On the thermal performance of a three-dimensional cross-ternary hybrid nanofluid over a wedge using a Bayesian regularization neural network approach

Nomenclature

I

Identity vector

k

Thermal conductivity (kg m·s⁻³·K⁻¹)

q r

Heat flux

k ⁎

Mean absorption coefficient (m⁻¹)

Q 0

Heat generation coefficient

We

Weissenberg number

Pr

Prandtl number

U

Free-stream velocity in the x-direction (m·s⁻¹)

f ′ ( η )

Dimensionless velocity in the x-direction

θ ( η )

Dimensionless temperature

g ′ ( η )

Dimensionless velocity in the y-direction

V

Free-stream velocity in the y-direction (m·s⁻¹)

C f x

Skin friction coefficient in the x-direction

C f y

Skin friction coefficient in the y-direction

Nu

Nusselt number

( x , y , z )

Cartesian coordinates (m)

( u , v , w )

Components of velocity in the x, y and z directions, respectively (m·s⁻¹)

n

Carreau fluid index

m

Power law index

M 2

Hartmann number

p

Pressure

A 1

First Rivlin-Ericksen tensor

T

Temperature of nanofluid (K)

T wa

Wall surface temperature (K)

T ∞

Constant temperature for the ambient/far-field (K)

1 Introduction

Rapidly increasing energy demands in various sectors of engineering have significant challenges in the discipline of thermal science. Therefore, developing effective heat transfer processes is essential for this purpose. The only way to solve this problem is to make fluids more thermally conductive. Thermal conductivity is a prominent and most important factor in heat transportation. Improved thermal conductivity enables more efficient heat transfer, reducing energy losses in conventional systems. Researchers reported that the role of hybrid nanofluids is crucial in such systems associated with heat transfer and thermal phenomena. Recently, ternary nanoparticles have gained more interest due to their excellent thermal proficiency and rapid heat transfer capabilities. Thus, a three-component nanofluid is used as an emerging source in flow dynamics. These fluids are prepared by a colloidal mixture of three distinct nanoparticles in the base fluid. This colloidal mixture exploits synergistic effects such as thermal conductivity, convective heat transfers, and heat dissipation and nourishes the conventional fluid. Many electronic cooling systems, aerospace technology, heat exchangers, and numerous engineering sectors are the primary consumers of thermal energy; thus, operating a rapid thermal transport without low energy is a keen interest. For instance, Pazarlioglu et al. [1] studied the NiO/Water nanofluid flow in the presence of a magnetic field to examine some fluid properties and heat transfer. The fluid engaged in their study was influenced by twisted ducts and varying twist ratios magnetic effect. Adun et al. [2] discussed some results of ternary hybrid nanofluid in thermal solar system performance with the help of numerical measurement and energy analysis. Saleem et al. [3] conducted research on ternary radiated blood incorporated with titled magnetic field action for the analysis of thermal effects cilia actuated transport. Adun et al. [4] researched the synthesis of ternary nanofluid and their use in solar-thermal systems. Furthermore, a comparative analysis is taken between single and hybrid nanofluids for energy and performance. Darvesh et al. [5] considered cross fluid over a Riga plate in the presence of entropy generation and melting process and made a numerical study of infinite shear rate viscosity by killer box numerical simulation. Moreover, a comprehensive analysis of the synthesis and stability of ternary hybrid nanofluid, together with critical thermophysical characteristics uses of heat transfer and several environmental effects, was uncovered by Adun et al. [6].

