Utilization of OHAM to investigate entropy generation with a temperature-dependent thermal conductivity model in hybrid nanofluid using the radiation phenomenon

Farwa Waseem; Muhammad Sohail; Nadia Sarhan; Emad Mahrous Awwad; Muhammad Jahangir Khan

doi:10.1515/phys-2024-0059

Article Open Access

Utilization of OHAM to investigate entropy generation with a temperature-dependent thermal conductivity model in hybrid nanofluid using the radiation phenomenon

Farwa Waseem , Muhammad Sohail , Nadia Sarhan , Emad Mahrous Awwad and Muhammad Jahangir Khan

Published/Copyright: July 26, 2024

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From the journal Open Physics Volume 22 Issue 1

Abstract

This investigation takes into account the flow of a hybrid copper–molybdenum disulfide ( Cu – MoS 2 ) /water nanofluid across a plane flat surface that has been nonlinearly extended in lateral directions. Suitable boundary conditions are used to characterize the nonlinear variants in the velocity and temperature profile of the sheet. The innovative aspect of this work is to examine the impact of thermal conductivity on temperature and entropy across an extended surface using hybrid nanofluids. We obtain numerical techniques of modified boundary layer ordinary differential equations using the effective and reliable optimal homotopy analysis technique (OHAM). A graphic depiction of the influence of several parameters is shown. In this case, the hybrid model takes into account 0.01 of copper ( Cu ) and 0.01 of molybdenum disulfide ( MoS 2 ) nanoparticles within base fluid water. The second principle of thermodynamics is used to compute the irreversibility factor. The performance of nanofluid and hybrid nanofluid was compared for pivotal velocity, temperature profile, and entropy formation. The estimated skin friction and Nusselt number are the significant physical parameters. It can be observed that when the values of the stretching rate ratio and power index law increase, the skin friction increases, but it can have the opposite behavior compared to the Nusselt number.

Nomenclature

a , b , A: positive constants
A: temperature gradient
B: magnetic field strength ( Ω 1 / 2 m − 1 s − 1 / 2 kg 1 / 2 )
Br: Brinkman number
C f x , C f y: skin friction
f , g: dimensionless stream function
k f: thermal conductivity of the base nanofluid ( W m − 1 K − 1 )
k hnf: thermal conductivity of the hybrid nanofluid ( W m − 1 K − 1 )
k(_hnf)(T) = k(_hnf)(1 + ∈θ): temperature-dependent thermal conductivity where ∈ is a positive constant
M: magnetic parameter ( T )
n: power-law index
N u *: Nusselt number
NG: ntropy generation
Pr: Prandtl number
Rd: thermal radiation parameter
Re: Reynolds number
T: fluid temperature
T ∞: ambient temperature
T w: wall temperature
u ˜ , v ˜ , w ˜: velocity components ( m s − 1 )
U w ˜ , V w ˜: velocities of the stretching sheet
x , y , z: Cartesian coordinate system
θ: dimensionless temperature
λ: ratio of the stretching sheet ( m )
ρ f: density of the base nanofluid ( kg m − 3 )
ρ hnf: density of the hybrid nanofluid ( kg m − 3 )
( ρ C P ) f: heat capacitance of the base nanofluid ( J kg 2 m 3 K − 1 )
( ρ C P ) hnf: heat capacitance of the hybrid nanofluid ( J kg 2 m 3 K − 1 )
ν f: kinematic viscosity of the base nanofluid ( kg m − 3 )
ν hnf: kinematic viscosity of the hybrid nanofluid ( kg m − 3 )
σ f: electricity conductivity of the base nanofluid ( Ω − 1 m − 1 )
σ hnf: electricity conductivity of the hybrid nanofluid ( Ω − 1 m − 1 )

1 Introduction

The core issue of two-dimensional flow resulting from the expansion of a flat surface, first explored by Crane [1], is a critical aspect of numerous manufacturing procedures, including the extrusion of aluminum and polymers, the drafting of plastic sheets, and manufacturing paper. Due to these practical uses, experts have examined this issue from multiple angles, including factors such as suction or injection, variable warming of the surface, mass transfer, and transpiration. The topic of three-dimensional (3-D) flow resulting from a flat, bi-directionally extended surface was initially explored by Wang [2]. In this situation, he was successful in deriving an accurate similarity result for the standard Navier–Stokes models. A numerical analysis of the unstable 3-D flow with mass and thermal transportation via an unstable stretched surface was then performed by Lakshmisha et al. [3]. Wang et al. [4] explored 3-D nanofluid motion with laminar boundaries and considered the influence of thermal radiation on the occurrence of a stretching surface. In this research, they focused on the activation energy’s role in inducing mass transfer within a couple of stress nanofluids near a stretching sheet. The numerical computations were carried out using a fourth-order Runge–Kutta–Fehlberg (R-K-F) scheme. In their study of the Darcy–Forchheimer unstable hybrid nanofluid flow across a bidirectional stretched surface, Gowda et al. [5] investigated particularly on convective transmission of heat. They extensively examined the heat transfer and flow properties of nanoparticle suspensions containing cadmium telluride and graphite particles, which were shaped like spheres and bricks. These suspensions were dispersed in water and subjected to conditions involving viscous dissipation and Joule heating effects.

