Modelling cross-diffusion in MHD Williamson nanofluid flow over a nonlinear stretching surface via Morlet wavelet neural networks

Abstract

In this paper, a new numerical technique was developed to investigate magnetohydrodynamic (MHD) flow of Williamson nanofluid past a nonlinear stretching surface imbedded in a porous medium laden with Soret and Dufour effects. The control equations, which are highly nonlinear partial differential equations, are first converted into ordinary differential equation (ODEs) using similarity transformation and then are solved effectively by the hybrid computational method applying Morlet Wavelet Neural Networks (MWNNs) combined with a heuristic optimizers neural network and particle swarm as MWNNs-PSO-NNA. The proposed MWNNs-PSO-NNA shows a very low mean square error and Theil’s Inequality Coefficient indicating that the accuracy of the model. To check the convergence and validation of the proposed approach, computing the hundred independent runs for statistical metrics. The fitness function, MSE and TIC values ranging from 10^–07 to 10^–05, 10^–09 to 10^–07 and 10^–06 to 10^–04 respectively. It is found that increasing the effects of the Williamson number, magnetic parameter, porosity and stretching index inhibit the velocity field while Brownian motion as well as the Williamson number enhances the temperature profile. The concentration rises with Soret and Brownian motion parameters but diminishes with intensified thermophoresis and magnetic influences. These findings confirm that the proposed hybrid model is not only computationally robust but also highly effective for solving complex fluid flow problems in engineering and applied sciences.

Introduction

Newton’s law of viscosity creates a direct proportionality between shear stress and shear rate. Though, this association does not hold for non-Newtonian fluids, in which viscosity differs with shear stress or shear rate, building a restriction of fluid to circulate dependent to dominate instances movement. Non-Newtonian fluids show diverse flow characteristics, considered into distinct types, including pseudoplastic fluids, dilatant fluids, Bingham plastic fluids, and viscoelastic fluids.

Magnetohydrodynamics (MHD) is an inter-disciplinary field that is a combination of electromagnetism and fluid dynamics to investigate the behavior of electrically conductive fluids under magnetic fields, and it is used extensively in engineering and astrophysics. Unsteady MHD movement of a fluid which is responding thermally over and movable plate with the properties of porosity, upward and inclination was studied by Raju et al.¹. More recent studied have analyzed various aspects of MHD flow^2,3, including thermal analysis of two-dimensional Darcy-Forchheimer flow in a Williamson fluid embedded with copper (Cu) nanoparticles. The interaction of transverse magnetic fields with electrically conducting fluids, which dampen motion through the Lorentz force, is described by magnetohydrodynamic (MHD) flows. Viscoelastic stresses and cross-diffusion of nanoparticles can partially offset magnetic damping and modify the momentum and thermal boundary layers in non-Newtonian and nanofluid systems. Our investigation of MHD Williamson nanofluid flow across a porous media with cross-diffusion processes is motivated by these effects.

The flow within the non- Newtonian fluids’ boundary layer is more significant due to their vast applications in industry. The Navier–Stokes equations themselves are not sufficient to predict and improve the rheological response of these materials. One better and recent solutions to overcome the aforementioned deficiency can be the Williamson fluid flow model (WFFM). The WFFM is characterized by the shear stress and shear rate power-law relation , where the power-law index defines the fluid’s rheological behavior. Specifically, exhibit shear-thinner. denotes the shear-thickening behaviour. The consistency index is the flow resistance of the fluid and is equal to its low shear-rate viscosity. In the Williamson fluid model, the viscosity is not constant but varies as a function of shear rate, and reduces with an increase in shear rate in shear-thinning fluids. The strength of WFFM lies in being able to capture shear-thinning and shear-thickening behaviors by altering the value of . This flexibility makes the model particularly appropriate designed for wide range of engineering applications involving fluids that aren’t Newtonian with shear-dependent viscosity characteristics. In porous-media and MHD contexts, the Williamson fluid flow model (WFFM), which is well-known for modelling pseudoplastic behavior through a shear-rate-dependent viscosity, has undergone various elegant extensions. Cross-diffusion effects in an MHD Williamson nanofluid across a nonlinear stretching sheet within a permeable medium were analyzed by Kumar et al.⁴ using WFFM, and they showed excellent agreement with benchmark solutions. The interaction between magnetic damping and porous resistance was then highlighted in a thorough numerical analysis of MHD Williamson nanofluid flow through a Darcy porous matrix by Khader et al.⁵. By adding ternary hybrid nanoparticles, Faizan et al.⁶ improved the model even more and demonstrated increased heat-transfer rates under various thermophoretic and Brownian conditions. Williamson fluid flow over a stretching sheet with Newtonian heating, thermal radiation, and source/sink terms was most recently studied by Kumar et al.⁷, highlighting the model’s adaptability in forecasting coupled thermal and momentum transport phenomena.

