Analysis of variable fluid properties for three-dimensional flow of ternary hybrid nanofluid on a stretching sheet with MHD effects

Nomenclature

(ρC _p)

fluid’s heat capacitance (J K⁻¹ m⁻³)

u ∼ , v ∼

velocity components in x and y direction ( m s , L / T )

B o

magnitude of magnetic field strength ( M / A T 2 )

M

magnetic parameter

k

thermal conductivity ( W m − 1 K − 1 )

C p

specific heat capacitance at constant temperature ( J kg − 1 K − 1 )

Pr

Prandtl number

Re x

local Reynold number (non-dimensional)

C fx , C fy

coefficient of skin friction along x and y direction (non-dimensional)

Nu x

Nusselt number (non-dimensional)

S

mass transportation parameter (non-dimensional)

T ∼

temperature ( K )

T 0

sheet ( K ) reference temperature

T ∞

ambient temperature ( K )

u w

wall velocity ( m s − 1 )

v w

wall velocity ( m s − 1 )

ax , by

linear velocities in x and y direction

a , b > 0

constants

Greek symbols

η

similarity variables

θ

dimensionless temperature

μ

dynamic viscosity (kg m⁻¹ s⁻¹)

ν

kinematic viscosity (m² s⁻¹)

ρ

density ( kg m − 3 )

Φ

nanoparticle volume fraction ( % )