The goal of this study was to increase the working fluid’s capacity for heat transmission with the least amount of resources and energy. Numerous studies have been conducted over the years using various nanofluids while taking into account various fluid models with external impacts [6,7,8,9,10]. However, to increase the effectiveness of the nanofluid, a special hybrid nanofluid was created. The presence of metallic solid nanoparticles in nanoparticles was discovered to improve the thermal conductivity of fluids. The thermal conductivity of fluids was found to be enhanced by the metallic solid nanoparticle-containing nanoparticles. With time, cooling equipment for MHD generators, nuclear engines, and digital gadgets has been used for this. Further advances in the nanoparticles lead to the formation of hybrid nanoparticles, a combination of two different nanoparticles dissolving in a base fluid to produce a fluid with increased conductivity. Mahanthesh et al. [11] suggested viscous heating and Joule heating impacts on the heat transfer efficiency of molybdenum–silver hybrid nanofluid across an isothermal edge to attain the required thermal efficiency for hybrid nanofluids. Chu et al. [12] examined the amplification in thermal energy and soluble materials by applying hybrid nanoparticles and utilizing stimulation energy across a parabolic surface. Nazir et al. [13] explored different hybrid nanoparticles combined in a Carreau Yasuda model for the enhancement of thermal efficiency. In order to assess the thermal characteristics of the Maxwell hybrid nanofluid border value issue in a vertically porous structure, Algehyne et al. [14] adopted a finite element method. Yaseen et al. [15] investigated the transmission of heat for mono- and hybrid nanofluids flowing across a pair of adjacent surfaces in a Darcy porous model involving thermal radiation and heat generation. In contrast to hybrid nanofluid flow, they found that nanofluid flow transported heat more quickly.

Magnetohydrodynamics (MHD), also known as magneto-fluid dynamics or hydromagnetic, is a discipline of physics that refers to the behavior of conducting fluids in the presence of magnetic fields. Some common examples of magnetofluids include electrolytes, saltwater, plasma, and liquid metals. The concept of MHD was first introduced by Michael Faraday in 1832. Faraday’s initial work on MHD was based on his study of the interaction involving the Earth’s magnetic field and the flow of seawater, which laid the foundation for understanding the principles of MHD. Abbas et al. [16] examined the impact of magnetic forces on the movement of liquid around a flexible cylinder. They showed how friction influences the temperature of the fluid changes based on its thickness. Suresh et al. [17] studied the effects of MHD radiated flow, heat, and mass transmission in the absence of chemical processes and thermal radiation. This analysis was conducted across an influencing vertical porous plate. Bilal et al. [18] examined a finite film thickness flow involving a pseudo-plastic MHD hybrid nanofluid. Their analysis considered factors like heat generation within the fluid and the variable thermal conductivity of the nanofluid. Chu et al. [19] investigated the influence of unstable viscous flow within a constriction route. They added engine oil as the foundation fluid, along with different-shaped silver–gold hybrid nanofluid particles. The study focused on examining the flow and heat transfer phenomena while taking into account the effects of MHD among two adjacent infinite surfaces.

