A numerical investigation of the two-dimensional magnetohydrodynamic water-based hybrid nanofluid flow composed of Fe3O4 and Au nanoparticles over a heated surface

Humaira Yasmin; Showkat Ahmad Lone; Hussam Alrabaiah; Zehba Raizah; Anwar Saeed

doi:10.1515/ntrev-2024-0010

Article Open Access

A numerical investigation of the two-dimensional magnetohydrodynamic water-based hybrid nanofluid flow composed of Fe₃O₄ and Au nanoparticles over a heated surface

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Published/Copyright: April 4, 2024

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From the journal Nanotechnology Reviews Volume 13 Issue 1

Abstract

In this research article, the viscous, steady, and incompressible two-dimensional hybrid nanofluid flow composed of Fe₃O₄ and Au nanoparticles on an extending sheet has been presented. An inclined magnetic field impact is used for evaluating the impacts of various factors in that case. Furthermore, the influences of porosity, Brownian motion, thermophoresis, thermal and space-dependent heat sources, and thermal radiation factors are also used in this work. The numerical analysis is done by using the bvp4c technique. Validation of the present results confirms that the present analysis is valid. The outcomes show that the higher magnetic factor reduces velocity distribution while increasing the frictional force at the surface due to Lorentz forces which oppose the fluid flow. The friction force at the sheet’s surface is higher when the sheet stretches as compared to the case when the sheet shrinks. Increase in the magnetic factor increases the skin friction of sheet’s surface which consequently increases the rate of thermal transmission at the surface along with thermal distribution. The higher values of thermal radiation and thermal-dependent heat source increase the thermal transportation rate of sheet’s surface. Insights from this investigation can improve electronics cooling systems, vital for devices prone to overheating. Optimizing heat transfer with magnetohydrodynamic water-based hybrid nanofluids containing Fe₃O₄ and Au nanoparticles ensures efficient heat dissipation, enhancing device performance and longevity.

Keywords: nanofluids; hybrid nanofluids; porous medium; Brownian motion and thermophoresis; activation energy; thermal convective surface

Nomenclature

B 0: strength of the magnetic field
B T: thermal Biot number
b: constant
C p: specific heat
D B: coefficient of Brownian diffusion
D T: thermophoretic coefficient
E a: coefficient of activation energy
Ec: Eckert number
K: porosity parameter
K p: coefficient of porous media
K r: chemical reaction parameter
k: thermal conductivity
k r: coefficient of chemical reaction
k ⁎: mean absorption coefficient
M: magnetic parameter
Nb: Brownian motion factor
Nt: thermophoresis parameter
N 1: first nanoparticle
N 2: second nanoparticle
η: similarity variable
Pr: Prandtl number
Q e: coefficient of space dependent heat source
Q Exp: space dependent heat source
Q I: thermal dependent heat source
Q t: coefficient of thermal dependent heat source
Rd: thermal radiation parameter
Re x: local Reynolds number
Sc: Schmidt number
T f: fluid temperature
T w: surface temperature
T ∞: free-stream temperature
u , v: stretching velocity
u w ( x ): stretching velocity
x , y: coordinates
γ: angle
ρ: density
μ: dynamic viscosity
σ: electrical conductivity
σ ⁎: Stefan–Boltzmann constant
λ: stretching factor
φ 1: volume fraction of the first nanoparticle
φ 2: volume fraction of the second nanoparticle

1 Introduction

Nanofluid flow is a complex phenomenon involving the movement and behavior of a suspension of nanoparticles within a fluid medium. Nanofluids are created by dispersing solid nanoparticles, typically ranging in size from a few nanometers to several hundred nanometers, into a pure fluid, like oil, water, etc. This idea of nanoparticle dispersal in the pure fluid was floated by Choi and Eastman [1] for augmenting the thermal flow performance of pure fluid. The flow of nanofluids presents unique characteristics and challenges due to the presence of nanoparticles. These tiny particles interact with the fluid on a molecular level, altering its properties and behavior [2]. One key aspect of nanofluid flow is its enhanced heat transfer characteristics. This property has led to the exploration of nanofluids in various applications, including cooling systems, heat exchangers, and thermal management in electronic devices [3,4]. Researchers and engineers investigate the flow behavior, rheology, and transport properties of nanofluids using experimental and computational methods [5]. This knowledge enables the development of tailored nanofluid formulations and the efficient utilization of nanofluids in various engineering applications, where their enhanced thermal conductivity and unique flow characteristics can be harnessed to achieve improved performance and energy efficiency [6]. Anjum et al. [7] discussed the significance of bio-convective nanofluid flow with the impact of microorganisms and activated energy and noted that microorganism panels have retarded with advancement in the Peclet factor. Khan et al. [8] inspected the bio-convective Maxwell nanofluid flow on a cylinder affected by activation energy and chemical reactivity and have noted that as the radiation factor augmenting the thermal distribution increased, the concentration profiles have reduced. Shamshuddin et al. [9] analyzed computationally the radiative magnetohydrodynamic (MHD) nanofluid flow on a sensor plate with the influence of thermo-solutal stratification and chemically reactive effects.