Φ ₁

nanoparticle volume fraction of Cu

Φ ₂

nanoparticle volume fraction of Fe₃O₄

Φ ₃

nanoparticle volume fraction of SiO₂

Subscripts

f

base fluid

nf

nanofluid

hnf

hybrid nanofluid

thnf

ternary hybrid nanofluid

s1

solid component for Cu

s2

solid component for Fe₃O₄

s3

solid component for SiO₂

1 Introduction

The study of the properties of electrically conducting fluids when subjected to the effects of magnetic fields, such as plasmas, liquid metals, and ionized gases, is known as magnetohydrodynamics (MHD). MHD is a key component of this study, indicating that the fluid under investigation is electrically conductive and that magnetic fields have an impact on how it behaves. The phrase “variable fluid properties” refers to the fact that a fluid’s properties, especially viscosity and thermal conductivity, are not constant but rather alter depending on the temperature or concentration. The study examines how the fluid’s behavior is impacted by these variable properties in this situation. In contrast to more straightforward one- or two-dimensional analyses, the concept of “thermally three-dimensional” indicates the analysis, which adopts heat transfer in three dimensions into account. This feature recognizes how intricately distributed heat is in three-dimensional (3D) systems. Ternary hybrid nanofluid denotes a mixture made up of three distinct elements. The term “hybrid nanofluid” suggests the presence of nanoparticles floating in the fluid that may substantially alter its characteristics and behavior. One of the boundary conditions is the stretching sheet. It symbolizes a stable surface on which the fluid is forced to flow and stretch. This situation frequently occurs in a variety of manufacturing procedures, such as the production of paper or plastic films. Industrial manufacturing processes aim to optimize a number of factors, including time, cost, space, and pollution, in order to satisfy global demand. In order to meet these needs in the industrial and engineering domains, the field of nanoscience and nanotechnology has become an invaluable resource. Significant progress has been made in this area by scientists and researchers, resulting in improvements and refinements. Nanofluid dynamics, a branch of nanoscience, examines how nanoparticle volume, energy distribution, and accumulation affect a base fluid. Understanding and utilizing the effects of nanoparticles in various applications depends heavily on this field of study. Nanotechnology is essential in many areas, including the repair of electronic equipment, lowering vehicle temperatures, and the use of nuclear catalysts. The ability of nanoparticles to improve the thermal conductivity of working fluids is one of their notable effects. Previous research, including the work of Das et al. [1], has shown the remarkable thermal enhancement effects of nanofluids, opening the door to real-world applications in the physical, chemical, and material sciences. Additionally, Wang and Mujumdar [2] have looked into how nanofluids affect flow characteristics and distribution in both free- and forced-convection problems. Murshed et al. [3] conducted a comparison of experimental and analytical results to look at the effects of adding nanofluid and increasing temperature. Wang and Mujumdar [4] presented numerical calculations related to thermal enhancement and the addition of nanofluids and discussed the theoretical aspects of these calculations. Wen et al. [5] contributed to the fascinating field of nanofluid research by investigating the use of nanofluids in heat distribution enhancement and its limitations. Yu and Xie [6] gave a brief description of nanofluids in their study, covering their development, their processes of stability, and a variety of applications. Taylor’s research [7] examined the enormous impact of tiny particles and provided a thorough overview of the numerous uses for nanofluids. Investigating how thermal radiation impacts MHD nanofluids was the main goal of Kumar et al. [8,9,10]. In the presence of a Lorentz force and variable heat sources/sinks, they simultaneously addressed the effects of micro-polar nanofluid flow throughout a surface going through convection. They also investigated, using a modified heat flux model, how viscous dissipation affects micropolar nanofluid. Hybrid nanofluids were thoroughly reviewed by Sarkar et al. [11], who covered current research advancements as well as uses in this area. In order to account for the uneven viscous impacts, Akbar et al. [12,13,14] developed a mathematical model to analyze the effects of nanofluids on a vertically positioned stretching surface under a magnetic field. They also emphasized the uses of carbon nanotubes (CNTs) in biotechnology and small electronic devices. With a focus on the enhanced heat transfer effects of nanofluids, Sheikholeslami and Chamkha [15] investigated the effects of asymmetric MHD conditions on the lid-driven cavity flow problem. The knowledge and applications of nanofluids in various fields are aided by these studies. A 3D analysis by Mahanthesh et al. [16], which specifically took nonlinear radiations into account, looked into the effect of nanofluids on the distribution of heat. In order to investigate the improved convective heat distribution effects brought about by the addition of nanofluid inside a wavy conduit containing aluminum oxide and ethylene glycol nanoparticles, Zhu et al. [17] used numerical analysis. The thermal conductivity of the silver-water nanofluid increases with the increasing nanoparticle volume fraction and temperature, while it decreases for larger sized nanoparticles, according to the numerical solutions they obtained. These investigations advance our knowledge of the 3D flow behavior and the impact of nanofluids on heat transfer. In a 3D flow analysis for nanofluids on a nonlinear stretching surface and a slandering stretching sheet, Nadeem et al. [18] developed a model that incorporates the effects of MHD. These studies provide knowledge about the intricate flow behavior of nanofluids on various stretching surfaces and provide information about their useful uses in business and nanotechnology. Sajid and Ali [19] reviewed the creation of hybrid nanofluids, focusing on methods for determining and enhancing their stability. They emphasized the significance of thermal conductivity because it has a direct impact on the rate of heat transfer in hybrid nanofluids. The importance of comprehending and enhancing hybrid nanofluids’ thermal conductivity was emphasized by the authors because it is essential for improving heat transfer efficiency. This research advances the use of hybrid nanofluids in a variety of applications by offering a thorough analysis of their preparation and stability enhancement. Ternary hybrid nanofluids have a lot of potential for use in a wide range of processes, such as heat transfer, lubrication, and catalysis. The analysis of unstable, incompressible flow near stagnation points in the 3D flow of ternary hybrid nanoparticles Al 2 O 3 , CuO , TiO 2 with a polymer base fluid was the main focus of Zafar Mahmood and Umar Khan’s study [20]. Khan et al. [21] examined the hydromagnetic flow of ternary nanomaterials while taking into account entropy, radiation, and dissipation effects. Numerous studies [22,23,24,25,26,27,28,29] on hybrid, and nanofluids have been carried out recently, including theoretical, experimental, and numerical investigations. Numerous researchers [30,31,32,33,34] recently discussed the ternary hybrid nanofluid for different geometries. The main goal of these research projects is to develop new technologies that improve base fluids’ capacity for heat transfer. To increase thermal conductivity, various kinds of nanoparticles are dispersed within the base fluid. Polymers have become one of these nanoparticles’ most widely used options, significantly improving the fluid’s thermal conductivity. These studies pave the way for the creation of enhanced heat transfer technologies and systems that make use of ternary, hybrid, and nanofluids’ advantages in a variety of applications. Monitoring fluid flow as well as heat transfer is crucial for the optimization of processes and the quality of the final product, so it is crucial to understand how these factors interact and affect one another in various industries. This analysis considers the combined impact of a magnetic field and three different nanoparticles ( Cu − Fe 3 O 4 − SiO 2 ) suspended in a base fluid polymer. It presents a 3D flow analysis for a stretching sheet. This analysis uses the Reynolds model of viscosity, which takes into account the temperature dependence of both thermal conductivity and viscosity. The lack of prior studies reporting on the combined analysis of a ternary hybrid nanofluid ( Cu − Fe 3 O 4 − SiO 2 / polymer ) with the Reynolds model of viscosity and temperature-dependent thermal conductivity is the driving force behind this work. The transport of sanitary fluids, blood pumps in heart-lung machines, and the handling of corrosive fluids are just a few industrial uses for this model. Given that fluids used in engineering and industrial settings frequently have variable fluid properties, this analysis offers important new information about the operation and behavior of such fluids.