The Williamson fluid flow model (WFFM) is an empirical illustration resulting from fitting experimental facts rather than fundamental physical principles. A mathematical background is offered for relating shear-rate-dependent viscosity, with a comparatively simple mathematical structure which improves usability and interpretability. This simplicity enables analytical solutions and increases computational efficiency, making it mainly advantageous in applications such as polymeric dispensation, nourishment production, and pharmacological invention. Nadeem et al.⁸ discussed flow behaviour of Williamson’s fluid past the stretched sheets, and Megahed⁹ discussed viscid evaporation as well as radiation of heat effect in a Williamson fluid under a non-linear stretching sheet. Bibi et al.¹⁰ have carried out numerical analysis of heat transfer in an MHD Williamson fluid past the porous stretched sheets, and Salahuddin et al.¹¹ have discussed the Cattaneo-Christov mass as well as heat diffusion process, including double stratification in a dissipative Williamson fluid.

Nanofluids have a various range of properties that concentrate them highly appropriate for several industrial applications. Nanofluids possess enhanced thermal conductance, enhanced heat transfer performance, enhanced lubrication and wear protection, stability in nanoparticle suspensions, and the capability to modify their properties over the selection of diverse nanoparticle–base fluid amalgamations. Moreover, nanofluids establish compatibility with existing thermal and mechanical systems while presenting the potential for multifunctionality. All these advantages come from the exclusive features of nanotechnology particles. Major uses of nanofluids are thermal transmission as well as cooling tools, tribological enhancements, oil as well as gas production and medical technologies.

The overview of nano-particles into a Williamson base fluid drastically changes the fluid properties, resulting in enhanced heat transfer and other desirable characteristics. Williamson nanofluid flow (WNFF) has stored significant attention due to its extensive applications. The heat transport features of WNFF were studied by Nadeem et al.¹², whereas the impacts of thermos diffusion and radiation of heat on a permeable stretchable sheets containing WNFF were observed by Bhatti et al.¹³. Furthermore, Reddy et al.¹⁴ evaluated mass and heat transfer in an MHD WNFF over a stretched sheet with variable thickness and thermal conductivity under the effect of thermal radiation. Mabood et al.¹⁵ investigated a finite difference method to observe the influence of heat radiation as well as the heat sources on MHD WNFF across a constantly affecting the layer that is heated. Mebarek-Oudina et al.¹⁶ examined a modified Buongiorno classic to explore the hydramagnetic performance in water with magnetite nano-fluid. Ramesh et al.¹⁷ executed a comprehensive analysis on the magnetized characteristics of radiant tiny Carreau fluid movement within the micro channel. The impacts of heat as well as diffusion of viscosity in the use of MHD-WNFF over a non-linearly stretched surfaces in a permeable standard¹⁸. In recent studies, the amount of hybrid studies in Nanozhida and Nevinsky liquid has increased, increasing ideas for thermal performance. Bouselsal et al.¹⁹ showed that the hybrid alignment of Alâ Oâmwcnt changed the shape of the tube to significantly increase the heat transfer efficiency of the tube and shell. Mebarek-oudina and Chabani²⁰ emphasized an important role in thermal energy and temperature control in intellectual heat systems, taking into account the development of materials with NEPCM (NASALIZED PHASE Change). On the other hand, Loganatan and Sangit²¹ studied the effects of Williamson parameters on the water flow under the vertical plate, showing how the viscosity changes the behaviour of the border and heat transport. This study strengthens the understanding of hybrid thermodynamics in nano hyds and the impact of Nevin skiing lacquate on the convection system.

