Statistical investigations and morphological aspects of cross-rheological material suspended in transportation of alumina, silica, titanium, and ethylene glycol via the Galerkin algorithm

Shuguang Li; Muhammad Sohail; Umar Nazir; El-Sayed M. Sherif; Ahmed M. Hassan

doi:10.1515/ntrev-2023-0169

Article Open Access

Statistical investigations and morphological aspects of cross-rheological material suspended in transportation of alumina, silica, titanium, and ethylene glycol via the Galerkin algorithm

Shuguang Li , Muhammad Sohail , Umar Nazir , El-Sayed M. Sherif and Ahmed M. Hassan

Published/Copyright: December 28, 2023

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From the journal Nanotechnology Reviews Volume 12 Issue 1

Abstract

Several industrial applications include the use of nanoparticles in base fluids to improve the thermal performance of numerous mechanisms. The current inspection covers the investigation of trihybrid nanoparticles mixed in the cross-fluid model part of a rotating stretched surface in the presence of a heat source/sink, radiation effect, and nonlinear/linear convection. A trihybrid fluid is a unique type of fluid that surpasses hybrid nanofluids, conventional fluids, and nanofluids. Moreover, acetone and engine oil interchange energy in comparison to other liquids. Trihybrid fluids are now widely used in facilities such as electrical chillers, biotechnology, power pumps, the automotive industry, and air cleaners. The flow presenting model equations are derived in a rotating frame to study the momentum and thermal transportation in a nanofluid. The modeled problem was simplified by considering the boundary layer principle, which resulted in the formation of complex coupled partial differential equations (PDEs). The derived PDEs were converted into the corresponding nonlinear ODEs using suitable similarity transformations. Then, the converted ordinary differential equations (ODEs) were solved numerically via a strong and authentic procedure called finite element procedure from the current inspection; it is recoded that finite element method is a powerful method to handle complex problems arising in modeling of several mechanisms.

Keywords: heat transfer; triadic hybrid nanoparticles; rotating flow; 3D model; quadratic and linear model; FEM; nanolayer thickness

1 Introduction

The improvement of the process of thermal enhancement and controlling energy losses in working fluids is an important mechanism. In recent times, analysis proposed that base fluids exhibit distinct properties after being reduced to small nanoparticles; therefore, the thermal proficiency of base fluids was enhanced with the inclusion of such nanoparticles. Choi (1995) was the first to examine the flow of nanofluids [1]. These nanometer-sized particles exhibit peculiarities among the physical and chemical demeanors. They demonstrate smoothness in their flow without clogging in the micro-channels. Nanofluids are highly diverse in terms of their physical applications in the heating and boiling processes of energy, electronic devices, thermal absorption, and nuclear reactions. Different approaches have been utilized to enhance thermal energy. The scattering of metallic nanoparticles has been studied as a new approach in recent development. Improvement in the thermal performance of the working fluids has been achieved. Hybrid nanomaterials are made up of several kinds of nanoparticles. The applications of hybrid nanomaterials include biomedicine, paper production, lubrication, nuclear production, electronic devices, solar devices, power systems, and welding processes. Several researchers have investigated hybrid nanoparticles. For example, Chu et al. [1] characterized four types of nanoparticles in a Newtonian fluid using thermal radiation across a vertical surface. They have adopted the Keller box approach for obtaining numerical solutions and found that the highest enhancement can be achieved using four kinds of nanoparticles. Considerable temperature differences among the surface and within the ambient fluid in numerous problems have been observed. This necessitates the fundamental consideration of temperature-reliant heat sources/sinks that may cause strong impact on the heat transport features. Physically, heat production/absorption study based on moving fluids is pivotal in numerous physical problems such as fluids under consideration of chemical reactions (endothermic or exothermic). Ramesh et al. [2] designed a flow model based on hybrid nanoparticles with a heat source toward a stretching surface involving particle deposition and thermophoresis. The heat transfer model of Williamson fluid in the presence of Lorentz force and heat source past a cylinder inserting nanoparticles has been investigated by Kavya et al. [3]. Swain et al. [4] studied characterizations of Newtonian fluid inserting suspension of MWCNT/Fe₃O₄ hybrid nanoparticles over a stretching surface under the implementation of a magnetic field and heat source. They have implemented a shooting approach to investigate temperature gradient and Sherwood number with different parameters. Khan et al. [5] discussed the influence of dust particles on the behavior of Casson liquid with hybrid nanofluid and Lorentz forces across a stretching frame. Fares et al. [6] estimated the performance of nanoparticles in entropy generation and heat transfer using the Darcy–Forchheimer theory in a square enclosure. Flow description as well as heat transmission toward the boundary layer bounded by stretched surface has numerous pivotal applications specifically in manufacturing procedures mainly in industry such as infinite metallic plate cooling through the cooling bath, cooling and also drying of paper, also textile and glass filer manufacturing, and many others. Xiu et al. [7] designed a model based on the motion of ethylene glycol in tri-hybrid nanoparticles with Reiner Philippoff martial and Lorentz forces past a stretching surface via finite element approach (FEA). They have estimated the highest enhancement in thermal conductivity for tri-hybrid nanoparticles rather than hybrid nanoparticles. Nazir et al. [8] developed influences of ternary hybrid nanomaterials in Sisko fluid in mass diffusion and thermal fields past the vertical plate utilizing an FEA. Sohail et al. [9] performed investigations of remarkable enhancement in thermal transport inserting mechanism of tri-hybrid nanoparticles and non-Fourier’s theory. Abbasi et al. [10] studied thermal mechanisms based on hybrid nanoparticles and blood flow in a wavy curved channel using slip conditions. Abbasi et al. [10] reviewed the influences of thermal performance via hybrid nanoparticles in fractional bioconvection flow within the suspension of a nanofluid filled in the channel. Some studies based on hybrid nanoparticles have been referred to in the literature [11–13].

