Advanced ANN computational procedure for thermal transport prediction in polymer-based ternary radiative Carreau nanofluid with extreme shear rates over bullet surface

Nomenclature

Physical parameter/Symbols

Representation

A 1

first Rivlin Erickson tensor

B 0

magnetic field strength

Cf

skin friction

Ec

Eckert number

I

identity tensor

k

thermal conductivity

k ⁎

mean absorption coefficient

L ⁎

velocity slip

M

magnetic parameter

N ⁎

velocity slip parameter

Nu

Nusselt number

n

Carreau index

P

pressure

Pr

Prandtl number

Q

heat source parameter

Q ⁎

uniform heat sink/source

R d

radiation parameter

Re

Reynolds number

T ∞ & T w

ambient and surface temperature

T

temperature

Γ

time relaxation constant

υ bf

kinematic viscosity

We

Weissenberg number

x & r

space coordinates

τ

Cauchy stress tensor

γ

shear rate

ε ⁎

infinite shear rate

ρ

density

ω

angle orientation

σ ⁎

Stefan-Boltzmann constant

δ

volume friction

ψ

stream function

η

dummy variable

ϑ

location parameter

μ

viscosity

σ

electrical conductivity

ρ c p

volumetric heat capacity

ε

stretching ratio parameter

1 Introduction

Infinite shear rate viscosity-based model of Carreau fluid shows that the behavior of all fluids lies in the category of non-Newtonian fluids, and that they experience extreme shear rate conditions. Carreau fluid model exhibits variable viscosity that changes with the rate of shear, and it is used to describe flow mechanisms of materials such as certain industrial liquids, biological fluids, and polymers. Infinite shear rate viscosity defines the limiting viscosity that the fluid approaches as the shear rate becomes very high. This characteristic is crucial for predicting the fluid behavior in processes such as in high-speed flows, extrusion, and mixing operations. By accounting for the infinite shear rate viscosity, engineers and scientists can better understand and predict how Carreau fluids will perform under different conditions. Ayub et al. [1] investigated the role of infinite shear rate viscosity for heat transport properties of magnetized Carreau nanofluid. This study highlighted the importance of considering both magnetic effects and viscosity variations in magnetized nanofluid. Their research found that the intensity of magnetic field and orientation affect the thermal characteristics of fluid. Khan and Sardar [2] focused on the steady fluid flow with Carreau fluid mathematical model and the flow configuration is induced through wedge. This study incorporates the infinite shear rate viscosity into models to accurately predict the fluid dynamics in such situation. Sardar and Khan [3] explored the impact of infinite shear rate viscosity in mixed convective thermal transport mechanisms in Carreau nanofluids. The study provided a comprehensive analysis of thermal transport dynamics in Carreau nanofluid by the aid of extreme shear rates, these interactions are crucial for understanding energy transport processes in advanced thermal management systems. Hassan et al. [4] made a computational examination of heat and mass transport in boundary layer regions within non-Newtonian fluid by employing Carreau fluid viscosity model. The research insights into the effectiveness of low and high shear (zero and infinite) shear rate viscosities on the heat and mass transportation in non-Newtonian Carreau fluid. Their study highlighted how the inclusion of these effects alters the fluid viscosity profile, especially at high shear rates. Ayub et al. [5] studied the heat transfer in magnetohydrodynamics (MHD) Carreau hybrid nanofluid in a peristaltic channel flow in the presence of dual chemical phenomenon such as homogeneous/heterogeneous reactions. The study established a physical link between the effects of inclined magnetic field and chemical activity by using the Carreau hybrid nanofluid flow through peristaltic channels with keeping the target focus on shear rate viscosity. Furthermore, Khan and Sardar [6] studied the impact of infinite shear rate viscosity in a two dimensional steady flow of Carreau fluid. They focused on the implementation of Carreau model with extreme shear rates to explore the intricate nature of complex fluid flow instead of using traditional fluid model, which could underestimate the role of viscosity for complex thermal and flow characteristics of non-Newtonian Carreau nanofluid. Moreover, the study of Carreau nanofluids with implementation of shear rates together with other additional facts were examined in previous studies [7,8].

Thermal prediction of hybridized nanofluid focuses on the heat transfer behavior of fluids with multiple types of nanoparticles. Tri-hybridized nanofluid usually combines three distinct nanoparticles within a base fluid possessing superior thermal properties compared to traditional single component nanofluids. One of the primary applications of hybridized nanofluids is advanced cooling systems, such as those used in microelectronics. Kanti et al. [9] made detailed exploration about the thermal conductivity of fly ash nanofluids with fly ash-Cu nanoparticles. This study shows that the addition of copper nanoparticles to fly ash nanofluids enhances thermal conductivity. Alizadeh et al. [10] implemented a unique approach by using machine learning to predict the thermal aspect of a hybrid nanofluid flow within porous media. This study has key findings, and they are relevant for applications in energy systems. Adun et al. [11] published the research work, which is based on the development of a neural network and prediction of thermal conductivity of hybrid nanofluids. This study revealed that hybrid nanofluids exhibit superior thermal conductivity due to the synergistic effects of different nanoparticles. Furthermore, Çolak et al. [12] exhibited the new aspect to predict the heat transport analysis keeping focus on specific heat in water-based hybrid nanofluid using an artificial neural network (ANN). This study has proposed a new correlation for predicting specific heat hybrid nanofluid behavior in thermal systems. Moreover, previous studies [13,14,15,16] explored the heat and mass transport characteristics of hybrid nanofluid by using different geometrical surfaces, which are very crucial in advanced thermal management systems.