1.1 Raising of the problem

The present issue entails a thorough analysis of the flow characteristics of a particular kind of fluid identified as a ternary hybrid nanofluid. This nanofluid is made up of a mixture of base fluid polymer, silicon dioxide ( Si O 2 ) , iron oxide ( Fe 3 O 4 ) , and copper ( Cu ) . This study’s main goal is to examine the analysis of variable fluid properties of flow behavior over a stretching surface that is horizontally oriented, as shown in Figure 1. The study uses boundary layer approximations as an aspect of its scientific strategy to accomplish this goal. In conclusion, the study hopes to:

• Pay attention to the nanofluid’s incompressibility and laminar flow properties.

• Investigate how the presence of cylinder-shaped nanoparticles in the fluid affects the flow characteristics.

• To affect the flow, apply a consistent magnetic field along the z-axis.

• Conduct the analysis using a Cartesian coordinate system ( x , y , z ) .

• Use boundary layer approximations to comprehend the system’s flow dynamics.

Figure 1

Problem geometry.

The objective of the study is to advance knowledge of the flow characteristics and potential uses of this special nanofluid by shedding light on how it responds to outside influences like a stretching surface and magnetic field.

The vector form of continuity, momentum, and energy are

(1) ∇ · V ̅ = 0 ,

(2) ρ thnf D V ̅ D t = ∇ ( μ thnf * ( T ) ∇ V ̅ ) − σ thnf B 0 2 V ̅ ,

(3) ( ρ C p ) thnf D T D t = ∇ ( k thnf * ( T ) ∇ T − q r ) .

According to the current marvels of the flow behavior of the problem, equations of continuity, momentum, and energy are as per previous studies [13].

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

(5) ρ thnf u ∼ ∂ u ∼ ∂ x + v ∼ ∂ u ∼ ∂ y + w ∼ ∂ u ∼ ∂ z = ∂ ∂ z μ thnf * ( T ) ∂ u ∼ ∂ z − σ thnf B 0 2 u , ∼

(6) ρ thnf u ∼ ∂ v ∼ ∂ x + v ∼ ∂ v ∼ ∂ y + w ∼ ∂ v ∼ ∂ z = ∂ ∂ z μ thnf * ( T ) ∂ v ∼ ∂ z − σ thnf B 0 2 v ∼ ,

(7) ( ρ C p ) thnf u ∼ ∂ T ∼ ∂ x + v ∼ ∂ T ∼ ∂ y + w ∼ ∂ T ∼ ∂ z = ∂ ∂ z k * thnf ( T ) ∂ T ∼ ∂ z .

The boundary conditions associated with the present problem are

(8) u ∼ = u w ( x ) = ax , v ∼ = v w ( y ) = by , w ∼ = 0 , T ∼ = T w , at z = 0 , u ∼ → 0 , v ∼ → 0 , T ∼ → T ∞ , as z → ∞ ,

where μ thnf , ρ thnf , k thnf , σ thnf , and ( ρ C p ) thnf are dynamic viscosity, density, thermal conductivity, heat capacity, and electric conductivity for ternary hybrid nanofluid, respectively. Variable viscosity and variable thermal conductivity are defined as [13,14] follows:

(9) μ thnf * ( T ) = μ f ( θ ) ( 1 − ϕ 1 ) − 2.5 ( 1 − ϕ 3 ) 2.5 ( 1 − ϕ 2 ) 2.5 ,

(10) k * thnf ( T ) = k thnf 1 + ϵ T − T ∞ T w − T ∞ ,

where α and ϵ are the viscosity and thermal conductivity parameters.