A material thermal conductivity is a key factor in heat transfer, and it is frequently assumed that it will not change. Multiple research investigations did, however, point out that different materials will respond to temperature changes differently in terms of how well they modify thermal conductivity. Specifically, nanofluids exhibit a strong temperature dependence that can drastically impact heat transmission owing to the enhanced aspect ratio provided by nanoparticles inside the base fluid. Thermal conductivity properties have been shown to greatly increase at higher temperatures. As a result, the thermal conductivity fluctuates over a large variation in temperature. Gbadeyan et al. [20] examined the influence of various factors, such as thermal conductivity, fluid viscosity, nonlinear thermal radiant energy, and MHD behavior, on the nanofluid flow. Additionally, their analysis considered velocity slip and convective heating as part of the study’s parameters. Usman et al. [21] conducted experiments involving the flow of Cu – Al 2 O 3 / H 2 hybrid nanofluid across a permeable medium, taking into account nonlinear radiant energy and varying thermal conductivity. On the other hand, Shamshuddin and Eid [22] addressed nanofluid flow while taking into account such as variable thermal conductivity and fluid viscosity, specifically in the context of flow from an adjacent extensible rotating surface. In the presence of thermal slip, an induced magnetic field was considered to impact viscosity dissipation and Darcy resistances by Abbas et al. [23]. They investigated if larger amounts of variable thermal conductivity led to an increase in the fluid temperature velocity. Saeed et al. [24] examined the steady flow of an electrically executing hybrid nanofluid under Darcy–Forchheimer conditions, taking into account MHD. Their analysis considered several factors, including Joule heating, viscous dissipation, heat generation, and variable fluid viscosity. In their study [25], Mandal and Pal focused on analyzing the behavior of a hybrid nanofluid made up of water-dispersed silver–molybdenum disulfide nanoparticles. The analysis took into account the impact of thermal radiant energy and the flow on a porous Riga layer that was diminishing exponentially. The study investigated the impact of various factors such as viscosity, thermal conductivity, slip velocity, and convective thermal boundary conditions on a shrinking Riga surface. They examined how these factors impacted velocity, temperature, skin friction coefficient, Nusselt number, and entropy production in the system.

Entropy generation, or entropy production, refers to the quantification of irreversibilities within a system, typically arising from friction forces and heat transport owing to temperature modifications. The study of entropy production in the context of natural convection heat transport is a captivating area of research. It has been explored extensively for both simple fluids [26,27] and nanofluids [28], shedding light on the thermodynamic aspects of heat transport mechanisms and the consequences of nanofluids on these irreversibilities. Entropy generation in the setting of alumina–water nanofluids in a slanted cavity was studied by Liu et al. [29]. They noticed that decreasing entropy generation occurred as the tilt inclination of the cavity increased. This finding suggests that the orientation or angle of the cavity can have a drastic effect on the thermodynamic irreversibilities associated with the nanofluid flow and heat transfer processes. Selimefendigil and Öztop [30] conducted a study on conjugate heat transfer involving an analysis of entropy production in copper oxide–water nanofluids within a cavity featuring a solid conductive partition. Their findings revealed that altering the geometric configuration of the cavity can significantly affect the distribution of entropy generation between the solid and liquid regions. This shows that the system’s thermodynamic behavior, particularly in terms of entropy generation, is significantly influenced by the form and structure of the cavity.

Indeed, the study of entropy production in the context of hybrid nanofluids is a relatively new and emerging research area. Hybrid nanofluids, which combine different types of nanoparticles or nanomaterials within a base fluid, present unique thermal and thermodynamic properties that can influence entropy generation in heat transfer processes. As researchers continue to explore and understand the behavior of these complex fluids, the investigation of entropy production in hybrid nanofluids becomes increasingly important, providing insights into their thermodynamic performance and potential applications. Alsabery et al. [31] focused on examining entropy generation, fluid flow, and transmission of heat in the context of Cu – Al 2 O 3 /water hybrid nanofluids within a complex-shaped containment that included a hot-half partition. The entropy production of an intersection nanofluid containing a heat source and sink in a porous substance was discovered by Hussain et al. [32]. Singh et al. [33] explored the conductivity and heat transfer properties of a specialized liquid with small particles as it moved between two electrically conductive walls. They also examined energy dissipation and the impact of the Hall effect on flow efficiency. Hayat et al. [34] examined how an induced magnetic field was used in the context of a hybrid nanoliquid flow. They likely studied the effects of applying a magnetic field on hybrid nanoliquid behavior, especially concerning entropy generation.

This work is innovative as it addresses the impacts of variable thermal conductivity despite the lack of viscous dissipation for calculating the heat transfer coefficient in a radiative hybrid nanofluid flow. According to our research, 3-D flows, as opposed to 2-D flows, tend to be valuable in providing more precise visual representations of actual situations. This study also assumes that the distribution of temperature throughout the sheet is not linear. Another novel aspect of this research is the existence of a linearly extensible surface via a hybrid nanofluid velocity slip. For the present research, entropy generation is essential to the improvement of heat transfer for both heating and cooling purposes in the engineering, biomedical, production, and technology sectors. The motivation behind this research is to explore the heat transport and entropy generation characteristics of a hybrid nanofluid flow passing over an extensible surface. The study accounts for several important factors like variable thermal conductivity, MHD, power-law index n and other parameters’ physical interpretation, which are the main points of discussion. The literature review is given in Section 1, physical scenarios are explained in Section 2, the irreversible occurrence is explained in Section 2.1, and the method used is explained in Section 3. The physical interpretation is explained in Section 4, followed by conclusion in Section 5.

