Thermal analysis of Fe3O4–Cu/water over a cone: a fractional Maxwell model

Hanifa Hanif; Muhammad Saqib; Sharidan Shafie

doi:10.1515/eng-2022-0600

Article Open Access

Thermal analysis of Fe₃O₄–Cu/water over a cone: a fractional Maxwell model

Hanifa Hanif , Muhammad Saqib and Sharidan Shafie

Published/Copyright: September 11, 2024

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From the journal Open Engineering Volume 14 Issue 1

Abstract

A hybrid nanofluid is a kind of nanofluid that is made by combining a base fluid with two distinct types of nanomaterials. Compared to nanofluids, they have been discovered to have better thermal properties and stability, which makes them viable options for thermal applications such as heat sinks, solar thermal systems, automotive cooling systems, and thermal energy storage. Moreover, the research of nanofluids is typically limited to models with partial differential equations of integer order, which neglect the heredity characteristics and memory effect. To overcome these shortcomings, this study seeks to enhance our understanding of heat transfer in hybrid nanofluids by considering fractional Maxwell models. In time-fractional problems, one of the most significant and useful tools is the Caputo fractional derivative. Therefore, the fractional-order derivatives are approximated using the Caputo derivative. However, the integer-order derivatives are discretized using an implicit finite difference method, namely, the Crank–Nicolson method. It is an unconditionally stable and a second-order method in time. The impact of pertinent flow parameters on fluid motion and heat transfer characteristics is examined and displayed in numerous graphs. The results indicate that the volume concentration of hybrid nanoparticles boosts temperature and Nusselt number. Moreover, increasing the magnetic parameter increases Lorentz’s resistive forces, which reduces the velocity and raises the temperature of the fluid, and these effects are more dominant at t = 5 .

Keywords: Hybrid nanofluid; Maxwell fluid model; fractional derivative; magnetohydrodynamics; Crank-Nicolson method

Nomenclature

: Roman letters
( u , v ): velocity components
ϕ 1 – ϕ 6: nanofluid constants
B: magnetic field strength
C p: specific heat capacity
E: electric field
g: gravitational acceleration
Gr: Grashof number
k: thermal conductivity
L: reference length
Nu x: Nusselt number
q w: heat flux at the surface of the cone
r: radius of the cone
T: temperature
t: time
Pr: Prandtl number
: Greek symbols
α: fractional order
β: volumetric thermal expansion
Δ t: time step
Δ x: grid size in x direction
Δ y: grid size in y direction
λ: relaxation time
μ: dynamic viscosity
ν: kinematic viscosity
ρ: density
σ: electrical conductivity
τ: stress tensor
φ: nanoparticles volume fraction
: Subscripts/Superscripts
*: nondimensional
f: base fluid
h n f: hybrid nanofluid
i: grid point in x direction
j: grid point in y direction
k: time level
n f: nanofluid
p: nanoparticles

1 Introduction

Nanofluids are becoming a critical element of heat transfer applications primarily for their compelling features and adaptability to different circumstances. Since diffusing nanoparticles has some significant advantages, mono nanofluids can be characterized by dispersing one type of nanoparticle. In the meantime, researchers introduced a new type of heat transfer fluid called hybrid nanofluids, a mixture of two different types of nanomaterials in the base fluid, to expand the thermal characteristics of nanofluids, which already have several significant features. Hybrid nanofluids might exhibit better thermal characteristics and heat transfer when compared to regular heat transfer fluids like water, ethylene glycol, and nanofluids containing only one type of nanoparticle. Scientific research indicates that hybrid nanofluids may replace nanofluids due to their advanced heat transfer rate, especially in the manufacture of autos, solar energy, and electro–mechanical sectors [1]. Several researches have been done on considering solo and hybrid nanofluid in different circumstances, and a few will be discussed here. Smaisim et al. [2] illustrated heat transfer of a solar collector using NiO, Al₂O₃, and CuO. The findings demonstrated that CuO/water nanofluid outperform Al₂O₃/water and NiO/water nanofluids, and with 1% CuO nanoparticles, the thermal efficiency increased by up to 7% compared to water. Zainal et al. [3] explored the convective flow of hybrid nanofluid across a porous moving surface while taking into account magnetohydrodynamic (MHD) and radiation impacts. According to what they determined, the hybrid nanofluid motion is decelerated through a magnetic field and suction due to Lorentz and resistive forces. The boost in the fluid’s heat transfer rate related to thermal radiation was observed in the meantime. Abdulwahid et al. [4] found noteworthy enhancement in the thermal performance of the impingement jet on adding CuO-Cu nanoparticles in the working fluid. Taking into consideration, viscous dissipation, MHD, and radiation, Waqas et al. [5] evaluated how heat transfer occurred while a hybrid nanofluid flowed toward a stretching cylinder that was implanted in a permeable medium. They discovered that heat transfer indicates a decline in contrast to the rising volume friction. On the basis of the Casson fluid model, Alkasasbeh [6] numerically analyzed heat transport in hybrid nanofluid past a stretching while being affected by the magnetic field. He noticed that as the magnetic and Casson parameters increase, the heat flux rises and the Nusselt number and hybrid nanofluid motion diminish. Ouyang et al. [7,8] presented the dual solution of tri-hybrid nanofluid flow in the presence of viscous dissipation. Taking into consideration heat source/sink, Waqas et al. [9] examined the implication of radiation on the convective flow of Jeffery hybrid nanofluid. Their findings indicated that hybrid nanofluid motion accelerated with growing Deborah number while decelerated with the strengthening magnetic number. They also noted that the heat flux grew with declining Biot number and radiation while dropping with rising Deborah numbers. In a stagnation point MHD flow of second-grade hybrid nanofluid through a heated shrinking/stretching sheet, Nadeem et al. [10] showed fuzzy triangular membership functions analysis, which not only assisted in overcoming the computational complexity but also provided results that were more reliable than the previous results. The literature has extensively studied the heat transport problem relating convective flow of mono and hybrid nanofluids in a range of geometrical settings [11–15].