Viscous dissipation is a broad mechanism in which heat is generated due to shear forces between adjacent fluid layers. In this process, mechanical work of fluid is transformed into heat due to shear forces. In other words, the viscous dissipation is a process in which conversion of mechanical energy into thermal energy takes place. Hussain et al. [17] investigated the effect of viscous dissipation on the flow of radiative 3D Carreau nanofluids with incorporating activation energy. This study focused on gyrotactic microorganisms and their interaction with viscous dissipation within the nanofluid flow. Li et al. [18] explored the combined effects of buoyancy and viscous dissipation on 3D MDH hybrid nanofluids with MgO−TiO₂ under slip conditions. This research explored the buoyancy effects coupled with viscous dissipation and heat transfer rates. Masthanaiah et al. [19] examined the impact of viscous dissipation and entropy generation in a cold liquid flowing through a channel with a porous medium. This study revealed that viscous dissipation contributes to increased entropy generation and the presence of a porous medium leads to higher energy losses and reduced system performance. Furthermore, Dharmaiah et al. [20] investigated the impact of viscous dissipation along with Joule dissipation and activation energy on heat and mass transfer in fluid systems. The research highlighted that viscous dissipation significantly affects the temperature distribution and substantial heat generation. Some of the latest published research are presented in previous literature [21,22,23].

The influence of a magnetic field on flow and heat transfer is of great importance in applications involving electrically conducting fluids. By applying a magnetic field, the flow of conducting fluids enable precise manipulation of velocity and heat transfer rates. In cooling systems for nuclear reactors or electronic devices, magnetic fields help suppress fluid instabilities and optimize heat dissipation. Furthermore, the magnetic field also plays a critical role in astrophysical and geophysical studies. Many research works focused on the heat transport mechanism under the influence of MHD. For instance, the study conducted by Rudraswamy et al. [24] focalized the importance of thermal transport in three dimensional Carreau fluid flow under the impact of MHD consequences. They examined the Carreau flow with effects of nonuniform heat sink/source together with buoyancy opposing/assisting flow over a bi-directionally moving surface. Sharma et al. [25] made a comprehensive and detailed analysis of ternary hybrid cross bio-nanofluid by integrating inclined magnetic field consequences. The study focused on the non-Newtonian flow of ternary hybrid bio-nanofluids through a bifurcated artery. They assumed Au-CuO-GO-based blood hybrid nanofluid along with gyrotactic microorganisms to explore the entropy generation. The computational data obtained by their study provide valuable aid in many bloodstream detection in medical interventions. Gopi and Golla [26] examined the effects of an inclined plate on a two-dimensional hybrid Cu-Fe₃O₄/EG nanofluid flow subject to an oblique magnetic flux in a magnetized environment. Furthermore, Vinutha et al. [27] delved into the computational examination of ternary nanofluid flow between parallel plates by integration of quadratic radiation and activation energy together with the influence of MHD consequences. They used response surface methodology and looked into the sensitivity measurement of MHD ternary nanofluid. Moreover, many other eye-catching facts and findings of different categories of fluids in the presence of radiative, chemically reactive, and MHD consequences were published in previous literature [28,29,30].

The study of flow and heat transfer over bullet-shaped geometry holds significant importance due to its unique aerodynamic profile. The bullet shape promotes smooth fluid flow with reduced turbulence and flow separation. In aerospace engineering, this geometry is critical in the design of high-speed vehicles, missiles, and spacecrafts. Furthermore, this geometry finds application in biomedical devices, such as stents and microfluidic systems. Overall, the contribution of bullet-shaped geometry to improve heat transfer and ensure operational reliability under flow conditions underscores its importance across a wide range of technological and industrial applications. Oke et al. [31] examined the key role of MHD consequences in a heat and mass transfer in a flow of Casson fluid with suspended single wall carbon nanotubes (SWCNT) and graphene nanoparticles. The numerical investigation of boundary layer flow (BLF) of non-Newtonian fluid over an exponentially stretching bullet-shaped surface was made by Ali and Alim [32]. They focused on the numerical assessment of the thermal transport process subjected to suction and injection over an exponentially stretching bullet-shaped object. Furthermore, Ali and Alim [33] studied the effectiveness of nonlinear stretching of the bullet-shaped object on the mixed convective heat transport mechanism in non-Newtonian BLF. In addition, they examined the impact of diverse facts such as shape factors and viscous dissipation as well as internal heat generation in a mixed convection heat transfer process. Moreover, previous studies [34,35,36] throw light on the key role of bullet-shaped geometry for heat transfer by means of different fluid flow configuration, which are crucial in many industrial and engineering applications.