Hybrid nanofluid flow describes the motion of a fluid that comprises the dispersion of two distinct nanoparticles in a pure fluid with unique properties that can be beneficial for various applications. The choice of nanoparticles depends on the desired properties and applications of these fluids. The study of this fluid involves investigating the behavior and interactions between nanoparticles and the surrounding fluid. Hybrid nanofluid flow has significant repercussions in different areas like microfluidics, biomedical engineering, thermal systems, etc. Khan et al. [10] inspected hydro-magnetic flow for hybrid nanofluid through two gyrating plates and noted that velocity distributions have retarded in all directions while thermal distribution has augmented with the increase in nanoparticles concentration. Dinarvand et al. [11] analyzed blood-based hybrid nanoparticles flow through a diverging as well as converging conduit using slip constraints. Qureshi et al. [12] discussed the impacts of nanolayers on MHD hybrid nanoparticles flow between two rotary disks and determined that the Nusselt number decreased with the progression in radius of nanoparticles. Yaseen et al. [13] investigated hybrid nanoparticles flow with microorganisms’ effects on fluid motion on various flow geometries and the thermal and mass flux model proposed by Cattaneo-Christov. Mahesh et al. [14] deliberated on the impacts of radiative MHD hybrid nanoparticles flow on a permeable sheet with couple stress and dissipative effects. Akhter et al. [15] optimized the generation of entropy for hybrid nanofluid flow on a partially and thermally heated surface comprising thermal conductive constraints. Mahmood et al. [16] computationally analyzed the flow of MHD hybrid nanoparticles on a curved shrinking/stretching sheet with suction effects and noted that thermal panels have boosted with a hike in the magnetic factor. Wang et al. [17] utilized computational simulations to analyze the behavior of hybrid Casson fluid flowing around an elongating cylinder. They investigated how this flow was affected by both magnetic field and the presence of microorganisms. To solve the complex equations governing this phenomenon, they employed a numerical method, namely, bvp4c approach. Naidu et al. [18] delved into the dynamics of hybrid nanofluid flow over a surface extending in exponential manner, scrutinizing the influence of convection and slip with a second-order consideration. Wang et al. [19] employed the bvp4c method to assess the naturally convective nanofluid flow on an inclined sheet, considering the influence of microorganisms. Their observations revealed a decrease in velocity distribution alongside an increase in thermal profiles as the Brownian factor increased. Babu et al. [20] investigated computationally the production of entropy for magneto nanofluid flowing on nonlinearly stretched surface with convective and slip constraints on the boundary and have proved that thermal transportation is supported more by surface stretched in nonlinear manner. Many such studies can be looked at in previous studies [21–26].

A porous medium contains pore spaces that can be filled with air, water, or other fluids. Porous media can be found in various natural and artificial materials, including rocks, soil, biological tissues, filters, sponges, and foams. Fluid flow through a porous medium is a phenomenon that occurs when a fluid moves through the interconnected voids or pores of a porous material. The flow behavior in porous media is governed by numerous aspects, like the properties of the fluid, the properties of the porous medium, and the driving forces that cause the flow. Darcy’s law is normally used to designate liquid flow in a permeable medium [27]. The word Forchheimer was suggested initially by Muskat and Wyckoff [28] to support the fluid flows with higher Reynolds number. The flow behavior in porous media can be categorized into two regimes, namely, laminar flow and turbulent flow [29]. The laminar flow arises when the fluid flows smoothly in parallel lines through the pores, while turbulent flow involves chaotic mixing and eddies within the pores. The flow regime depends on factors such as fluid velocity, pore size, and fluid viscosity [30]. Kumar et al. [31] studied the impacts of thermal absorption and generation on mixed convective fluid flow through a vertical porous channel.