Rotating flows have been visualized as a basic framework in recent developments and fluid dynamics. The pioneering work in this way was investigated by Kármán [14]. Sohail et al. [15] provided the influence of rotational flow in Prandtl fluid considering Hall currents and ion slip forces over-stretching cone implementing finite element methods. Javaid et al. [16] modeled rotating flow in Burgers’ fluid in terms of fractional utilizing unsteady flow within the power law kernel. Ali et al. [17] used an FEA to drive numerical consequences in magnetohydrodynamics (MHD) flow tangent hyperbolic and Maxwell fluid over a 3D frame via the Cattaneo Christov model. Hussain et al. [18] estimated the numerical results of rotational flow in MHD flow using hybrid nanoparticles.

Numerous liquids have properties and nature related to radiating thermal energy because of electromagnetic waves. Such kinds of fluids are termed radiative fluids. Stefan–Boltzmann theory is used in radiative fluids. This theory has been used in radiative modeling during the process of heat transfer. Thermal radiations are considered substantial in numerous mechanisms specifically in thermal engineering and also in material industries. With the rise in temperature, radiative heat transmission demonstrates interesting behavior. Practically, its applications include design of distinct devices for aircrafts and numerous space vehicles. Works related to thermal radiation in flow problems are discussed here. For example, Khan et al. [19] performed modeling of Williamson fluid considering thermal radiation and Lorentz forces. They have also used nanoparticles (gold) in base fluid on the surface (curved). Lund et al. [20] drove the consequences of thermal radiations and viscous dissipation with various nanoparticles via a single‐phase approach. Nazeer et al. [21] discussed the consequences of thermal radiation in heat transfer using Lorentz force and porous media. Farooq et al. [22] studied the impacts of thermal radiations in nanofluid caused by Lorentz force, including entropy generation. Sreedevi and Reddy [23] performed a model based on the thermal radiation inclusion of nanoparticles in a square cavity. Further investigations regarding thermal radiations are mentioned in previous studies [24–29] and references therein. Figure 1 predicts effective approach of nanofluid on thermal conductivity.

Figure 1

Affective mechanism of nanofluid on thermal conductivity.

According to the literature, there is no study on the suspension of triadic hybrid nanomaterials of cross nanomaterial due to quadratic and linear thermal convections past rotating 3D surfaces. The mechanism of heat energy has been carried out by heat sink and thermal radiation. Five sections are prepared for this development. The first, second, third, four, and five teams are associated with modeling, literature review, numerical procedure, discussion of results and conclusions, etc.