2 Mathematical modeling

Consider the extendable surface flow in three dimensions at z = 0 , as shown in Figure 1. The flow is caused by two transverse stretching of the sheet. The velocities and temperature of the extensible surface U w ( x ) = a ( x + y ) n and V w ( x ) = b ( x + y ) n are velocities across the x , y directions, with a , b , n > 0 being constants. The variable surface temperature T w = T ∞ + A ( x + y ) n , where T ∞ is the fluid’s ambient temperature and A > 0 is a value that remains constant. Under conventional boundary layer conditions, the equations regulating 3-D flow, heat transmission, thermal conductivity, and internal heat generation/absorption can be expressed without requiring viscous dissipation.

Figure 1

Geometrical arrangement in association with the issue.

The regulating equations for continuity, conservation of momentum, energy, and internal heat generation are given below [35]. All the aforementioned presumptions, the boundary layer approximation, and the empirical relationship between the characteristics of nanoparticles are depicted in Table 1, and their numerical values are mentioned in Table 2.

(1) ∂ u ∼ ∂ x + ∂ v ∼ ∂ y + ∂ w ∼ ∂ z = 0 ,

(2) u ∼ ∂ u ∼ ∂ x + v ∼ ∂ u ∼ ∂ y + w ∼ ∂ u ∼ ∂ z = v hnf ∂ 2 u ∼ ∂ z 2 − σ hnf ρ hnf B 2 u ∼ ,

(3) u ∼ ∂ v ∼ ∂ x + v ∼ ∂ v ∼ ∂ y + w ∼ ∂ v ∼ ∂ z = v hnf ∂ 2 v ∼ ∂ z 2 − σ hnf ρ hnf B 2 v ∼ ,

(4) u ∼ ∂ T ∂ x + v ∼ ∂ T ∂ y + w ∼ ∂ T ∂ z = 1 ( ρ C P ) hnf ∂ ∂ z k hnf ( T ) ∂ T ∂ z + 16 δ * T ∞ 3 3 K * ( ρ C P ) hnf ∂ 2 T ∂ z 2 .

Table 1

Thermophysical characteristics of nanofluids and hybrid nanofluids [25]

Parameters	Nanofluid with hybrid nanofluid
Density	ρ nf ρ f = ( 1 ‒ φ 1 ) + φ 1 ρ s 1 ρ f , ρ hnf ρ f = ( 1 ‒ φ 2 ) ( 1 ‒ φ 1 ) + φ 1 ρ s 2 ρ f + φ 2 ρ s 2 ρ f
Electrical conductivity	σ nf σ f = ( 1 + 2 φ 1 ) σ 1 + ( 1 ‒ 2 φ 1 ) σ f ( 1 ‒ φ 1 ) σ 1 + ( 1 + φ 1 ) σ f , σ hnf σ nf = ( 1 + 2 φ 2 ) σ 2 + ( 1 ‒ 2 φ 2 ) σ nf ( 1 ‒ φ 2 ) σ 2 + ( 1 + φ 2 ) σ nf , σ hnf σ f = σ hnf σ nf × σ nf σ f
Thermal conductivity	k nf k f = k s 1 + 2 k f ‒ 2 φ 1 ( k f ‒ k s 1 ) k s 1 + 2 k f + φ 1 ( k f ‒ k s 1 ) , k hnf k f = k s 2 + 2 k nf ‒ 2 φ 2 ( k nf ‒ k s 2 ) k s 2 + 2 k nf + φ 2 ( k nf ‒ k s 2 ) , k hnf k f = k hnf k nf × k nf k f .

Table 2

Cu and MoS 2 thermal characteristics in the base liquid [25,36]

Nanoparticles/base fluid	Cu	MoS 2	H 2 O
ρ	8933	5060	997.1
C P	385	397.21	4179
k	401	904.4	0.613
σ	5.977 × 10 5	2.09 × 10 4	0.0500

Considering appropriate boundary conditions [35]

(5) u ∼ = U w ( x ) = a ( x + y ) n , T = T w = T ∞ + A ( x + y ) n , v ∼ = V w ( y ) = b ( x + y ) n , w ∼ = 0 : z = 0 , u ∼ = 0 , v ∼ = 0 , T → T ∞ : z → ∞ .

However,

(6) k hnf ( T ) = k hnf ( 1 + ∈ θ ) .

According to similarity transformation [35], the stated problem is

(7) u ∼ = a ( x + y ) n f ′ ( η ) , v ∼ = b ( x + y ) n g ′ ( η ) , w ∼ = − a v nf ( x + y ) n − 1 2 n + 1 2 ( f + g ) + n − 1 2 η ( f ′ + g ′ ) , η = a v nf ( x + y ) n − 1 2 z , θ ( η ) = T − T ∞ T w − T ∞ , λ = b a .