Nanofluids are becoming a critical element of heat transfer applications primarily for their compelling features and adaptability to different circumstances. Since diffusing nanoparticles has some significant advantages, mono nanofluids can be characterized by dispersing one type of nanoparticle. In the meantime, researchers introduced a new type of heat transfer fluid called hybrid nanofluids, a mixture of two different types of nanomaterials in the base fluid, to expand the thermal characteristics of nanofluids, which already have several significant features. Hybrid nanofluids might exhibit better thermal characteristics and heat transfer when compared to regular heat transfer fluids like water, ethylene glycol, and nanofluids containing only one type of nanoparticle. Scientific research indicates that hybrid nanofluids may replace nanofluids due to their advanced heat transfer rate, especially in the manufacture of autos, solar energy, and electro–mechanical sectors [1]. Several researches have been done on considering solo and hybrid nanofluid in different circumstances, and a few will be discussed here. Smaisim et al. [2] illustrated heat transfer of a solar collector using NiO, Al₂O₃, and CuO. The findings demonstrated that CuO/water nanofluid outperform Al₂O₃/water and NiO/water nanofluids, and with 1% CuO nanoparticles, the thermal efficiency increased by up to 7% compared to water. Zainal et al. [3] explored the convective flow of hybrid nanofluid across a porous moving surface while taking into account magnetohydrodynamic (MHD) and radiation impacts. According to what they determined, the hybrid nanofluid motion is decelerated through a magnetic field and suction due to Lorentz and resistive forces. The boost in the fluid’s heat transfer rate related to thermal radiation was observed in the meantime. Abdulwahid et al. [4] found noteworthy enhancement in the thermal performance of the impingement jet on adding CuO-Cu nanoparticles in the working fluid. Taking into consideration, viscous dissipation, MHD, and radiation, Waqas et al. [5] evaluated how heat transfer occurred while a hybrid nanofluid flowed toward a stretching cylinder that was implanted in a permeable medium. They discovered that heat transfer indicates a decline in contrast to the rising volume friction. On the Casson fluid model, Alkasasbeh [6] numerically analyzed heat transport in hybrid nanofluid past a stretching while being affected by the magnetic field. He noticed that as the magnetic and Casson parameters increase, the heat flux rises and the Nusselt number and hybrid nanofluid motion diminish. Ouyang et al. [7,8] presented the dual solution of tri-hybrid nanofluid flow in the presence of viscous dissipation. Taking into consideration heat source/sink, Waqas et al. [9] examined the implication of radiation on the convective flow of Jeffery hybrid nanofluid. Their findings indicated that hybrid nanofluid motion accelerated with growing Deborah number while decelerated with strengthening magnetic number. They also noted that the heat flux grew with declining Biot number and radiation while dropping with rising Deborah numbers. In a stagnation point MHD flow of second-grade hybrid nanofluid through a heated shrinking/stretching sheet, Nadeem et al. [10] showed fuzzy triangular membership functions analysis, which not only assisted in overcoming the computational complexity but also provided results that were more reliable than the previous results. The literature has extensively studied the heat transport problem relating convective flow of mono and hybrid nanofluids in a range of geometrical settings [11–15].