Table 1 presents the numerical values and thermophysical characteristics for Cu , Fe 3 O 4 , Si O 2 , and for base fluid use polymer. Thermo-physical properties of ternary hybrid nanofluid are listed in Table 2, where ϕ 1 , ϕ 2 , and ϕ 3 are the volume fractions of nanoparticles Cu , Fe 3 O 4 , and Si O 2 , respectively. The properties of Cu , Fe 3 O 4 , and Si O 2 nanoparticles denoted by subscripts s 1 , s 2 , and s 3 . The nanoparticles are in cylindrical form, so m = 4.9 .

Table 1

Investigational values of Cu , Fe 3 O 4 , Si O 2 , and polymer base [20,22]

Properties	ρ ( kg m 3 )	Cp ( J kg K )	k ( W mK )	σ ( Ω m ) ‒ 1	Pr
Cu ( ϕ 1 )	8 , 933	385	401	5.96 × 10 7
Fe 3 O 4 ( ϕ 2 )	5 , 180	670	9.7	2.5 × 10 ‒ 4
Si O 2 ( ϕ 3 )	2 , 650	730	1.5	1.0 × 10 ‒ 18
P olymer	1 , 060	3 , 770	0.429	4.3 × 10 ‒ 5	2.36

Table 2

Thermochemical properties of ternary hybrid nanofluid [20,22]

Density ρ thnf	ρ thnf = ( 1 ‒ ϕ 3 ) { ( 1 ‒ ϕ 2 ) [ ϕ 1 ρ s 1 + ( 1 ‒ ϕ 1 ) ρ f ] + ϕ 2 ρ s 2 } + ϕ 3 ρ s 3
Dynamic viscosity μ thnf	μ thnf = μ f ( 1 ‒ ϕ 1 ) ‒ 2.5 ( 1 ‒ ϕ 3 ) 2.5 ( 1 ‒ ϕ 2 ) 2.5
Thermal conductivity	k thnf k hnf = k s 3 + ( m ‒ 1 ) k hnf ‒ ( m ‒ 1 ) ϕ 3 ( k hnf ‒ k s 3 ) k s 3 + ( m ‒ 1 ) k hnf + ϕ 3 ( k hnf ‒ k s 3 ) , where, k hnf k nf = k s 2 + ( m ‒ 1 ) k nf ‒ ( m ‒ 1 ) ϕ 2 ( k nf ‒ k s 2 ) k s 2 + ( m ‒ 1 ) k nf + ϕ 2 ( k nf ‒ k s 2 ) , and k nf k f = k s 1 + ( m ‒ 1 ) k f ‒ ( m ‒ 1 ) ϕ 1 ( k f ‒ k s 1 ) k s 1 + ( m ‒ 1 ) k f + ϕ 1 ( k f ‒ k s 1 ) .
Heat capacity	( ρ C p ) thnf = ( 1 ‒ ϕ 3 ) { ( 1 ‒ ϕ 2 ) [ ( 1 ‒ ϕ 1 ) ( ρ C p ) f + ϕ 1 ( ρ C p ) s 1 ] + ϕ 2 ( ρ C p ) s 2 } + ϕ 3 ( ρ C p ) s 3
Electrical conductivity	σ thnf σ hnf = k s 3 + 2 σ hnf ‒ 2 ϕ 3 ( σ hnf ‒ σ s 3 ) σ s 3 + 2 σ hnf + ϕ 3 ( σ hnf ‒ σ s 3 ) , where σ hnf σ nf = σ s 2 + 2 σ nf ‒ 2 ϕ 2 ( σ nf ‒ σ s 2 ) σ s 2 + 2 σ nf + ϕ 2 ( σ nf ‒ σ s 2 ) , and σ nf σ f = σ s 1 + 2 σ f ‒ 2 ϕ 1 ( σ f ‒ σ s 1 ) σ s 1 + 2 σ f + ϕ 1 ( σ f ‒ σ s 1 ) .