ANNs play a transformative role in the study of flow and heat transfer in fluids, and it offers tools for predicting thermal and fluid dynamic behavior with high accuracy. ANNs have the ability to learn from simulated data and it captures the nonlinear relationships and intricate patterns in fluid flow and heat transfer phenomena. ANNs reduce computational time by providing instant predictions after training, enabling real-time monitoring and control in industrial applications. Recently, many scholars are focusing on the integration of some novel ANN architectures and computational frameworks for computational fluid dynamics analysis, specifically heat and mass transfer simulations in many heat transfer fluids. For instance, the novel investigation of thermal transport predictions of three-dimensional cross-ternary hybrid nanofluid was made by Shah et al. [37]. They implemented a Bayesian regularization neural network approach for the thermal transport analysis of a ternary flow over a wedge geometry. The result of their study presents good predictions of thermal performance, which are crucial in many thermal transport processes. Ramesh et al. [38] built a neural network algorithm for computational analysis of heat transport mechanism in the non-Newtonian flow of ternary nanofluids. They delved into the detailed and comprehensive assessment of heat transfer mechanism of ternary hybrid nanofluid by incorporating chemical reaction and Arrhenius kinetics in ternary flow. Ayub et al. [39] discussed the effectiveness of magnetic field orientation in quadratic-convection-based thermal transport mechanism of non-Newtonian fluid. The study involves the enhanced thermal transport process of ternary-radiative-blood-based nanofluids by integration of novel ANN framework for computational procedures, i.e., intelligent neural networks based on two hidden layers’ mechanism. Furthermore, some novel applications of ANN for thermal transport prediction in non-Newtonian fluid is presented by Sheikholeslami et al. [40]. They delved into the heat and mass transfer estimation in a flow of Al₂O₃-H₂O-based nanofluid through a channel. The results of their assessment are useful in the management of heat-based systems. Moreover, previous literature [41,42,43,44] are useful in the heat and mass transfer simulation and provide valuable information for advanced thermal management tools.

2 Flow configuration

Figure 1 provides a visual understanding of the flow dynamics of ternary hybrid nanofluid over a bullet-shaped surface. The flow is assumed to be BLF and influenced by MHD. In addition, the flow configuration involves a laminar, steady flow of a nanofluid influenced by an oblique magnetic field applied at an inclination angle relative to the surface. The flow occurs over a stretching, non-isothermal surface where both the surface velocity and temperature vary exponentially with spatial position. Moreover, the influence of viscous dissipation and non-uniform heat sink source effect is incorporated in the physical model for thermal transport account. The radius of bullet is R ( x ) while the expressions u w ( x ) = a e x L and u ∞ ( x ) = b e x L demonstrate the stretching and ambient velocity, respectively. In the velocity expression, L represents the reference length and (a, b) are the positive constants. Furthermore, T w ( x ) = e x 2 L signifies the surface temperature related to assumed geometry and T ∞ has been chosen for free stream temperature. The physical model incorporates the Carreau model and the stress tensor.

Figure 1

Physical interpretation.

2.2 Thermal and physical properties of ternary hybrid nanofluid

This subsection presents the rheological and thermophysical characteristics of ternary hybrid nanofluids, which are under consideration. The ternary hybrid nanofluid comprises of three different types of nanoparticles such as (Cu, Fe₃O₄, and SiO₂) in a base fluid (polymer). The thermophysical attributes such as thermal radiation account for radiative heat transport. Heat generation within the fluid is included to simulate scenarios where internal energy sources impact the thermal distribution, whereas viscous dissipation accounts for high shear rates or non-Newtonian fluids. The empirical values of the base fluid and involved ternary nanoparticles are shown in Table 1, whereas important thermal and physical characteristics are associated with ternary hybrid nanofluids are presented in Table 2. These thermal and physical attributes are taken into consideration based on the previous research [14,15]. The following key terms reflect the notion of co-rations for improved viscosity, density, thermal conductivity, specific heat, and electrical conductivity, respectively. In addition, the co-ration of thermal conductivity and electrical conductivity for different categories of fluids such as nanofluid, di-hybrid, and ternary hybrid are stated in equations (7) and (8).

Table 1

Numerical values of base fluid and ternary nanoparticles

Base fluid and nanoparticles	ρ ( kg m ‒ 3 )	c p ( j / kg K )	k ( W / mK )	σ ( S m ‒ 1 )	Pr
Polymer	1,060	3,770	0.429	4.3 × 10 ‒ 5	2.36
Cu	8,933	385	401	5.96 × 10 7	—
SiO2	2,650	730	1.5	1 × 10 ‒ 18	—
Fe3O4	5,180	670	9.7	2.5 × 10 ‒ 4	—

Table 2

Mathematical expression for enhanced thermal and physical properties of ternary hybrid nanofluids

V R = μ tri ‒ hnf μ bf = [ ( 1 ‒ δ Cu ) ( 1 ‒ δ SiO 2 ) ( 1 ‒ δ Fe 3 O 4 ) ] ‒ 2.5

D R = ρ tri ‒ hnf ρ bf = ( 1 ‒ δ Fe 3 O 4 ) ( 1 ‒ δ Cu ) ( 1 ‒ δ SiO 2 ) + δ SiO 2 ρ SiO 2 ρ bf + δ Cu ρ Cu ρ bf + δ Fe 3 O 4 ρ Fe 3 O 4 ρ bf

( TC ) R = k tri ‒ hnf k bf = k tri ‒ hnf k hnf . k hnf k nf . k nf k bf

( SH ) R = ( ρ c p ) tri ‒ hnf ( ρ c p ) bf = ( 1 ‒ δ Fe 3 O 4 ) ( 1 ‒ δ Cu ) ( 1 ‒ δ SiO 2 ) + δ SiO 2 ( ρ c p ) SiO 2 ( ρ c p ) bf + δ Cu ( ρ c p ) Cu ( ρ c p ) bf + δ Fe 3 O 4 ( ρ c p ) Fe 3 O 4 ( ρ c p ) bf

( EC ) R = σ tri ‒ hnf σ bf = σ tri ‒ hnf σ hnf . σ hnf σ nf . σ nf k bf

The co-ratios of thermophysical characteristics in terms of assumed co-ratios are depicted in Table 3, whereas Table 4 shows the numerical outcomes for different categories of fluid such as nano-fluid, di-hybrid, and trihybrid nanofluids with varying nanoparticles volume fractions. Moreover, Figure 2 describes the pictorial display of fundamental properties in respect of Table 2.