A mathematical model is a robust framework for solving numerous fluidic models. A Carreau mathematical model integrates multiple essential features that enhance its efficiency and capability to tackle the fluidics in power law region issues. This mathematical model is more effective for a wide range of environmental, industrial, and biomedical fluidics problems due to its versatility and ability to accommodate a variety of fluidic scenarios. Its capacity to capture the fluidic behavior at very high and very low shear rates makes it a valuable tool for researchers and practitioners in fluidics modeling. Waqas et al. [7] discussed a Carreau-mathematical model for bio-nanofluid flow in an expanding/contracting cylinder in an inclined magnetic field. Ayub et al. [8] studied the Carreau-fluid mathematical model with an inclined magnetic dipole for the nature analysis of the fluid flow. Akbar et al. [9] made a numerical study for heat transfer in MHD stagnation point flow over a stretched surface using a Carreau-mathematical model. Hussain et al. [10] researched fluid flow containing nanoparticles and motile gyrotactic organisms in a three-dimensional cylinder with the Carreau model. Nabwey et al. [11] made nanoscale heat transport and studied inclined magnetized Carreau fluid passed over the cylinder. Azam et al. [12] studied Carreau-fluid by an expanding-contracting cylinder with a revised nanofluid model in the stagnation region for simulation. Further studies [13,14] are made by different scholars about the stagnation-point flow of a Carreau nanofluid in the presence of thermophoresis and hydromagnetic stagnation point flow of Carreau nanomaterials.

Fluid flow over a wedge is a fascinating topic for researchers across the globe because it is important for many different scientific and engineering fields. Aerodynamics is one of the big disciplines where designing streamlined aircraft and spacecraft requires understanding the airflow patterns over wedges. Moreover, this geometry is considered in multiple scenarios because of one prominent application. For instance, the fluid flow over wedges is crucial for various thermal systems, such as heat exchangers and cooling systems. This system contributes to the creation of effective thermal management techniques that influence a variety of industries, including electronics and power generation. Moreover, the knowledge of natural phenomena like sediment transport and riverbed erosion requires an understanding of fluid flow over wedges, which is crucial in civil and environmental engineering. This study of the wedge is very valuable and interdisciplinary because the results not only show the advanced basic knowledge in fluid dynamics but also have practical uses for many applications that enable scientists to optimize vehicle shapes, improve overall performance and fuel efficiency, etc. Using a numerical deformation model, Strayer et al. [15] studied fluid flow in an evolving thrust wedge. Sajid et al. [16] detailed a study of ternary hybrid nanofluid flow over a wedge under the influence of Arrhenius energy and the effects of chemical reactions in ternary nanoparticles. Abdal et al. [17] analyzed tangent hyperbolic nanofluid flow over a Riga wedge in the presence of a heat source accompanied by a chemical reaction with activation energy. Khan et al. [18] studied hybrid nanofluids [(Cu-Al₂O₃)/water] by numerical simulations for a heat transport mechanism over a wedge geometry. Mahdy et al. [19] considered a stretching wedge geometry with an ohmic heating effect for detecting the MHD mixed convective heat transfer in a Cu-Al₂O₃ water hybrid nanofluid.

Bayesian neural network (BNN) is a superficial technique using a probabilistic method to represent neural network uncertainty. The entire process of building a BNN differs from building neural networks. The traditional neural networks (NNs) only provide point estimation, but BNNs handle weights and biases as probability distributions. In the BNN scheme, prior distributions over the network parameters are initially set as a first step representing our initial assumptions regarding their values. In a new phase, new data are introduced by a like hood function, which captures the relationship between inputs and outputs. Then, Bayes’ theorem incorporates the observed data into the posterior distribution. A few standard techniques, such as variation inference (VI) or Markov Chain Monte Carlo (MCMC), are mainly adopted to sample from the posterior distribution and approximate the true uncertainty in the model. The predictions are made in the last stage by averaging several samples from the rear, and thus, finally, a point estimate accompanied by an uncertainty measure for each prediction is gained. BNNs can produce more accurate and nuanced predictions, especially when there is a lot of uncertainty or little data. Thus, it concludes that BNNs have superior conjoint associations with all-encompassing methods and are a good choice rather than a traditional neural network. Mueller et al. [20] used this novel idea of BNNs, making stochastic predictions for fluid simulations. Hirschen and Schäfer [21] studied a few optimizing fluid flow processes by Bayesian regularization NNs. Geneva and Zabaras [22] reviewed the Reynolds-averaged turbulence model from uncertainty quantification through the Bayesian deep NNs scheme. Liu et al. [23] made a detailed analysis of the optimization of airfoil aerodynamic performance and made some quantitative predictions with the help of a deep neural network coupled Bayesian method. Moreover, Maulik et al. [24] researched fluid flow for data recovery and surrogate modeling using probabilistic NNs.