Activation energy is the least quantity of energy that must be overcome for a chemical reaction to occur. Svante Arrhenius was the pioneer in introducing the concept of activation energy, which holds significant importance in various applications, including geothermal energy storage and nuclear cooling [32]. It is the energy barrier that separates the reactants from the transition state or the activated complex state. In a chemical reaction, reactant molecules collide with each other. Qureshi et al. [33] inspected the performance of activated energy for MHD fluid flow using the impact of a cluster of nano-layers and established that magnetic factor has played a crucial role in controlling the thermal and momentum behavior of the system. Higher activation energy generally indicates a slower reaction rate, as it requires more energy for a significant fraction of reactant molecules to possess the necessary energy for the reaction to occur. Ayub et al. [34] debated the significance of activated energy on nanofluid flow through a cylindrical conduit with an infinite shearing rate and noted that the inclined magnetic effects demonstrated lower velocity but higher thermal energy, while the unsteadiness parameter was associated with the time factor, resulting in a reduction in the magnitude of velocity whereas a positive change in magnetic factor reduced the surface drag phenomenon. Raizah et al. [35] examined fluid flow using activation energy through a channel. Mandal et al. [36] studied bio-convective nanofluid flow using activated energy and microorganisms and observed that with growth in magnetic factor from 0.1 to 0.5, the Nusselt number declined by 28.36%, while the density of microorganisms increased by 3.53%. Irfan [37] inspected the transportation phenomenon for thermal flow through Joule heating regarding Maxwell fluid flow with activation energy effects.

Brownian motion and thermophoresis processes occur at the microscopic level and are related to the motion of particles in a fluid. While they are distinct phenomena, they can interact with each other in certain situations. The motion of particles in Brownian motion is characterized by erratic, random movements. The particles move in a zigzag pattern and undergo continuous changes in direction and speed. This motion arises due to the random bombardment of fluid molecules, which imparts momentum to the suspended particles [38]. It plays a decisive role in the diffusion of substances, the dispersion of pollutants, the behavior of colloids, and the motion of small particles such as dust particles or pollen grains in the air. Thermophoresis, also known as the Soret effect, is a phenomenon that occurs when particles in a fluid experience a directed motion in response to a temperature gradient [39]. Unlike Brownian motion, which is driven by random molecular collisions, thermophoresis arises due to thermal gradients in the fluid. When there is a temperature difference in a fluid, the particles within the fluid practice a force that propels them from regions of higher temperature to lower temperature [40]. This effect is caused by a difference in the molecular interactions between the particles and the fluid at different temperatures. Mittal and Patel [41] debated the impact of Brownian motion and thermophoresis on convective MHD fluid flow.

We have shown the two-dimensional flow of a hybrid nanofluid on an expanding/shrinking sheet, which was motivated by the aforementioned literature review. Fe₃O₄ and Au nanoparticles are combined with water, which serves as the base fluid, to create a hybrid nanofluid. To study the flow under the influence of an inclined magnetic field, a magnetic field having an acute angle is used. The Brownian motion and thermophoresis phenomena are also used while examining the random movement of nanoparticles. Thermal transportation at the sheet’s surface is maximized and examined by the hybrid nanofluid flow under the thermally dependent and space-dependent heat phenomena. The problem description is described in Section 2 and presents additional assumptions. In Section 3, a computational analysis using bvp4c is provided. Section 4 presents published data that validate the current findings. A description of the results is included in Section 5. The closing observations about this study are enumerated in Section 6.

2 Formulation of problem

Take the viscous, steady, and two-dimensional flow of a hybrid nanofluid composed of Fe₃O₄ and Au nanoparticles over an inclined extending surface having stretching velocity u w ( x ) = b x along x-direction that is parallel to fluid flow. The y-axis is normal to the fluid flow. An inclined magnetic field having strength B 0 with acute angle γ is practiced in normal direction to the fluid flow. It is important to mention that when γ = 90 0 , then magnetic field acts in a transverse manner. Figure 1 shows the geometrical representation of the flow problem.

Figure 1

Problem’s geometrical view.

Further restrictions are applied as:

The hybrid nanofluid is taken on a convectively heated sheet ( T f > T w > T ∞ ) so that the rate of thermal transportation at the hot is determined. Here T f , T w , and T ∞ are, respectively, the reference, surface, and ambient temperatures.
The effects of buoyancy forces and slip among pure fluid and nanoparticles are neglected.
The exponential and thermal-dependent heat sources are taken to maximize the rate of heat transfer at the sheet.
Thermophoresis and Brownian motion phenomena are taken to study the variation in a hybrid nanofluid flow due to the collision of nanoparticles.
The stretching and shrinking case of the sheet is also taken into consideration.