(7) k nf k bf = k Cu + 2 k bf − 2 δ Cu ( k bf − k Cu ) k Cu + 2 k bf + δ Cu ( k bf − k Cu ) , k hnf k nf = k SiO 2 + 2 k nf − 2 δ SiO 2 ( k nf − k SiO 2 ) k SiO 2 + 2 k nf + δ SiO 2 ( k nf − k SiO 2 ) , k tri − hnf k hnf = k Fe 3 O 4 + 2 k hnf − 2 δ Fe 3 O 4 ( k hnf − k Fe 3 O 4 ) k Fe 3 O 4 + 2 k hnf + δ Fe 3 O 4 ( k hnf − k Fe 3 O 4 ) ,

(8) σ nf σ bf = σ Cu + 2 σ bf − 2 δ Cu ( σ bf − σ Cu ) σ Cu + 2 σ bf + δ Cu ( σ bf − σ Cu ) , σ hnf σ nf = σ SiO 2 + 2 σ nf − 2 δ SiO 2 ( σ nf − σ SiO 2 ) σ SiO 2 + 2 σ nf + δ SiO 2 ( σ nf − σ SiO 2 ) , σ tri − hnf σ hnf = σ Fe 3 O 4 + 2 σ hnf − 2 δ Fe 3 O 4 ( σ hnf − σ Fe 3 O 4 ) σ Fe 3 O 4 + 2 σ hnf + δ Fe 3 O 4 ( σ hnf − σ Fe 3 O 4 ) ,

Table 3

Symbolic representation of thermal and physical attributes (co-ratios)

V R

Viscosity

D R

Density

(TC)R

Thermal conductivity

(SH)R

Specific heat

(EC)R

Electrical conductivity

Table 4

Thermophysical features of different fluids with varying volume fractions

δ Cu	Nanofluid	( δ Cu , δ SiO 2 )	Hybrid nanofluid	( δ Cu , δ SiO 2 , δ Fe 3 O 4 )	Trihybrid nanofluid
Ratio of viscosity of nanofluid, hybrid nanofluid, and trihybrid nanofluid
δ Cu = 0.01	1.0254	δ Cu = δ SiO 2 = 0.01	1.0515	δ Cu = δ SiO 2 = δ Fe 3 O 4 = 0.01	1.0783
δ Cu = 0.02	1.0518	δ Cu = δ SiO 2 = 0.02	1.1063	δ Cu = δ SiO 2 = δ Fe 3 O 4 = 0.02	1.1636
δ Cu = 0.03	1.0791	δ Cu = δ SiO 2 = 0.03	1.1645	δ Cu = δ SiO 2 = δ Fe 3 O 4 = 0.03	1.2566
δ Cu = 0.03	1.1074	δ Cu = δ SiO 2 = 0.04	1.2264	δ Cu = δ SiO 2 = δ Fe 3 O 4 = 0.04	1.3582
Ratio of density of nanofluid, hybrid nanofluid, and trihybrid nanofluid
δ Cu = 0.01	1.0743	δ Cu = δ SiO 2 = 0.01	1.1128	δ Cu = δ SiO 2 = δ Fe 3 O 4 = 0.01	1.1272
δ Cu = 0.02	1.1485	δ Cu = δ SiO 2 = 0.02	1.2247	δ Cu = δ SiO 2 = δ Fe 3 O 4 = 0.02	1.2526
δ Cu = 0.03	1.2228	δ Cu = δ SiO 2 = 0.03	1.3359	δ Cu = δ SiO 2 = δ Fe 3 O 4 = 0.03	1.3761
δ Cu = 0.03	1.2971	δ Cu = δ SiO 2 = 0.04	1.4463	δ Cu = δ SiO 2 = δ Fe 3 O 4 = 0.04	1.4980
Ratio of thermal conductivity of nanofluid, hybrid nanofluid, and trihybrid nanofluid
δ Cu = 0.01	1.0302	δ Cu = δ SiO 2 = 0.01	1.0576	δ Cu = δ SiO 2 = δ Fe 3 O 4 = 0.01	1.0720
δ Cu = 0.02	1.0610	δ Cu = δ SiO 2 = 0.02	1.1179	δ Cu = δ SiO 2 = δ Fe 3 O 4 = 0.02	1.1484
δ Cu = 0.03	1.0925	δ Cu = δ SiO 2 = 0.03	1.1811	δ Cu = δ SiO 2 = δ Fe 3 O 4 = 0.03	1.2296
δ Cu = 0.03	1.1246	δ Cu = δ SiO 2 = 0.04	1.2473	δ Cu = δ SiO 2 = δ Fe 3 O 4 = 0.04	1.3156

Figure 2

Visualization of assumed empirical data based on thermophysical characteristics.

The following transformations satisfy the desired criteria and are quite compatible with assumed geometry of flow. Using these transformations, the governing system of PDEs is transformed into system of ODEs.