2 The mathematical formulation of the physical problem

It is assumed that the electrically conducting laminar viscous boundary-layer flow fluid is considered over a constant wedge [25]. Cartesian coordinates ( x , y , z ) in the system are taken and the flow is bounded in the space z > 0 ; the x- and y-axes are measured along the direction of the flow, and the z-axis is measured along the normal to the flow. μ ( γ ) is the Carreau viscosity model and ρ is the density of the fluid. U ( x , y ) = U ∞ x + y l m and V ( x , y ) = V ∞ x + y l m are chosen as a variable outer freestream velocity in both x-and y directions, where U ∞ , V ∞ and m are constants and l is the characteristic length. A uniform magnetic field of strength B = B 0 x + y l m − 1 2 is taken inclined through angle ω . The induced magnetic field produced is negligible for small magnetic Reynolds number. Heat transport analysis is made through heat generation coefficient and nonlinear thermal radiations (Figure 1).

Figure 1

Physical sketch of the flow problem.

Wedge geometry is applied to investigate the behavior of fluid flow and heat transfer in geometrically complex environments and this mimics real-world applications such as natural formations, chemical reactors, and cooling systems. This geometry is effective for analyzing the effects of fluid flow over surfaces with varying angles and inclinations. The inspiration for this geometry comes from its prevalence in engineering and natural systems, where fluids often interact with non-uniform surfaces. The unique interactions between nanoparticles and the base fluid within the wedge geometries offer insights into enhanced thermal conductivity and convective heat transfer.

3 Fundamental equations

The equations [25,26,27,28,29,30] of continuity, momentum, heat, flow tensor, and Carreau fluid model are given as follows:

After utilizing all the information in equation (2) using the Carreau model and the addition of three nanoparticles, the momentum equation and the heat equation are as follows:

The subjected boundary constraints are as follows:

Initially, the velocity components of the fluid are zero at the surface of the wedge (z = 0), and this indicates that there is no motion along the x, y, and z axes, which represents a no-slip condition. In the far field, velocity components become u → U , v → V at ( z → ∞ ) , with the velocity in the z-direction approaching zero which indicates a steady-state flow unaffected by the wedge at a distance. Fluid temperature is T = T w at z = 0 and T = T ∞ at z = ∞ . These conditions are crucial for modeling real-life scenarios such as the flow over aircraft wings and spacecraft nose cones, where understanding fluid and thermal dynamics is essential for efficient cooling and structural integrity during high-speed flight. Additionally, the study benefits electronics cooling by improving the performance of heat sinks with wedge features, preventing overheating and extending device life. Environmental and civil engineering applications also benefit, such as designing water management systems and erosion prevention measures by understanding water flow over structures. Tables 1 and 2 represent the correlations and experimental data of the thermo-physical properties of the ternary hybrid nanofluids, respectively.

The way of obtaining the velocity components from stream functions is given as follows:

Stream functions are

By using all equations (13)–(15) in equations (8)–(10), we obtain

f ( η ) , g ( η ) , and θ ( η ) are magnitude of velocity along x axis, y axis and dimensionless temperature distribution, respectively, for nanofluid over the three-dimensional wedge. β = 2 m m + 1 is pressure gradient, M 2 = σ f B 0 2 l ρ f U ∞ is Hartmann number, W e 2 = Γ 2 U ∞ 3 ( m + 1 ) ( x + y ) 3 m − 1 2 υ f l 3 m is Weissenberg number, α = V ∞ U ∞ is the velocity ratio parameter and Pr = υ f α f is Prandtl number. Physical quantities are as follows:

4 Numerical scheme

Bvp4c is a numerical process for boundary value problems related to ODEs in MATLAB. It is explicitly designed to solve boundary value problems where the solution is sought subject to certain conditions at both the initial and final points of the domain. A complete description of this scheme is given as follows:

4.1 ANN approach

BRNNs are used to train NNs with a probabilistic approach to prevent overfitting and improve generalization. The mathematical steps involved in training and predicting BRNNs are demarcated as follows:

Let { ( x i , y i ) } i = 1 N is given data set having x i as an input and y i is output, the goal is to train a neural network f ( x , W ) , where W is weights for training adjustment. The purpose is to define the neural network with an input layer, hidden layers with neurons, and an output layer that produces predictions y = f ( x , W ) . The objective function combines the mean-squared error (MSE) and a regularization term to control the complexity of the network:

(26) E ( W ) = E D ( W ) + λ E W ( W ) ,

(27) E D ( W ) = 1 2 ∑ i = 1 n ( y i − f ( x i , W ) ) 2 & E W ( W ) = 1 2 ∑ j = 1 M w j 2 .

Here, E D ( W ) , λ , E W ( W ) are the sum of squared errors, regularization parameter, and regularization term, respectively. In the Bayesian framework, the weights W are treated as random variables with prior distributions. The objective is to find the posterior distribution of the weights given to the data. Assume Gaussian priors and Gaussian noise for the weights are as follows, respectively:

(28) p ( W , α ) = α 2 π N 2 exp − α 2 W T W ,

(29) p ( y | x , W , β ) = β 2 π N 2 exp − β 2 ∑ i = 1 n ( y i − f ( x i , W ) ) 2 .

The posterior distribution for weights is then given as follows:

(30) p ( W | x , y , α , β ) ∞ p ( y | x , W , β ) p ( W | α ) ,

The regularization parameters α and β are determined by maximizing the evidence (marginal likelihood):

(31) p ( W | x , y , α , β ) = ∫ p ( y | x , W , β ) p ( W | α ) d W .

Iterative re-estimation of the weights is made through optimization techniques such as gradient descent or second-order methods. Adjusting ( α , β ) , the balance trade-off between the data fit and regularization, often using evidence maximization or empirical Bayes methods. Training involves minimizing the objective function through iterative steps:

• Initialize weights

• Compute gradients

• Update weights (e.g., backpropagation with Bayesian regularization).

• Re-estimate based on the current state of the network.

• Repeat until convergence criteria are met.

Once trained, the neural network is used to make predictions on new data. The Bayesian approach also allows for estimating the uncertainty in predictions, providing confidence intervals for the predicted values based on the posterior distribution of the weights. The BR-NN involves formulating the problem with an objective function that balances data fitting and model complexity, training the neural network using Bayesian principles to adjust weights and regularization parameters, and making predictions with an understanding of the associated uncertainties. This methodology ensures robust and reliable predictions, which are particularly useful in complex fluid dynamics and heat transfer problems. Figure 2 shows the hidden layer, output layer, weights, and bias mechanism and Figure 3 displays the training setup for the current study.

Figure 2

Layer structure with 10 neurons and three outputs.

Figure 3

Fixation of one hidden and one output layer and 10 neurons for ANN training.

5 Validity of the current scheme

Table 3 presents a comprehensive comparison of skin friction coefficients f ″ ( 0 ) , g ″ ( 0 ) , in both directions based on our study with old literature [27]. The investigation focuses on the absence of the Hartmann number and considers stagnation point flow with a pressure gradient of 1. This table justifies our findings with those from previous literature and provides a detailed analysis of the discrepancies and similarities.

6 Results and discussion

A new class of solutions of three-dimensional boundary-layer Carreau ternary nanofluid flows has been discovered with several physical effects such as inclined magnetic field and thermal radiation. A magnetic field on a three-dimensional boundary layer is applied to the same pressure gradient. This section concludes the impact of several parameters involved in this study on the velocity of the Carreau nanofluid over the three-dimensional wedge. In this study, the numerical treatment of the Carreau-mathematical model and related PDEs are transformed into ODEs, and the bvp4c Matlab function has been utilized to get a numerical solution and trained through a Bayesian regularization neural network (BR-NN). All other results are predicted BR-NN and found smooth agreement with bvp4c. In this study, the thermal proficiency is checked through thermal radiation and inclusion of nanoparticles [Ag, Cu, MoS₂]/H₂O in the base fluid (water) and impact of several physical parameters such as magnetic parameter ( M ) , inclined magnetic field angle ( ω ) , pressure gradient parameter ( β ) , shear-to-strain rate parameter ( α ) , Weissenberg Number ( We ) , Carreau fluid index parameter ( n ) , thermal radiation parameter ( R d ) , and Prandtl number on velocity and temperature profile over a three-dimensional wedge. Figure 4 shows the whole story of the current study.