The leading equations can be expressed as follows using these assumptions [26,42]:

(1) u x + v y = 0 ,

(2) ρ hnf ( u u x + v u y ) = μ hnf u y y − μ hnf u K p − σ hnf B 0 2 u sin 2 γ ,

(3) ( ρ C p ) hnf ( u T x + v T y ) = k hnf + 16 σ ⁎ T ∞ 3 3 k ⁎ T y y + σ hnf B 0 2 u 2 sin 2 γ + Q t ( T − T ∞ ) + Q e ( T f − T ∞ ) exp − n a ν f y + ( ρ C p ) np D B T y C y + D T T ∞ ( T y ) 2 ,

(4) u C x + v C y = D B C y y + D T T ∞ T y y − k r 2 ( C − C ∞ ) T T ∞ m − E a k B T ,

with constraints at boundaries

(5) u = u w ( x ) λ , v = 0 , − k hnf T y = h f ( T f − T ) , C = C w at y = 0 , u → 0 , C → C ∞ , T → T ∞ as y → ∞ . ,

where λ > 0 shows the stretching case of the sheet and λ < 0 shows the shrinking case of the sheet.

The following are listed as the effective qualities of nanofluids and hybrid nanofluids:

(6) μ hnf = μ f ( 1 − φ 1 ) 2.5 ( 1 − φ 2 ) 2.5 ,

(7) ρ hnf = { ρ f ( 1 − φ 1 ) + ρ N 1 φ 1 } ( 1 − φ 2 ) + φ 2 ρ N 2 ,

(8) ( ρ C p ) hnf = { ( ρ C p ) f ( 1 − φ 1 ) + ( ρ C p ) N 1 φ 1 } ( 1 − φ 2 ) + φ 2 ( ρ C p ) N 2 ,

(9) k nf k f = k N 1 + 2 k f − 2 φ 1 ( k f − k N 1 ) k N 1 + 2 k f + φ 1 ( k f − k N 1 ) , k hnf k nf = k N 2 + 2 k f − 2 φ 2 ( k f − k N 2 ) k N 2 + 2 k f + φ 2 ( k f − k N 2 ) ,

(10) σ nf σ f = σ N 1 + 2 σ f − 2 ( σ f − σ N 1 ) φ 1 σ N 1 + 2 σ f + ( σ f − σ N 1 ) φ 1 , σ hnf σ nf = σ N 2 + 2 σ f − 2 φ 2 ( σ f − σ N 2 ) σ N 2 + 2 σ f + φ 2 ( σ f − σ N 2 ) ,

where N 1 and N 2 , respectively, represent the first and second nanoparticles, and φ 1 and φ 2 characterize the volume fractions of first and second nanoparticles. The thermo-physical features of the nanoparticles and pure fluid are defined in Table 1.

Table 1

Thermophysical properties of H 2 O , Fe 3 O 4 , and Au [43]

Physical property	H 2 O	Fe 3 O 4	Au
ρ	997.1	5200.0	19,282
C p	4,179	670.0	129.0
k	0.613	6.0	310.0
σ	5.5 × 10⁻⁶	2500.0	4.1 × 10⁶

To reduce the above system of Eqs. (1)–(5), the following variables (similarity) are defined [26,42]:

(11) u = b x f ′ ( η ) , v = − b ν f f ( η ) , T = ( T f − T ∞ ) θ ( η ) + T ∞ , C = ( C w − C ∞ ) ϕ ( η ) + C ∞ , η = b ν f .

Using equation (11), we have from above

(12) Ω 1 Ω 2 g ‴ + g g ″ − g ′ 2 − Ω 3 Ω 2 M sin 2 γ g ′ + Ω 1 Ω 2 K g ′ = 0 ,

(13) 1 Ω 5 ( Ω 4 + Rd ) ϑ ″ + Pr Ω 5 ( g ϑ ′ + ξ 3 Ec M sin 2 γ g ′ 2 + Q Exp exp ( − n η ) + Q I ϑ + Nb ϑ ′ ψ ′ + Nt ϑ ′ 2 ) = 0 ,

(14) ψ ″ + Sc g ψ ′ − Sc K r ψ + Nt Nb ϑ ″ = 0 ,

with boundary constraints

(15) g ′ ( 0 ) = λ , g ( 0 ) = 0 , g ′ ( ∞ ) → 0 Ω 4 ϑ ′ ( 0 ) = − B T ( 1 − ϑ ( 0 ) ) , ϑ ( ∞ ) → 0 ψ ( 0 ) = 0 , ψ ( ∞ ) → 0 .

The parameters involved in the above equations are described as

(16) Ω 1 = μ hnf μ f , Ω 1 = ρ hnf ρ f , Ω 3 = σ hnf σ f , Ω 4 = k hnf k f , Ω 1 = ( ρ C p ) hnf ( ρ C p ) f , Sc = ν f D B , Pr = ν f ( ρ C p ) f k f , M = σ f B 0 2 b ρ f , K = μ f b ρ f K p , Rd = 16 σ ⁎ T ∞ 3 3 k ⁎ k f , K r = k r b , Ec = u w 2 ( C p ) f ( T f − T ∞ ) , Q Exp = Q t b ( ρ C p ) f , Q I = Q e b ( ρ C p ) f , B T = h f k f ν f b , Nb = ( ρ C p ) np D B ( C w − C ∞ ) ( ρ C p ) f ν f , Nt = ( ρ C p ) np D T ( T f − T ∞ ) ( ρ C p ) f T ∞ ν f

The values of these factors are defined in Table 2.