Non-linearly stretched sheet discussed as non-linearly stretchable sheets, give unique mechanical properties that purify them highly valuable for an extensive range of industrial applications. These sheets are categorised by the straining stresses that are not directly proportionate with the displacements, demanding specialized mathematical models to explain their behaviour. Such models show a key role in the study of transport of heat and fluid motion for numerous scientific procedures. Das²² considered how fractional slippage impacted on the nanofluid’s boundary surface that travel over the non-linearly stretched sheets (NLSS) under quantified ambient temperatures. Bilal et al.²³ inspected the magnetohydrodynamic (MHD) three-dimensional bordering zone movement to a Williamson-fluids subjected to non-linearly stretched in both directions. Furthermore, Abo-Dahab et al.²⁴ discussed the influence of injector and sucking on the use of MHD Casson’s nanofluid flowing through an NLSS within a heating porosity layer. Hayat et al.²⁵ scrutinised the MHD bordering sheet motion of Powell-Eyring nanofluid past a non-linearly stretched sheets with mutable wideness. Additionally, Qayyam et al.²⁶ considered an MHD flow of an inferior nanofluid technology over a non-linearly stretched measureable of varying wideness, integrating the impacts of radiating heat and chemicals processes. These research highlight the wide-ranging industrial applications of nonlinear stretching sheets, containing their consumption in electronic stretching, packaging, energy garnering, automation, and soft robotics. Ahmed et al.²⁷ has formulated effective similarity to analyse Williamson’s nanozid’s self -fluid mechanics (MHD) stream along the nonlinear stretch surface. Their research shows how much the nonlinearity of surface stretching affects the speed and thermal boundaries, giving a deeper understanding of the undergraduate vision simulation.

The study of nanofluid flows combined with hybrid nanofluid research has led to important developments in grasping heat and mass transfer dynamics within magnetohydrodynamic (MHD) systems and porous media environments. The study by Ragulkumar et al.^28,29 explored dissipative MHD nanofluid flows along vertical cones and examined their behavior with thermal radiation effects alongside chemical reactions and mass/heat flux while emphasizing the significance of geometrical and thermophysical factors. Likewise, Choudhary et al.³⁰ developed a mass-based hybrid nanofluid model to study thermal radiation effects in MHD flow across a wedge located in a porous medium. An unsteady MHD hybrid nanofluid flow model over nonlinear porous surfaces with gyrotactic microorganisms was researched by Choudhary et al. The work of Choudhary et al.³¹ yielded significant understanding of how biological processes interact with thermal dynamics within complex geometrical structures.

Loganathan et al.³² analyzed Williamson nanofluid flow using an entropy framework that integrates bioconvective mechanisms and entropy generation over a Riga plate. An entropy framework to examine Williamson nanofluid flow over a Riga plate while considering triple stratification effects, radiation impact, and swimming microorganisms was used. This method highlights the importance of thermodynamic irreversibility within nanofluidic systems. Eswaramoorthi and Sivasankaran³³ performed entropy optimization analysis on MHD Casson-Williamson fluid flow across a convectively heated stretching surface and utilized the Cattaneo-Christov dual-phase lag model which provided additional insights to theoretical heat transport models in non-Newtonian fluids. Together, these investigations provide a strong basis for investigating sophisticated nanofluid models, particularly those that incorporate radiative transport, hybrid nanoparticles, MHD effects, porous media, and bio-convective processes.

The development of multi-component nanofluids such as ternary, tri, and tetra hybrid nanofluids, is becoming more popular in recent research due to their higher thermal conductivity and tunable properties. Farayola et al.³⁴ studied the thermophysical behavior of a temperature-dependent TiO₂–SiO₂–ZnO–Fe₂O₃/PAO tetra-hybrid nanofluid, considering suction effects along a vertical porous surface. His results highlighted the importance of surface permeability and variable properties regarding heat transport optimization. In³⁵, Haq et al. analysed mixed convection nanofluid flow over an inclined irregular surface with internal heat sources and chemical reaction, explaining the influence of geometry and reactive phenomena on flow and temperature fields. As part of the rotational systems, Patgiri et al.³⁶ studied the impact of Soret and Dufour effects on viscoelastic tetra-hybrid nanofluid flow across a stretchable rotatory disk, whereas Paul et al.³⁷ examined the behaviour of Casson ternary hybrid nanofluid over a porous rotating disk in the presence of Hall current. Also, Paul et al.³⁸ studied the Darcy–Forchheimer effects of flow in Ag–ZnO–CoFe₂O₄/H₂O ternary hybrid liquid nanofluids with EMHD interactions on a spinning disk and discussed its complicated momentum transfer characteristics. Their other work³⁹ on tri-hybrid engine oil nanofluids emphasized the importance of porosity as well as the effect of rotation on the thermal and flow behaviour of the system. All these studies form a basis for the non-isothermal flow of multi-component nanofluids with rotation and suction, revealing additional electromagnetic and reactive transport effects.