2 Mathematical procedure and modeling

The three-dimensional flow of a cross-hybrid nanofluid is considered across a 3D surface in the presence of quadratic and linear convection mechanisms. Body forces, Hall current, and induced magnetic field have been ignored using the concept of low Reynolds number. Single phase mechanism was performed by adding various kinds of nanoparticles. Furthermore, base fluid based on engine oil is considered for the development of a single phase of hybrid nanomaterial. The use of quadratic form of thermal radiation is considered along with heat source/absorption. A triadic kind of nanomaterial is used in the presence of metallic particles of copper. A 3D rotational flow in terms of physical boundary conditions is carried out in Figure 2.

Figure 2

Physical diagram and coordinate system.

Constitutive expressions are defined as:

(1) U x + U y + U z = 0 ,

(2) U U x + V U y + W U z − 2 ω V = ν Th U zz [ 1 + Γ m ( U z ) m ] − 1 − m ν Th [ 1 + Γ m ( U z ) m ] − 2 + [ G ( ρ B 0 ) Th ( T − T ∞ ) + G ( ρ B 1 ) Th ( T − T ∞ ) 2 ] ,

(3) U V x + V V y + W V z + 2 ω U = ν Th V zz [ 1 + Γ m ( V z ) m ] − 1 − m ν Th [ 1 + Γ m ( V z ) m ] − 2 Γ m ( V z ) m + A 1 A 3 G ( ρ B 0 ) Th ( T − T ∞ ) + A 2 A 3 G ( ρ B 1 ) Th ( T − T ∞ ) 2 ,

(4) ( ρ C P ) Th ( U T x + V T y + W T z ) = K Th T zz + Q 0 ( T − T ∞ ) − σ * 32 ( T ∞ ) 3 3 k * T zz + 24 σ * ( T ∞ ) 3 3 k * ( T 2 ) zz .

Boundary conditions for 3D flow are as follows:

(5) U = ax , V = 0 , w = 0 : z = 0 , T → T ∞ , V → 0 , U → 0 : z → ∞ .

Transformation variables are defined as:

(6) Θ = T − T ∞ T w − T ∞ , U = ax F ′ , V = axG , W = − ( a ν Th ) 1 2 F , η = z a ν Th 1 2 .

Dimensionless models in terms of ordinary differential equations (ODEs) along with boundary conditions are derived using transformation variables which are defined as:

(7) ( 1 + ( 1 − m ) We m ( F '' ) m ) ( 1 + We m ( F '' ) m ) − 2 F ''' + ν f ν Thb F F '' − ν f ν Thb F ′ F ′ + ν f ν Thb 2 Ω G + A 1 A 3 Γ Θ + A 2 A 3 Q Θ 2 = 0 ,

(8) ( 1 + ( 1 − m ) We m ( G ′ ) m ) ( 1 + We m ( G ′ ) m ) − 2 G '' + ν f ν Thb F G ′ − ν f ν Thb F ′ G + ν f ν Thb 2 Ω G + A 1 A 3 Γ Θ + A 2 A 3 Q Θ 2 = 0 ,

(9) 1 − 8 R 3 Θ '' + 2 Q Pr ( Θ w − 1 ) [ 1 + ( Θ w − 1 ) Θ '' ] '' + A 4 ( Pr F Θ ′ + Pr G Θ ′ ) = 0 ,

(10) F ( 0 ) = 0 , Θ ( 0 ) = 1 ; F ′ ( 0 ) = 1 , G ( 0 ) = 0 , G ′ ( 0 ) = 0 , F ( ∞ ) = 0 , G ( ∞ ) = 0 ; Θ ( ∞ ) = 0 .

Correlations of triadic hybrid nanomaterial for thermal properties are defined as follows and their values are listed in Table 1.