Table 2

Default values of the embedded factor

M	magnetic factor	0.5
K	porosity factor	1.4
Rd	radiation factor	0.5
Pr	Prandtl number	6.2
Ec	Eckert number	0.5
Q Exp	exponential heat source factor	0.4
Q I	heat source	0.1
B T	thermal Biot number	1.0
Nb	Brownian motion factor	0.4
Nt	thermophoresis factor	0.3
K r	chemical reaction factor	1.0
Sc	Schmidt number	1.0
λ	stretching factor	0.8

Important quantities like skin friction ( C f x ) , local Nusselt number ( Nu x ) , and Sherwood number ( Sh x ) are defined as:

(17) C f x = τ w ρ f ( u w ( x ) ) 2 , Nu x = x q w k f ( T f − T ∞ ) , Sh x = x q m D B ( C w − C ∞ ) ,

where

(18) τ w = μ hnf u y ∣ y = 0 , q w = − k hnf T y ∣ y = 0 − 16 σ ⁎ 3 k ⁎ T y 4 ∣ y = 0 , q m = − D B C y ∣ y = 0 .

Thus, equation (17) reduces as

(19) Re x C f x C f = Ω 1 g ″ ( 0 ) , Nu = − ( Ω 4 + Rd ) ϑ ′ ( 0 ) , Sh = − ψ ′ ( 0 ) ,

where C f = Re x C f x , Nu = Nu x Re x , Sh = Sh x Re x , and Re x = a x 2 ν f is Reynolds number (local).

3 Numerical solution

The numerical solution of the nonlinear coupled ODEs in Eqs. (12)–(14) with boundary constraints (15) is performed using the shooting technique and the computer program MATLAB. At this stage, the first-order set of equations replaces the higher-order system. The first initial guess is required for bvp4c’s computations later on. To achieve the desired precession, variations in the step size are incorporated keeping in mind the guess proposed at the first mesh point. On the basis of the values of the utilized parameters, it is required to choose the suitable initial guess and thickness of the layer at the boundary. In this instance, a tolerance of 10⁻⁶ is considered. The current technique’s numerical process is as follows:

Assume that

(20) ℵ ( 1 ) = g ( ξ ) , ℵ ( 2 ) = g ′ ( ξ ) , g ″ ( ξ ) = ℵ ( 3 ) , g ‴ ( ξ ) = ℵ ′ ( 3 ) , ϑ ( ξ ) = ℵ ( 4 ) , ϑ ′ ( ξ ) = ℵ ( 5 ) , ϑ ″ ( ξ ) = ℵ ′ ( 5 ) , ψ ( ξ ) = ℵ ( 6 ) , ψ ′ ( ξ ) = ℵ ( 7 ) , ψ ″ ( ξ ) = ℵ ′ ( 7 ) .

Then, equations (12)–(14) can be written as

(21) ℵ ′ ( 3 ) = − ℵ ( 1 ) ℵ ( 3 ) − ( ℵ ( 2 ) ) 2 − ξ 3 ξ 2 M sin 2 γ ℵ ( 2 ) + ξ 1 ξ 2 K ℵ ( 2 ) ξ 1 ξ 2 ,

(22) ℵ ′ ( 5 ) = − Pr ξ 5 ℵ ( 1 ) ℵ ( 5 ) + ξ 3 M Ec sin 2 γ ( ℵ ( 2 ) ) 2 + Q Exp exp ( − n η ) + Q I ℵ ( 4 ) + Nb ℵ ( 5 ) ℵ ( 7 ) + Nt ( ℵ ( 7 ) ) 2 1 ξ 5 ( ξ 4 + Rd ) ,

(23) ℵ ′ ( 7 ) = − Sc ℵ ( 1 ) ℵ ( 7 ) − Sc K r ( 1 + σ ℵ ( 5 ) ) m exp − E 1 + σ T ℵ ( 5 ) ℵ ( 6 ) + Nt Nb ℵ ′ ( 5 ) ,

with constraints at boundaries

(24) ℵ a ( 1 ) − 0 , ℵ a ( 2 ) − λ α , ℵ b ( 2 ) − 0 , k hnf k f ℵ a ( 5 ) + B T ( 1 − ℵ a ( 4 ) ) , ℵ b ( 4 ) − 0 , ℵ a ( 6 ) = 0 , ℵ b ( 6 ) − 0 , .