Using Eq. (7) to Eqs. (1)–(4), ODEs appear as

(8) f ‴ − n f ′ ( f ′ + g ′ ) − A 1 A 2 M f ′ + n + 1 2 f ″ ( f + g ) = 0 ,

(9) g ‴ − n g ′ ( f ′ + g ′ ) − A 1 A 2 M g ′ + n + 1 2 g ″ ( f + g ) = 0 ,

(10) A 3 Pr θ ″ ( 1 + ϵ θ ) + Rd Pr θ ″ + A 3 Pr ( θ ′ ) 2 + θ ′ n + 1 2 ( f ′ + g ′ ) = 0 .

Using specified boundary requirements [35],

(11) f ( 0 ) = 0 , f ′ ( 0 ) = 1 , g ( 0 ) = 0 , g ′ ( 0 ) = λ , θ ( 0 ) = 1 , : η = 0 , f ′ ( ∞ ) → 0 , g ′ ( ∞ ) → 0 , θ ( ∞ ) → 0 : η → ∞ .

The fluid characteristics are distinguished as

(12) M = B 0 2 σ nf ρ nf a , Pr = ν α nf , Rd = 16 δ * T ∞ 3 3 K * ( k ) f .

The variables A 1 , A 2 , and A 3 are described as follows:

(13) A 1 = σ hnf σ nf , A 2 = ρ hnf ρ f , A 3 = k hnf k f .

Indeed, the execution of physical measurements n in the engineering curriculum is of utmost importance. Inspections of various physical parameters provide crucial insights into the behavior of systems and processes. Skin friction and local Nusselt number are two significant physical parameters that are frequently employed in relation to fluid flow and heat transfer:

(14) Re x 1 / 2 C f x = A 1 A 2 f ″ ( 0 ) ,

(15) Re y 1 / 2 C f y = A 1 A 2 g ″ ( 0 ) ,

(16) Re x − 1 / 2 Nu * = { A 3 ( 1 + ∈ θ ) + Rd } θ ′ ( 0 ) .

2.1 Entropy generation

Entropy generation, also referred to as entropy production, is a fundamental concept in thermodynamics and plays a significant role in various industrial and technical processes. It represents the amount of energy lost or wasted in a system due to irreversibility associated with heat transfer and fluid flow:

(17) E G = k f T ∞ 2 k hnf k f + 16 δ * T ∞ 3 3 k f k * ∂ T ∂ z 2 + σ hnf T ∞ ( u ∼ 2 + v ∼ 2 ) B 2 .

The following equation shows an overview of dimensionless entropy generation:

(18) NG = T ∞ 2 a 2 E G k f ( T w − T ∞ ) 2 .

Using Eq. (17), we were able to obtain an alternative dimensionless version of the entropy equation.

(19) NG = Re k hnf k f ( 1 + ∈ θ ) + Rd θ ′ 2 + M Br A σ hnf σ nf ( f ′ 2 + g ′ 2 ) .

Here,

(20) Re = a 2 U w ν f x , Br = μ nf U w 2 k f ( T w − T ∞ ) , A = T ∞ T w − T ∞ .

3 Optimal homotopy analysis method (OHAM)

The optimization of a homotopy scheme for solving a system of differential equations is indeed a valuable approach, particularly in dealing with nonlinear systems. The proposed homotopy scheme offers a robust and flexible approach to solving differential equation systems. Its parameter-free or large nature, stability, and applicability to diverse problem types make it a valuable tool for researchers and engineers in various fields. However, the careful selection of an appropriate initial guess remains crucial for its success in solving complex problems [37]:

(21) f ˆ ( η ) = 1 − exp ( − η ) ,

(22) g ˆ ( η ) = λ { 1 − exp ( − η ) ,

(23) θ ˆ ( η ) = exp ( − η ) ,

(24) L f ˆ = d 3 f ˆ d η 3 − d f ˆ d η , L g ˆ = d 3 g ˆ d η 3 − d g ˆ d η , L θ ˆ = d 2 θ ˆ d η 2 − θ ˆ .

Such operators simulate the essential quality:

(25) L f ˆ { Z 1 * + Z 2 * exp ( η ) + Z 3 * exp ( − η ) } = 0 ,

(26) L g ˆ { Z 4 * + Z 5 * exp ( η ) + Z 6 * exp ( − η ) } = 0 ,

(27) L θ ˆ { Z 7 * exp ( η ) + Z 8 * exp ( − η ) } = 0 ,

where Z j * ( j = 1 – 8 ) are arbitrary constants.

