MHD radiative mixed convective flow of a sodium alginate-based hybrid nanofluid over a convectively heated extending sheet with Joule heating

Humaira Yasmin; Rawan Bossly; Fuad S. Alduais; Afrah Al-Bossly; Anwar Saeed

doi:10.1515/ntrev-2024-0132

Article Open Access

MHD radiative mixed convective flow of a sodium alginate-based hybrid nanofluid over a convectively heated extending sheet with Joule heating

Humaira Yasmin , Rawan Bossly , Fuad S. Alduais , Afrah Al-Bossly and Anwar Saeed

Published/Copyright: January 10, 2025

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From the journal Nanotechnology Reviews Volume 14 Issue 1

Abstract

The study of hybrid nanofluids is significant in thermal management applications by optimizing heat transfer through pioneering materials, mainly the flow of sodium alginate-based hybrid nanofluids. This work addresses the demand for effective cooling solutions in a variety of industrial processes and uses the unique characteristics of non-Newtonian fluids and their inferences for rheological modeling and heat transform enhancement. Inspired by the progressive properties of the non-Newtonian Casson fluid and its applied significance in the rheological modeling and heat transfer characteristics of different dynamic fluids, there is limited knowledge of their response in the mixed convective flow, particularly when influenced by factors like thermal radiation, joule heating, and thermal relaxation time. The present study aims to investigate the heat transfer enhancement of the mixed convective flow of a sodium alginate-based hybrid nanofluid on an extending sheet concentrating on the interaction of different physical parameters that affect thermal performance. The physical phenomena are modeled in a nonlinear partial differential equation, which is then converted into ordinary differential equations with the help of suitable similarity variables. Tables and figures are constructed to show the behavior of the physical parameters involved in the momentum and temperature equations. Premilinary assumptions applied to the flow are electrically conducting, rotating, dissipative, and thermal boundary conditions. A semi-analytical approach homotopy analysis method is employed to obtain the solution of the problem. The outcome witnessed that the velocity profiles show a diminishing behavior through a magnetic parameter; however, the temperature profile shows an escalating behavior. Similarly, the thermal plot intensifies with the Eckert number and thermal radiation. In addition, the numerical data from the tables portrayed that the skin friction decreases with the Casson parameter and the rotation parameter; however, an improvement behavior is noticed in the Nusselt number through thermal radiation and thermal Biot number. From the numerical data, it is concluded that the hybrid nanofluid has superior heat transfer characteristics compared to nanofluid. The finding of this result has also been compared with available results in the literature through a comparative study.

Keywords: MHD; thermal radiation; mixed convection; hybrid nanofluid; stretching sheet; Joule heating

Nomenclature

u , v , w: velocity components
x , y , z: coordinates
a: constants
h T: heat transfer coefficient
Ω: angular velocity
T: temperature
T f: reference temperature
T ∞: far away temperature
C p: specific heat
α: rotation parameter
λ: mixed convective parameter
Ec: Eckert number
M: magnetic parameter
Q: heat source parameter
Rd: thermal radiation parameter
β 1: thermal Biot number
Pr: Prandtl number
k: thermal conductivity
q r: radiative heat flux
β: Casson factor
β 0: strength of magnetic field
ρ: density
σ: electrical conductivity
σ ⁎: Stefan–Boltzmann constant
μ: dynamic viscosity
C p: heat capacitance
ξ: similarity variable
ν = μ / ρ: kinematic viscosity
Φ 1 , Φ 2: nanoparticles volume fraction

Subscripts

f: base fluid
nf: nanofluid
hnf: hybrid nanofluid

1 Introduction

Nanofluid flow, a cutting-edge area of research in fluid dynamics, involves the dispersion of nanoscale particles within a base fluid, creating a composite material with unique thermal properties. The introduction of nanoparticles, typically metallic or oxide in nature, significantly alters the thermal flow features of the fluid. One of the key factors contributing to this transformation is the substantial increase in thermal conductivity conferred by the suspended nanoparticles as described first by Choi [1]. Moreover, nanofluids exhibit enhanced convective heat transfer, which is crucial for applications where efficient cooling is paramount, as studied by Bhatti et al. [2]. The convective heat transference improvement was endorsed by the increased thickness of the thermal layer at the boundary and augmented fluid velocity near the solid surfaces. These effects collectively contribute to heightened heat transfer rates, positioning nanofluids as promising candidates for various industrial and technological applications. The impact of the nanofluid flow on heat transfer extends to diverse fields, including electronics cooling, renewable energy systems, and thermal management in manufacturing processes [3,4]. In electronics, nanofluids have shown potential for enhancing the efficiency of heat dissipation in microelectronic devices, thereby preventing overheating and improving overall device performance and reliability, as studied by Moita et al. [5]. In solar thermal collectors, nanofluids can enhance the absorption of solar radiation and improve the efficiency of energy conversion, as studied by Chen et al. [6]. Industrial heat exchangers can also benefit from incorporating nanofluids, leading to more compact and efficient heat exchange systems examined by Huq et al. [7]. Acharya et al. [8] inspected the motion of nanofluid with variations in hydrothermal features using monolayers and nanoparticles’ diameter. Hussain et al. [9] used chemical reactivity and thermal features for the thermal enhancement of the Maxwell nanofluid flow on a solar collector and noted that thermal panels have declined with the growth in the Schmidt number and chemical reactivity parameter.