The similarity conversions availed in this analysis are given as follows [13]:

η = a v f 1 / 2 z , u ∼ = ax f ′ ( η ) , v ∼ = by g ′ ( η ) , w ∼ = v f a 1 / 2 ( af ( η ) + bg ( η ) )

(11) θ = T ∼ − T ∞ T w − T ∞ , M = B 0 a σ ρ u w ,

Equations (4)–(8) convey the following reduced form after utilizing the equations (9)–(11).

(12) ( 1 − ϕ 1 ) − 2.5 ( 1 − ϕ 3 ) 2.5 ( 1 − ϕ 2 ) 2.5 { μ f ( θ ) f ‴ + μ ′ f ( θ ) f ″ } + ρ thnf ρ f { ( f + λ g ) f ″ − f ′ 2 } + σ thnf σ hnf M 2 f ′ = 0 ,

(13) ( 1 − ϕ 1 ) − 2.5 ( 1 − ϕ 3 ) 2.5 ( 1 − ϕ 2 ) 2.5 { μ f ( θ ) g ‴ + μ ′ f ( θ ) g ″ } + ρ thnf ρ f { ( f + λ g ) g ″ − g ′ 2 } + σ thnf σ hnf M 2 g ′ = 0 ,

(14) k thnf Pr k hnf ( 1 + ϵ θ ) θ ″ + k thnf Pr k hnf ϵ θ ′ 2 + ( ρ C p ) thnf ( ρ C p ) f ( f θ ′ + λ g ) = 0 ,

(15) f ( 0 ) = 0 , f ′ ( 0 ) = 1 , f ′ ( ∞ ) = 0 , g ( 0 ) = 0 , g ′ ( 0 ) = λ , g ′ ( ∞ ) = 0 , θ ( 0 ) = 1 , θ ( ∞ ) = 0 , .

The dimensionless variables appearing here are provided as

Pr = ( μ C p ) f k f , λ = b a ,

where λ is the ratio of stretching velocities in x and y directions. The Reynolds model for viscosity is

(16) μ f ( θ ) = e − ( α θ ) = 1 − ( α θ ) + O ( α 2 ) .

Skin friction and Nusselt number

(17) c f x = μ thnf ( T ) ρ f u w 2 ∂ u ∼ ∂ z z = 0 , c f y = μ thnf ( T ) ρ f u w 2 ∂ v ∼ ∂ z z = 0 , Nu x = − x k thnf ( T ) k f ( T w − T ∞ ) ∂ T ∼ ∂ z z = 0 .

Equation (17) in dimensionless form is given as

(18) ( Re x ) 1 2 C f x = μ f ( θ ( 0 ) ) ( 1 − ϕ 1 ) − 2.5 ( 1 − ϕ 2 ) 2.5 ( 1 − ϕ 3 ) 2.5 f ″ ( 0 ) , ( Re y ) 1 2 C f y = λ y x μ f ( θ ( 0 ) ) ( 1 − ϕ 1 ) − 2.5 ( 1 − ϕ 2 ) 2.5 ( 1 − ϕ 3 ) 2.5 g ″ ( 0 ) , ( Re x ) 1 / 2 N u x = − k thnf k f θ ′ ( 0 ) .

2 Novel resolution

The RK-IV technique and a shooting technique are used in this study to examine the ternary hybrid nanofluid’s flow patterns under various parameter configurations. Equations (12)–(14) and certain boundary conditions (15) are computationally solved in particular using the shooting method. Although the underlying issue is essentially a boundary value problem, the shooting technique is used to approach it as an initial value problem. In order to accomplish this, the fourth-order accurate RK method is used to successfully tackle the present challenge and produce extremely precise numerical solutions. In our study, we assumed that by taking into account a sufficiently large finite value, which we denoted as η → ∞ , a useful far field boundary condition could be established. The far field boundary condition was chosen to be this finite value, denoted by η ∞ . We were able to calculate the values of f ′ ′ ( η ) , g ′ ′ ( η ) , and θ ′ ( η ) using η ∞ . After careful consideration, we came to the conclusion that 20 was the right value to set for η ∞ in the circumstances of our particular case. It is important to note that although we kept the Prandtl number, abbreviated as Pr , at 2.36 throughout our research, we enabled other physical variables to change in order to conduct our analysis. This choice was made in order to thoroughly examine the system’s behavior and its reaction to changes in various parameters while keeping a constant Prandtl number as a benchmark.