(9) ψ = υ bf e x 2 L F ( η ) , T = T ∞ H ( η ) e x 2 L , V 1 = 1 r ∂ ψ ∂ r , V 2 = − 1 r ∂ ψ ∂ x , η = a r 2 e x 2 L 2 υ bf L ,

The expression R ( x ) = a r 2 e x 2 L 2 υ bf L denotes the radius of the bullet. Utilization of equation (9) gives non-dimensional form of ODEs as follows;

(10) V R D R ( ε ⁎ + ( 1 − ε ⁎ ) + n ( 1 − ε ⁎ ) ( We ( F ″ ) 2 ) ) × ( ε ⁎ + ( 1 − ε ⁎ ) ( We ( F ″ ) 2 ) ) n − 3 2 η F ″ ′ + F F ″ e ϑ 2 + F ″ + e ϑ 2 ( ε 2 − F ′ 2 ) + M Sin 2 ω ( EC ) R D R e − ϑ 2 ( ε − F ′ ) = 0 ,

(11) ( TC ) R + 4 3 R d η H ″ + H ′ 2 + Pr ( SH ) R e ϑ 2 F H ′ − F ′ H 2 + e − 2 ϑ Pr Q H + 2 η V R ( SH ) R Ec F ″ 2 + M ( EC ) R V R sin 2 ( ω ) Ec F ′ 2 e − 3 ϑ 2 + ( TC ) R ( SH ) r ( J ⁎ F ′ + K ⁎ G ) = 0 ,

and the associated boundary conditions are as follows:

(12) F ′ ( 0 ) = 1 + N ⁎ F ′ ′ , H ( 0 ) = 1 , F ( 0 ) = 1 , F ′ ( ∞ ) → ε , H ( ∞ ) → 0 .

The important physical quantities such as skin friction (C_f) and Nusselt number (N_u) together with their non-dimensional form are given as follows:

(13) Cf = − 2 τ w ρ bf ( u w ) 2 ; Nu = L ⁎ q w k bf ( T w − T ∞ ) ,

(14) τ w = μ tri − hnf ∂ V 1 ∂ r ε ⁎ + ( 1 − ε ⁎ ) Γ ∂ V 1 ∂ r 2 r = R n − 1 / 2 ; q w = k tri − hnf + 16 σ ⁎ ⁎ T ∞ 3 3 k ⁎ ∂ T ∂ r r = R ,

(15) R e 0.5 C ζ = − 2 ϑ ( V r ) F ′ ′ ( 0 ) ( ε ⁎ + ( 1 − ε ⁎ ) ( We ( F ′ ′ ( 0 ) ) 2 ) ) n − 1 / 2 } ,

(16) Re − 0.5 Nu = − ϑ 2 { ( TC ) R + ( 4 / 3 ) R d } H ′ ,

Table 5 gives the description of non-dimensional form of governing parameters in mathematical formulation.

Table 5

Non-dimensional parameters

Name	Mathematical expression	Name	Mathematical expression
Prandtl number	Pr = μ bf c p / k	Magnetic parameter	M = σ bf L B 0 / ρ bf a
Stretching ratio parameter	ε = u w / u ∞	Velocity slip parameter	N ⁎ = L ⁎ a r
Defined Reynolds number (Local)	Re = u w L / υ bf	Heat generation parameter	Q = Q ⁎ L ( T w ‒ T ∞ ) / a ρc p
Thermal radiation	R d = 16 σ ⁎ T ∞ 3 / 3 k bf k ⁎	Weissenberg number	We = 2 Γ r a 2 e 3 x 2 L / υ bf L
Location parameter	ϑ = x / L	Eckert number	Ec = u w 2 / ( c p ) bf ( T w ‒ T ∞ )

3 Methodology

This section gives the detailed description of the numerical procedure of the current analysis which is based on the altered system of ODEs. The current analysis delves into the joint computational examination of the heat transport dynamics in Carreau ternary nanofluids. The simulation is performed through computer programing, which is based on bvp4c solver strategy with the combination of ANNs. The explanation of both the strategies is discussed below.

3.2 ANN computational procedure

In this study, we used a general ANN scheme for the considered problems. For this, 60% of data obtained from bvp4c are taken for training with one input layer having 8 neurons, 20% of data are chosen for testing, and 20% data are allocated for validation process in the network. Figure 3(a) and (b) is displayed for neural architecture and internal structure for one hidden and one output layer. Figure 4 shows internal neural architecture for one input and one output layer. Figure 5 explains the internal neural architecture for weights for one input and one output layer and Figure 6 is for output layer with 2 neurons.

1. Network architecture:

   • Input: η ∈ R 1 .

   • First hidden layer: 08 neurons, sigmoid activation function

   • Output: There are two output variables F ′ ( η ) , G ( η ) .

2. Input to the first hidden layer:

   • Weights: W 1 ∈ R 08 × 1

   • Biases: b 1 ∈ R 8

   • Activation function: Sigmoid function σ ¯ ( z ) = 1 1 + e − z

   • Pre-activation: z 1 = W 1 η + b 1

   • Activation (output of first hidden layer): a 1 = σ ¯ ( z 1 ) = σ ¯ ( W 1 η + b 1 ) . Here a 1 ∈ R 8 .

3. Output layer:

   • Weights: W 2 ∈ R 1 × 8

   • Biases: b 2 ∈ R 2

   • Outputs: y ¯ = W 2 a 1 + b 2 ; where y ¯ = F ′ ( η ) G ( η )

Figure 3

(a) and (b) ANN-based architecture. (a) Hidden layer with neurons disterbution; and (b) Internal structure for one input and one output layer.

Figure 4

Internal neural architecture for weights for one input and one output layer.

Figure 5

Weights for first hidden layer with 8 neurons.

Figure 6

Output layer with 2 neurons.