Hybrid nanofluid flow, an advanced and evolving domain within the realm of heat transfer and fluid dynamics, introduces a new dimension by combining multiple types of nanoparticles or a mixture of nanoparticles and traditional fluids [10]. This synergistic approach aims to exploit the complementary characteristics of different nanomaterials to improve convective heat transfer and thermal conductivity beyond what is achievable with single-component nanofluids [11]. The hybrid nature allows for a fine-tuning of the nanofluid properties, enabling tailoring for specific applications and operating conditions [12]. Hussain et al. [13] assumed the convective and slip conditions to study the flow of hybrid nanofluid across an extending sheet. The impact of hybrid nanofluid flow on heat transfer is profound, offering superior thermal performance compared to conventional fluids or single-component nanofluids [14]. The synergistic effects lead to enhanced heat transference rates, making hybrid nanofluids particularly favorable for applications demanding high thermal efficiency, such as in nuclear reactors, electronic cooling systems, and concentrated solar power technologies [15,16]. In concentrated solar power systems, the tailored combination of nanoparticles can optimize heat transfer fluid properties, leading to increased efficiency in energy conversion processes. While the potential benefits of hybrid nanofluid flow are substantial, ongoing research focuses on overcoming challenges related to stability, dispersion, and the scalability of production methods. As this innovative field continues to evolve, the rational design of hybrid nanofluids and their application-specific customization hold great promise for revolutionizing heat transfer technologies across various industrial sectors. Elattar et al. [17] assessed the impacts of chemical reactions and Hall current on hybrid nanofluid flow on slender elongating surfaces.

1.1 Non-Newtonian fluid

Non-Newtonian fluids have extensive applications in various fields such as cosmetics, the food industry, biomedical, and manufacturing. The phenomenon of non-Newtonian fluid has a substantial influence on the innovation of renewable and substantial energy processes for the development of contemporary trends. The Casson model is a non-Newtonian fluid model that shows shear-thinning behavior and stress. Ullah et al. [18] studied the non-Newtonian fluid flow to analyze the effect of chemical reaction and thermal radiation over a stretching sheet with aspect to porous medium. Wajihah and Sankar [19] discussed the rheological behavior of blood, dilations, and the emergence of the solid layer fluid of non-Newtonian fluid flow in constricted arteries. It was investigated that the pulsatile nature of blood movement produces a dynamic environment that poses a slew of intriguing and unstable fluid mechanical states. Shahmuddin et al. [20] have examined the non-Newtonian fluid Prandtl hybrid nanofluid flow over a stretching surface with multislips and varying chemical reactions. Jeelani and Abbas [21] have determined the alumina copper ethylene-based magnetohydrodynamic (MHD) hybrid nanofluid flow over a stretching sheet with effects of solar radiation and plate suction. From this inquiry, it was revealed that the higher solar radiation and magnetic parameters boost the temperature of the hybrid nanofluid flow. Hussain et al. [22] have explored the non-Newtonian hybrid nanofluid flow over a stretching sheet with slip effects. The result revealed that the fluid velocity has a declining behavior for the Casson parameter. Reddy et al. [23] discussed the non-Newtonian fluid behavior in the presence of the transverse magnetic field over an elongating surface. Meenakumari et al. [24] have examined the Casson nanofluid flow to examine the convective heat and mass transfer feathers over a stretching sheet in the presence of activation energy and chemical reaction. Ilango and Lakshminarayana [25] inspected the Brownian motion and the thermophoresis effect on Casson nanofluid flow over a stretching sheet under the induced magnetic field.

Fluid flow characterized by mixed convection, a phenomenon resulting from the simultaneous influence of forced and natural convection, has profound impacts on heat transfer in various engineering applications [26]. In mixed convection, both buoyancy-driven flow and externally imposed flow coexist, leading to complex fluid dynamics (CFD) and heat transfer behavior [27]. The relationship between these two modes of convection often results in nonuniform temperature and velocity distributions within the fluid, creating intricate thermal boundary layers. The impact on heat transfer is substantial, as mixed convection can either enhance or impede the overall heat exchange depending on the relative strengths of forced and natural convection components [28]. In scenarios where forced convection dominates, such as in high-speed flows, mixed convection tends to augment heat transfer rates due to increased fluid motion [29]. Conversely, in situations where natural convection prevails, such as in low-speed or buoyancy-driven flows, mixed convection can hinder heat transfer efficiency by disrupting the natural flow patterns. Mixed convection phenomena are crucial in the design of heat exchangers, electronics cooling systems, and renewable energy devices, where optimizing heat transfer under varying conditions is essential [30,31]. Consequently, researchers and engineers strive to unravel the intricacies of mixed convection to harness its potential for improved thermal performance in diverse technological applications. Mixed convection is encountered in various engineering applications [32], including electronics cooling, heat exchangers, and renewable energy systems, where understanding and manipulating the interplay between forced and natural convection is crucial for optimizing heat transfer performance under diverse operating conditions.

MHD fluid flow, a specialized branch of fluid dynamics, explores the behavior of electrically conducting fluids, using effects of magnetic fields. The impact of MHD on heat transmission phenomena is profound, as the interaction between magnetic fields and flowing conductive fluids induces additional forces and constraints on the fluid motion [33]. The Lorentz force, arising from the interaction of magnetic fields and electric currents within the fluid, alters the velocity and temperature distributions, influencing the overall heat transfer characteristics. In some cases, MHD can suppress turbulence, affecting the convective heat transfer rates, while in others, it may enhance heat transfer through increased fluid mixing and circulation [34]. Applications of MHD fluid flow in heat transfer range from electromagnetic processing and liquid metal cooling in advanced nuclear systems to space propulsion technologies [35]. Controlling the complexities of MHD fluid flow is crucial for optimizing heat transfer processes and advancing innovative technologies in areas such as energy generation, space exploration, and materials science. However, challenges such as fluid stability, electromagnetic interactions, and system complexity must be carefully addressed to fully exploit the potential benefits of MHD in heat transfer applications. Bejawada et al. [36] examined radiative magnetized fluid flow on a sheet with chemical reactivity and impacts of the Forchheimer medium. Mahabaleshwar et al. [37] inspected MHD fluid flow and thermal transportation with carbon nanotubes and thermally radiative effects on a shrinking and elongating surface. In terms of heat transfer, MHD has a substantial impact on convective heat transfer processes.

