence of a magnetic dipole,²⁹ brought up the concept of the important impact of Stefan blowing on nanocomposite motion across a stretchable medium. Ariel et al.³⁰ used the HPM technique and examined the fluid flow over an elongating surface. Using two simultaneous extended permeable surfaces with slipping velocities,³¹ investigated a 3D nonlinear radiative hybrid nanofluid flow. Dutta et al.³² analyzed the temperature field inflow across an expanding surface with heat flux and the Prandtl effect. To model, the movement of nanofluids over a bent stretchable surface,³³ looked into the use of convective energy transmission along with KKL correlation. Fang et al.³⁴ explained the slip of MHD ferrofluid flow over a spreading texture and investigated the exact solution with the combined effect of slip conditions and magnetic field. Punith et al.³⁵ looked explored the impact of thermophoretic particle deposition and magnetic dipole on heat and mass transport in ferromagnetic fluid flow across a stretching sheet. ’According to Andersson et al.,³⁶ who researched the power law model of fluid flow across an expanding sheet under the influence of a magnetic field, the thinner boundary layer in this article is due to the magnetic field impact, which increases wall friction.

The hybrid flow of nanofluids over a permeable porous surface has become the focus of many academics due to its wide range of applications. The fields of nuclear engineering, solar sciences, and material sciences could “all” benefit from the use of flow-through permeable porous surfaces. Darcy’s law is incorrect and frequently applied to comprehend the porous surface flow. Deutsch et al.³⁷ developed the second-order polynomial in the momentum equation based on the permeability when calculating the impact of inertia to pinpoint the Forchheimer component, see the study.³⁸ Many hybrid nanofluid regularities utilize Darcy Forchheimer’s theories, and numerous academics have explored a porous medium flow. Here many of them are cited in this manner. Sahu et al.³⁹ studied the Carreau nanofluid’s hydrothermal stagnation point flow across a moveable tinny needle using non-uniform Navier’s slip condition along with chemical processes in a Darcy-Forchheimer surface. Eid et al.⁴⁰ investigated and incorporated homogeneous and heterogeneous catalysis on electromagnetic radiational Prandtl fluid flow with the theory of the Darcy-Forchheimer substance scheme. Sahu et al.⁴¹ examined the thermal implications with SWCNT/MWCNT suspensions and the Darcy-Forchheimer flow behavior as a result of the rotating disk’s shrinkage. In this regard, many researchers have studied the Darcy Forchheimer fluid flow for different industrial and engineering fields for heat transfer applications.^42–44

In this work, we demonstrate the results of a complex fluid over an exponentially stretching sheet that contains nanomaterials of Copper and Aluminum Oxide suspended in water. Our inspiration for this study came from the aforementioned viewpoints. The investigation of many elements affecting the hybrid flow, and nanofluid’s heat transport form the basis of the suggested model. Additionally, graphic and statistical sketches of the impact of several parameters on the kinetics of hybrid fluid phases are presented. The current analysis is a novelization of the previously published work.⁴⁵

The following research gaps are addressed by this study.

➢ How does the magnetic field cause the (HNF) to flow over the slippery surface?

➢ How can one observe the impact of the porous space in the existence of suction/injection terminologies?

➢ What consequences does natural convection have on (HNF) heat transfer?

➢ How does the thermal radiation, and heat absorption/omission affect the (HT) of (HNF)?

Applications

By combining the velocity slip, temperature convective conditions, and cross-diffusion analysis, it is possible to obtain information about the behavior of HNF flow via a stretched surface. The results of this analysis can lead to better uses in different fields, such as refrigeration, industries, chemical treatment, solar energy systems, heat exchangers, biomedical engineering, energy production, etc.

Novelty

The recent work aims to investigate the effect of hybrid nanoparticles (Cu-Al₂O₃) in the form of (HNF) to analyze the (HT) rate. The problem is based on differential equations which have been solved through the (HAM) technique.

The structure of the paper is displayed below.

❖ Recent work is extended to describe the impacts of solid volume fraction, suction, and injection, viscous dissipation, and Newtonian heating.

❖ The most current research relies on a hybridized fluid that incorporates Cu (copper) and Al₂O₃ (aluminum oxide) nanoparticles into the H₂O base fluid.

❖ Current findings have incorporated the influence of thermal enlargement, energy source, Darcy- Forchheimer model as well as thermal, and velocity slip of nanofluids.

❖ The magnetic field effect is inclined to the flow of fluid.

❖ Filling the spaces in the literature is the basic aim of this analysis, the modified set of differential equations is attempted with the help of the HAM technique using the advanced package of BVPh.2.

❖ The influence of the physical parameters is compared for the nanofluids and hybrid nanofluids. The obtained results show that hybrid nanofluids are more appropriate for the enhancement of heat transfer.

Materials and methods

The analysis of the heat transmission of a hybrid nanofluid flow through an exponentially extending surface with electrical conductivity and incompressibility has been taken into consideration. The physical configuration of the suggested model is presented in Figure 1. With the following presumptions, the influence of velocity, and thermal slip conditions on the flow is explored.

The flow of fluid is permeated over a stretchable sheet as a result of the stretching velocity $ε_{1} (x)$ .

H₂O is contained of Cu and Al₂0₃ nanostructure, where it has to be supposed that these materials are in the form of thermal equilibrium, among these there are free slippage effects.

The induced magnetic field is despite, that is very weak, instigated by the angle $θ$ to the flow.

$B_{0}$ is the strength of the magnetic field applying to the fluid flow and $F_{0} = \frac{C_{b} x}{\sqrt{K^{*}}}$ is the non-uniform inertia coefficient of the permeable surface.

$T_{λ},$ represent the energy of the hybridized fluid on the surface of the stretch surface film, while $T_{\infty},$ representing the surrounding fluid energy.

The polarization effect is disregarded because there is no outside induce electric field.

The energy exchange rate is enlarged by integrating the ideas of optically high radiating effect, absorption of energy, and viscous dissipation.