Figure 4

Overall glimpse of the current study.

The purpose of training the model, 70% of the numerical data was allocated for training, 15% for testing and 15% for validation. 10 number of neurons are utilized in this network. Four physical parameters were examined: the shear-to-strain rate parameter, pressure gradient parameter, magnetic parameter, and thermal radiation parameter. Table 4 makes the illustration related to four parameters, each of which was tested in three different scenarios. Figure 5(a)–(d) displays the numerical results for performance validation, Mu, gradient, error histogram, regression, and function fitting curves for variations in the shear-to-strain rate parameter. The best performance validation was found 1.2459 × 10⁻⁸ for training, validation, and testing at 93 epochs, with a gradient of 9.9932 × 10⁻⁸, Mu of 1 × 10⁻⁸, and zero validation checks at 209 epochs. The error in the error histogram was 5.46 × 10⁻⁶. Regression analysis for training, validation, and testing showed a value of 1, indicating a linear relationship between input and output, with the best error bounds visible in the fitness graph. Figure 6(a)–(d) presents similar results for variations in the pressure gradient parameter. 1.2459 × 10⁻⁸, 9.8693 × 10⁻⁸, 1 × 10⁻⁸ and 1.12 × 10⁻⁵ best performance, (for training, validation, testing at 93 epochs), gradient, Mu, and error histogram, respectively. Figure 7(a)–(d) shows the results for variations in the magnetic parameter. The best performance validation was 1.9086 × 10⁻⁸ for training, validation, and testing at 199 epochs, with a gradient of 9.8284 × 10⁻⁸, Mu of 1 × 10⁻⁹, and zero validation checks at 199 epochs. The error in the error histogram was −6.7 × 10⁻⁸. Regression analysis for training, validation, and testing showed a value of 1, indicating a linear relationship between input and output, with the best error bounds visible in the fitness graph. Finally, Figure 8(a)–(d) presents the results for variations in the thermal radiation parameter. The best performance validation was 1.6627 × 10⁻⁹ for training, validation, and testing at 187 epochs, with a gradient of 9.9159 × 10⁻⁸, Mu of 1 × 10⁻⁹, and zero validation checks at 187 epochs. The error in the error histogram was −1.13 × 10⁻⁵. Regression analysis for training, validation, and testing showed a value of 1, indicating a linear relationship between input and output, with the best error bounds visible in the fitness graph.

Figure 5

(a)–(d) Pictorial facts related to the training process α .

Figure 6

(a)–(d) Pictorial facts related to the training process β .

Figure 7

(a)–(d) Pictorial facts related to the training process M .

Figure 8

(a)–(d) Pictorial facts related to the training process R d .