(11) μ Th = μ Bf ( 1 + 0.1008 { ( Φ 1 ) 0.69574 ( DP 1 ) 0.44708 + ( Φ 2 ) 0.69574 ( DP 1 ) 0.44708 } ) ,

(12) A 1 = ( 1 − Φ 1 − Φ 2 ) + Φ 1 ( ρ β 0 ) s 1 ( ρ β 0 ) bf + Φ 2 ( ρ β 0 ) s 2 ( ρ β 0 ) bf , A 2 = ( 1 − Φ 1 − Φ 2 ) + Φ 1 ( ρ β 1 ) s 1 ( ρ β 1 ) bf + Φ 2 ( ρ β 1 ) s 2 ( ρ β 1 ) bf ,

(13) A 3 = ( 1 − Φ 1 − Φ 2 ) + Φ 1 ( ρ ) s 1 ( ρ ) bf + Φ 2 ( ρ ) s 2 ( ρ ) bf , A 4 = ( 1 − Φ 1 − Φ 2 ) + Φ 1 ( ρ C p ) s 1 ( ρ C p ) bf + Φ 2 ( ρ C p ) s 2 ( ρ C p ) bf ,

(16) ( ρ C p ) Th = Φ 1 ( ρ C p ) s 1 + ( 1 − Φ 1 − Φ 2 ) ( ρ C p ) bf , ρ Th = Φ 1 ρ s 1 + Φ 2 ρ s 2 + ( 1 − Φ 1 − Φ 2 ) ρ bf ,

(17) ( k ) Th = k s 2 + ( N − 1 ) k mbf + Φ 2 ( k mbf − k s 2 ) k s 2 + ( N − 1 ) k mbf + ( N − 1 ) Φ 2 ( k mbf − k s 2 ) k bf , k bf = k s 1 + ( N − 1 ) k bf + Φ 1 ( k bf − k s 1 ) k s 1 + ( N − 1 ) k bf + ( N − 1 ) Φ 1 ( k bf − k s 1 ) k f .

Table 1

Thermal properties of ρ , k , and c p

Base fluids	Thermal conductivity ( k )	Specific heat capacity ( c p )	Density ( ρ )
Cu	401	385	8933
Ti O 2	8.9538	686.2	4250
Fe 3 O 4	9.7	670	5180
Al 2 O 3	40	765	3970
Engine oil	0.1404	2048	863

Wall shear stresses and thermal transfer rate are defined as:

(18) Cf = τ zx | z = 0 ρ TH ( u w ) 2 , τ w = μ Thb ∂ U ∂ z 1 + Γ ∂ U ∂ z m , Cg = τ zy | z = 0 ρ TH ( u w ) 2 , τ w = μ Thb ∂ V ∂ Z 1 + Γ ∂ V ∂ z m ,

(19) Re 1 / 2 Cf = 1 + 0.1008 { ( Φ 1 ) 0.69574 ( DP 1 ) 0.44708 + ( Φ 2 ) 0.69574 ( DP 1 ) 0.44708 } Φ 1 ρ s 1 ρ bf + Φ 2 ρ s 2 ρ bf + ( 1 − Φ 1 − Φ 2 ) F '' ( 0 ) 1 + We m ( F '' ( 0 ) ) m ,

(20) Re 1 / 2 Cg = 1 + 0.1008 { ( Φ 1 ) 0.69574 ( DP 1 ) 0.44708 + ( Φ 2 ) 0.69574 ( DP 1 ) 0.44708 } Φ 1 ρ s 1 ρ bf + Φ 2 ρ s 2 ρ bf + ( 1 − Φ 1 − Φ 2 ) G ′ ( 0 ) 1 + We m ( G ′ ( 0 ) ) m ,

(21) Nu = − zK TH ∂ T ∂ z z = 0 ( T w − T ∞ ) K f , Nu Re − 1 / 2 = − K TH K f θ ′ ( 0 ) .

3 Numerical results

Numerical results of formulated ODEs are simulated by numerical approach finite element method (FEM). Finite element methodology is discussed below.

Step-I: The required problem domain is discretized into the specified number of elements in step-I. The idea of weighted residual was used to create the weak form. Shape functions based on linear-type polynomials are derived.

Unknowns ( N , F , Θ , ϕ ) are defined as:

(22) N = ∑ j = 1 2 ( N j ψ j ) , F = ∑ j = 1 2 ( F j ψ j ) , Θ = ∑ j = 1 2 ( Θ j ψ j ) ,

(23) ϕ = ∑ j = 1 2 ( ϕ j ψ j ) , ψ j = ( − 1 ) j − 1 η − η j − 1 η j − η j − 1 .