\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0,

(1)

\begin{matrix} u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} = υ_{hnf} (\frac{\partial^{2} u}{\partial y^{2}}) - \frac{υ_{hnf}}{K^{*}} u - \frac{1}{ρ_{hnf}} F_{0} u^{2} \\ + g \frac{{(ρ β_{T})}_{hnf}}{ρ_{hnf}} (T - T_{\infty}) - \frac{σ_{hnf}}{ρ_{hnf}} B_{0}^{2} u \sin^{2} (θ), \end{matrix}

(2)

\begin{matrix} u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} = \frac{k_{hnf}}{{(ρ C_{p})}_{hnf}} \frac{\partial^{2} T}{\partial y^{2}} - \frac{1}{{(ρ C_{p})}_{hnf}} \frac{\partial q_{r}}{\partial y} \\ + \frac{μ_{hnf}}{{(ρ C_{p})}_{hnf}} {(\frac{\partial u}{\partial y})}^{2} - \frac{Q_{0}}{{(ρ C_{p})}_{hnf}} \\ (T - T_{\infty}) + \frac{σ_{hnf}}{{(ρ C_{p})}_{hnf}} B_{0}^{2} u^{2} \sin^{2} (θ), \end{matrix}

(3)

Figure 1.

The physical configuration of the model: (a) Hybrid nanofluids and (b) Geometry.

The Rosseland approximation allows for the following representation of the radiation heat flux $q_{r}$ :

q_{r} = \frac{- 4 σ^{*}}{3 k^{*}} \frac{\partial T^{4}}{\partial y},

(4)

Where in equation (4), $σ^{*}$ (Stefan-Boltzmann constant), and $k^{*}$ (heat absorption constants). By implementing Taylor’s successions and ignoring the higher powers term, we get:

T^{4} ≅ T^{3} (4 T - 3 T_{\infty}),

(5)

Equations (5) and (6) are employed in equation (3) getting the following form:

\begin{matrix} u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} = (\frac{k_{hnf}}{{(ρ C_{p})}_{hnf}} + \frac{16 T_{\infty}^{3} σ^{*}}{3 {(ρ C_{p})}_{hnf} k^{*}}) \frac{\partial^{2} T}{\partial y^{2}} + \frac{μ_{hnf}}{{(ρ C_{p})}_{hnf}} {(\frac{\partial u}{\partial y})}^{2} \\ - \frac{H_{0}}{{(ρ C_{p})}_{hnf}} (T - T_{\infty}) + \frac{σ_{hnf}}{{(ρ C_{p})}_{hnf}} B_{0}^{2} u^{2} Si n^{2} (θ), \end{matrix}

(6)

The initial and boundary conditions as given follow:

\begin{matrix} \begin{matrix} u = c \frac{\partial u}{\partial y} + ε_{1} (x), v = ε_{2} (x), T = δ \frac{\partial T}{\partial y} + T_{λ} (x), at y = 0, \\ u = v \to 0, T \to T_{\infty}, as y \to \infty \end{matrix}} \end{matrix}

(7)

In Table 1, the experimental values of the supposed materials are presented, whereas various thermos-physical features of the hybrid nanofluid are shown in Table 2. Here $ρ \to$ (density), $C_{p} \to$ (heat capacity), $k \to$ (thermal conductivity), $σ \to$ (electrical conductivity), $μ \to$ (dynamic viscosity), and $ν \to$ (kinematic viscosity). The subscript $f \to$ (base fluid), $nf \to$ (nanofluid), $hnf \to$ (hybrid nanofluid), one for the $Cu$ nanoparticles, and two for the $A l_{2} O_{3}$ nanoparticles.

Table 1.

The tentative values of copper, aluminum oxide, and water.

System	$C_{p} (j / kgK)$	$ρ (kg / m^{3})$	$k (W / mK)$	$σ (S / m)$
H₂O	4179	997.1	0.613	$5.5 \times 10^{- 6}$
Cu	385	8933	401	$5.96 \times 10^{7}$
Al₂O₃	700	4907	3.7	$1.1 \times 10^{7}$

Table 2.

The thermochemical possessions of the hybrid nanofluid.

Properties	Models
Viscosity	$\frac{μ_{hnf}}{μ_{f}} = \frac{1}{{(1 - ϕ_{1})}^{2.5} {(1 - ϕ_{2})}^{2.5}}$
Density	$\frac{ρ_{hnf}}{ρ_{f}} = (1 - ϕ_{2}) [(1 - ϕ_{1}) + ϕ_{1} (\frac{ρ_{1}}{ρ_{f}})] + ϕ_{2} (\frac{ρ_{2}}{ρ_{f}})$
Thermal capacity	$\frac{{(ρ c_{p})}_{hnf}}{{(ρ c_{p})}_{f}} = (1 - ϕ_{2}) [(1 - ϕ_{1}) + ϕ_{1} (\frac{{(ρ c_{p})}_{1}}{{(ρ c_{p})}_{f}})] + ϕ_{2} (\frac{{(ρ c_{p})}_{2}}{{(ρ c_{p})}_{f}})$
Thermal conductivity	$\frac{k_{hnf}}{k_{nf}} = \frac{k_{2} + 2 k_{nf} - 2 ϕ_{2} (k_{nf} - k_{2})}{k_{2} + 2 k_{nf} + ϕ_{2} (k_{nf} - k_{2})}, \frac{k_{nf}}{k_{f}} = \frac{k_{1} + 2 k_{f} - 2 ϕ_{1} (k_{f} - k_{1})}{k_{1} + 2 k_{f} + ϕ_{1} (k_{f} - k_{1})},$
Electrical conductivity	$\frac{σ_{hnf}}{σ_{f}} = [1 + \frac{3 (\frac{σ_{2}}{σ_{nf}} - 1) ϕ_{2}}{(\frac{σ_{2}}{σ_{nf}} + 2) - (\frac{σ_{2}}{σ_{nf}} - 1) ϕ_{2}}], \frac{σ_{nf}}{σ_{f}} = [1 + \frac{3 (\frac{σ_{1}}{σ_{f}} - 1) ϕ_{1}}{(\frac{σ_{1}}{σ_{f}} + 2) - (\frac{σ_{1}}{σ_{f}} - 1) ϕ_{1}}]$