4 Results and discussion

4.1 Analysis of velocity and temperature profile

In this section, graphical results are presented that depict the fluid velocity and temperature fields as they relate to various influential variables by comparing nanofluid and hybrid nanofluid. These influential variables likely correspond to parameters or factors that play a significant role in the behavior of the system described by the transformed system of ordinary differential equations (ODEs). Overall, presenting graphical results is a common and effective way to communicate the findings of a study and enhance the understanding of complex systems and their responses to various influential variables. In this research, we investigate the effects of several influential factors on the velocity, temperature, and entropy generation within a system. These influential variables include the power index law, the magnetic parameter ( M ) , stretching rate ratio ( λ ) , the Prandtl number ( Pr ) , subject to temperature thermal conductivity ( ∈ ) , the radiation parameter ( Rd ) , the Brinkman number ( Br ) , Reynolds number ( Re ) , and temperature gradient ( A ) . The volume fractions of the nanoparticles, φ 1 = φ 2 = 0.01 , have been taken into consideration in every situation. We have taken into consideration φ 2 = 0 , which denotes the absence of molybdenum in the base fluid compared to the ordinary nanofluid. This comparison reveals an enormous variation in the velocity description across each case. Figures 2a and 3a show the impact of changing the magnetic parameter ( M ) on velocity profiles and behaviors when comparing hybrid and general nanofluids. Figures 2a and 3a show how the velocities in both directions decay as a result of the fluid speed falling and M effectively reacting to the resistive Lorentz force.

$Figure 2 (a) Impact of M M on f ′ ( η ) f\left^{\prime} \left(\eta ) , (b) results of λ \lambda on f ′ ( η ) f\left^{\prime} \left(\eta ) , and (c) effect of n n on f ′ ( η ) . {f}^{^{\prime} }(\eta ).$

Figure 2

(a) Impact of M on f ′ ( η ) , (b) results of λ on f ′ ( η ) , and (c) effect of n on f ′ ( η ) .

$Figure 3 (a) Impact of M M on g ′ ( η ) g\left^{\prime} \left(\eta ) , (b) results of λ \lambda on g ′ ( η ) g\left^{\prime} \left(\eta ) , and (c) effect of n n on g ′ ( η ) g\left^{\prime} \left(\eta ) .$

Figure 3

(a) Impact of M on g ′ ( η ) , (b) results of λ on g ′ ( η ) , and (c) effect of n on g ′ ( η ) .

Figures 2b and 3b depict the modifications in the horizontal and vertical constituents of velocity with an increase in the stretching rate ratio ( λ ) . These observations indicate that changes in the stretching rates ratio ( λ ) have a significant influence on the velocity components within the system. Increasing ( λ ) leads to a stronger stretching effect along the y-direction, resulting in increased vertical velocity but reduced horizontal velocity. This behavior is relevant in various applications involving stretching surfaces, such as the production of thin films, polymer processing, or coating processes, where the stretching rate ratio can be a crucial factor in controlling material properties and fluid flow. A vertical downhill flow will occur due to the bi-directional extended surface. It follows that, in this case, a negative vertical component at the distant field border is anticipated. When λ increases, we observe that shear stresses at the wall also increase. Furthermore, the cold fluid moves more quickly at ambient temperatures with larger values of λ. Thus, an increasing value of λ determines the entrainment velocity.

Figures 2c and 3c show the behavior of the power index on the velocity profile compared to the hybrid and nanofluid. This shows that when the value of n increases, then the velocity along the horizontal and vertical directions decreases.

The temperature profile increases when the value of ∈ increases, as shown in Figure 4(a). This is because more heat may be transmitted from the plate to the liquid as a result of increasing ∈ and its associated higher thermal conductivity. Owing to the inverse relationship between the Prandtl number ( Pr ) and thermal diffusivity, as shown in Figure 4(b), the system is shown to cool down. The fluid temperature and the corresponding boundary layer thickness decrease against higher values of the Prandtl number. The thermal distribution is reduced by the opposite thermal diffusion contact. Figure 4c illustrates the behavior of the power index on the temperature profile compared to the hybrid and nanofluid. This shows that when the value of n increases, then the temperature profile also increases.

$Figure 4 Upshot of (a) ϵ \epsilon on θ ( η ) \theta \left(\eta ) , (b) Pr \Pr on θ ( η ) \theta \left(\eta ) , and (c) n n on θ ( η ) \theta \left(\eta ) .$

Figure 4

Upshot of (a) ϵ on θ ( η ) , (b) Pr on θ ( η ) , and (c) n on θ ( η ) .