Fluid flow with Joule heating represents a dynamic interaction of electricity, fluid mechanics, and heat transfer, with profound implications for thermal transportation. Joule heating arises when an electric current passes in a conductive fluid, which as a result converges the electrical energy into heat energy [38]. This phenomenon impacts thermal transportation in multifaceted ways. The resistive heating induces changes in fluid properties such as viscosity, density, and thermal conductivity, consequently altering the fluid dynamics. In forced convection scenarios, where an external force propels the fluid, Joule heating can intensify convective heat transfer by creating additional thermal gradients and enhancing fluid mixing. Applications are diverse, spanning fields like electromagnetically processed materials, electrothermal actuators, and microfluidic systems. The precision control of thermal transportation enabled by fluid flow with Joule heating is particularly valuable in microscale applications where conventional heating methods may be impractical [39]. Naseem et al. [40] inspected Joule heating and viscously dissipative impacts on fluid flow on a sheet. Waqas et al. [41] examined bioconvective electrically conductive fluid flow with Joule heating on a radiating stretchable surface and observed that thermal transportation scattering has increased with an expansion in Brownian motion and thermophoretic factor. Vinodkumar Reddy et al. [42] have explored the convective MHD Casson nanofluid flow over a permeable stretching sheet under the effects of joule heating and heat source. From their study, it was found that the velocity profile decreases with the Casson parameter. Manshadi and Beskok [43] inspected the effect of Joule heating on the thermal transportation phenomenon in a conduit with the impacts of electro-thermal effects. Islam et al. [44] inspected the radiative flow of Maxwell nanofluid on an elongating cylinder with impacts of Joule heating.

The inclusion of thermal radiation effects is significant in fluid flow dynamics, particularly in the realm of heat transfer processes. In the polymer processing industry, thermal radiation plays a crucial role in governing heat transfer mechanisms. Hossain and Thakar [45] explored the performance of thermal radiation in heat transfer phenomena. Shoaib et al. [46] investigated how radiation impacts the rotating flow of fluid under the influence of MHD. Their findings indicated that the fluid’s temperature rose as the radiation factor escalated. Ali et al. [47] considered the stretching cylinder to examine radiative impacts and heat sources on flow of fluid. Wahid et al. [48] investigated flow characteristics for a magnetized convective flow of a nanofluid on constraints at the boundary. Their research focused on analyzing convective boundary conditions. The findings indicated that increased thermal radiation levels led to enhanced heat transfer efficiency of nanofluids. Hussain et al. [49] studied radiative effects for the flow of a hybrid nanofluid on a sheet and noticed that thermal panels have augmented with escalation magnetic and radiative factors. Thabet et al. [50] discussed the enhancement of thermal transportation for fluid flow on an inclined surface with impacts of thermophoresis and Brownian motion. Idres et al. [51] analyzed the flow of a hybrid nanofluid on an elongating sheet with impacts of radiation and Joule heating and thermal slip constraints and have proved that thermal flow escalated from 79 to 9.35% with an escalating value of magnetic and radiative factors.

One of the most noteworthy and widely used substances is alumina and copper. Compared to other substances, alumina has a great temperature, higher chemical stability, and thermal conductivity [52,53]. Alumina nanoparticles are widely used in medical, industry, and household items [54]. Copper metal nanoparticles have attracted applications due to their improved catalytic, optical, thermal, and electrical characteristics at the nano level. It has various applications in heat transfer, electronics instruments such as industrial cooling, lithium batteries, and heat transfer enhancements. It is highly reactive and rapidly oxidized.

Alumina is a common ceramic material named for its high thermal conductivity and stability at high temperature, while copper that has excellent thermal conductivity is composed of elemental copper. In preparation of nanofluid with copper and alumina nanoparticles at volume fraction ranging from 0.1 to 0.5% with sodium alginate concentration ranging from 0.1 to 1% by weight relative to the base fluid. It influences the stability, thermal conductivity, and other feathers of nanofluid in experimental studies. Alumina and copper nanoparticles have various promising applications in stretching the surface. The copper nanoparticles are used in coating surfaces in hospitals and public places. By the incorporation of copper nanoparticles in coating, the surface can effectively reduce the spread of infection. Alumina nanoparticles are used in heat transfer properties due to their high thermal conductivity. It is used in coating of electronic devices. Both nanoparticles are used as catalysts in different chemical reactions. Alumina nanoparticles are added to polymers and composite materials to increase their thermal conductivity and strength. Due to these expensive applications of copper and nanoparticles, we have considered these nanoparticles with base fluid sodium alginate to create a hybrid nanofluid.

The utilization of copper and alumina nanoparticles in sodium alginate base fluid has tremendous applications mainly in extending surface functionalities.

Biomedical coating: It is used to coating medical implants such as dental and orthopedic implants.

Food packing: The coating containing copper and alumina nanoparticles in sodium alginate can be used in food packing. Alumina enhances the durability of water repellency of fabrics, improving their performances. The coating is also used in environmental remediation efforts, etc.

Inspired by the progressive relaxation properties of the non-Newtonian Casson and its applied significance in the rheological modeling and heat transfer characteristics of different dynamic fluids, the present study aims to investigate the heat transfer enhancement of the mixed convective flow of a sodium alginate-based hybrid nanofluid on an extending sheet.

The preliminary assumptions are rotational flow, dissipative, thermally heated, electrically conductive, and convective.

1.2 Problem assumptions

The flow is considered incompressible,
Laminar, steady,
Non-Newtonian,
Joule heating and thermal radiation effects are considered in the analysis,
The flow is considered over a stretching sheet.

At the end of this study, we can answer the following research questions:
How does the velocity profile affect via rotation and magnetic parameters?
What impact does the Casson and Eckert number have on temperature and heat transfer rate?
How does the skin friction along primary and secondary directions behave for magnetic and rotation parameters?