By implementing the following non-dimensional factors:

\begin{matrix} \begin{matrix} u = \frac{\partial φ}{\partial y}, = \frac{υ_{f} Re}{l} e^{x / l} F' (η), v = - \frac{\partial φ}{\partial x} = \frac{υ_{f} \sqrt{2 Re}}{l} \\ (η F' (η) + F (η)) e^{x / 2 l}, \\ η (x, y) = \sqrt{Re} e^{x / 2 l} (\frac{y}{\sqrt{2} l}), ψ (x, y) = υ_{f} e^{x / 2 l} \sqrt{2 Re} F (η), \\ T (x, y) = T_{\infty} - ((T_{\infty} - T_{0}) e^{ax / 2 l}) Θ (η) . \end{matrix}} \end{matrix}

(8)

Equation identically satisfied, while the equations (1–4) yield the form:

\begin{matrix} (1 + \frac{1}{β}) F ″' (η) + \frac{ρ_{hnf} / ρ_{f}}{μ_{hnf} / μ_{f}} [F ″ (η) F (η) - 2 F' {(η)}^{2}] \\ - 2 ξ F' (η) - \frac{1}{μ_{hnf} / μ_{f}} \\ (2 e^{- Λ} {Sin}^{2} θ \frac{σ_{hnf}}{σ_{f}} MF' (η) + 2 FrF' {(η)}^{2} + \frac{{(ρ β_{T})}_{hnf}}{{(ρ β)}_{f}} Gr Θ (η)) = 0, \end{matrix}

(9)

\begin{matrix} \frac{1}{\Pr (\frac{k_{hnf}}{k_{f}} + Tb) ″ (η) {(\frac{μ_{hnf}}{μ_{f}} e^{Λ} F ″ {(η)}^{2} + \frac{σ_{hnf}}{σ_{f}} 2 M \sin^{2} θ F' {(η)}^{2})}^{(2 - a) Λ}} \\ + \frac{{(ρ C_{p})}_{hnf}}{{(ρ C_{p})}_{f}} (a Θ (η) F' (η) - F (η) Θ' (η)) + 2 Ha e^{- Λ} Θ (η) = 0, \end{matrix}

(10)

Boundary conditions are:

\begin{matrix} \begin{matrix} F (η) = S, F' (η) = 1 + LF ″ (η), and Θ (η) \\ = 1 + d Θ' (η) at η = 0, \\ F' (η) \to 0, and Θ (η) \to 0 at η \to \infty . \end{matrix}} \end{matrix}

(11)

Here $Λ$ denotes dimensionless coordinate, M denotes magnetic term, Pr denotes Prandtl number, Tb denotes radiation constraint, Ec denotes Eckert number, Re denotes Reynolds number, L denotes velocity slip, S denotes injection/suction, Ha denotes heat absorption, d denotes thermal slip parameters, Fr denotes Forchheimer factor, Gr denotes Grashof-number, and $ξ$ denotes porosity term respectively.

\begin{matrix} Λ = \frac{x}{l}, M = \frac{{(B_{0} l)}^{2} σ_{f}}{μ_{f} Re}, Tb = \frac{16}{3} \frac{T_{\infty}^{3} σ^{*}}{k_{f} k^{*}}, \Pr = \frac{{(ρ C_{p} υ)}_{f}}{k_{f}}, Sc = \frac{υ_{f}}{D_{B}}, \\ K_{c} = \frac{e^{- Λ} K_{0} l^{2}}{Re υ_{f}}, Re = \frac{l ε_{1}}{υ_{f}}, Ec = \frac{{ε_{1}}^{2} = U_{w}^{2}}{{(C_{p})}_{f} (T_{0} - T_{\infty})}, L = \frac{c}{l} \sqrt{\frac{Re}{2}}, \\ Ha = \frac{H_{0} l^{2}}{{(ρ C_{p})}_{f} υ_{f} Re}, Fr = \frac{F_{0} l}{ρ_{f}}, S = \frac{l ν_{0}}{υ_{f}} \sqrt{\frac{2}{Re}}, D = \frac{δ_{1}}{l} \sqrt{\frac{Re}{2}}, \\ ξ = \frac{l^{2} e^{- Λ}}{Re K^{*}}, Gr = \frac{g l^{3} {(ρ β_{T})}_{f} (T_{0} - T_{\infty}) e^{Λ (\frac{a}{2} - 2)}}{υ_{f} μ_{f} {Re}^{2}} . \end{matrix}}

(12)

Expressions of $N u_{x} \to$ (Local Nusselt Number), $S F_{x} \to$ (Skin Friction Coefficient) is calculated in dimension form with dimensionless form. After derivation, the dimension forms of $N u_{x}$ and $S F_{x}$ as mentioned below:

N u_{x} = \frac{k_{hnf} x τ_{w}}{k_{f} (T_{λ} - T_{\infty})}, S F_{x} = \frac{F_{w}}{ε_{1}^{2} ρ_{f}},

(13)

Where are the wall heat flux, and the wall shear stress are given below:

τ_{w} = - k_{hnf} {(\frac{\partial T}{\partial y})}_{y = 0} + {(q_{r})}_{y = 0}, F_{w} = μ_{hnf} {(\frac{\partial u}{\partial y})}_{y = 0}

(14)

Therefore, $N u_{x}$ , $S F_{x}$ dimensionless forms are stated below:

\sqrt{2 {Re}_{x}} S F_{x} = \frac{μ_{hnf}}{μ_{f}} F ″ (0), \frac{N u_{x}}{\sqrt{\frac{χ {Re}_{x}}{2}}} = - (\frac{k_{hnf}}{k_{f}} + Tb) Θ' (0), .