4.2 Analysis of entropy generation

The entropy generation in a system refers to the quantity of entropy that develops during the irreversibility process as a result of heat flow through thermal resistance, joule heating, fluid viscosity, etc. Figure 5(a) and (b) depicts graphical representations of entropy generation NG as a function of specific parameters. The increase in entropy generation is associated with an increase in the radiation parameter ( Rd ) and the Brinkman number ( Br ) . These parameters represent contributions from thermal radiation and Joule heating, respectively. The increase in entropy generation implies a greater degree of irreversibility and instability within the system, which can have implications for system efficiency and performance. Understanding these effects is crucial for optimizing system design and operation, especially in applications where heat transfer and fluid flow are significant factors. Entropy generation via joule heating increases in the magnetic field, which increases the overall system entropy generation, as shown in Figure 5(c). Figure 5(d) shows the change in NG on the temperature’s thermal conductivity. In this instance, an increasing impact is seen. Figure 5(e) shows the variation in NG caused by the temperature differential parameter A . An increasing impact is displayed here. Figure 5(f) shows that when the Reynold number increases, the entropy decreases ( Re ) . The ratio of inertial to viscous forces is represented by the Reynolds number. As a result, the higher Reynolds number indicates that inertial forces predominate and that entropy formation is increasing.

$Figure 5 (a) The impact of (a) Br {\rm{Br}} on NG ( η ) {\rm{NG}}\left(\eta ) , (b) Rd {\rm{Rd}} on NG ( η ) {\rm{NG}}\left(\eta ) , (c) M M on NG ( η ) {\rm{NG}}\left(\eta ) , (d) ∈ \in on NG ( η ) {\rm{NG}}\left(\eta ) , (e) A A on NG ( η ) {\rm{NG}}\left(\eta ) , and (f) Re \mathrm{Re} on NG ( η ) {\rm{NG}}\left(\eta ) .$

Figure 5

(a) The impact of (a) Br on NG ( η ) , (b) Rd on NG ( η ) , (c) M on NG ( η ) , (d) ∈ on NG ( η ) , (e) A on NG ( η ) , and (f) Re on NG ( η ) .

Impressive comparable responses are displayed in Tables 3 and 4, which further provide competence in the methodology used in this study.

Table 3

Numerical values for the velocity distribution along f ′ ′ ( 0 ) and g ′ ′ ( 0 ) for different values of n and λ

n	λ	Present results	Khan et al. [35]		Present results	Khan et al. [35]
		f ″ ( 0 )	f ″ ( 0 )		g ″ ( 0 )	g ″ ( 0 )
		OHAM	Shooting	bvp 5 c	OHAM	Shooting	bvp 5 c
1	0	‒ 1.00000	‒ 1	‒ 1	0	0	0
	0.5	‒ 1.22728	‒ 1.224745	‒ 1.224742	‒ 0.628582	‒ 0.612372	‒ 0.612371
	1	‒ 1.41049	‒ 1.414214	‒ 1.414214	‒ 1.42403	‒ 1.414214	‒ 1.414214
3	0	‒ 1.62169	‒ 1.624356	‒ 1.624356	0	0	0
	0.5	‒ 1.98233	‒ 1.989422	‒ 1.989422	‒ 0.98405	‒ 0.994711	‒ 0.994711
	1	‒ 2.29728	‒ 2.297186	‒ 2.297182	‒ 2.29630	‒ 2.297186	‒ 2.297182

Table 4

Numerical values for temperature distribution along ‒ θ ′ ( 0 ) for different values of Pr , n , and λ

n	Pr	λ	Present result	Khan et al. [35]
			‒ θ ′ ( 0 )	‒ θ ′ ( 0 )	‒ θ ′ ( 0 )
			OHAM	Shooting	bvp 5 c
1	0.7	0	0.791489	0.793668	0.793668
		0.5	0.961545	0.972033	0.972029
		1	0.13252	1.122406	1.122321
	1	0	1.0000	1.00000	0.999990
		0.5	1.205923	1.224745	1.224742
		1	1.414549	1.414214	1.414214
	7	0	3.067901	3.072250	3.072251
		0.5	3.763663	3.762723	3.762724
		1	4.351980	4.344818	4.344779
3	0.7	0	1.290612	1.292193	1.292194
		0.5	1.579578	1.582607	1.582607
		1	1.822310	1.827437	1.827427
	1	0	1.624878	1.624356	1.624356
		0.5	1.973610	1.989422	1.989422
		1	2.294094	2.297186	2.297182
	7	0	4.96672	4.968777	4.968777
		0.5	6.077149	6.085484	6.085485
		1	7.02747	7.026912	7.026913

The skin friction coefficients and local Nusselt numbers for various parameter values are presented numerically in Tables 5 and 6 using OHAM. Table 5 illustrates the effects of the magnetic parameter, the power index law, and the stretching rate ratio on skin friction, Re x 1 / 2 C f x = A 1 A 2 f ″ ( 0 ) , and Re y 1 / 2 C f y = A 1 A 2 g ″ ( 0 ) . It can be seen that as the values of M , n , and λ increase, the skin friction coefficients across both the horizontal and vertical components of velocity for the first solution increase. It is also observed that for the second solution, they increase. Additionally, while holding other parameters constant, Table 6 shows the local Nusselt number Re x − 1 / 2 Nu * = { A 3 ( 1 + ∈ θ ) + Rd } θ ′ ( 0 ) evolution for various values of the Prandtl number, power index law, thermal conductivity, and stretching rate ratio. It demonstrates that the local Nusselt number increases for higher values of n , ∈ , and λ despite decreasing for Pr .