2 Statement of the problem

Consider the mixed convective sodium alginate-based hybrid nanofluid flow of Cu and Al₂O₃ nanoparticles on an elongating surface in the presence of thermal radiation behavior. At the boundary, convective heat flux conditions are employed. The Cartesian coordinates x-, y-, and z- axis are considered along which the flow components are, respectively, described as u , v , w , such that the sheet is stretched with velocity u w ( x ) = a x along x -axis as depicted in Figure 1. The flow is rotated with Ω as angular velocity along the z-axis. The flow is magnetized with a magnetic field strength B 0 , which is applied perpendicularly to the flow direction and g is the gravitational acceleration. Moreover, T f and T w are the reference and surface temperatures such that T f > T w . The principal equations are as follows [55,56]:

(1) ∂ u ∂ x + ∂ v ∂ y + ∂ w ∂ z = 0 ,

(2) u ∂ u ∂ x + v ∂ u ∂ y + w ∂ u ∂ z − 2 Ω v = μ hnf ρ hnf 1 + 1 β ∂ 2 u ∂ z 2 − σ hnf ρ hnf β 0 2 u + g ( ρ β T ) hnf ρ hnf ( T − T ∞ ) ,

(3) u ∂ v ∂ x + v ∂ v ∂ y + w ∂ v ∂ z + 2 u Ω = μ hnf ρ hnf 1 + 1 β ∂ 2 v ∂ z 2 − σ hnf ρ hnf β 0 2 v ,

(4) u ∂ T ∂ x + v ∂ T ∂ y + w ∂ T ∂ z = k hnf ( ρ C p ) hnf ∂ 2 T ∂ z 2 + σ hnf ( ρ C p ) hnf β 0 2 ( u 2 + v 2 ) + μ hnf ( ρ C p ) hnf 1 + 1 β ∂ u ∂ z 2 + ∂ v ∂ z 2 + Q 0 ( ρ C p ) hnf ( T − T ∞ ) − 1 ( ρ C p ) hnf ∂ q r ∂ z .

Figure 1

Graphical view of flow problem.

The value of ( q r ) is defined as follows:

(5) q r = − 4 σ ∗ 3 k ∗ ∂ T 4 ∂ z , T 4 ≅ 4 T ∞ 3 T − 3 T ∞ 4 .

2.1 Constraints at boundaries

The constraints at boundaries are as follows [57,58]:

(6) u = a x , v = 0 , w = 0 , − ( k hnf ) ∂ T ∂ z = − h T ( T − T f ) , at z = 0 , u → 0 , T → T ∞ , v → 0 , } as z → ∞ .

Table 1 shows the thermophysical properties of the hybrid nanofluid.

Table 1

Thermophysical properties of the hybrid nanofluid [59,60]

Properties	Hybrid nanofluid
Dynamic viscosity	μ hnf = μ f ( 1 ‒ Φ 1 ‒ Φ 2 ) 2.5
Heat capacity	( ρ C p ) hnf = { [ ( 1 ‒ Φ 1 ) ( ρ C p ) f + Φ 1 ( ρ C p ) p 1 ] ( 1 ‒ Φ 2 ) } + Φ 2 ( ρ C p ) p 2
Density	ρ hnf = { [ ( 1 ‒ Φ 1 ) ρ f + Φ 1 ρ p 1 ] ( 1 ‒ Φ 2 ) } + Φ 2 ρ p 2
Electrical conductivity	σ hnf σ nf = σ p 2 + 2 σ nf ‒ 2 ( σ nf ‒ σ p 2 ) Φ 2 2 σ nf + σ p 2 + ( σ nf ‒ σ p 2 ) Φ 2 , where σ nf σ f = 2 σ f + σ p 1 ‒ 2 Φ 1 ( σ f ‒ σ p 1 ) 2 σ f + σ p 1 + Φ 1 ( σ f ‒ σ p 1 )
Thermal conductivity	k hnf k nf = k p 2 + k nf ( n ‒ 1 ) ‒ ( n ‒ 1 ) Φ 2 ( k nf ‒ k p 2 ) k p 2 + k nf ( n ‒ 1 ) + Φ 2 ( k nf ‒ k p 2 ) , where k bf k f = k p 1 + k f ( n ‒ 1 ) ‒ ( n ‒ 1 ) ( k f ‒ k p 1 ) Φ 1 k p 1 + k f ( n ‒ 1 ) + ( k f ‒ k p 1 ) Φ 1

Table 2 indicates the thermophysical properties of nanoparticles and pure fluid.

Table 2

The thermophysical properties of nanoparticles and pure fluid [61–63]

Physical properties	SA	Al 2 O 3	Cu
ρ [ kg/m 3 ]	989	3,970	8,933
C ˜ p [ J K/gK ]	4,175	765	385
σ [ S/m ]	2.60 × 10⁻⁴	1 × 10⁻⁷	5.96 × 10⁷
k [ W/m K ]	0.6376	40	401
β [ / K ]	21 × 10⁻⁵	0.85 × 10⁻⁵	1.67

Here, Φ 1 and Φ 2 show the Cu and Al₂O₃ nanoparticles volume fractions. p 1 shows Cu nanoparticles and p 2 shows Al₂O₃ nanoparticles. The symbol μ is dynamic viscosity, σ is electrical conductivity, C p is specific heat, ρ is density, and k of the fluid.

The set of appropriate similarity variables is given as follows [58,59]:

(7) ξ = a v f z , u = a x f ′ ( ξ ) , v = a x g ( ξ ) , w = − f ( ξ ) a ν f , θ ( ξ ) = T − T ∞ T f − T ∞ .

By using equation (7), we obtain the following equation:

(8) μ hnf / μ f ρ hnf / ρ f 1 + 1 β f ‴ − f ′ 2 + f f ″ + 2 α g − σ hnf / σ f ρ hnf / ρ f M f ′ + ( ρ β T ) hnf / ( ρ β T ) f ρ hnf / ρ f λ θ = 0 ,

(9) μ hnf / μ f ρ hnf / ρ f 1 + 1 β g ″ − g f ′ + f g ′ − 2 α f ′ − σ hnf / σ f ρ hnf / ρ f M g = 0 ,

(10) ( ρ C p ) f ( ρ C p ) hnf k hnf k f + Rd θ ″ + σ hnf / σ f ( ρ C p ) hnf / ( ρ C p ) f Pr M Ec ( f ′ 2 + g 2 ) + Pr f θ ′ + Pr ( ρ C p ) f ( C p ) hnf Q θ + ( μ hnf / μ f ) ( ρ C p ) hnf / ( ρ C p ) f 1 + 1 β Pr Ec ( f ″ 2 + g ′ 2 ) = 0 .