The quantities of interest can be written as

(25) Re x C f x = μ hnf μ f ℵ a ( 3 ) , Nu x Re x = − ( Ω 4 + Rd ) ℵ a ( 5 ) , Sh x Re x = − ℵ a ( 7 ) .

4 Validation

To compare and validate the results of the present analysis with that published by Hamad [44], and Reddy Gorla and Sidawi [45], Table 3 is displayed. With variations in Prandtl number and taking all other parameters constant, a fine promise is established between the past and present results. This confirms that the applied technique to the present mathematical model is correct and applicable.

Table 3

Comparative analysis of present values with established outcomes

Pr	Hamad [44]	Wang [45]	Reddy Gorla and Sidawi [45]	Current Results
0.07	0.06556	0.06560	0.06560	0.0655710239
0.2	0.16909	0.16910	0.16910	0.1690894383
0.7	0.45391	0.45390	0.45390	0.4539099472
2.0	0.91136	0.91140	0.91140	0.9113632118
7.0	1.89540	1.89540	1.89540	1.8954126306
20.0	3.35390	3.35390	3.35390	3.3538024529
70.0	6.46220	6.46220	6.46220	6.4621925730

5 Results and discussion

Specifically, the consequences of the current study have been presented in depth in this section. The current study is based on the two-dimensional, viscous, steady, and incompressible flow of hybrid nanofluid made up of Fe₃O₄ and Au nanoparticles across an expanding surface. To analyze the effects of different components in such a situation, an inclined magnetic field influence is taken into account. Figures 2–9 are drawn in order to describe each of the results one by one. Figure 2(a) and (b) show the impression of the magnetic parameter ( M ) on velocity distribution ( g ′ ( η ) ) and skin friction ( C f ) . Clearly a higher ( M ) decreases g ′ ( η ) while increases C f . Actually, increase in M produces Lorentz forces that oppose the fluid flow. This opposing force increases the frictional force at the sheet’s surface as shown in Figure 2(b). The higher opposing force decreases the motion of fluid particles, which decreases the velocity. By this fact, we concluded that the higher M reduces g ′ ( η ) for higher friction force at sheet’s surface. It is important to note that a magnetic effect does not always reduce the fluid motion at all. But there is a case of stagnation point flow (which is not included in this study) in which the fluid motion accelerates for the lower frictional force at the sheet’s surface. Also, the fluid flow in a finite domain like flow between two parallel disks and between two parallel plates, the fluid velocity accelerates for the higher magnetic factor. Overall, the meaning of this explanation is to point out that the consequences of the magnetic field also depend on the surface over which the fluid is considered to be studied. Figure 2(c) and (d) portray the effect of M on ϑ ( η ) and Nu . From these figures we see that ϑ ( η ) and Nu both are the increasing functions of magnetic factor. Actually, increase in M increases the skin friction at the sheet’s surface which consequently heightens the rate of heat transfer at the surface (Figure 2(d)) and temperature profile (Figure 2(d)) as well. Therefore, the higher M increases the thermal rate of transportation at the sheet’s surface and temperature profiles. Figure 3 depicts the influence of Ec (Eckert number) on ϑ ( η ) . The figure shows that when Ec increases, the values of ϑ ( η ) increases. This happens because when Ec increases, heat is produced in the fluid due to frictional heating. The Eckert number is a physical quantity made up of the specific enthalpy differential amid of fluid and the wall as well as kinetic energy. Escalation in Ec corresponds to the work that is done as a consquence of strains of viscous fluid and exchanges the kinetic and internal energies. As a consequence of this, ϑ ( η ) increases with the increase in Ec . Figure 4(a) and (b) show the variation in temperature profiles ( ϑ ( η ) ) and local Nusselt number (Nu) via exponential heat source factor ( Q Exp ) , respectively. Nusselt number and temperature are the increasing functions of Q Exp . Physically, the addition of heat source factor ( Q Exp ) increases the rate of thermal transportation at the sheet’s surface as shown in Figure 4(b) which consequently increases the width of thermal layer at the boundary and boosts the temperature distribution. Therefore, the increase in Q Exp increases both Nu and ϑ ( η ) . Figure 5(a) and (b) reveal the deviation in temperature ( ϑ ( η ) ) and concentration ( ψ ( η ) ) panels via thermophoresis factor (Nt) . It is perceived from these figures that the higher value of Nt increases both ϑ ( η ) and ψ ( η ) . Physically, the increase in Nt induces a faster movement of the nanoparticles from hotter to colder region by the thermophoretic forces which are created in the flow system of the thermophoretic phenomena. Thus, the increase in Nt increases the width of the temperature layer at boundary and as a result ϑ ( η ) increases. Also, the increase in Nt increases the layer width at the boundary of concentration profile which results in an augmenting impact on ψ ( η ) . Actually, the increase in Nt signifies an increased