Step-II: Each component’s stiffness elements are computed in relation to the broken problem domain. A global stiffness matrix is also obtained. To create a linear system from a nonlinear system, Picard’s method is used.

The definition of the residual view is

(24) [ R ] = [ M ( F ( r − 1 ) , N ( r − 1 ) , Θ ( r − 1 ) , ϕ ( r − 1 ) ) ] F r Θ r ϕ r = [ F ] .

(25) ∑ i = 1 N ( | ω r − ω r − 1 | ) 2 1 2 ∑ i = 1 N | ω r | 2 < 10 − 8 .

Step-III: This step involves solving the system of linear equations

(26) M ( F , N , Θ , ϕ ) F Θ ϕ = [ F ] .

Step-IV: Code for FEM is developed using Maple 18. The computation domain is set to [0, 8], and Table 2 contains the results of the grid size analysis.

Table 2

Grid size analysis of F' ( η ) , G ( η ) , and Θ ( η ) at mid of each element

e	F ′ η Max 2	G η Max 2	Θ η Max 2
30	0.6461024647	0.03305768338	0.5433185144
60	0.6057350926	0.03015930018	0.5266440113
90	0.5926481375	0.02925851687	0.5210803476
120	0.5861752284	0.02881976657	0.5182984229
150	0.5823165277	0.02856045358	0.5166277343
180	0.5797550268	0.02838907952	0.5155146785
210	0.5779326147	0.02826761636	0.5147195319
240	0.5765679979	0.02817693262	0.5141214095
270	0.5755132473	0.02810681937	0.5136580884
300	0.5746710118	0.02805100694	0.5132876380

4 Results and discussion

An implementation of triadic nanomaterials is utilized to find an enhancement of thermal distribution in cross nanomaterial involving quadratic convection and linear convection on a 3D rotating flow of surface. The motion of ethylene glycol (EG) into metallic nanomaterials of copper, aluminum, Fe 3 O 4 , and titanium. Variable thermal radiation and heat sink/source are shown in the energy equation. Compression investigations among alumina, silica, and titanium in EG and silica and titanium in EG on the thermal field and velocity field are studied against different physical parameters. The detailed consequences are mentioned below.

4.1 Velocity fields

In this section, the mechanism of velocity fields is measured versus We , Q , and n , as shown in Figure 3a–c. It is essential to mention that a comparison study between alumina, silica/EG and alumina, silica, and titanium/EG is observed on the motion of EG shown in Figure 3a–c. The motion of alumina, silica, and titanium/EG is shown by point dot curves, and the motion of silica and titanium/EG is captured as dot curves. Figure 3a shows the movement in y- and x-directions of alumina–silica–titanium/EG and silica–titanium/EG with a change in We . It is shown that the motion of alumina–silica–titanium/EG and silica–titanium/EG decreased with increase in We . We and relaxation time are directly proportional to each other. For upsurge values of We , there is increase in relaxation time. On the basis of this fact, motion of fluid particles slowdowns. Physically, the concept of We is captured using the ratio between elastic force and viscous force, while viscous force is directly proportional to We . Therefore, the motion of alumina–silica–titanium/EG and silica–titanium/EG decreased. Moreover, the movement of silica–titanium/EG is less than the motion of alumina–silica–titanium/EG Motion of alumina–silica–titanium/EG and that of silica–titanium/EG in y- and x-directions with different values of n are shown in Figure 3b. Similar behavior was observed in the motion of alumina–silica–titanium/EG and silica–titanium/EG. It was visualized that n is termed a dimensionless number and power law index number. Figure 3c shows the estimation of Q on the motion of alumina–silica–titanium/EG and silica–titanium/EG in both directions (horizontal and vertical). The occurrence of Q is estimated utilizing the concept of quadratic convection and linear convection in the momentum equation. The highest motion of Q on the movement of alumina–silica–titanium/EG and silica–titanium/EG in both directions (horizontal and vertical) is achieved with different values of Q . Moreover, the motion of alumina–silica–titanium/EG is higher than the motion of silica–titanium/EG in both directions (horizontal and vertical).