Figures 9(a) and (b)–24(a) and (b) show the impact of several physical parameters on f ′ ( η ) , g ′ ( η ) profiles. Each figure is classified with ternary [Ag, Cu, MoS₂] and bi-hybrid [Ag, Cu] nanofluids, and the numerical solution is obtained by trained BNN technique and compared with the bvp4c method and error plots are also shown for each parameter. Figures 9(a) and (b)–12(a) and (b) are related to velocity ( f ′ ( η ) , g ′ ( η ) ) profiles of ternary [Ag, Cu, MoS₂] and bi-hybrid [Ag, Cu] nanofluids attached with Weissenberg number. Figures 9(a) and (b)–11(a) and (b) show the impact of Weissenberg number on f ′ ( η ) , g ′ ( η ) , while Figures 10(a) and (b)–12(a) and (b) are related to error estimation with respect to f ′ ( η ) , g ′ ( η ) . The velocity of nanofluids (copper, copper oxide, and aluminum oxide in water) on a three-dimensional (3D) wedge decreases at high Weissenberg numbers. This parameter measures the competition between the viscous and elastic properties of the fluid. When Weissenberg Numbers increase, then elastic forces become more dominant, and this dominancy leads to enhanced flexible effects such as stretching and alignment of nanoparticles along the flow direction, and anisotropic viscosity is produced. Hence, overall, the velocity of the fluid lowers. Furthermore, a high Weissenberg number strongly influences nanofluid viscosity and increases resistance to flow. Figures 13(a) and (b)–16(a) and (b) are related to velocity ( f ′ ( η ) , g ′ ( η ) ) profiles of ternary [Ag, Cu, MoS₂] and bi-hybrid [Ag, Cu] nanofluids attached with a Carreau fluid index number. Figures 13(a) and (b)–15(a) and (b) show an impact of Carreau fluid index number on f ′ ( η ) , g ′ ( η ) , while Figures 14(a) and (b)–16(a) and (b) are related to error estimation with respect to f ′ ( η ) , g ′ ( η ) . The high Carreau fluid index parameter classifies the fluid into shear-thinning ( n > 1 ) and shear-thickening ( n < 1 ) fluid. Velocity of shear-thinning ( n > 1 ) fluid increases and velocity of shear-thickening ( n < 1 ) fluid decreases with increasing this parameter. Increased parameter gives low viscosity, fluid becomes thinning, and increase the overall flow velocity.

Figure 9

(a) and (b) Velocity of ternary [Ag, Cu, MoS₂] and bi-hybrid [Ag, Cu] Carreau nanofluids attached with Weissenberg number.

Figure 10

Error estimation between bvp4c and ANN technique with variation of Weissenberg number on the velocity f ′ ( η ) profile.

Figure 11

(a) and (b) Velocity of ternary [Ag, Cu, MoS₂] and bi-hybrid [Ag, Cu] nanofluid attached with Weissenberg number.

Figure 12

Error estimation between bvp4c and ANN technique with a variation of Weissenberg number on the velocity g ′ ( η ) profile.

Figure 13

(a) and (b) Velocity of ternary [Ag, Cu, MoS₂] and bi-hybrid [Ag, Cu] nanofluids attached with the Carreau fluid index parameter.

Figure 14

Error estimation between bvp4c and ANN technique with a variation of the fluid index on the velocity f ′ ( η ) profile.

Figure 15

(a) and (b) Velocity of ternary [Ag, Cu, MoS₂] and bi-hybrid [Ag, Cu] nanofluids attached with the Carreau fluid index parameter.

Figure 16

Error estimation between bvp4c and ANN technique with a variation of the fluid index on the velocity g ′ ( η ) profile.

Figure 17

(a) and (b) Velocity of ternary [Ag, Cu, MoS₂] and bi-hybrid [Ag, Cu] nanofluids attached with Hartmann parameter.

Figure 18

Error estimation between bvp4c and ANN technique with a variation of the Hartmann number on the velocity f ′ ( η ) profile.

Figure 19

(a) and (b) Velocity of ternary [Ag, Cu, MoS₂] and bi-hybrid [Ag, Cu] nanofluid attached with Hartmann parameter.

Figure 20

Error estimation between bvp4c and ANN technique with a variation of the Hartmann number on the velocity g ′ ( η ) profile.

Figure 21

(a) and (b) Velocity of ternary [Ag, Cu, MoS₂] and bi-hybrid [Ag, Cu] nanofluids attached with a pressure gradient parameter.

Figure 22

Error estimation between bvp4c and ANN technique with a variation of the pressure gradient on the velocity f ′ ( η ) profile.

Figure 23

(a) and (b) Velocity of ternary [Ag, Cu, MoS₂] and bi-hybrid [Ag, Cu] nanofluids attached with a pressure gradient parameter.