propensity for particles to migrate toward regions of temperature gradient within a fluid medium. This augmentation in ψ ( η ) can be physically interpreted as a greater tendency for particles to accumulate in regions of varying temperatures. Essentially, as Nt increases, particles experience stronger thermophoretic forces, compelling them to move toward regions of either upper or inferior temperature depending on their properties. This phenomenon can have significant implications in various applications such as nanofluidics, where controlling particle distribution based on temperature gradients is crucial for optimizing processes like heat transfer and particle manipulation. Figure 6(a) and (b) show the deviation in ϑ ( η ) and ψ ( η ) via Brownian motion factor ( Nb ) . It is perceived from these figures that higher Nb increases ϑ ( η ) while reduces ψ ( η ) . An increase in the values of Nb accelerates the random movement of fluid from the hot plate to the surrounding fluid and causes the particles to delve deeper into the fluid, according to one theory. The nanofluid’s temperature is raised as a result. In this phenomenon, the random movement of nanoparticles lowers their kinetic energy, which in turn lowers the rate of heat transmission. While as Nb increases, the fluid particle collisions increase and reduced mass transmission from the heated sheet to the cool fluid is ensured. Hence, ψ ( η ) exhibits diminishing behavior. Figure 7(a) shows the effect of Schmidt number ( Sc ) on ψ ( η ) . It is noticed from the figure that escalation in Sc reduces ψ ( η ) . Since by definition, the mass diffusivity has inverse relation with Sc . So, the higher Schmidt number diminishes the mass diffusions and decreases ψ ( η ) . Therefore, the higher Sc reduces ψ ( η ) . Figure 7(b) depicts the influence of chemically reactive factor ( K r ) on ( ψ ( η ) ) . It is evident from this graphical view that higher values of K r reduces ψ ( η ) . Actually when K r increases, extra solutal molecules are involved in the reaction, that results in reduction in the concentration field. Thus, a destructive chemical reaction drastically lowers the thickness of the solutal border layer. The reduction in the solutal boundary layer results in reduction in the concentration profile. Consequently, a corrosive chemical reaction significantly diminishes the thickness of the solutal boundary layer. This reduction in the width of the layer at boundary leads to a decrease in the concentration of fluid. Figure 8(a) shows the impact of porosity factor ( K ) on the skin friction coefficient ( C f ) . From this figure, we see that the higher values of K increases C f . Actually, increase in K signifies a greater presence of void spaces or pores within the material through which fluid flows. Physically, this leads to an increase in C f , as the presence of pores introduces additional surface area for fluid–solid interactions, resulting in heightened drag forces exerted on the fluid. Essentially, the fluid encounters more resistance as it flows through the porous medium, leading to an augmentation in C f . Figure 8(b) shows the effect of stretching/shrining factor ( λ ) on C f . From this figure, it is evident that the higher estimations of λ increases frictional force at sheet’s surface. It is important to mention that the friction force at the surface of the sheet is higher when the sheet stretches as compared to the case when the sheet shrinks. Figure 9(a) depicts the impact of (Rd) on (Nu) . It is detected from this figure that the higher values of Rd increases N u . The reason is that when Rd augments more, heat is generated in the system which increases the rate of heat transfer. Physically, this indicates that as Rd increases, there is a greater contribution of radiative heat transfer compared to conductive heat transfer. In practical terms, this could mean that surface exposed to higher levels of radiation, such as those in high-temperature environments or subjected to intense electromagnetic radiation, will experience increased heat transfer rates. This phenomenon is crucial in various engineering applications, such as thermal management systems in aerospace or power generation, where understanding and controlling heat transfer mechanisms are essential for optimizing efficiency and performance. Thus, a higher impact of Rd on Nu is observed. Figure 9(b) shows the impression of thermal dependent heat factor ( Q I ) on (Nu) . It is evident from this figure that the higher values of Q I increases Nu . Physically, this implies that as Q I increases, the contribution of temperature-dependent heat transfer processes, such as convection or thermophoresis, becomes more significant compared to other factors like conduction. In practical terms, this suggests that systems experiencing pronounced variations in temperature gradients will exhibit enhanced heat transfer rates due to the intensified effect of temperature-dependent mechanisms, potentially influencing applications ranging from thermal management in electronic devices to heat exchangers in industrial processes.