Figure 3

(a) Graphical simulation of We on velocity field in y- and x-directions, (b) graphical simulation of n on velocity field in y- and x-directions, and (c) graphical simulation of Q on velocity field in y- and x-directions.

4.2 Thermal fields

Figure 4a–d is plotted to determine the temperature of alumina–silica–titanium/EG and silica–titanium/EG with different values of R , Q , n , and Γ . The heat energy of alumina, silica, and titanium/EG is shown by point dot curves, and the motion of silica and titanium/EG is shown by dot curves. Figure 4a shows graphical simulations of Q on thermal field inserting suspension of alumina–silica–titanium/EG and silica–titanium/EG. Here, Q (mix convection number) is the dimensionless parameter which is implemented to characterize thermal distribution among alumina–silica–titanium/EG and silica–titanium/EG. Mathematically, mix convection number ( Q ) has direct proportional relation versus the thermal field. Generally, mixed convection is defined as forced as well as free convection combination when flow is examined by both outer (i.e. outer energy given to the fluid-streamlined shaped body) system and also inner volumetric forces through the fluid medium density distribution across a gravity field. Figure 4b demonstrates graphical visualizations of R on the thermal field involving aspects of alumina–silica–titanium/EG and silica–titanium/EG. Thermal field declines versus higher values of radiation parameter ( R ). In a physics point of view, heat transfers through radiations from the surface. Also, R occurs in the denominator in the energy equation. Hence, an inverse proportional relation exists versus the thermal field. The thermal field for alumina–silica–titanium/EG is higher than the thermal field for silica–titanium/EG. Thermal production for R = 0 is more higher than thermal production for R ≠ 0 . Graphical simulations of Γ on the thermal area are shown in Figure 4c, involving alumina–silica–titanium/EG and silica–titanium/EG. It was investigated that θ w is produced. The effect of power law index number ( n ) on the heat energy profile is shown in Figure 4d. Temperature decreases with increasing values of n . The temperature of alumina–silica–titanium/EG has been increased than that of silica–titanium/EG. Figure 5a and b reveals an interpolation of We and n on skin friction coefficients. In this figure, divergent velocity increases versus an enhancement of power law index number and Weissenberg number. Effects of Q and R on Nusselt number are shown in Figure 5c and d. Nusselt number decreases when Q and R are increased. Furthermore, Nusselt number for tri-hybrid nanoparticles is higher than that of hybrid nanofluid. Figure 5e shows change in thermal conductivity against hybrid nanofluid using various values of volume fraction. It was shown that thermal conductivity increases with an increase in volume fraction. Figure 5f shows comparative study of nanofluid, tri-hybrid nano-structures, and hybrid nanofluid. It is included that maximum heat transfercan be achieved for the case of tri-hybrid nanoparticles.

Figure 4

(a) Graphical simulation of Q on the thermal field, (b) graphical simulation of R on the thermal field, (c) graphical simulation of Θ _w on the thermal field, and (d) graphical simulation of n on the thermal field.

$Figure 5 (a and b) Interpolation of We {\rm{We}} and n n on skin friction coefficients, (c and d) interpolation of Q Q and R R on Nusselt number, (e) numerical consequences of thermal conductivity on nanolayer thickness, and (f) an effect regarding volume fraction on heat transfer among tri-hybrid nanofluid, nanofluid, and hybrid nano-structures.$

Figure 5

(a and b) Interpolation of We and n on skin friction coefficients, (c and d) interpolation of Q and R on Nusselt number, (e) numerical consequences of thermal conductivity on nanolayer thickness, and (f) an effect regarding volume fraction on heat transfer among tri-hybrid nanofluid, nanofluid, and hybrid nano-structures.