Figure 24

Error estimation between bvp4c and ANN technique with a variation of the pressure gradient on the velocity g ′ ( η ) profile.

Figures 17(a) and (b)–20(a) and (b) are related to velocity ( f ′ ( η ) , g ′ ( η ) ) profiles of ternary [Ag, Cu, MoS₂] and bi-hybrid [Ag, Cu] nanofluids attached with Hartmann number. Figures 17(a) and (b)–19(a) and (b) show the impact of Hartmann number on f ′ ( η ) , g ′ ( η ) , while Figures 18(a) and (b)–20(a) and (b) are related to error estimation with respect to f ′ ( η ) , g ′ ( η ) . Geometrical interpretation shows that the velocity of a nanofluid comprising [Ag, Cu, MoS₂] nanoparticles dispersed in water for high Hartmann numbers on a 3D wedge decreases. The Hartmann number is based on the strength of the applied magnetic field, and it significantly alters the fluid dynamics. When this parameter gets high, magnetic field strength becomes dominant and Lorentz force is produced and this makes resistance within flow and velocity decrease. Figures 21(a) and (b)–24(a) and (b) are related to velocity ( f ′ ( η ) , g ′ ( η ) ) profiles of ternary [Ag, Cu, MoS₂] and bi-hybrid [Ag, Cu] nanofluids attached with pressure gradient. Figures 21(a) and (b)–23(a) and (b) show impact of pressure gradient on f ′ ( η ) , g ′ ( η ) , while Figures 22(a) and (b)–24(a) and (b) are related to error estimation with respect to f ′ ( η ) , g ′ ( η ) . The decrease in the velocity of nanofluid around a three-dimensional wedge is found by increasing the pressure gradient parameter. The pressure gradient parameter is basically rate of change of pressure along the flow direction. In the context of nanofluids, the presence of nanoparticles rheological behavior becomes modified and under high-pressure gradients on a 3D wedge, the nanofluid experiences enhanced resistance and increased drag force and this leads to a reduction in velocity. The understanding of these dynamics is crucial for applications where nanofluid behavior is influenced by pressure gradients, aiding in the optimization and design of systems involving three-dimensional wedges under varying flow conditions. Figures 25(a) and (b) and 26(a) and (b) are related to temperature profiles of ternary [Ag, Cu, MoS₂] and bi-hybrid [Ag, Cu] nanofluids attached with thermal radiation parameters. Figure 25(a) and (b) shows impact of thermal radiation parameters on θ ( η ) , while Figure 26(a) and (b) is related to error estimation with respect to θ ( η ) . The elevation in the temperature of nanofluids at higher radiation parameters on a three-dimensional (3D) wedge gives a higher temperature distribution. High radiation parameters show that heat transport is augmented through radiative heat transfer mechanisms. Nanoparticles in fluids are capable of enhancing the absorption and emission characteristics, and this results in an increased capacity to absorb and emit thermal radiation. This process causes for elevated temperatures within the nanofluid on the 3D wedge surface. The practical significance of this result is used in applications for energy management and thermal control in various engineering systems like improved performance in solar collectors, cooling in electronic devices, and optimized thermal management in aerospace applications. Figures 27(a) and (b) and 28(a) and (b) are related to temperature profiles of ternary [Ag, Cu, MoS₂] and bi-hybrid [Ag, Cu] nanofluid attached with Prandtl number. Figure 27(a) and (b) shows the impact of the Prandtl number on θ ( η ) , while Figure 28(a) and (b) is related to error estimation with respect to θ ( η ) . Increased Prandtl number indicates a higher ratio of momentum diffusivity to thermal diffusivity. According to this capacity of the fluid to conduct heat becomes more powerful as compared to its ability to transfer momentum. This is the main reason for more efficient dissipation of thermal energy and this leads to a decrease in the nanofluid temperature. The practical implication of this phenomenon is in thermal systems engineering and heat exchanger design. Furthermore, this result has applications in various industries, including electronics for efficient cooling, device performance and longevity. Figures 29(a) and (b) and 30(a) and (b) are related to temperature profiles of ternary [Ag, Cu, MoS₂] and bi-hybrid [Ag, Cu] nanofluids attached with