$Figure 2 (a) Impact of M M on g ′ ( η ) g^{\prime} (\eta ) . (b) Impact of M M on C f {C}_{\text{f}} . (c) Impact of M M on ϑ ( η ) {\vartheta }(\eta ) . (d) Impact of M M on Nu \text{Nu} .$

Figure 2

(a) Impact of M on g ′ ( η ) . (b) Impact of M on C f . (c) Impact of M on ϑ ( η ) . (d) Impact of M on Nu .

$Figure 3 Impact of Ec \text{Ec} on ϑ ( η ) {\vartheta }(\eta ) .$

Figure 3

Impact of Ec on ϑ ( η ) .

$Figure 4 (a) Impact of Q Exp {Q}_{\text{Exp}} on ϑ ( η ) {\vartheta }(\eta ) . (b) Impact of Q Exp {Q}_{\text{Exp}} on Nu \text{Nu} .$

Figure 4

(a) Impact of Q Exp on ϑ ( η ) . (b) Impact of Q Exp on Nu .

$Figure 5 (a) Impact of Nt \text{Nt} on ϑ ( η ) {\vartheta }(\eta ) . (b) Impact of Nt \text{Nt} on ψ ( η ) \psi (\eta ) .$

Figure 5

(a) Impact of Nt on ϑ ( η ) . (b) Impact of Nt on ψ ( η ) .

$Figure 6 (a) Impact of Nb \text{Nb} on ϑ ( η ) {\vartheta }(\eta ) . (b) Impact of Nb \text{Nb} on ψ ( η ) \psi (\eta ) .$

Figure 6

(a) Impact of Nb on ϑ ( η ) . (b) Impact of Nb on ψ ( η ) .

$Figure 7 (a) Impact of Sc \text{Sc} on ψ ( η ) \psi (\eta ) . (b) Impact of K r {K}_{\text{r}} on ψ ( η ) \psi (\eta ) .$

Figure 7

(a) Impact of Sc on ψ ( η ) . (b) Impact of K r on ψ ( η ) .

$Figure 8 (a) Impact of K K on C f {C}_{\text{f}} . (b) Impact of λ \lambda on C f {C}_{\text{f}} .$

Figure 8

(a) Impact of K on C f . (b) Impact of λ on C f .

$Figure 9 (a) Impact of Rd \text{Rd} on Nu \text{Nu} . (b) Impact of Q I {Q}_{\text{I}} on Nu \text{Nu} .$

Figure 9

(a) Impact of Rd on Nu . (b) Impact of Q I on Nu .

6 Conclusion

A computational study of a water-based hybrid nanofluid made up of Fe₃O₄ and Au nanoparticles flowing over an extended surface was carried out. To analyze the effects of different components in such situation, an inclined magnetic field influence is taken into account. The effects of variables such as thermal radiation, Brownian motion, thermophoresis, thermal and space dependent heat sources, and porosity are also taken into account. The following essential points are derived from the observed results.

The higher magnetic factor reduces velocity distribution while increasing the frictional force at the surface due to Lorentz forces which oppose the fluid flow.
The increase in magnetic factor increases the skin friction which consequently increases the rate of heat transfer at the surface and also the thermal distribution.
The fluid temperature increases as Eckert number increases. Both the temperature distribution and Nusselt number are the augmenting functions of space dependent heat source.
The increase in thermophoresis factor induces a faster movement of the nanoparticles from hotter to colder region by the thermophoretic forces which are created in the flow system of the thermophoretic phenomena which as a result reduces the thermal and concentration distributions. Also, the higher Brownian motion factor augments the thermal distribution and reduces the concentration distribution.
The higher Schmidt number reduces the mass diffusivity which results in reduction in the width of concentration layer at boundary and declines the concentration panels. Also, higher chemical reactivity factor reduces the concentration distribution.
The friction force at the sheet’s surface is higher when the sheet stretches as compared to the case when the sheet shrinks.
The higher values of thermal radiation and thermal dependent heat source increases the heat transfer rate at the sheet’s surface.
In future, the impacts of nonlinear convection will be considered in the proposed model along with variable porous characteristics on the surface of sheet.

Funding information: The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University, Abha, Saudi Arabia, for funding this work through the Research Group Project under Grant Number (RGP.2/505/44). 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. 6031).
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 data that support the findings of this study are available from the corresponding author upon reasonable request.

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Received: 2023-09-07

Revised: 2024-02-14

Accepted: 2024-03-11

Published Online: 2024-04-04

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

Articles in the same Issue

https://doi.org/10.1515/ntrev-2024-0010

Keywords for this article

nanofluids; hybrid nanofluids; porous medium; Brownian motion and thermophoresis; activation energy; thermal convective surface

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