(15)

Solution methodology

The homotopy analysis (HAM) method is a semi-numerical serial solution used to solve high non-linear problems.^46–48 The trial solution is obtained from the higher-order linear terms with the help of physical conditions. Then the MATHEMATICA software is used to find the standard solution using the initial solution. The HAM scheme is employed to get the analytical outcome of the proposed model. The transformed model (ODEs) is highly non-linear, to obtain the solution through HAM Mathematica 12 software has been used. For the current analysis, the preliminary guess and linear operators are:

L_{f} (F) = F ″' - F', L_{Θ} (Θ) = Θ ″ - Θ .}

(16)

The extended form of $L_{F}, L_{Θ}$ are:

L_{F} (l_{1} + l_{2} e^{- η} + l_{3} e^{- η}) = 0, L_{Θ} (l_{4} e^{- η} + l_{5} e^{- η}) = 0,}

(17)

Where $l_{k} (k = 1, 2, . . ., 7)$ clarify the random constants. The problems have been constructed for zeroth and mth order distortion as given the above linear operators. In Taylor’s expansion form

Employing Taylor’s expansion

F (η; ρ) = F_{0} (η) + \sum_{x = 1}^{\infty} F_{x} (η) ρ^{x},

(18)

Θ (η; ρ) = Θ_{0} (η) + \sum_{x = 1}^{\infty} Θ_{x} (η) ρ^{x},

(19)

Now

F_{x} (η) = \frac{1}{x!} \frac{dF (η; ρ)}{d η} |_{ρ = 0},

(20)

Θ_{x} (η) = \frac{1}{x!} \frac{d Θ (η; ρ)}{d η} |_{ρ = 0},

(21)

After the above implementations take the following form

\begin{matrix} L_{F} (F_{x} (η) - N_{x} F_{x - 1} (η)) = L_{F} R_{x}^{F} (η), \\ L_{Θ} (Θ_{x} (η) - N_{x} Θ_{x - 1} (η)) = L_{Θ} R_{x}^{Θ} (η), \end{matrix}

(22)

Where in the above equations $R e_{x} = \frac{E_{1} (x) x}{ν_{f}}$ (local Reynolds number) for simplifications.

Optimal control parameters for Convergence

In homotopical outcomes, the $h_{F}$ , $h_{Θ}$ (non-zero auxiliary factors) regulate convergence region as well as the rate of homotopy expressions examined. Using the concept of optimization to achieve the optimum magnitude of auxiliary factors ( $h_{f}$ , $h_{Θ}$ ) by employing the average squared residual errors:

E_{i}^{F} = \frac{1}{k + 1} \sum_{j = 0}^{k} {[N_{F} [\sum_{m = 0}^{i} F {(η)}_{η = j δ η}]]}^{2},

(23)

E_{i}^{Θ} = \frac{1}{k + 1} \sum_{j = 0}^{k} {[N_{Θ} [\sum_{m = 0}^{i} F {(η)}_{η = j δ η}, \sum_{m = 0}^{i} Θ {(η)}_{η = j δ η},]]}^{2},

(24)

According to Liao⁴⁶:

E_{i}^{t} = E_{i}^{F} + E_{i}^{Θ},

(25)

Where $E_{i}^{t}$ (total squares residual error). For the hybrid fluid model, the optimum magnitude of converging control factor at the second order of approximation, when L = 0.4 is calculated as $h_{F} = - 0.617154, h_{Θ} = - 1.51497$ . The total averaged squared residual error is become as by equation (25), ${\tilde{ε}}_{i}^{t} = 2.38 \times 10^{- 2}$ . The associated total residual error graph is displayed in Figure 2.

Figure 2.

Total residual error of slip factor $L = 0.4$ for hybrid fluid.

Result and discussion

Graphic and tabular representations of the hybrid nanofluid flow and energy transport features were made to better understand the current problem. In this portion, we deliberate the impact of several nan-dimensional physical factors on hybridized fluid flow, heat, and concentration distribution. Individual average-squared residual errors are shown in Table 3 using the optimum convergent control parameters. The average squared residual errors for higher approximations are apparent. Table 4 demonstrates a strong similarity between the bvp4c (MATLAB 2021b) numerical findings and the HAM.

Table 3.

The individual average squared residual errors for the slip factor $L = 0.4$ of hybrid nanofluid.

$m$	$E_{i}^{F}$	$E_{i}^{Θ}$
2	$5.11 \times 10^{- 3}$	$8.20 \times 10^{- 3}$
4	$3.47 \times 10^{- 5}$	$3.84 \times 10^{- 3}$
10	$7.28 \times 10^{- 6}$	$1.05 \times 10^{- 3}$
14	$5.31 \times 10^{- 7}$	$6.81 \times 10^{- 4}$
18	$1.96 \times 10^{- 7}$	$4.81 \times 10^{- 4}$
24	$0.48 \times 10^{- 7}$	$1.74 \times 10^{- 4}$
30	$1.79 \times 10^{- 8}$	$2.97 \times 10^{- 5}$

Table 4.

Comparative outcomes of M for $\frac{μ_{hnf}}{μ_{f}} F ″ (0)$ .

$\frac{μ_{hnf}}{μ_{f}} F ″ (0)$
S.No.	M	bvp4c	HAM
1	0	1.00004	1.00004
2	0.1	1.00432	1.00294
3	0.2	1.00468	1.00419
4	0.3	1.00087	1.00074
5	0.5	1.00007	1.00015

Figure 3(a) depicts the relationship between the temperature profile and Eckert number $Ec$ for slip- and no-slip hybrid nanofluid conditions. This graphic demonstrates how an increase in Eckert number reduces the thermal profile. The basic cause behind this scenario is $Ec$ reduces the stresses in viscous fluids by translating kinetic energy to internal heat since it physically signifies the relationship between kinetic energy and enthalpy. The improvement in the Eckert number raises the internal energy, which improves the fluid temperature profile.