According to the published literature, the flows of hybrid nanofluid related to convective heat transfer are extensively studied using several models based on partial differential equations of integer order. However, during the earlier period, the experimental measuring methods and instruments achieved a high degree of precision. Thus, experimental finding discloses notable inconsistencies between the theoretical and experimental outcomes for several thermal processes. For instance, there appear to be many discrepancies between the theoretical and experimental findings provided by Cattaneo’s and Jeffery’s type models for non-Fourier heat conduction processes [16]. To get over the limitations of classical mathematical models, fractional derivatives are proposed because a fractional mathematical model can accurately depict a variety of complex transport phenomena. For example, the exponential relaxation moduli of a classical model cannot accurately capture the algebraic decay throughout the relaxation process of many materials. However, experiments demonstrate that fractional models are capable of accurately capturing and correlating these occurrences [17]. Depending on the memory kernel that is being utilized in the fractional operator, these models can accurately describe heat transfer phenomena [18–21]. Since then, it has come to light that fractional derivatives and integrals may be utilized to accurately describe the complex characteristics, including memory, nonlocality, and fractionality [22]. Meanwhile, in the formulation of the constitutive relationship of viscoelastic materials (fluids), fractional derivatives have been demonstrated to be incredibly adaptable. The memory characteristics of viscoelastic materials (fluids) can be accurately depicted by the nonlocal character of fractional differential operators, which can predict stress relaxation [23].

Multiple mathematical models for viscoelastic fluids have already been offered in the published literature. The fundamental viscoelastic fluid model among many is the Maxwell fluid that can predict stress relaxation, introduced by Maxwell [24]. Being the first and simplest viscoelastic fluid model, the Maxwell fluid model has received a great deal of attention. It continues to be frequently used, particularly to explain how some polymeric fluids interact. However, the Maxwell fluid model does have significant limitations like other models. For instance, in a simple shear flow, this model does not accurately capture the typical relationship between shear rate and shear stress [25]. Due to the fact that the Maxwell fluid model could not match the data to be investigated, the fractional Maxwell model was developed. Friedrich [26] was the first who fractionalized the regular Maxwell model with multiple order derivatives of stress and strain, which indicated that viscoelastic fluid behaved like a fluid when the strain derivative is taken into consideration. The fractional Maxwell model can show better agreement with empirical observations compared to the regular Maxwell fluid model [27].

Maxwell fluid has gained significant attention due to its significance as one of the fundamental viscoelastic fluids. By using the second-order slip model, Yang et al. [28] explored the flow and heat transfer of dual fractional Maxwell fluid. According to their findings, the fractional Maxwell fluid shows greater viscosity/elasticity for various fractional order, and oscillation behavior gradually declines as the slip parameter grow. Zhang et al. [29] proposed a novel heat conduction model with time and spatial fractional derivatives to convection heat transfer in Maxwell nanofluid. They argued that the new heat conduction model reduces the thickness of the thermal boundary resulting in less heat loss. Besides this, Shen et al. [30] reported on how the shape of nanoparticles affects the motion and heat transfer in fractional Maxwell nanofluid. They demonstrated that the spherical shape nanoparticles provide the optimum boost for heat conduction and the lowest for Nusselt number. Madhura and Makinde [31] examined fractional Maxwell carbon nanotubes (CNTs) nanofluid while accounting for electric, magnetic, heating, and buoyancy impacts. They disclosed that the Maxwell nanofluid with single wall CNTs has higher surface drag than nanofluid with multi wall CNTs, whereas the opposite pattern was observed for the ascending Grashof number. Several authors including [32–36] have also studied the topic of fractional Maxwell nanofluid flow with the influence of various physical factors.