The related constraints are as follows:

(11) f ( 0 ) = 0 , g ( 0 ) = 0 , f ′ ( 0 ) = 1 , k hnf k f θ ′ ( 0 ) = − β 1 ( 1 − θ ( 0 ) ) , f ′ ( ∞ ) → 0 , g ( ∞ ) → 0 , θ ( ∞ ) → 0 .

Various parameters used in the aforementioned equations are described as follows:

M = σ f β 0 2 ρ f a , Rd = 16 σ ∗ T ∞ 3 3 k ∗ k f , Pr = ( μ C p ) f k f , Gr x = g ( β T ) f ( T f − T ∞ ) x 3 v f 2 , α = Ω a , β 1 = h T k f v f a , Q = Q 0 a ( ρ C p ) f , λ = Gr x Re x 2 , Re x = u w ( x ) x ρ f a , Ec = u w 2 ( C p ) f ( T f − T ∞ ) . .

Quantities of interest:

(12) C fx = τ wx ρ f u w 2 , C fy = τ wy ρ f u w 2 , Nu x = x q w k f ( T f − T ∞ ) ,

where

(13) τ wx = μ hnf ∂ u ∂ z z = 0 , τ wy = μ hnf ∂ v ∂ z z = 0 , q w = − k hnf ∂ T ∂ z z = 0 + q r ∣ z = 0 ,

(14) Re 1 2 C fx = μ hnf μ f f ″ ( 0 ) , Re 1 2 C fy = μ hnf μ f g ′ ( 0 ) , Nu x Re x = − k hnf k f + Rd θ ′ ( 0 ) ,

where Re x = u w ( x ) x v f is the local Reynolds number.

3 Homotopy analysis method (HAM) solution

The homotopy analysis method is a very powerful semi-analytical method, which is very applicable to the solution of nonlinear differential equations. HAM has many advantages over other techniques. Some advantages of the method are discussed as follows: HAM provides a systematic framework for obtaining analytical approximations of solutions. It allows flexibility in choosing auxiliary linear operators and parameters. Which can be applied to control the convergence of the solution. HAM needs no linearization like other perturbation methods.

The initial guesses are defined as follows:

(15) f 0 = 1 − exp ( − ξ ) , g 0 = 0 , θ 0 = β 1 k hnf / k f + β 1 exp ( − ξ ) .

The linear operators are defined as follows:

(16) L f = f ‴ − f , L g = g ‴ − g , L θ = θ ″ − θ .

With properties

(17) L f [ A 1 + A 2 ( exp ( − ξ ) ) + A 3 ( exp ( ξ ) ) ] = 0 , L g [ A 4 ( exp ( − ξ ) ) + A 5 ( exp ( ξ ) ) ] = 0 , L θ [ A 6 ( exp ( − ξ ) ) + A 7 ( exp ( ξ ) ) ] = 0 ,

where A 1 − A 7 are the fixed values.

The corresponding nonlinear operators are given as follows:

(18) N f [ f ( ξ ; z ) , g ( ξ ; z ) , θ ( ξ ; z ) ] = μ hnf / μ f ρ hnf / ρ f 1 + 1 β ∂ 3 f ( ξ ; z ) ∂ ξ 3 − ∂ f ( ξ ; z ) ∂ ξ 2 + f ( ξ ; z ) ∂ 2 f ( ξ ; z ) ∂ ξ 2 + 2 α g ( ξ ; z ) − σ hnf / σ f ρ hnf / ρ f M ∂ f ( ξ ; z ) ∂ ξ + ( ρ β T ) hnf / ( ρ β T ) f ρ hnf / ρ f λ θ ( ξ ; z ) ,

(19) N g [ f ( ξ ; z ) , g ( ξ ; z ) ] = μ hnf / μ f ρ hnf / ρ f 1 + 1 β ∂ 2 g ( ξ ; z ) ∂ ξ 2 − g ( ξ ; z ) ∂ f ( ξ ; z ) ∂ ξ − 2 α ∂ f ( ξ ; z ) ∂ ξ − σ hnf / σ f ρ hnf / ρ f M g ( ξ ; z ) ,

(20) N θ [ θ ( ξ ; z ) , f ( ξ ; z ) , g ( ξ ; z ) ] = k hnf k f + Rd ( ρ C p ) f ( ρ C p ) hnf ∂ 2 θ ( ξ ; z ) ∂ ξ 2 + Pr ( ρ C p ) f ( ρ C p ) hnf Q θ ( ξ ; z ) + Pr f ( ξ ; z ) ∂ θ ( ξ ; z ) ∂ ξ + σ hnf / σ f ( ρ C p ) hnf / ( ρ C p ) f Pr M Ec ∂ f ( ξ ; z ) ∂ ξ 2 + g ( ξ ; z ) 2 + μ hnf / μ f ( ρ C p ) hnf / ( ρ C p ) f 1 + 1 β Pr Ec ∂ 2 f ( ξ ; z ) ∂ ξ 2 2 + ∂ f ( ξ ; z ) ∂ ξ 2 .

The zeroth-order problem can be written as follows:

(21) ( 1 − z ) L f [ f ( ξ ; z ) − f 0 ( ξ ) ] = z h f N f [ f ( ξ ; z ) , g ( ξ ; z ) , θ ( ξ ; z ) ] ,

(22) ( 1 − z ) L g [ g ( ξ ; z ) − g 0 ( ξ ) ] = z h g N g [ g ( ξ ; z ) , f ( ξ ; z ) ] ,

(23) ( 1 − z ) L θ [ θ ( ξ ; z ) − θ 0 ( ξ ) ] = z h θ N θ [ θ ( ξ ; z ) , g ( ξ ; z ) , f ( ξ ; z ) ] .