Figure 3.

(a): Influence of $Ec$ , (b): $d$ on $Θ (η)$ .

The $Ec$ is the link between the kinetic energy and the enthalpy difference of the flows. Higher $Ec$ values cause the thermal field to expand. This is due to the increased thermal transfer resulting from the higher heat capacity, which is influenced by the increased conductivity of NFs. Moreover, trihybrid NFs were found to exhibit greater enhancements in thermal fields compared to regular and hybrid NFs. In addition, the enhancement in the thermal profile is more progressive in the case of (HNF). Figure 3(b), shows how the thermal slip parameter $d$ disturbs the thermal distribution of slip and non-slip hybrid nanofluidic conditions. This is because an intensification in the temperature sliding factor improves the thermal profile by allowing the heat to flow from the expandable sheet to the (HNF). This impact is comparatively larger in the case of the HNFs.

Figure 4(a) shows how the thermal radiation parameter $Tb$ affects the curves of the temperature profile under slips and no-slips conditions. The rising value of thermal radiation explains the rising value of the temperature profile. The temperature profile of the environment is the reason why thermal radiation is liberated as a form of electromagnetic wave. Radiation is therefore likely to depend on thermal distribution. But the thermal effect reinforces the conduction phenomena of the hybrid nanofluid, and as a result, the boundary film thickens. Consequently, an increase in thermal distribution is observed. In this regard, Figure 4(b) illustrates the temperature profile against the absorption parameter $Ha$ for both slip and no-slip scenarios. The increase in uptake produces a downward trend in thermal distribution. Physically, the uptake parameter of the increment greatly increases the ability of the hybridized fluid to absorb heat, which causes the temperature profile to disappear.

Figure 4.

(a): Influence of $Tb$ , (b) $Ha$ on $Θ (η)$ .

Figure 5(a) shows the relationship between the dimensionless magnetic parameters $M$ and the velocity profile $F' (η)$ in slip and non-slip conditions. Improving the strength of $M$ reduces the behavior $F' (η)$ . Figure 5(b) shows the relationship between the dimensionless magnetic parameter and thermal distribution for both slip scenarios (slip and non-slip). It can be seen that the thermal distribution is magnified by the increase in the magnitude of the magnetic parameter. Physically, when the magnetic parameter gradually improves, a resistive type force (Lorentz force) is generated by the implementation of $M$ provided to the flow that opposes the flow of fluid and increases the frictional drag force. For both slip and no-slip situations, Figure 6(a) and (b) is sketched to show the effect of the suction parameter, which is denoted as $(S > 0)$ for both $F' (η)$ and $Θ (η)$ profiles. In Figure 6(c) and (d), It is evident that these curves bend to the opposite trend when the seizure parameter is dropped below zero, while the aspiration parameter has to decrease values for $F' (η)$ and $Θ (η)$ profiles. In this case, when suction occurs, the motion boundary film physically attaches with a stretchable foil sheet, disrupting the momentum of the flow, and lowering the $F^{'} (η)$ and $Θ (η)$ profiles. Also, the injection $(S < 0)$ case, enhances the motion of fluid by creating a lateral flow across the stretch film, giving more momentum to the fluid flow. The $F' (η)$ and $Θ (η)$ profiles of hybrid nanofluids are accordingly enhanced.

Figure 5.

(a) Influence of $M$ on $F^{'} (η)$ and (b) $Θ (η)$ .

Figure 6.

(a) Influence of $S$ on $F' (η)$ , (b) $Θ (η)$ , (c) Influence of $- S$ on $F' (η)$ and (d) $Θ (η)$ .

Figure 7a represents the bond between the $F' (η)$ profile of the hybrid nanofluid and $Fr$ (The forchheimer factor) under slip and non-slip conditions. The Forchheimer parameter (Fr) is a useful tool for analyzing fluid flow on a porous medium. The forchheimer parameter has a diverse collection of applications in a variety of technical and scientific fields. This is defined as the ratio of the non-linear (square) drag force to the linear drag force in a fluid that flows through a porous medium. As the magnitude of the Forchheimer parameter improves, the $F' (η)$ profile declines. Physically, increasing the Forchheimer parameter increases inertial resistance, which acts as a barrier to the velocity profile, causing it to disappear. Variation of the thermal profile of fluids for changing the thermal distribution of the hybrid nanofluid for changing the Forchheimer parameter is shown in Figure 7(b). The Forchheimer factor increases when the hybrid nanofluid has a strong temperature profile and a thicker thermal layer.

Figure 7.

(a): Influence of $Nt$ on $F' (η)$ , and (b) $Fr$ on $Θ (η)$ .

It displays all of these Figures 3 to 7, that $Θ (η)$ profile has increasing behavior on no-slip condition $(L = 0.5)$ as equated to slip condition $(L = 0)$ , while in $F' (η)$ opposite trend can be seen.

Table 5 shows the variation of $S F_{x}$ via $S$ and $- S$ parameters for both conditions (slip and no-slip). Skin friction is clearly growing against to suction parameter while reducing against to injection parameter. In such context, Table 6 depicted the impact of $S F_{x}$ against $M$ and $Gr$ factors for both conditions (slip and no-slip), as seen that improving the magnitude of these parameters $M$ and $Gr$ enhances skin friction. Table 7 shows the impact of $N u_{x}$ against $Ec$ and $Ha$ parameters for both conditions (slip and no-slip). It is noticeable that the augmented value of $Ec$ the parameter increases the Nusselt Number while $Ha$ declining the heat transfer rate. In this regard, Table 8 is calculated for the $N u_{x}$ against $S$ and $- S$ parameter for both conditions (slip and no-slip). The increasing value of the aspirate parameter increases the number of Nusselt while the injection parameter decreases the number of Nusselt. Table 9 calculated the numerical values of the $N u_{x}$ against $ϕ_{1} + ϕ_{2}$ for cases (slip and no-slip) conditions. It is noticed that the Nusselt number enhances with the increasing values of the nanoparticle volume fraction. The heat transfer rate is comparatively higher for hybrid nanofluids, as evidenced by the numerical results. The comparison of current results with published work is presented in Table 10, considering the common parameters. The results obtained are consistent.