3 New findings and results discussion

The analysis of the findings shown in the figures and table has yielded important revelations. The ternary hybrid nanofluid systems, which are composed of Cu / polymer , Fe 3 O 4 / polymer , and Cu − Fe 3 O 4 − Si O 2 / polymer , have been supported by this study. Volume fraction values ϕ for these systems, which range from 0.005 to 0.02 are something one should keep in the forefront. These results highlight the applicability and potential of these particular ternary hybrid nanofluid formulations, which combine nanoparticles and polymers. In essence, the study has shown that these nanofluid systems are promising and could be used successfully in a variety of real-world applications. Figures 2–12 visually illustrate the main findings regarding the velocity profiles in the x and y directions along with the patterns of temperature under various physical conditions. Figure 2(a)–(c) specifically examines the impact of volume fractions ϕ on velocity profile for nanofluid, hybrid nanofluid, and ternary hybrid nanofluid. It is clear that the velocity appearance in the x-direction reduces as the volume fractions rise. The maximum velocity appearance is seen at η = 0 and gradually decreases as it gets closer to infinity. The fact that the boundary layer thickness declines as the volume fraction of nanoparticles rises is an intriguing finding. This shows that the presence of nanoparticles significantly affects the boundary layer’s properties, with a thinner boundary layer being produced when nanoparticle volumes are higher. These results shed important light on how volume fractions affect the velocity profile and boundary layer thickness in the system under investigation. The effect of the Hartmann number M on the velocity appearance in the x -direction is shown in Figure 3a–c for nanofluid, hybrid nanofluid, and ternary hybrid nanofluid. The relationship of electromagnetic force to viscous force is represented by the Hartmann number. As the Hartmann number M rises in this context, it is seen that the velocity increases as well. The predominance of viscous forces over electromagnetic forces is responsible for this trend. The velocity profile in the x -direction is improved as the Hartmann number increases because electromagnetic forces have less of an impact than viscous forces do. The results basically show that the significance of electromagnetic and viscous forces in the structure is changed by changes in the Hartmann number, which has a direct impact on the velocity profile. The relationship within the flow appearance and the viscosity parameter α is shown in Figure 4. It is obvious that the flow velocity in the x -direction reduces as the numerical values of α rise. In this case, a model of viscosity that is dependent on temperature is taken into account, in which the viscosity is not constant but changes with the temperature outline. Further determined viscous effects are introduced when we improve the value of α , which in turn raises the temperature and the flow profile as a whole. As a result, Figure 4 shows that changes in the viscosity parameter, which directly affects flow velocity, have an obvious and significant effect on the dynamics of the flow in the system. The combined value of stretching velocities in the x and y directions, represented as λ , is shown in Figure 5. The graph clearly shows that the velocity appearance declines as the values of λ rise. This finding suggests that a reduced velocity profile is caused by a high stretching velocity ratio between the x and y directions. In other words, a drop in the overall velocity profile results from an elevated stretching velocity in the y -direction compared to the x -direction. Figure 6 investigates the effects of volume fractions ϕ , on the other hand. Here it is seen that as the volume fractions rise, the velocity appearance in the y -direction decreases. This shows that the velocity appearance along the y -direction decreases as the volume of tiny particles in the nanofluid increases. Another interesting finding is that the boundary layer thickness appears to be trending downward as the volume of tiny particles in the y -direction increases.

Figure 2

(a) Nanofluid: f ′ ( η ) in x -direction for different values of ϕ . (b) Hybrid nanofluid: f ′ ( η ) in x -direction for different values of ϕ . (c) Ternary hybrid nanofluid: f ′ ( η ) in x -direction for different values of ϕ .