4 Agreement of validation study

In this section, comparison analysis of the proposed results with the earlier published results is done. The comparison is made with the study conducted by Ntantis et al. [36] for F ″ ( 0 ) in case of varying values of physical parameter M. Table 6 shows the validation outcomes of F ″ ( 0 ) with previously published data. In addition, the comparison is made under some specific constraints by fixing the values of some physical parameters, i.e., We = 0 , n = 1 , δ Cu = δ SiO 2 = δ Fe 3 O 4 = ω = Ec = ε ⁎ = 0 . It is found that the results of the proposed study are greatly matched with the existing study, which ensures the efficacy and reliability of the proposed model.

5 Results interpretation

This section presents a detailed and comprehensive explanation of the facts and findings reported in the study. The proposed study focused on the ternary hybrid nanofluid flow over bullet-shaped objects because of its significances in many industrial and engineering applications such as in automotive, aerospace, and microfluidic devices. Numerical computations are obtained by using an advanced computational procedure such as neural network and bvp4c. This study delved into the thermal transport prediction in Carreau ternary hybrid nanofluids and its solution is predicted by ANNs. This study investigated how these fluids responded to the combined effects of viscous dissipation and infinite shear rate viscosity when interacting with a bullet-shaped geometry. Velocity and temperature profiles of the Carreau ternary hybrid nanofluid are analyzed for several parameters like Weissenberg number, infinite shear rate viscosity parameter, inclined angle, location parameter, stretching ratio parameter, velocity slip parameter, power law index, and magnetic parameter. All velocity plots are sketched for two cases, i.e., presence and absence of infinite shear rate viscosity parameter. Moreover, the ANN illustrations such as histogram errors, gradient, Mu, and validation checks are also captured for each effective parameters during the neural training.

Velocity profile is most important in fluid flow problem. It enables us to understand the underlying behavior of the complex flow pattern of the fluid flow, which is crucial in many scientific and engineering processes. In Addition, it is worth noting that, to examine how fluid moves within and over a surface is absolutely essential in many practical implications. In many fluid flow problems, velocity profile impacts the heat transfer rate, with steeper profiles near boundaries enhancing local convection rates such as reactors or heat exchangers. Therefore, it is necessary to analyze the velocity profile, because engineers and scientists can build more efficient systems by optimizing flow conditions, minimizing wear, and reducing energy consumption. Due to these attentions, some prominent parameters are depicted for velocity profile of Carreau ternary hybrid nanofluid. These parameters include stretching ratio parameter (ε), inclined location parameters (ϑ), velocity slip parameter (N ^*), Carreau index parameter (n), and magnetic parameter (M).

Figure 7(a)–(f) demonstrates the effectiveness of Weissenberg number (We) and inclined angle orientation (ω) on velocity profile of Carreau ternary hybrid nanofluid with two cases of infinite shear rate (ε ^*), i.e., presence (ε ^* equal to zero) and absence (ε ^* not equal to zero) of infinite shear rate viscosity. In addition, ANN illustrations such as error histogram together with gradient, mu, and validation checks are also given. It is noted that velocity magnitude becomes lower for higher Weissenberg numbers (We) due to viscoelastic nature in both cases, i.e., presence and absence of infinite shear viscosity parameter. Higher values of Weissenberg numbers (We) indicate that the elastic effects of the fluid become more dominant and this fact leads to increased resistance to deformation. Another aspect for reduction in velocity is that the relaxation time becomes significant when this parameter becomes numerically high and it means the nanofluid requires more time to respond to changes in flow conditions, so this velocity decreases. On the other hand, the velocity profile of Carreau ternary hybrid nanofluid decreases for higher numeric values of inclined angle (ω) due to stronger interaction between the magnetic field in a conducting fluid and enhanced Lorentz force generation. The inclination of the magnetic field disrupts the primary flow direction, diverting energy away from forward motion and decreasing the overall velocity. The best validation performance is noted at 215 epochs and histogram is noted for 20 bins. Figure 8(a)–(f) demonstrates the effectiveness of stretching ratio parameter (ε) and inclined location parameters (ϑ) on velocity profile of Carreau ternary hybrid nanofluid with two cases of infinite shear rate (ε ^*), i.e., presence (ε ^* equal to zero) and absence (ε ^* not equal to zero) of infinite shear rate viscosity. In addition, ANN illustrations such as error histogram together with gradient, mu and validation checks are also given. The increase in velocity magnitude with higher values of the stretching ratio parameter (ε) is due to enhanced stretching forces, greater momentum transfers in the boundary layer, and reduced fluid thickness near the surface. Greater momentum transport propels the fluid more effectively and small fluid thickness allows for a streamlined flow. On the other hand, the velocity profile of Carreau ternary hybrid nanofluid decreases with higher values of location parameters due to the increased resistance of the flow region for this parameter. It is concluded that the trends are consistent in both cases of infinite shear rate viscosity parameter. The best performance validation is noted at 117 epochs and histogram is noted for 20 bins.

Figure 7

(a)–(f) Effectiveness of Weissenberg number (We) and angle orientation (ω) on the velocity profile.

Figure 8

(a)–(f) Effectiveness of stretching ratio parameter (ε) and location parameter (ϑ) on the velocity profile.