4.3 Effects of different parameters on thermal rate and velocity gradients

Table 3 shows the effects of the diameter of nanoparticles and volume fractions on skin friction coefficients (divergent velocities). In Table 3 velocity gradients have been magnified by higher values of DPI , DP 2 , Φ 1 , and Φ 2 . Moreover, velocity gradients for the case of alumina–silica–titanium/EG are greater than velocity gradients for silica–titanium/EG and comparative numerical consequences among silica–titanium/EG and alumina–silica–titanium/EG for thermal rates versus applications of R , Q , Q h , and θ w are recorded in Table 4. Thermal rates decrease when Q h and θ w are increased. But thermal rates increase with change in Q h and thermal radiation number. It was essentially mentioned that thermal rates for silica–titanium/EG are less than thermal rates for alumina–silica–titanium/EG.

Table 3

Effect of different parameters on velocity gradients (skin friction coefficients) of alumina–silica–titanium/EG and silica–titanium/EG

		Alumina–silica–titanium/EG		Silica–titanium/EG
		‒ Re 1 / 2 Cf	‒ Re 1 / 2 Cg	‒ Re 1 / 2 Cf	‒ Re 1 / 2 Cg
	1	1.3927878934	1.07502772684	0.568040141	0.8865885774
DP 1	2	1.3992110929	1.07628476892	0.563466986	0.8874078565
	3	1.4025381184	1.07770593486	0.562737209	0.8863789140
	2	1.4072263603	1.01296811937	0.562390587	0.8663651209
DP 2	3	1.4147819383	1.02092572963	0.571924961	0.8763466397
	4	1.4519022812	1.04993653246	0.585333654	0.8994269344
	0.01	1.3859253520	1.05087978298	0.2870768205	0.5597429643
Φ 1	0.02	1.3918242016	1.06500346640	0.3042953376	0.5872700205
	0.04	1.3934196337	1.07683411524	0.3397825848	0.6349731369
	0.01	1.4030575588	1.05812068567	0.3495972204	0.6461830680
Φ 2	0.02	1.4124931969	1.05936224974	0.3837294993	0.6796012272
	0.04	1.4217466666	1.06056285508	0.4130286822	0.7027848453

Table 4

Effect of different parameters on Nusselt number of alumina–silica–titanium/EG and silica–titanium/EG

		Alumina–silica–titanium/EG	Silica–titanium/EG
		Nu Re ‒ 1 / 2	Nu Re ‒ 1 / 2
	0.0	2.1263986039	0.1266006793
Q	0.4	2.1271044645	0.1366037450
	0.8	2.1294076759	0.1466070725
	0.0	2.1263984247	0.1173045923
R	2.0	2.1364062400	0.1284409648
	4.0	2.1365072598	0.2999880625
	‒1.5	2.1215771587	0.59190450599
Q h	0.3	2.1078877602	0.44785174801
	1.5	2.0993272461	0.31524958032
	0.0	2.11006836122	0.08093700132
θ w	0.7	2.08531484507	0.07201161121
	1.8	2.06017384823	0.04295706013

5 Key findings

The mechanism of triadic nanoparticles in cross nanomaterial is implemented on 3D rotating sheet involving quadratic convection and linear convection in ethylene glycol. Heat sinks/sources involving variable thermal radiation have been investigated. A complex model is prepared and numerically handled using an FEA. Key findings are mentioned as follows.

The motion of nanoparticles decreases with an increase in We and n , but motion increases with an increase in mix convection number;
Thermal radiation and mixed convection parameters play a remarkable role in adjusting the width of thermal layers’ thickness;
Motion and thermal energy for alumina–silica–titanium/EG are higher than those for silica–titanium/EG;
A remarkable enhancement in thermal production has been studied rather than thermal production for silica–titanium/EG;
The highest thermal rates have been achieved for alumina–silica–titanium/EG as compared to thermal rates for silica–titanium/EG.

Funding information: This research was funded by Researchers Supporting Project Number (RSP2023R33), King Saud University, Riyadh, Saudi Arabia.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflicts of interest: The authors state no conflict of interest.
Data availability statement: The datasets used and/or analyzed during the current study available from the corresponding author on reasonable request.

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Received: 2023-07-28

Revised: 2023-10-25

Accepted: 2023-11-19

Published Online: 2023-12-28

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

Articles in the same Issue

https://doi.org/10.1515/ntrev-2023-0169

Keywords for this article

heat transfer; triadic hybrid nanoparticles; rotating flow; 3D model; quadratic and linear model; FEM; nanolayer thickness

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