Despite the enormous importance and frequent application in science and engineering, according to earlier research, the flow of fractional Maxwell hybrid nanofluid past an inverted cone has not yet been examined. To fill this research gap, the flow of fractional Maxwell hybrid nanofluid under the impact of a magnetic field is taken into account in this research article. In mathematical modeling of the flow phenomenon, the fractional constitutive equations of Friedrich for shear stress are employed. The L1 algorithm and Crank–Nicolson numerical techniques are used to tackle the problem for numerical solutions. It is important to note that solving the Navier–Stokes equations is a challenging task, and an analytical solution is not always available. Therefore, numerical approaches such as the Crank–Nicolson method can be used to estimate the numerical solutions [37]. Furthermore, the Crank–Nicolson method for solving the Navier–Stokes equations has the benefit of being unconditionally stable since an explicit approach can become unstable if the time step size is too great. The results are presented in graphs and the impacts of pertinent flow parameters on the fluid motion and temperature distribution are addressed. This study aims to address:

How do a fractional derivative and relaxation time influence the flow and heat transfer behavior of a fluid?
How does a combination of hybrid nanoparticles improve the thermal performance of Maxwell fluid?
How does Maxwell hybrid nanofluid correlate with the magnetic field?

2 Mathematical formulation

The detailed mathematical modeling of the Maxwell hybrid nanofluid with time-fractional derivative is the emphasis of this section.

2.1 Stress tensor

The extra stress tensor τ for Maxwell fluid is described mathematically as follows [26]:

(1) ( τ + λ α ∂ t α τ ) = μ ϒ ˙ ,

where ϒ ˙ represents shear stress, λ is time relaxation, and μ is the viscosity of the fluid. The well-known Caputo derivative ∂ t α is defined as follows:

(2) ∂ t α f ( t ) = ∂ α ∂ t α f ( t ) = 1 Γ ( n − α ) ∫ 0 t ( t − ω ) n − α − 1 ∂ n ∂ ω n f ( ω ) d ω ,

n − 1 < α < n , n ∈ N . The Gamma function Γ ( ⋅ ) is

(3) Γ ( ξ ) = ∫ R e − ω ω ξ − 1 d ω , ∀ ξ ∈ C , ℜ ξ > 0 .

2.2 Maxwell equations

Let J be the current density, E denotes the electric field, and B = B 0 + B 1 is the magnetic field strength then Maxwell’s set of equations may be used to define the Lorentz force:

(4) ∇ ⋅ B = 0 , ∇ × B = μ 0 J , ∇ × E = − ∂ B ∂ t .

If a fluid with electrical conductivity σ experiences an electric field, then the relationship between current density J and electric field E is given by Oham’s law as follows:

(5) J = σ E .

In the presence of magnetic field B , an additional term must be introduced to account for the current produced by the Lorentz force, which leads us to

(6) J = σ ( E + U × B ) .

2.3 Hybrid nanofluid properties

Let us add two distinct nanoparticles p 1 and p 2 of volume fraction φ p 1 and φ p 2 , respectively, in a water-based fluid. Then the thermal physical properties of the resultant fluid, i.e., hybrid nanofluid are modified as follows [38,39]:

(7) ρ h n f = ( 1 − φ p 2 ) ρ n f + φ p 2 ρ p 2 , μ h n f = μ f ( 1 − φ p 1 ) 2.5 ( 1 − φ p 2 ) 2.5 , σ h n f σ n f = ( σ p 2 + 2 σ n f ) + 2 φ p 2 ( σ p 2 − σ n f ) ( σ p 2 + 2 σ n f ) − φ p 2 ( σ p 2 − σ n f ) , ( ρ β ) h n f = ( 1 − φ p 2 ) ( ρ β ) n f + φ p 2 ( ρ β ) p 2 , ( ρ C p ) h n f = ( 1 − φ p 2 ) ( ρ C p ) n f + φ p 2 ( ρ C p ) p 2 , k h n f k n f = ( k p 2 + 2 k n f ) + 2 φ p 2 ( k p 2 − k n f ) ( k p 2 + 2 k n f ) − φ p 2 ( k p 2 − k n f ) .

In this study, we considered Fe₃O₄ and Cu nanoparticles and water base fluid, see Table 1.

Table 1

Thermo-physical properties of base fluid and nanoparticles [35]

Materials	ρ	σ	β	C p	k
	( kg m ‒ 3 )	( S m ‒ 1 )	( K ‒ 1 )	(J ( kg K ) ‒ 1 )	( W ( mK ) ‒ 1 )
Water	997.1	0.05	21 × 1 0 ‒ 5	4,179	0.613
Fe 3 O 4	5,200	2.5 × 1 0 4	1.3 × 1 0 ‒ 5	670	6
Cu	8,933	5.96 × 1 0 7	1.67 × 1 0 ‒ 5	385	401

2.4 Governing equations

The governing equations of fluid flow along a cone in the presence of electromagnetic forces are described as follows:

(8) ∂ ρ ∂ t + ∇ ⋅ ( ρ r U ) = 0 .