Table 5

Analysis of skin friction for velocity distribution along f ″ ( 0 ) and g ″ ( 0 ) for different values of M , n , and λ

M	n	λ	f ″ ( 0 )	f ″ ( 0 )	g ″ ( 0 )	g ″ ( 0 )
			φ 1 , φ 2 ≠ 0	φ 1 ≠ 0 , φ 2 = 0	φ 1 , φ 2 ≠ 0	φ 1 ≠ 0 , φ 2 = 0
0.1	0.5	0.5	‒ 0.931843	‒ 0.931543	‒ 0.465899	‒ 0.465836
0.2			‒ 0.973068	‒ 0.967328	‒ 0.484485	‒ 0.483627
0.3			‒ 1.01331	‒ 1.00311	‒ 0.506583	‒ 0.501419
0.4			‒ 1.0526	‒ 1.0389	‒ 0.526208	‒ 0.519211
0.5	0.1		‒ 0.873468	‒ 0.877826	‒ 0.43673	‒ 0.439128
	0.2		‒ 0.93133	‒ 0.92704	‒ 0.465627	‒ 0.463596
	0.3		‒ 0.986765	‒ 0.976253	‒ 0.493316	‒ 0.488065
	0.4		‒ 1.03993	‒ 1.02547	‒ 0.519873	‒ 0.512534
	0.5	0.1	‒ 0.995467	‒ 0.990314	‒ 0.0995368	‒ 0.0990113
		0.2	‒ 1.01989	‒ 1.01141	‒ 0.203952	‒ 0.202217
		0.3	‒ 1.04394	‒ 1.0325	‒ 0.313134	‒ 0.309618
		0.4	‒ 1.06763	‒ 1.05359	‒ 0.426977	‒ 0.421213

Table 6

Analysis of local Nusselt number for temperature distribution along ‒ θ ′ ( 0 ) for different values of Pr , n , ∈ , and λ

Pr	n	ϵ	λ	‒ θ ′ ( 0 )	‒ θ ′ ( 0 )
				φ 1 , φ 2 ≠ 0	φ 1 ≠ 0 , φ 2 = 0
1.0	0.5	0.5	0.5	0.318844	0.324116
1.5				0.714146	0.748099
2.0				0.799276	0.960091
1.0	0.1			0.502065	0.464534
	0.2			0.462427	0.429429
	0.3			0.418872	0.394325
	0.4	0.1		0.473024	0.431032
		0.2		0.466892	0.419813
		0.3		0.449208	0.404104
		0.4		0.417993	0.383907
		0.5	0.1	0.404485	0.39983
			0.2	0.39778	0.389725
			0.3	0.388989	0.379557
			0.4	0.380102	0.369388

5 Conclusions

This investigation yielded the following significant features:

When the magnetic ( M ) parameter value is increased, the fluid velocity in the horizontal and vertical components decreases.
As the stretching rate ratio ( λ ) increases, the shear stress across the wall increases. As a result of the bi-directional extended surface, the entrainment velocity is negative, signifying that it flows down in the vertical direction.
When the value of power index law ( n ) increases, the velocity along horizontal and vertical directions decreases, but the opposite behavior is observed with temperature.
Increasing the thermal conductivity ( ϵ ) and the power index law ( n ) causes the system to heat up, whereas increasing the Prandtl number ( Pr ) has a reverse effect.
The Brinkman number ( Br ) , thermal conductivity ( ϵ ) , temperature gradient ( A ) , radiation parameter ( Rd ) , Reynolds number ( Re ) , and magnetic parameter ( M ) all have an increasing effect on entropy generation ( NG ) .

Acknowledgments

The authors thank King Saud University for funding this research through the Researchers Supporting Program number (RSPD2024R1052), King Saud University, Riyadh, Saudi Arabia.

Funding information: The authors thank King Saud University for funding this research through the Researchers Supporting Program number (RSPD2024R1052), King Saud University, Riyadh, Saudi Arabia.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: The datasets generated and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Received: 2023-11-23

Revised: 2024-06-11

Accepted: 2024-07-01

Published Online: 2024-07-26

This work is licensed under the Creative Commons Attribution 4.0 International License.

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