Table 5.

Presentation of $S F_{x}$ against suction $S$ / injection $(- S)$ factors.

Suction parameter	Injection parameter	No slip condition $L = 0$		Slip condition $L = 0.4$
$S$	$- S$	$Cu + A l_{2} O_{3}$	$Cu$	$Cu + A l_{2} O_{3}$
0.4	−0.4	1.236843	1.10494	0.67608
0.8		1.381708	1.19669	0.724987
1.2		1.540183	1.38182	0.774442
1.6		1.711289	1.59866	0.823438
0.4	−0.4	1.236843	1.10494	0.67608
	−0.8	1.286451	1.18835	0.638252
	−1.2	1.328479	1.26864	0.580928
	−1.6	1.449526	1.35672	0.520865

Table 6.

Presentation of $S F_{x}$ against $M$ and $Gr$ factors.

Magnetic parameter	Grashof number	No slip condition $L = 0$		Slip condition $L = 0.4$
$M$	$Gr$	$C u + A l_{2} O_{3}$	$Cu$	$Cu + A l_{2} O_{3}$
0.4	0.4	1.236843	1.10494	0.67608
0.8		1.289547	1.148341	0.749376
1.2		1.373508	1.319534	0.814398
1.6		1.495024	1.425453	0.932753
0.4	0.4	1.236843	1.10494	0.67608
	0.8	1.309423	1.235244	0.707353
	1.2	1.398436	1.307565	0.753642
	1.6	1.470322	1.419648	0.887529

Table 7.

Presentation of $N u_{x}$ against $Ec$ and $Ha$ factors.

Eckert number	Heat absorption	No slip condition $L = 0$		Slip condition $L = 0.4$	Slip condition $L = 0.4$
$Ec$	$Ha$	$Cu + A l_{2} O_{3}$	$Cu$	$Cu + A l_{2} O_{3}$	$Cu$
0.2	0.2	1.458319	1.427616	1.373757	1.362341
0.6		1.485502	1.468355	1.396493	1.3734211
1.0		1.518453	1.4867474	1.417461	1.3841746
1.4		1.546745	1.5039546	1.4312645	1.4031264
0.2	0.2	1.458319	1.427616	1.373757	1.362341
	0.6	1.434951	1.4090944	1.354689	1.342394
	1.0	1.418313	1.38519	1.333575	1.322835
	1.4	1.383594	1.3691948	1.313163	1.28320316

Table 8.

Presentation of $N u_{x}$ against Suction $S$ / Injection $(- S)$ .

Suction parameter	Injection parameter	No slip condition $L = 0$		Slip condition $L = 0.4$
$S$	$- S$	$Cu + A l_{2} O_{3}$	$Cu$	$Cu + A l_{2} O_{3}$
0.2	−0.2	1.458319	1.427616	1.533757
0.6		1.748416	1.768368	1.775493
1.0		2.009374	2.246961	2.257653
1.4		2.485641	2.653297	2.989452
0.2	−0.2	1.458319	1.427616	1.533757
	−0.6	1.148902	1.034567	1.284672
	−1.0	0.893545	0.830761	0.986345
	−1.4	0.649361	0.589463	0.864463

Table 9.

Presentation of $N u_{x}$ against $ϕ_{1} + ϕ_{2}$ factors.

Nanoparticle concentration	No slip condition $L = 0$	No slip condition $L = 0$		Slip condition $L = 0.4$
$ϕ_{1} + ϕ_{2}$	$Cu + A l_{2} O_{3}$	$Cu$	$Cu + A l_{2} O_{3}$	$Cu$
0.0	1.4521324	1.4521324	1.45033757	1.45033757
0.01	1.4752148	1.4652148	1.47175493	1.46231471
0.02	1.492310	1.472310	1.49257653	1.47238792
0.03	1.515623	1.485623	1.5089452	1.4810308
0.04	1.5358321	1.4958321	1.5233757	1.49215233
0.05	1.5521763	1.5021763	1.53284672	1.501323284

Bold value signifies the maximum range that is 5%.

Table 10.

Skin friction validation in consideration of common parameters.

$γ$	$θ$	$Cfl$ Hussain¹⁸ $L = 0$	$Cfl$ Hussain¹⁸ $L = 0.5$	$Cfl$ [Present] $L = 0$	$Cfl$ [Present] $L = 0.5$
0.1	$\frac{π}{3}$	0.9732178	0.532165	0.97421675	0.5332653
0.5		1.0328921	0.6127621	1.03327651	0.6135417
1		1.2321045	0.6632162	1.2332561	0.6642721
	$\frac{π}{6}$	1.0643130	0.5526713	1.0654217	0.5537862
	$\frac{π}{4}$	1.1532102	0.6241563	1.1543297	0.6252761

Conclusion

The current investigation showed the energy conversion using a hybrid Cu and Al₂O₃ nanoparticle liquid flow over a large surface sheet. Also taken into account are the thermal and velocity slip conditions. By adding Cu and Al₂O₃ nanoparticles to the water as a base fluid, a hybrid nanofluid is produced. A hybrid substance merges the physical and chemical characteristics of sundry materials and subsequently presents these features in an augmented and standardized manner. The field of HT, surrounding transportation, and the medical sciences, can reap benefits from employing this model. In light of all of the above-mentioned findings, hybrid NF exhibits superior HT characteristics compared to ordinary NFs owing to their elevated concentration of NPs.

The main conclusions are:

Increasing the angle of the magnetic field and effect reduces a hybrid nanofluid’s velocity contour while enhancing its energy profile.

As the injection limitation is improved, the velocity and energy profiles are enhanced, however, the Darcy-Forchhemier effect along with the suction term has the opposite effect.