Figure 3

(a) Nanofluid: f ′ ( η ) in x -direction for various values of M . (b) Hybrid nanofluid: f ′ ( η ) in x -direction for various values of M . (c) Ternary hybrid nanofluid: f ′ ( η ) in x -direction for various values of M .

Figure 4

f ′ ( η ) for different values of α in x direction.

Figure 5

f ′ ( η ) for different values of λ in x direction.

Figure 6

g ′ ( η ) in y direction for various values of ϕ .

Figure 7

g ′ ( η ) in y direction for different values of M .

Figure 8

(a) Nanofluid: g ′ ( η ) in y direction for different values of α . (b) Hybrid nanofluid: g ′ ( η ) in y direction for different values of α . (c) Ternary hybrid nanofluid: g ′ ( η ) in y direction for different values of α .

Figure 9

(a) Nanofluid: g ′ ( η ) in y direction for different values of λ . (b) Hybrid nanofluid: g ′ ( η ) in y direction for different values of λ . (c) Ternary hybrid nanofluid: g ′ ( η ) in y direction for different values of λ .

Figure 10

Temperature distribution θ ( η ) for different values of ϕ .

Figure 11

Temperature distribution θ ( η ) for different values of ϵ .

Figure 12

Temperature distribution θ ( η ) for various values of λ .

Figures 7 and 8(a)–(c) show graphical illustrations of the velocity profile g ′ ( η ) in the y -direction taking into account different variations in the Hartmann number M and viscosity parameter α . The results show that the y -direction velocity profile behaves similar to what we saw in the x -direction when we make modifications in both the Hartmann value and the viscosity parameter. This indicates that the y-direction velocity profile is affected by changes in the Hartmann number M and the viscosity parameter α . Changes in M consequently lead to modifications in the velocity profile along the y -axis. The uniformity of the aforementioned parameters’ impacts on the flow features in various directions is noteworthy in this case. The consistency of their influence on flow patterns in different directions is highlighted by the similarities in the responses of the x and y -direction velocity curves when subjected to alterations in the Hartmann number and viscosity parameter. We focus on a parameter called λ , which denotes the proportion of stretching velocities in the x and y directions, as we look at the velocity profile in the y -direction in Figure 9(a)–(c) for nanofluid, hybrid nanofluid, and ternary hybrid nanofluid. Velocity profile is seen to be increasing with the increasing values of involved parameters. This shows that the velocity profile in the y -direction also increases as the stretching velocity ratio between the x and y directions is increased. The velocity appearance in the y -direction, however, exhibits a different behavior, i.e., rises above 0.8 and goes on to infinity. In this case, a rise in the stretching velocity ratio results in a fall in the velocity appearance along the y -axis. In conclusion, increasing causes an enhanced velocity profile in the y -direction for values between 0 and 0.8, but velocity appearances decrease in the y -direction for values greater than 0.8.

The influence of various physical factors on the temperature variations represented by θ ( η ) is visually demonstrated in Figures 10–12. Figure 10 specifically investigates the impact of volume fractions ϕ on this temperature distribution. It becomes clear that the temperature distribution θ ( η ) rises proportionately to the volume fractions of tiny particles ϕ . In other words, as the volume fractions of tiny particles increase, so does the temperature distribution inside the nanofluid. It is noteworthy that the temperature distribution is highest at η = 0 and progressively reduces as it gets closer to infinity.

The effect of the thermal conductivity variable, represented as ϵ , on the temperature outlines θ ( η ) is the main subject of Figure 11. In this study, it is considered that temperature changes affect thermal conductivity. The findings show that as the value of ϵ rises, the temperature distribution also rises. In plainer terms, higher values are connected to wider temperature distribution. This implies that the temperature-dependent variation in thermal conductivity has a significant impact on the temperature distribution throughout the system. The relationship between the temperature outlines and the ratio of stretching velocities in the x and y directions, denoted by λ , is explored in Figure 12. According to the analysis, the temperature outlines θ ( η ) decreases as λ rises. In plainer terms, a decrease in the temperature distribution results from an increase in the stretching velocity ratio. This shows that variations in the stretching velocity ratio directly affect the temperature outlines, with lower temperature distributions being correlated with higher values of λ .