(9) ρ ∂ U ∂ t + U ⋅ ∇ U = − ∇ p + ∇ ⋅ τ + J × B + F g .

The following energy equation helps us to characterize the heat transfer of the fluid

(10) ( ρ C p ) ∂ T ∂ t + U ⋅ ∇ T = k Δ T + τ : ∇ U .

2.5 Problem description

In this study, we consider that an incompressible fluid ( ρ = constant ) is moving along a vertical cone. The considered fluid is a dilute suspension of Fe₃O₄ and Cu in pure water, named a hybrid nanofluid. The surface of the cone is taken as the x -axis and the y -axis is considered normal to the cone (Figure 1). A magnetic field is imposed in the y direction assuming that the induced magnetic field B 1 = 0 and electric field E = 0 . Moreover, the viscous dissipation effects are supposed to be insignificant, and no pressure gradient is applied. Bearing the aforementioned assumptions in mind, the boundary layer and Boussenisq approximations yield us to

(11) ∂ ∂ x ( r u ) + ∂ ∂ y ( r v ) = 0 ,

(12) ρ h n f ∂ u ∂ t + u ∂ u ∂ x + v ∂ u ∂ y = ∂ τ x y ∂ y − σ h n f B 0 2 u + g ( ρ β ) h n f ( T − T ∞ ) cos γ ,

(13) ( ρ C p ) h n f ∂ T ∂ t + u ∂ T ∂ x + v ∂ T ∂ y = k h n f ∂ 2 T ∂ y 2 .

For the considered hybrid nanofluid the Equation (1) provides us

(14) ( 1 + λ α ∂ t α ) τ x y = μ h n f ∂ u ∂ y .

Evaluating shear stress τ x y from Equations (12) and (14) leads us to

(15) ρ h n f ( 1 + λ α ∂ t α ) ∂ u ∂ t + u ∂ u ∂ x + v ∂ u ∂ y = μ h n f ∂ 2 u ∂ y 2 − σ h n f B 0 2 ( 1 + λ α ∂ t α ) u + g ( ρ β ) h n f ( 1 + λ α ∂ t α ) ( T − T ∞ ) cos γ .

The imposed initial and boundary conditions are as follows:

(16) u ( x , y , 0 ) = 0 , v ( x , y , 0 ) = 0 , T ( x , y , 0 ) = T ∞ , u ( 0 , y , t ) = 0 , T ( 0 , y , t ) = T ∞ , u ( x , 0 , t ) = 0 , v ( x , 0 , t ) = 0 , T ( x , 0 , t ) = T w ( x ) , u ( x , ∞ , t ) = 0 , T ( x , ∞ , t ) = T ∞ .

Figure 1

Graphical representation.

2.6 Nondimensional problem

To completely comprehend the mechanics of flow, we require a nondimensional version of our suggested model. Therefore, we will introduce a set of nondimensional parameters [35]:

(17) x * = x L , y * = y L Gr 1 4 , r * = r L , t * = ν f t L 2 Gr 1 2 , u * = u L ν f Gr − 1 2 , v * = v L ν f Gr − 1 4 , T * = T − T ∞ T w − T ∞ , λ * = ν f λ L 2 Gr 1 2 , Gr = g β f ( T w − T ∞ ) L 3 ν f 2 .

Employing the hybrid nanofluid properties (7) and the nondimensional variables (17) in Equations (11), (13), and (15) yields us to (after removing * ):

(18) ∂ ∂ x ( r u ) + ∂ ∂ y ( r v ) = 0 ,

(19) ϕ 1 ( 1 + λ α ∂ t α ) ∂ u ∂ t + u ∂ u ∂ x + v ∂ u ∂ y = ϕ 2 ∂ 2 u ∂ y 2 − ϕ 3 M ( 1 + λ α ∂ t α ) u + ϕ 4 cos γ ( 1 + λ α ∂ t α ) T ,

(20) ϕ 5 ∂ T ∂ t + u ∂ T ∂ x + v ∂ T ∂ y = ϕ 6 Pr ∂ 2 T ∂ y 2 ,

subject to initial boundary conditions:

(21) u ( x , y , 0 ) = 0 , v ( x , y , 0 ) = 0 , T ( x , y , 0 ) = 0 , u ( 0 , y , t ) = 0 , T ( 0 , y , t ) = 0 , u ( x , 0 , t ) = 0 , v ( x , 0 , t ) = 0 , T ( x , 0 , t ) = x n , u ( x , ∞ , t ) = 0 , T ( x , ∞ , t ) = 0 .