By increasing the radiation effect, energy source, and dissipation, while decreasing the thermal slip factor, the thermal profile of the hybrid nanofluid rose.

The dispersion of copper and aluminum oxide nanoparticles in the base fluid significantly enhances the velocity and energy transmission rate.

Thermal radiation and heat absorption contributions increase the thermal profile and boost the energy transmission rate.

Footnotes

Appendix

Acknowledgements

Handling Editor: Chenhui Liang

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Taza Gul

Data availability statement

References

Tripathy

Regulagadda

Lam

CWE

, et al. Ultrathin durable organic hydrophobic coatings enhancing dropwise condensation heat transfer. Langmuir 2022; 38: 11296–11303.

Tripathy

Lam

CWE

Davila

, et al. Ultrathin lubricant-infused vertical graphene nanoscaffolds for high-performance dropwise condensation. ACS Nano 2021; 15: 14305–14315.

Memos

Kokkoris

Constantoudis

, et al. The role of shadowed droplets in condensation heat transfer. Int J Heat Mass Transf 2022; 197: 123297.

Choi

SUS

Zhang

, et al. Anomalous thermal conductivity enhancement in nanotube suspensions. Appl Phys Lett 2001; 79: 2252–2254.

Punith Gowda

Naveen Kumar

Jyothi

, et al. Impact of binary chemical reaction and activation energy on heat and mass transfer of Marangoni driven boundary layer flow of a non-Newtonian nanofluid. Process 2021; 9: 702. 2021.

Punith Gowda

Al-Mubaddel

Naveen Kumar

, et al. Computational modelling of nanofluid flow over a curved stretching sheet using Koo–Kleinstreuer and Li (KKL) correlation and modified Fourier heat flux model. Chaos Solitons Fractals 2021; 145: 110774.

Sheikholeslami

Ganji

. Heat transfer of Cu-water nanofluid flow between parallel plates. Powder Technol 2013; 235: 873–879.

Umavathi

Prakasha

Alanazi

, et al. Magnetohydrodynamic squeezing Casson nanofluid flow between parallel convectively heated disks. Int J Mod Phys B 2023; 37(4): 23500315. DOI: 10.1142/S0217979223500315

Hamad

Wakif

Alshehri

. Towards the dynamics of a radiative-reactive magnetized viscoelastic nanofluid involving gyrotactic microorganisms and flowing over a vertical stretching sheet under multiple convective and stratification constraints Waves Random Complex Media. Epub ahead of print 22 July 2022. DOI: 10.1080/17455030.2022.2100944.

10.

Ghalandari

Mirzadeh Koohshahi

Mohamadian

, et al. Numerical simulation of nanofluid flow inside a root canal. Eng Appl Comput Fluid Mech 2019; 13: 254–264.

11.

Shoaib

Zubair

Nisar

, et al. Ohmic heating effects and entropy generation for nanofluidic system of Ree-Eyring fluid: intelligent computing paradigm. Int Commun Heat Mass Transf 2021; 129: 105683.

12.

Suresh

Venkitaraj

Selvakumar

, et al. Effect of Al2O3–Cu/water hybrid nanofluid in heat transfer. Exp Therm Fluid Sci 2012; 38: 54–60.

13.

Baghbanzadeh

Rashidi

Rashtchian

, et al. Synthesis of spherical silica/multiwall carbon nanotubes hybrid nanostructures and investigation of thermal conductivity of related nanofluids. Thermochim Acta 2012; 549: 87–94.

14.

Mishra

Upreti

. A comparative study of Ag–MgO/water and Fe3O4–CoFe2O4/EG–water hybrid nanofluid flow over a curved surface with chemical reaction using Buongiorno model. Partial Differ Equ Appl Math 2022; 5: 1–11. DOI: 10.1016/J.PADIFF.2022.100322

15.

Tlili

Nabwey

Ashwinkumar

, et al. 2020, 3-D magnetohydrodynamic AA7072-AA7075/methanol hybrid nanofluid flow above an uneven thickness surface with slip effect. Sci Rep 2022; 10, https://www-nature-com-s.web.bisu.edu.cn/articles/s41598-020-61215-8 (accessed 1 April 2022).

16.

Varun Kumar

Alhadhrami

Punith Gowda

, et al. Exploration of Arrhenius activation energy on hybrid nanofluid flow over a curved stretchable surface. ZAMM Z fur Angew Math und Mech 2021; 101: 12.

17.

Sahu

Shaw

Thatoi

, et al. A thermal management of Darcy-Forchheimer SWCNT–MWCNT cross hybrid nanofluid flow due to vertical stretched cylinder with and without inertia effects. Waves Random Complex Media. Epub ahead of print 23 June 2022. DOI: 10.1080/17455030.2022.2088889.

18.

Hussain

. Dynamics of ethylene glycol-based graphene and molybdenum disulfide hybrid nanofluid over a stretchable surface with slip conditions. Sci Rep 2022; 12: 1–17.

19.

Sarada

Gamaoun

Abdulrahman

. Impact of exponential form of internal heat generation on water-based ternary hybrid nanofluid flow by capitalizing non-Fourier heat flux model. Case Studies Thermal Eng 2022; 38: 102332.,

20.

Sen

SSS

Das

Mahato

, et al. Entropy analysis on nonlinear radiative MHD flow of diamond-Co3O4/ethylene glycol hybrid nanofluid with catalytic effects. Int Commun Heat Mass Transf 2021; 129: 105704.

21.

Waini

Ishak

Groşan

, et al. Mixed convection of a hybrid nanofluid flow along a vertical surface embedded in a porous medium. Int Commun Heat Mass Transf 2020; 114: 1–5. DOI: 10.1016/j.icheatmasstransfer.2020.104565

22.

Shaw

Samantaray

Misra

. Hydromagnetic flow and thermal interpretations of Cross hybrid nanofluid influenced by linear, nonlinear and quadratic thermal radiations for any Prandtl number. Int Commun Heat Mass Transfer 2022; 130: 105816.