4 Verification of results with the existing literature

In terms of skin friction and Nusselt number for pure fluid, Tables 3 and 4 present a detailed comparison between the results of the current study and the prior literature. These tables give a thorough overview and analysis of the findings of the current work, enabling a direct comparison to be made with earlier studies in the field. The goal of these tables is to make it easier to understand how the current study and earlier research compare and contrast in terms of skin friction and Nusselt number.

The calculated values for skin friction in the x- and y-directions as well as the Nusselt number for a ternary hybrid nanofluid made up of a base fluid polymer are shown in Table 5. For a number of flow parameters, including ϕ , α , ϵ , and M , these solutions are given.

The skin-friction values represent the flow resistance that the fluid comes across in the corresponding x and y directions, whereas the Nusselt number illustrates the convective heat transfer in the system. Skin friction is found to increase as the volume fractions ϕ , viscosity parameter α , and Hartmann number M boost in both the x and y directions when a ternary hybrid nanofluid is put into consideration. In particular, increased skin friction in both directions is brought on by a higher Hartmann number. This is provided by the fact that the electromagnetic force begins to exceed the viscous force as the Hartmann number increases. As the electromagnetic force gets stronger, skin friction gets worse. Further, it is observed that enhancing the temperature-dependent viscosity variable α increases skin friction in both the x and y directions. It is believed that the higher values of increased fluid resistance are what caused the enhancement in skin friction. Since the fluid becomes more viscous as it rises, the skin friction coefficient rises as well. Skin friction in both directions increases as the temperature-dependent thermal conductivity parameter ϵ is increased. Higher values of ϵ denote a fluid’s greater ability to conduct heat, which increase the flow resistance and, consequently, increase the skin friction value. The Nusselt number shows a tendency to decrease as the volume fraction ϕ increases. This suggests that higher volume fractions cause the system to experience less convective heat transfer. The Nusselt number, on the other hand, falls as the Hartmann number M rises. This demonstrates that when a strong magnetic field exists and putting an overpowering effect over the viscous effects, convective heat transfer is hindered, resulting in a lower Nusselt number. Furthermore, the numerical values of the Nusselt number increases as temperature-dependent viscosity α and thermal conductivity ϵ parameters rise. This suggests that when α and ϵ have higher values, the system’s convective heat transfer increase. The comparison between nanofluid ( Cu / polymer ) , hybrid nanofluid ( Cu − Fe 3 O 4 / polymer ) , and ternary hybrid nanofluids ( Cu − Fe 3 O 4 − SiO 2 / polymer ) are shown in Table 6 for different parameters.

5 Conclusion

This study examines the effects of variable fluid viscosity and thermal conductivity on 3D flows of ternary hybrid nanofluids with a polymer-based fluid over a stretchable surface. The research has numerous engineering and industrial applications, especially in the printing and production of paper. Significant findings are produced by numerical solutions, which offer insights:

• Velocity profiles: The velocity profiles, denoted by f ′ ( η ) and g ′ ( η ) , of nanoparticles decrease in both the x and y directions with the increasing volume fractions ϕ and stretching velocities λ . While increase in velocities outlines with the increase in M .

• Temperature distribution: A rise in the temperature distribution θ ( η ) is caused by elevated volume fractions ϕ and the thermal conductivity parameter ϵ . While temperature outlines decrease with the increase in λ .

• Skin friction: As the volume fraction ϕ , viscosity parameter α , and Hartmann number M increase in a ternary hybrid nanofluid, skin friction increases in both the x and y directions. More significant values of these variables therefore correlate with higher fluid flow resistance and higher skin friction.

• Nusselt number: At higher nanoparticle concentrations, convective heat transfer is improved as the Nusselt number Nu grows alongside the rising fluid volume fractions ϕ .

• Impact of α and ϵ : The Nusselt number Nu increases with the increase in temperature-dependent viscosity α , and thermal conductivity ϵ parameters. Impact of α and ϵ shows more convective heat transfer.

6 Future study

In the present work, viscous fluid model of ternary hybrid nanofluid with one type of base fluid polymer with variable fluid properties was considered. In future, this work can be extended for other nanoparticles and different types of base fluids, e.g., water, C 2 H 6 O 2 , etc.