Hartmann number. Figure 29(a) and (b) shows impact of Hartmann number on θ ( η ) , while Figure 30(a) and (b) is related to error estimation with respect to θ ( η ) . The rise in the temperature of nanofluid is found with an increase in the Hartmann number on a three-dimensional (3D) wedge because due to this electromagnetic forces are produced and they are suppressing fluid motion. Hartmann number signifies strong magnetic field strength and that leads to enhanced magnetic damping of fluid flow and hence reduction in fluid motion diminishes convective heat dissipation. The practical significance of this result is found in applications where controlled temperature regulation is essential. Industries involving magnetic field manipulation, such as MHD systems can leverage this phenomenon for precise thermal management. Figure 31(a) and (b) and 32(a) and (b) are related to 3D visuals of velocities ( f ′ ( η ) , g ′ ( η ) ) profiles with respect to η with variation of Weissenberg number, Carreau fluid index, Hartmann number, and pressure gradient. Figure 31(a) and (b) show impact of Weissenberg number, Carreau fluid index on f ′ ( η ) , g ′ ( η ) , while Figure 32(a) and (b) show the influence of Hartmann number and pressure gradient parameter. Table 5 shows the numerical outcome of f ″ ( 0 ) and g ″ ( 0 ) when α is varied and Hartmann number set at 5. This examination aids in understanding the influence of magnetic field strength on the skin friction coefficients, offering valuable insights for applications involving magnetohydrodynamics. In Table 6, extension related to our exploration with shear stresses, f ″ ( 0 ) and g ″ ( 0 ) , for a large range α and value of Hartmann number is fixed at a value of 1. This nuanced investigation helps elucidate the sensitivity of skin friction coefficients to varying α , providing essential information for further theoretical and practical considerations. Tables 7–10 present a series of tables that are dedicated to the estimation of skin friction and Nusselt numbers for different nanofluid compositions specifically about ternary hybrid nanofluid and other related formulations. These tables are evidence for variation of skin friction and Nusselt numbers concerning diverse physical parameters. These tables contribute to their role in understanding the mechanism of heat transfer characteristics in fluid dynamics in the context of nanofluid applications. The increasing heat transport of ternary hybrid nanofluid is seen for several parameters like rising numerical values of thermal radiation and volumetric fractions of [Ag, Cu, MoS₂]. Higher numerical values of thermal radiation parameters indicate greater radiative heat exchange. Thermal radiation plays a crucial role in enhancing heat transfer within the nanofluid system. The incorporation of copper [Ag, Cu, MoS₂] in the nanofluid (water) contributes significantly to the enhancement of thermal conductivity. Volumetric fraction determines their concentration in the fluid, and as it increases, so does the overall thermal conductivity of the nanofluid. The effect of high thermal radiation and increased volumetric fractions of these nanoparticles leads to a notable augmentation.

Figure 25

(a) and (b) Temperature of ternary [Ag, Cu, MoS₂] and bi-hybrid [Ag, Cu] nanofluid attached with the thermal radiation parameter.

Figure 26

Error estimation between bvp4c and ANN technique with a variation of the radiation parameter on the temperature profile.

Figure 27

(a) and (b) Temperature of ternary [Ag, Cu, MoS₂] and bi-hybrid [Ag, Cu] nanofluids attached with Prandtl number.

Figure 28

Error estimation between bvp4c and ANN technique with a variation of the Prandtl number on the temperature profile.

Figure 29

(a) and (b) Temperature of ternary [Ag, Cu, MoS₂] and bi-hybrid [Ag, Cu] nanofluid attached Hartmann number.

Figure 30

Error estimation between bvp4c and ANN technique with a variation of the Hartmann number on the temperature profile.

Figure 31

The variation of velocity components along the x-axis and y-axis direction against η for the different values of (a) Weissenberg number and (b) power-law index parameter.

Figure 32

The variation of velocity components along the x-axis and y-axis direction against η for the different values of (a) Hartmann number and (b) pressure gradient parameter.