The required boundary conditions are as follows:

(24) f ( ξ ; z ) ∣ ξ = 0 = 0 , ∂ f ( ξ ; z ) ∂ ξ ξ = 0 = 1 , ∂ f ( ξ ; z ) ∂ ξ ξ → ∞ = 0 , g ( ξ ; z ) ∣ ξ = 0 = 0 , g ( ξ ; z ) ∣ ξ → ∞ = 0 , k hnf k f ∂ θ ( ξ ; z ) ∂ ξ ξ = 0 = − β 1 ( 1 − θ ( ξ ; z ) ) ∣ ξ = 0 , θ ( ξ ; z ) ∣ ξ → ∞ = 0 ,

where z ∈ [ 0 , 1 ] the entrenching is a factor and h is the auxiliary parameter,

For z = 0 and z = 1 , we have:

(25) f 0 ( ξ ) = f ( ξ ; 0 ) , f ( ξ ) = f ( ξ ; 1 ) , g 0 ( ξ ) = g ( ξ ; 0 ) , g ( ξ ) = g ( ξ ; 1 ) , θ ( ξ ; 0 ) = θ 0 ( ξ ) , θ ( ξ ) = θ ( ξ ; 1 ) .

Expanding by Taylor series

(26) f ( ξ ; z ) = f 0 ( ξ ) + ∑ q = 1 ∞ f q ( ξ ) z q , where f q ( ξ ) = 1 q ! ∂ q f ( ξ ; z ) ∂ z q z = 0 ,

(27) g ( ξ ; z ) = g 0 ( ξ ) + ∑ q = 1 ∞ g q ( ξ ) z q , where g q ( ξ ) = 1 q ! ∂ q g ( ξ ; z ) ∂ z q z = 0 ,

(28) θ ( ξ ; z ) = θ 0 ( ξ ) + ∑ q = 1 ∞ θ q ( ξ ) z q , where θ q ( ξ ) = 1 q ! ∂ q θ ( ξ ; z ) ∂ z q z = 0 .

The qth-order problems can be inscribed as follows:

(29) L f [ f q ( ξ ) − X q f q − 1 ( ξ ) ] = h f z q f ( ξ ) , L g [ g q ( ξ ) − X q g q − 1 ( ξ ) ] = h g z q g ( ξ ) , L θ [ θ q ( ξ ) − X q θ q − 1 ( ξ ) ] = h θ z q θ ( ξ ) ,

(30) f q ( 0 ) = 0 , f q ′ ( 0 ) = 1 , f ′ q ( ∞ ) = 0 , g q ( 0 ) = 0 , g q ( ∞ ) = 0 , k hnf k f θ q ′ ( 0 ) = − β 1 ( 1 − θ q ( 0 ) ) , θ q ( ∞ ) = 0 .

Here,

(31) z q f ( ξ ) = μ hnf / μ f ρ hnf / ρ f 1 + 1 β f q − 1 ‴ − ∑ n = 0 q − 1 f q − 1 − j ′ ( f q − 1 ′ ) + ∑ n = 0 q − 1 f q − 1 − j ″ ( f q − 1 ) + 2 α ( g q − 1 ) − σ hnf / σ f ρ hnf / ρ f M f q − 1 ′ + ( ρ β T ) hnf / ( ρ β T ) f ρ hnf / ρ f λ θ q − 1 ,

(32) z q g ( ξ ) = μ hnf / μ f ρ hnf / ρ f 1 + 1 β g q − 1 ″ − ∑ n = 0 q − 1 f q − 1 − j ′ ( g q − 1 ) − 2 α ( f q − 1 ′ ) − σ hnf / σ f ρ hnf / ρ f M g q − 1 ,

(33) R q θ ( ξ ) = ( ρ C p ) f ( ρ C p ) hnf k hnf k f + Rd θ q − 1 ″ + μ hnf / μ f ( ρ C p ) hnf / ( ρ C p ) f 1 + 1 β Pr Ec ∑ n = 0 q − 1 f q − 1 − j ″ f q − 1 ″ + ∑ n = 0 q − 1 g q − 1 − j ′ g q − 1 ′ + σ hnf / σ f ( ρ C p ) hnf / ( ρ C p ) f Pr M Ec ∑ n = 0 q − 1 f q − 1 − j ′ f q − 1 ′ + ∑ n = 0 q − 1 g q − 1 − j g q − 1 + Pr ( ρ C p ) f ( ρ C p ) hnf Q θ q − 1 + Pr ∑ n = 0 q − 1 θ q − 1 − j ′ f i − 1 ,

where

(34) X q = 0 , if z ≤ 1 1 , if z > 1 .

4 Convergence of HAM

In this section, the convergence solution for the problem is considered in Figures 2–4. The controlling factors ℏ f , ℏ g , and ℏ θ are used for the regulatory and adjusting of the series solution of their problem. The acceptable region of convergence for f ′ ( ξ ) , g ′ ( ξ ) , and θ ′ ( ξ ) are − 1.75 ≤ h f ≤ 0.45 , − 1.75 ≤ h g ≤ 0.45 , and − 1.73 ≤ h θ ≤ 0.45 , respectively.

$Figure 2 ℏ \hslash curve graph for f ″ ( 0 ) f^{\prime\prime} (0) .$

Figure 2

ℏ curve graph for f ″ ( 0 ) .

$Figure 3 ℏ \hslash curve graph for g ′ ( 0 ) g^{\prime} (0) .$

Figure 3

ℏ curve graph for g ′ ( 0 ) .

$Figure 4 ℏ \hslash curve graph for θ ′ ( 0 ) \theta ^{\prime} (0) .$

Figure 4

ℏ curve graph for θ ′ ( 0 ) .

5 Code validation

In this section, the author constructs the table for skin friction with various values of rotation parameters. The numerical data from Table 3 verified that the solution of our model has a fine promise with already published dataset, which shows the accuracy of the suggested model.