Here, ϕ 1 − ϕ 6 , Pr, and M , are the nanofluid constants, Prandtl number, and magnetic parameter defined as follows:

(22) ϕ 1 = ρ h n f ρ f , ϕ 2 = μ h n f μ f , ϕ 3 = σ h n f σ f , ϕ 4 = ( ρ β ) h n f ( ρ β ) f , ϕ 5 = ( ρ C p ) h n f ( ρ C p ) f , ϕ 6 = k h n f k f , Pr = ( μ C p ) f k f , M = σ f B 0 2 L 2 μ f Gr 1 2 .

2.7 Physical quantities

This section aims to discuss some important physical quantities like shear stress and Nusselt number.

2.7.1 Wall shear stress

Shear stress is described as a form of stress that acts coplanar with a specific material cross-section and is extremely important because it explains material and fluid behavior. It is worthwhile to investigate how the active parameters affect the wall shear stress. For an ordinary integer system, wall shear stress is defined as follows:

(23) τ s = μ h n f ∂ u ∂ y y = 0 .

In reference to Equation (1), shear stress τ w in fractional form is given by:

(24) ( 1 + λ α ∂ t α ) τ s = μ h n f ∂ u ∂ y y = 0 .

If τ w = L 2 τ s μ f ν f Gr 1 4 , then with the help of nondimensional variables (17), we can have

(25) ( 1 + λ α ∂ t α ) τ w = ϕ 2 ∂ u ∂ y y = 0 .

2.7.2 Nusselt number

The Nusselt number is a dimensionless quantity that represents the rate of energy conversion at the surface [40]. It is an essential metric that can help improve the rate of heat exchange. Therefore, it is worth examining how the Nusselt number is influenced by the active parameters. Mathematically, it can be expressed as follows:

(26) Nu = x q w k f ( T w − T ∞ ) ,

where q w = − k h n f ∂ T ∂ y y = 0 . Equation (17) helps us to obtained the nondimensional form of Equation (26):

(27) Nu x = Nu Gr − 1 4 = − ϕ 6 x ∂ T ∂ y y = 0 .

3 Discrete model

The discrete form of nondimensional Equations (18)–(20) are obtained using the L1 algorithm of Caputo time fractional derivative and Crank–Nicolson method, and for more details, please see [21,27]:

(28) u i , j − 1 k + 1 − u i − 1 , j − 1 k + 1 + u i , j k + 1 − u i − 1 , j k + 1 + u i , j − 1 k − u i − 1 , j − 1 k + u i , j k − u i − 1 , j k 4 Δ x + v i , j k + 1 − v i , j − 1 k + 1 + v i , j k − v i , j − 1 k 2 Δ y = 0 .

(29) u i , j k + 1 − u i , j k Δ t + u i , j k u i , j k + 1 − u i − 1 , j k + 1 + u i , j k − u i − 1 , j k 2 Δ x + v i , j k u i , j + 1 k + 1 − u i , j − 1 k + 1 + u i , j + 1 k − u i , j − 1 k 4 Δ y − χ 1 ∑ m = 1 k b m u i , j k + 1 − m − u i , j k − m Δ t + ∑ m = 1 k b m u i , j k − m u i , j k + 1 − m − u i − 1 , j k + 1 − m 2 Δ x + ∑ m = 1 k b m v i , j k − m u i , j + 1 k + 1 − m − u i , j − 1 k + 1 − m 4 Δ y + ∑ m = 1 k − 1 b m u i , j k − m u i , j k − m − u i − 1 , j k − m 2 Δ x + ∑ m = 1 k − 1 b m v i , j k − m u i , j + 1 k − m − u i , j − 1 k − m 4 Δ y = χ 2 u i , j + 1 k + 1 − 2 u i , j k + 1 + u i , j − 1 k + 1 + u i , j + 1 k − 2 u i , j k + u i , j − 1 k 2 Δ y 2 − χ 3 u i , j k + 1 + u i , j k 2 + χ 4 T i , j k + 1 + T i , j k 2 + χ 5 ∑ m = 1 k b m u i , j k + 1 − m 2 + ∑ m = 1 k − 1 b m u i , j k − m 2 − χ 6 ∑ m = 1 k b m T i , j k + 1 − m 2 + ∑ m = 1 k − 1 b m T i , j k − m 2 .