23.

Kumar

Gamaoun

Abdulrahman

, et al. Heat transfer analysis in three-dimensional unsteady magnetic fluid flow of water-based ternary hybrid nanofluid conveying three various shaped nanoparticles: A comparative study. Int J Mod Phys B 2022; 36: 22501703. DOI: 10.1142/S0217979222501703

24.

Nayak

Pandey

Shaw

. (eds.). Thermo-fluidic significance of non Newtonian fluid with hybrid nanostructures. Case Studies Thermal Eng 2021; 26: 101092.

25.

Nadeem

Abbas

Malik

. Inspection of hybrid based nanofluid flow over a curved surface. Comput Methods Programs Biomed 2020; 189: 1–6. DOI: 10.1016/j.cmpb.2019.105193

26.

Crane

. Flow past a stretching plate. Z für angew Math und Phys ZAMP 1970; 21: 645–647.

27.

Punith Gowda

Naveen Kumar

Jyothi

, et al. Kkl correlation for simulation of nanofluid flow over a stretching sheet considering magnetic dipole and chemical reaction. ZAMM Z fur Angew Math und Mech 2021; 101: 1–12. DOI: 10.1002/ZAMM.202000372

28.

Vajravelu

Cannon

. Fluid flow over a nonlinearly stretching sheet. Appl Math Comput 2006; 181: 609–618.

29.

Kumar

Gowda

(eds.). Stefan blowing effect on nanofluid flow over a stretching sheet in the presence of a magnetic dipole. In: Micro and nanofluid convection with magnetic field effects for heat and mass transfer applications using MATLAB. Netherlands: Elsevier, 2022, pp. 91–111.

30.

Ariel

Hayat

Asghar

. Homotopy perturbation method and axisymmetric flow over a stretching sheet. Int J Nonlinear Sci Numer Simul 2006; 7: 399–406.

31.

Nayak

Mahanta

Das

, et al. Entropy analysis of a 3D nonlinear radiative hybrid nanofluid flow between two parallel stretching permeable sheets with slip velocities. Int J Ambient Energy 2022; 43: 8710–8721.

32.

Dutta

Roy

Gupta

. Temperature field in flow over a stretching sheet with uniform heat flux. Int Commun Heat Mass Transf 1985; 12: 89–94.

33.

Kumar

Gowda

Alam

. (eds.). 2021, Inspection of convective heat transfer and KKL correlation for simulation of nanofluid flow over a curved stretching sheetInt Commun Heat Mass Transfer 2021; 126: 105445.

34.

Fang

Zhang

Yao

. Slip MHD viscous flow over a stretching sheet – an exact solution. Commun Nonlinear Sci Numer Simul 2009; 14: 3731–3737.

35.

Punith Gowda

Baskonus

Naveen Kumar

, et al. Evaluation of heat and mass transfer in ferromagnetic fluid flow over a stretching sheet with combined effects of thermophoretic particle deposition and magnetic dipole. Waves Random Complex Media. Epub ahead of print 9 September 2021. DOI: 10.1080/17455030.2021.1969063

36.

Andersson

Bech

Dandapat

. Magnetohydrodynamic flow of a power-law fluid over a stretching sheet. Int J Non Linear Mech 1992; 27: 929–936.

37.

Forchheimer P. Wasserbewegung durch boden. Z. Ver. Deutsch, Ing, 1901; 45: 1782–1788. https://ci.nii.ac.jp/naid/10010395788/ (accessed 9 April 2022)

38.

Muskat

Wyckoff

. The flow of homogeneous fluids through porous media. York, PA: The Maple Press, 1946, https://events.interpore.org/event/2/contributions/880/contribution.pdf (accessed 9 April 2022).

39.

Sahu

Rout

Shaw

, et al. Hydrothermal stagnation point flow of Carreau nanofluid over a moving thin needle with non-linear Navier’s slip and cubic autocatalytic chemical reactions in Darcy-Forchheimer medium. J Indian Chem Soc 2022; 99: 100741.

40.

Eid

Mahny

Al-Hossainy

. Homogeneous-heterogeneous catalysis on electromagnetic radiative Prandtl fluid flow: Darcy-Forchheimer substance scheme. Surf Interfaces 2021; 24: 101119.

41.

Sahu

Shaw

Thatoi

, et al. Darcy-Forchheimer flow behavior and thermal inferences with SWCNT/MWCNT suspensions due to shrinking rotating disk. Waves Random Complex Media. Epub ahead of print 12 July 2022. DOI: 10.1080/17455030.2022.2094496.

42.

Muhammad

Mahanthesh

, et al. Significance of Darcy-Forchheimer porous medium in nanofluid through carbon nanotubes. Commun Theor Phys 2018; 70: 361.

43.

Eid

. Thermal characteristics of 3D nanofluid flow over a convectively heated Riga surface in a darcy–forchheimer porous material with linear thermal radiation: an optimal analysis. Arab J Sci Eng 2020; 45: 9803–9814.

44.

Eid

Mabood

. Two-phase permeable non-Newtonian cross-nanomaterial flow with arrhenius energy and entropy generation: Darcy-Forchheimer model. Phys Scr 2020; 95: 105209.

45.

Mustafa

Hayat

Ioan

, et al. Stagnation-point flow and heat transfer of a casson fluid towards a stretching sheet. Z fur Naturforsch Sect J Phys Sci 2012; 67: 70–76.

46.

Liao

. The proposed homotopy analysis technique for the solution of nonlinear problems. PhD thesis, Shanghai Jiao Tong University, China, 1992.

47.

Liao

. An optimal homotopy-analysis approach for strongly nonlinear differential equations. Commun Nonlinear Sci Numer Simul 2010; 15: 2003–2016.

48.

Ullah

Zari

Gul

, et al. Darcy-Forchheimer hybrid nanofluids flow with quadratic convection over a stretched tube. Adv Mech Eng 2023; 15: 16878132231180866.