Table 3

Comparison of − f ″ ( 0 ) with several values of rotation parameter

α	− f ″ ( 0 )
α	Hassain et al. [64]	Nazir et al. [58]	Dawar et al. [65]	Current results
0.0	1.00436	1.000431019	1.000431	1.0040
0.5	1.17199	1.171217113	1.171210	1.1876
1.0	1.35320	1.358181503	1.358182	1.35270
2.0	1.698033	1.681031011	1.681031	1.6750

6 Results and discussion

This section deals with the behavior of various physical factors on different flow profiles as well as quantities of interest as shown in Figures 5–16. Table 4 demonstrates the consequence of volume fraction on C fx , C fy , and Nu of nanofluid and hybrid nanofluid. C fx , C fy , and Nu show a growing behavior through increasing values of volume fractions. The results portrayed that C fx , C fy , and Nu have increasing behavior for hybrid nanofluid compared to nanolfuid. Figure 5 shows the physical behavior of λ on velocity profile. The increase in λ significantly increases f ′ ( ξ ) because an increase in λ escalates the buoyancy force, which influences the speed of the fluid flow due to the inertial force and the ratio of the temperature difference. Figures 6–8 express the consequences of M on velocities and temperature distribution. It can be noticed that the higher magnetic parameter declines the velocity profile and increases θ ( ξ ) . It is because the larger value of the magnetic parameter produces Lorentz force. The Lorentz force acts like an opposing force. This resistive force slows the motion of the nanoparticles and also decreases the width of the layer at boundary. Thus, the flow of the nanofluid diminishes. Clearly, θ ( ξ ) increases with M because the strong Lorentz force associated with joule heating effect increases the width of thermal layer at boundary and thermal transfer rate, which subsequently increases θ ( ξ ) . From Table 5, it is also substantiated that growing M increases frictional force. Figures 9 and 10 display the impacts of Casson factor β on velocity distribution. Increasing β corresponds to a decrease in the velocities f ′ ( ξ ) and g ′ ( ξ ) . Physically, this can be interpreted as a reduction in the fluid’s ability to deform and flow under shear stress. Since β represents the yield stress required for the fluid to begin moving. So as β increases, the yield stress becomes larger, necessitating a higher applied force to initiate flow. Consequently, the fluid exhibits lower velocities as it resists deformation more strongly. This decrease in velocities can have implications for various industrial processes, affecting phenomena such as mixing, pumping, and coating, where fluid flow dynamics play a critical role. The same effect can be seen in skin friction through the Casson parameter as displayed in Table 5. Figures 11 and 12 express the outcome of the rotation parameter on the velocities plot. The velocity profiles show a reducing trend through the rotation parameter. An escalation in rotation factor escalates the surface friction and the boundary layer thickness. Additionally, a centrifugal force is produced to the nanofluid due to an increase in rotation factor. The sheet that stretches along the x direction acts against the stretching of the surface, which efficiently decreases f ′ ( ξ ) and g ′ ( ξ ) . Similarly, the surface drag force decreases through the rotation parameter as shown in Table 5. Figure 13 shows the influence of the heat source factor on the thermal plot, which shows an improvement behavior. Physically extra heat is produced in the system of nanofluid which heightens the thermal plot. Hence, the larger Q heightens θ ( ξ ) . Figure 14 expresses the impact of Rd on θ ( ξ ) . Temperature plot increases an increase in Rd . Physically, the higher Rd causes the increase of the width of thermal layer at boundary, which enhances the rate of heat transmission. Thus, as a result of this θ ( ξ ) increases. The same effects are observed in Table 6 on Nusselt number through Q and Rd . It is because the higher Q and Rd yields interior heat in the system, which increases the thermal transmission rate. Figure 15 shows the impression of Eckert number Ec on θ ( ξ ) . Physically the ratio of kinetic energy to the change in enthalpy is equal to the Eckert number. Thus, as Ec increases the kinetic energy increases. It produces significant internal fluid motion friction, which enlarges the temperature of the nanofluid system. Therefore, θ ( ξ ) boosts with higher Ec . Figure 16 expresses the outcome of thermal Biot number on the temperature plot. The temperature plot display an increasing trend. Physically the increase in β 1 increases the thermal and heat transfer rate, which helps the thermal transfer of nanoparticles toward boundary layers. This transferring of nanoparticles from area of greater temperature to the boundary layer region causes the improvement of θ ( ξ ) . A similar behavior is also seen on heat transfer rate through β 1 as shown in Table 6. Physically the estimation of β 1 produces more resistive force to motion of fluid, for which the heat transfer rate increases. Also, the convective heat transfer increases within an increase in the thermal Biot number because the thermal Biot number is the ratio of convective heat transfer to conductive heat transfer resistance.Figure 17 demonstrate the consequence of the Casson factor on θ ( ξ ) . An increasing behavior is observed in θ ( ξ ) with higher values of β because growth in β enlarges the plastic dynamic viscosity, and subsequently, the yield stress diminishes. This results in a hindrance to the motion of the fluid and increases θ ( ξ ) . Figures 18 and 19 demonstrate the behavior of β , M on C fx and C fy . The surface drag force diminishes with amplified in β , M . Figure 20 shows the effect of the similar effects of Rd and Ec on Nu . The Nusselt number intensifies with an increase in Ec and Rd . The greater heat source, Eckert number, and thermal radiation parameter lead to the increment of the convective heat transfer, which subsequently increases the heat transfer rate.

$Figure 5 Variation in f ′ ( ξ ) f^{\prime} (\xi ) via λ \lambda .$

Figure 5

Variation in f ′ ( ξ ) via λ .

$Figure 6 Variation in f ′ ( ξ ) f^{\prime} (\xi ) via M M .$

Figure 6

Variation in f ′ ( ξ ) via M .

$Figure 7 Variation in g ′ ( ξ ) g^{\prime} (\xi ) via M M .$

Figure 7

Variation in g ′ ( ξ ) via M .

$Figure 8 Variation in θ ( ξ ) \theta (\xi ) via M M .$

Figure 8

Variation in θ ( ξ ) via M .

$Figure 9 Variation in f ′ ( ξ ) f^{\prime} (\xi ) via β \beta .$

Figure 9

Variation in f ′ ( ξ ) via β .

$Figure 10 Variation in g ′ ( ξ ) g^{\prime} (\xi ) via β \beta .$

Figure 10

Variation in g ′ ( ξ ) via β .

$Figure 11 Variation in f ′ ( ξ ) f^{\prime} (\xi ) via α \alpha .$

Figure 11

Variation in f ′ ( ξ ) via α .

$Figure 12 Variation in g ′ ( ξ ) g^{\prime} (\xi ) via α \alpha .$

Figure 12

Variation in g ′ ( ξ ) via α .

$Figure 13 Variation in θ ( ξ ) \theta (\xi ) via Q Q .$

Figure 13

Variation in θ ( ξ ) via Q .