(30) T i , j k + 1 − T i , j k Δ t + u i , j k T i , j k + 1 − T i − 1 , j k + 1 + T i , j k − T i − 1 , j k 2 Δ x + v i , j k T i , j + 1 k + 1 − T i , j − 1 k + 1 + T i , j + 1 k − T i , j − 1 k 4 Δ y = χ 7 T i , j + 1 k + 1 − 2 T i , j k + 1 + T i , j − 1 k + 1 + T i , j + 1 k − 2 T i , j k + T i , j − 1 k 2 Δ y 2 .

Here, the superscript k represents the time level, and the subscripts ( i , j ) denote the regularly spaced grid point in the ( x , y ) -directions with mesh size ( Δ x , Δ y ) , respectively. The constant coefficients χ 1 – χ 7 are given as follows:

(31) χ 1 = ϕ 1 R 1 R 2 , χ 2 = ϕ 2 R 2 , χ 3 = ϕ 3 M R 2 , χ 4 = ϕ 4 cos γ R 2 , χ 5 = χ 3 R 1 , χ 6 = χ 4 R 1 , R 1 = λ 1 α Δ t − α Γ ( 2 − α ) , R 2 = ϕ 1 ( 1 + R 1 ) .

4 Results and discussion

The primary objective of this section is to demonstrate the variability of fluid motion, temperature, shear stress, and Nusselt number for varying pertinent flow parameters including fractional parameter α , relaxation time λ , magnetic parameter M , volume fraction φ , and power index parameter n .

4.1 Velocity

To demonstrate how nanoparticle volume fraction affects the velocity profile, Figure 2 is plotted at different times t = 2 and t = 5 . The results show that the velocity of the fluid decreases when the concentration of Fe₃O₄–Cu increases. This is partially due to the viscosity of the fluid, which increases by adding more nanoparticles. It is worth mentioning that fluids with high viscosity are frequently employed in applications that need flow resistance, such as lubricating oils. Figure 3 displays the velocity profile for varying α when t = 2 and t = 5 . For both t = 2 and t = 5 , it is seen that the fluid motion decreases uniformly with growing values of α . However, under greater α , the momentum boundary layer is bigger, demonstrating that the viscoelasticity enhances flow resistance as α grows. This finding is significant because if we wish to have a certain velocity at a given time in a specified position/point (say a 0 ), then the model with fractional derivatives is beneficial. For varying λ , Figure 4 illustrates the fluid motion at t = 2 and t = 5 . When λ rises, the fluid motion accelerated for both t = 2 and t = 5 . The impact of the magnetic parameter M on the fluid motion is depicted in Figure 5. As an increment in M strengthens the Lorentz forces opposing the flow direction, as a result, it demonstrates a reduction in fluid velocity.

$Figure 2 Velocity profile for various values of φ \varphi ; left: t = 2 t=2 , right: t = 5 t=5 .$

Figure 2

Velocity profile for various values of φ ; left: t = 2 , right: t = 5 .

$Figure 3 Velocity profile for various values of α \alpha ; left: t = 2 t=2 , right: t = 5 t=5 .$

Figure 3

Velocity profile for various values of α ; left: t = 2 , right: t = 5 .

$Figure 4 Velocity profile for various values of λ \lambda ; left: t = 2 t=2 , right: t = 5 t=5 .$

Figure 4

Velocity profile for various values of λ ; left: t = 2 , right: t = 5 .

Figure 5

Velocity profile for various values of M ; left: t = 2 , right: t = 5 .

4.2 Temperature

Figure 6 depicts the effects of φ on the temperature field when t = 2 and t = 5 . From this figure, it is perceived that the increased thermal conductivity increases the temperature of the fluid. In general, the thermal conductivity of nanoparticles is higher than that of fluids, and obviously, the thermal conductivity of the fluid increases upon adding nanoparticles to it. Figure 7 illustrates how magnetic parameter M affects the temperature field when t = 2 and t = 5 . A rise in the temperature profile is seen to be caused by a growth in M . The physical explanation for this phenomenon is that when M grows, the resistive forces are generated, which causes the temperature profile to increase. Figure 8 provides the temperature profile for various values of n . It is clear from the findings that increasing the n causes the thermal boundary layer to shrink, which drives the temperature profile to tend to drop.

$Figure 6 Temperature profile for various values of φ \varphi ; left: t = 2 t=2 , right: t = 5 t=5 .$

Figure 6

Temperature profile for various values of φ ; left: t = 2 , right: t = 5 .

Figure 7

Temperature profile for various values of M ; left: t = 2 , right: t = 5 .