Figure 9(a)–(f) demonstrates the effectiveness of velocity slip parameter (N ^*) and Carreau index parameter (n) on velocity profile of Carreau ternary hybrid nanofluid with two cases of infinite shear rate (ε ^*), i.e., presence (ε ^* equal to zero) and absence (ε ^* not equal to zero) of infinite shear rate viscosity. It is examined that the velocity profile of the Carreau ternary hybrid nanofluids decreases with higher values of the velocity slip parameter (N ^*) because the slip condition reduces the interaction between the fluid and the surface. The less friction at the boundary, driving force for fluid acceleration, and reduced boundary layer thickness are major causes for the declining profile of Carreau ternary hybrid nanofluids. On other hand, opposite behavior is seen in the velocity profile of Carreau ternary hybrid nanofluid with augmented values of Carreau index parameter (n). The best performance validation is noted at 248 epochs and histogram is noted for 20 bins. Figure 10(a) and (b) demonstrates the effectiveness of magnetic parameter (M) on velocity profile of Carreau ternary hybrid nanofluid with two cases of infinite shear rate (ε ^*), i.e., presence (ε ^* equal to zero) and absence (ε ^* not equal to zero) of infinite shear rate viscosity. It is observed that numeric growth in the values of magnetic parameter (M) produces strong Lorentz force within the fluid which hinders the motion of the fluid very badly and reduction occurs in the flow; hence, velocity profile of Carreau ternary hybrid nanofluid decreases.

Figure 9

(a)–(f) Effectiveness of velocity slip parameter (N ^*) and Carreau index parameter (n) on the velocity profile.

Figure 10

(a) and (b) Effectiveness of magnetic parameter (M) on the velocity profile.

The temperature profile in a fluid flow is very essential because it describes how temperature varies throughout the flow field, impacting numerous physical and chemical properties of the fluid. Good understanding of temperature profile is crucial for better analyzing, accurately modeling, and optimizing flow systems in many industrial and engineering processes. In general, the temperature profile is very vital in heat and mass transfer fluid-based problems because it could be very helpful in understanding environmental impacts, controlling reaction rates, and ensuring material durability. It can support in better design, improved safety, and enhanced efficiency in numerous applications across industrial and engineering fields. Due to these attentions, the temperature profile of Carreau ternary hybrid nanofluids is depicted for several parameters like radiation parameter (R _d), heat generation parameter (Q), location parameter (ϑ), magnetic parameter (M), Eckert number (Ec), and space-dependent and heat source/sink parameters (J ^*, K ^*). All temperature plots are sketched for two cases, one is presence and second is absence of space-dependent and heat source/sink characteristics. Moreover, the ANN illustrations such as histogram errors, gradient, Mu, and validation checks are also captured for each effective parameters during the neural training.

Figure 11(a)–(f) is used to illustrate the impact of thermal radiation (R _d) and heat generation parameter (Q) on temperature profile of Carreau ternary hybrid nanofluids in the presence and absence of space-dependent and heat source/sink characteristics with error histogram and ANN gradient, mu, and validation checks with calculated number of epochs. It is examined that the decrease in nanofluid temperature with higher values of the radiation parameter is seen due to the enhanced radiative heat loss and reduced internal thermal energy. Additionally, increased radiation leads to more heat transfer from the fluid to the surroundings. In contrast, the rise in nanofluid temperature with an increase in the heat generation number occurs due to the addition of internal heat. This effect remains consistent whether space-dependent heat source/sink characteristics are present or absent, as heat generation continuously supplies energy, compensating for any losses and causing an overall temperature increase. Figure 12(a)–(f) is used to illustrate the impact of location parameter (ϑ) and magnetic parameter (M) on the temperature profile of Carreau ternary hybrid nanofluids in the presence and absence of space-dependent and temperature-dependent heat source/sink characteristics with error histogram and ANN gradient, mu, and validation checks with calculated number of epochs. The decrease in the temperature for higher values of the location parameter is found to be due to reduction in the boundary layer thickness, the heat dissipation increments, and lower thermal gradients. On the other hand, the temperature increases with the magnetic number in both the presence and absence of space-dependent heat sources or sinks due to the Lorentz force.

Figure 11

(a)–(f) Effectiveness of radiation parameter (M) and heat generation parameter (Q) on the temperature profile.

Figure 12

(a)–(f) Effectiveness of location parameter (M) and magnetic parameter (Q) on the temperature profile.

Figure 13(a)–(f) is used to illustrate the impact of volumetric fraction (δ) of nanoparticles and Eckert number (Ec) on the temperature profile of Carreau ternary hybrid nanofluids in presence and absence of space-dependent and heat source/sink characteristics with error histogram and ANN gradient, mu, and validation checks with calculated number of epochs. It is noted that increase in the temperature of a nanofluid with higher values of the volumetric fraction of nanoparticles is due to enhancement in the thermal conductivity, facilitation of better energy absorption, and storage within the fluid and increased interaction between nanoparticles. Conversely, the temperature decreases with higher Eckert numbers in both the presence and absence of space-dependent heat sources or sinks. This number creates stronger convective cooling due to the dissipation of mechanical energy into heat and due to this, temperature becomes lower. Furthermore, Figure 14(a)–(d) is used to show the effectiveness of infinite shear rate viscosity parameter (ε ^*) and radiation parameter (R _d) on the velocity and temperature profile of Carreau ternary hybrid nanofluids, respectively. Both the profiles (velocity and temperature) of Carreau ternary hybrid nanofluid are depicted by varying concentrations of nanoparticles volume frictions with focusing presence and absence of space-dependence and heat source/sink characteristics. It is examined that both velocity and temperature distribution intensified in all types of fluids such as (nanofluid, hybrid nanofluid, and ternary hybrid nanofluids) with augmentation in the numerical values of both parameters, i.e., infinite shear rate viscosity parameter (ε ^*) and radiation parameter (R _d). It is worth noting that the rate of temperature increment is highest in trihybrid nanofluids due to the multiple nanoparticles incorporation and lowest in standard nanofluids because of their simpler composition and lower thermal conductivity enhancement. All three parameters enhance the temperature profile. Moreover, Figure 15(a)–(d) show the different visuals of streamlines.