$Figure 14 Variation in θ ( ξ ) \theta (\xi ) via Rd \text{Rd} .$

Figure 14

Variation in θ ( ξ ) via Rd .

$Figure 15 Variation in θ ( ξ ) \theta (\xi ) via Ec \text{Ec} .$

Figure 15

Variation in θ ( ξ ) via Ec .

$Figure 16 Variation in θ ( ξ ) \theta (\xi ) via β 1 {\beta }_{1} .$

Figure 16

Variation in θ ( ξ ) via β 1 .

Table 4

The impact of volume fraction on C fx , C fy , and Nu for nanofluid and hybrid nanofluid

Φ 1	Φ 2	Φ 1 = Φ 2	C fx	C fy	Nu
0.01			−0.795838	0.783846	0.407401
0.02			−0.706098	0.793059	0.409723
0.03			−0.629543	0.803067	0.412084
0.04			−0.892481	0.813888	0.414482
	0.01		−0.885522	0.773666	0.407711
	0.02		−0.88083	0.784666	0.41032
	0.03		−0.878313	0.805575	0.412952
	0.04		−0.868313	0.805785	0.415613
		0.01	−0.79162	0.793711	0.410054
		0.02	−0.707931	0.815256	0.415089
		0.03	−0.644804	0.840213	0.420251
		0.04	−0.598173	0.868783	0.425575

Table 5

The impact of M , α , and β on skin friction coefficients C fx and C fy of hybrid nanofluid

M	α	β	C fx	C fy
1			−0.7315	−0.5051
2			−0.5866	−0.4696
3			−0.6572	−0.4487
	0.1		−0.7616	−0.5812
	0.2		−0.7981	−0.6250
	0.3		−0.8323	−0.6616
		0.1	−0.7160	−0.5203
		0.2	−0.8262	−0.6151
		0.3	−0.8763	−0.6570

Table 6

The impact of β 1 , Ec , and Rd on Nusselt number Nu of hybrid nanofluid

β 1	Ec	Rd	Nu
0.1			0.5368
0.2			0.6533
0.3			0.7007
	0.1		1.3539
	0.3		1.1004
	0.5		0.8971
		1	1.6243
		2	2.4871
		3	3.2377

$Figure 17 Variation in θ ( ξ ) \theta (\xi ) via β \beta .$

Figure 17

Variation in θ ( ξ ) via β .

$Figure 18 Effect of β \beta , M M on C fx {C}_{\text{fx}} .$

Figure 18

Effect of β , M on C fx .

$Figure 19 Effect of β \beta , M M on C fy {C}_{\text{fy}} .$

Figure 19

Effect of β , M on C fy .

$Figure 20 Effect of Rd \text{Rd} , Ec \text{Ec} on Nu \text{Nu} .$

Figure 20

Effect of Rd , Ec on Nu .

7 Final remarks

In this section, the author discussed the incompressible MHD steady three-dimensional Casson hybrid nanofluid flow across an extending surface. The viscous dissipation and convective boundary conditions are analyzed in the mathematical model. A complete study of heat and transport characteristics is assumed considering the thermal radiation, mixed convection, and heat source effects. The physical model is considered in the form of partial differential equations, which are then changed into ordinary differential equations using the appropriate resemblance transformation. Noteworthy investigations based on the influence of various parameters that affected the flow, temperature, skin friction, and heat transfer are discussed below:

The velocities along x - and y -directions are reduced with the effects of rotation parameters by enhancing the surface friction.
The higher Casson factor escalates the flow plot; however, the strong Lorentz force created by the increasing magnetic parameter causes the reduction of primary velocity.
The heat transfer rate and temperature plot as well as the heat transfer rate get enlarged with growing values of thermal Biot number.
The skin friction of hybrid nanofluid flow escalates with larger magnetic parameters.
The highest sensitivity in heat transport is perceived with maximum values of Eckert number, thermal radiation, and heat source factor.
The result revealed that the heat transfer rate of hybrid nanofluid is higher than nanofluid for increasing values of volume fraction.
Interestingly, the augmented values of the Eckert number and thermal radiation parameter increase the rate of heat transfer.

8 Future directions

There is a need to analyze the techniques such as genetic algorithms, gradient-based methods, or machine learning approaches to maximize heat transfer rates or minimize energy consumption. We need to explore practical applications of the findings in industrial processes such as cooling systems for electronics, heat exchangers, or energy-efficient technologies. Validation of the numerical results with experimental data from real-world scenarios or with CFD. This validation process ensures the accuracy and reliability of the numerical model.

8.1 Theoretical implications

This study helps in understanding the nonlinear dynamics of fluid flow, electromagnetic induction, and heat transfer in complex systems and provides a framework for analyzing couple MHD and convective heat transform performances. The interaction of joule heating and thermal radiation effects provides theoretical foundations for designing effective thermal management systems.

8.2 Practical implications

This study utilized engineering applications such as optimizing the design of heat exchangers, cooling systems, etc. Provide a framework to understanding nanofluid performance stimulate research in nanotechnology and in material sciences, renewable energy systems and microelectronic cooling systems.

Acknowledgments

This study is supported via funding from Prince Sattam bin Abdulaziz University project number (PSAU/2024/R/1446). This work was supported by the Deanship of Scientific Research, the Vice Presidency for Graduate Studies and Scientific Research, King Faisal University Saudi Arabia (Grant No. KFU242558).

Funding information: This study is supported via funding from Prince Sattam bin Abdulaziz University project number (PSAU/2024/R/1446). This work was supported by the Deanship of Scientific Research, the Vice Presidency for Graduate Studies and Scientific Research, King Faisal University Saudi Arabia (Grant No. KFU242558).
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 analysed during the current study are available from the corresponding author on reasonable request.

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Received: 2024-05-06

Revised: 2024-09-29

Accepted: 2024-11-26

Published Online: 2025-01-10

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

Articles in the same Issue

https://doi.org/10.1515/ntrev-2024-0132

Keywords for this article

MHD; thermal radiation; mixed convection; hybrid nanofluid; stretching sheet; Joule heating

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