Figure 8

Temperature profile for various values of n ; left: t = 2 , right: t = 5 .

4.3 Wall shear stress

Figure 9 demonstrates that the wall shear stress develops with growing α because the friction among the nearby layer reduces. Besides this, Figure 10 indicates that wall shear stress diminishes with rising λ . In addition, Figure 11 presents that wall shear stress diminishes with increasing M , and the Lorentz forces strengthen causing resistance among the neighboring layers.

$Figure 9 Wall shear stress for various values of α \alpha ; left: t = 2 t=2 , right: t = 5 t=5 .$

Figure 9

Wall shear stress for various values of α ; left: t = 2 , right: t = 5 .

$Figure 10 Wall shear stress for various values of λ \lambda ; left: t = 2 t=2 , right: t = 5 t=5 .$

Figure 10

Wall shear stress for various values of λ ; left: t = 2 , right: t = 5 .

Figure 11

Wall shear stress for various values of M ; left: t = 2 , right: t = 5 .

4.4 Nusselt number

A graph for the Nusselt number that correlates to the variation in φ is displayed in Figure 12. The thermal conductivity is related to φ . The thermal conductivity rises as φ rises, resulting in a boost in the Nusselt number. It is worth noting that high thermal conductivity fluids are utilized to increase efficiency and production in a variety of sectors such as heat exchangers, solar energy, and thermal management. Variation in Nusselt number for varying M is presented in Figure 13 when t = 2 and t = 5 . The temperature of the fluid increases when M grows, which ultimately reduces the Nusselt number. This tells us that the maximum heat transfer rate of a hybrid nanofluid can be achieved in the absence of a magnetic field for this particular problem. Likewise, the exponent n diminishes the Nusselt number (Figure 14).

$Figure 12 Nusselt number for various values of φ \varphi ; left: t = 2 t=2 , right: t = 5 t=5 .$

Figure 12

Nusselt number for various values of φ ; left: t = 2 , right: t = 5 .

Figure 13

Nusselt number for various values of M ; left: t = 2 , right: t = 5 .

Figure 14

Nusselt number for various values of n ; left: t = 2 , right: t = 5 .

5 Conclusion

Investigations are made into the boundary layer flow of fractional Maxwell hybrid nanofluid with variable temperature when a magnetic field is present. To provide numerical solutions for the proposed problem, the Crank–Nicolson method and the L1 algorithm are used. In various figures, the effects of relevant flow parameters on fluid velocity, heat transfer, wall shear stress, and Nusselt number are explored. Following is a list of the study’s key findings:

The friction between the adjacent layers of a moving fluid increases as the φ rises, increasing the viscosity, which is important in the lubrication process. Furthermore, the thermal conductivity boosts as the φ increases, which increases the Nusselt number or heat transfer rate. High heat transfer rates can assist in increasing industrial process efficiency by lowering the amount of energy required to get a given output. This can result in considerable economic savings as well as environmental advantages.
Higher α causes the momentum boundary layer to grow, indicating that when viscoelasticity rises, fluid motion is reduced, and wall shear stress increases.
The fluid motion speeds up when λ rises. Since the momentum boundary layer thickens with an increase in λ , the fluid motion increases and wall shear stress decreases.
An increase in M causes a decrease in fluid velocity and wall shear stress because it strengthens the Lorentz forces that oppose the direction of the flow. Besides this, the temperature profile increases along with the thermal boundary layer’s expansion as M grows.

This study can be expanded in the future for a ternary nanofluid with the second law of thermodynamics.

Acknowledgement

The authors would like to acknowledge the financial support from Universiti Teknologi Malaysia for the funding under UTM Fundamental Research (UTMFR: Q.J130000.3854.23H22).

Funding information: The authors would like to acknowledge the financial support from Universiti Teknologi Malaysia for the funding under UTM Fundamental Research (UTMFR: Q.J130000.3854.23H22).
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results and approved the final version of the manuscript. HH: Conceptualization, Methodology, Investigation, Writing original draft. MS: Formal Analysis, Writing original draft. SS: Supervision, Resources.
Conflict of interest: Authors state no conflict of interest.
Data availability statement: No data associated with the article.

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Received: 2023-07-12

Revised: 2023-11-10

Accepted: 2024-02-14

Published Online: 2024-09-11

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

Articles in the same Issue

https://doi.org/10.1515/eng-2022-0600

Keywords for this article

Hybrid nanofluid; Maxwell fluid model; fractional derivative; magnetohydrodynamics; Crank-Nicolson method

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