Figure 13

(a)–(f) Effectiveness of varying volume friction (δ) and Eckert number (Ec) on the temperature profile.

Figure 14

(a)–(d) Effectiveness of infinite shear rate (ε ^*) and radiation parameter (R _d) on the velocity the temperature profile.

Figure 15

(a)–(d) Streamline visuals. (a) Streamline visuals for 50 lines. (b) Streamlines with 500 lines. (c) Streamlines with 150 lines. (d) 3D view for stream lines.

The numerical outcomes of the physical quantities such as skin fraction (Cf) and Nusselt number (Nu) are computed for some key parameters by both computational technique, i.e., bvp4c and ANN. Table 7 presents the behavior of skin fraction (Cf), which is computed for varying numerical values in an appropriate range of some physical parameters. These parameters include Weisenberg number (We), Carreau index parameter (n), velocity slip parameter (N ^*), stretching ratio parameter (ε), infinite shear rate parameter (ε ^*), magnetic parameter (M), and location parameter (ϑ). Skin friction (Cf) is seen to gradually reduce in both types of fluid (hybrid nanofluids and ternary hybrid nanofluid) by augmented values of Weisenberg number (We), velocity slip parameter (N ^*), magnetic parameter (M), and location parameter (ϑ). Whereas it demonstrates opposite trend for Carreau index parameter (n), stretching ratio parameter (ε), and infinite shear rate parameter (ε ^*). On the other hand, in Table 8, Nusselt number (Nu) is computed with their varying numerical values in an appropriate range computed for some physical parameters such as radiation parameter (R_d), Eckert number (E_c), infinite shear rate parameter (ε ^*), magnetic parameter (M), location parameter (ϑ), as well as space-dependent and temperature-dependent heat source/sink characteristics (J ^* & K ^*). The rate of heat transport improved with the augmented values of infinite shear rate parameter (ε ^*), magnetic parameter (M) as well as space-dependent and temperature-dependent heat source/sink characteristics (J ^* & K ^*). Whereas the opposite behavior is noted for radiation parameter (R _d), Eckert number (Ec), and location parameter (ϑ). In addition, Figure 16(a)–(f) gives the physical appearance of behavior of physical quantities, i.e., Nusselt number (Nu) and skin friction (Cf). Figure 16(a)–(c) show the mutual influence of Nusselt number (Nu) and skin friction (Cf) with the varying values of Weisenberg number (ω), radiation parameter (R _d), and angle orientation (ω). Figure 16(d) is plotted to visualize the deviation of physical quantities in different categories of fluids, i.e., nanofluids, hybrid nanofluids, and ternary hybrid nanofluids by varying values of heat source parameter (Q), while Figure 16(e) –(f) illustrates the influence of Nusselt number (Nu) and skin friction (Cf) with the combined effectiveness via varying values of (n, N ^*) and (ε, ε ^*) in the same range of values.

Figure 16

(a)–(f) Trend of physical quantities with varying values of physical parameters. (a) Impact of angle orentation. (b) Impact of Wiessenberg number. (c) Impact of radiation parameter. (d) Impact of angle orientation. (e) Impact of heat source parameter. (f) Impact of model index along with velocity slip parameter and Impact of infinte shaer rate along with stretching ratio parameter.

6 Conclusion

This study investigated the thermal transport prediction in the flow of Carreau ternary hybrid nanofluid over bullet-shaped geometry by implementation of an advanced computational ANN procedure. Effective parameters such as Weissenberg number, infinite shear rate viscosity parameter, location parameter, stretching magnetic parameter, and velocity slip parameter etc., are analyzed for velocity and temperature profiles of Carreau ternary hybrid nanofluid. The key findings of the proposed study are presented as follows

• The velocity profile is intensified with augmentation in numerical values for infinite shear rate parameter, whereas it shows decreasing trend for Weissenberg number.

• The velocity profile of ternary hybrid nanofluid increases due to shear thinning behavior, which occurred due to varying augmented values of Carreau index parameter.

• Velocity profile decreases with higher values of location parameters, whereas it gives the opposite trend with higher values of the velocity slip parameter because the slip condition reduces the interaction between the fluid and the surface.

• The temperature profile decreases due to numeric growth in the values of radiation parameter because of enhanced radiative heat loss and reduction in its internal thermal energy.

• Rise in nanofluid temperature with an increase in the heat generation number occurs due to the addition of internal heat.

• The decrease in the temperature for higher values of the location parameter is found to be due to reduction in the boundary layer thickness, the heat dissipation increments, and lower thermal gradients.

• The increase in the temperature of a nanofluid with higher values of the volumetric fraction of nanoparticles enhances the thermal conductivity, facilitation of better energy absorption, and storage within the fluid.

• The rate of temperature increment is highest in trihybrid nanofluids due to the multiple nanoparticles incorporation and lowest in standard nanofluids because of their simpler composition of nanoparticles.

7 Future work

The current study should be extended in a multiple aspect. For instance, various other non-Newtonain fluid models can be implemented for thermal transport analysis in a ternary based hybrid nanofluid over the same geometry by involving some variable conditions. Furthermore, several facts can be incorporated into the physical model such as non-uniform heat source, dynamics of nanoparticles shape. Moreover, advance computational framework such as multilayer supervised neural network procedure based on different neurons can be integrated.