Modern studies bring attention to the non-Newtonian behaviour of fluids as well as the thermodiffusion (Soret) and diffusion–thermo (Dufour) effects, and also to the interplay of different effects in nanofluids transport phenomena. Abhijith et al.⁴⁰ studied MHD flow of Reiner–Rivlin nanofluids with nonlinear chemical reaction and Soret–Dufour effects calculating how reactive and cross-diffusive processes affect transport mechanisms. Taka et al.⁴¹ employed non-similar approach to examine Carreau nanofluid flow under quadractic thermal radiation and Soret–Dufour coupling, noting the action of non-Newtonian viscosity under changing magnetic and thermal fields. Areekara et al.⁴² analysed the EMHD flow of Casson blood-gold nanofluids with non-linear heat source and Arrhenius type kinetics in the framework of biofluidics and metallic nanofluids, demonstrating the importance of nanofluid particle radius on entropy and energy profiles. Their complementary work⁴³ included the second-order velocity slip boundary condition, demonstrating the impact of nanoscale boundary condition and electromagnetic forces on heat transfer increase in Casson nanomaterials. Seid et al.⁴⁴ concentrated at the movement of oscillation of a certain fluid that is viscous over the curled, deformable layer while including the diffusion across boundaries effects.These investigations add to the expanding corpus of research that models high-fidelity nanofluid flows controlled by multi-scale thermal processes, non-Newtonian effects, slip dynamics, and radiative transmission.

Zafar et al.⁴⁵ studied the effects of active as well as passive control schemes with respect to bioconvection flow of Carreua nanofluid over thermal radiation and Cattaneo–Christov double diffusion with non-Fourier heat and mass transfer laws. Kamal et al.⁴⁶ surveyed Marangoni convection and Girotaxic Microorganism’s Maxwell undercut stream in the rotary disk, including the analysis of the second law (entropy) to evaluate the reversal reversibility. In relevant studies of Ali et al.⁴⁷ showed a remarkable improvement in bio configuration transportation by analyzing the flow of satterbi underwear built into the microorganisms and chemical reactions of the magnetic field. Also Shah et al.⁴⁸. It is converted to a chemical reactive-hybrid stream of nano-item streams under the influence of heat radiation near the porous plate to stretch exponentially using the Laplace transformation method for heat and mass transfer. Ahmed et al.⁴⁹ showed how the characteristics of the congestion point affect the transmission controlled by heat radiation by using a semi-analytical approach to study the behavior of Walter-B Nanofluids on the stretching surface under the heat source/immersion condition. Such contributions reflect advanced modelling of hybrids and bio kerner’s nano hyds with complex physics, providing ideas for multiple fluids was studied^{50, 51, 52–53}. This study uses a hybrid computational approach based on Morlet Wavelet Neural Networks (MWNNs) combined with Particle Swarm Optimization (PSO) and the Neural Network Algorithm (NNA) to handle the nonlinear complexity of the governing equations. The recent relevant literature of machine learning approaches to intricate heat and mass transfer phenomena has advanced significantly in recent literature^{52, 53, 54, 55, 56, 57–58}.

Modeling complex fluid flow with cross-diffusion and porous effects remains challenging due to nonlinear interactions and coupled transport phenomena. This motivates the development of a robust, intelligent solver that ensures high accuracy and computational efficiency. The novelty of this study lies in the integration of Soret–Dufour cross-diffusion effects and porous medium characteristics within the framework of non-Newtonian Williamson fluid dynamics, solved using a hybrid machine learning-based approach. By combining Morlet Wavelet Neural Networks (MWNNs), Particle Swarm Optimization (PSO), and a Neural Network Algorithm (NNA), the proposed method offers a powerful alternative to conventional numerical schemes. This innovative approach not only enhances accuracy and convergence but also addresses the complex coupling of thermal and solutal transport mechanisms in porous media, marking a substantial contribution to advanced computational fluid dynamics modelling.

Research questions

The current study seeks to answer the following important research issues in light of the aforementioned motivations.

1. How can complex non-Newtonian nanofluid flow models involving MHD and porous media be solved more efficiently and accurately using machine learning-based solvers like MWNNs-PSO-NNA?

2. How can cross-diffusion effects (thermophoresis and Brownian motion) affect the performance of heat and mass transfer in viscoelastic nanofluids, especially when nonlinear stretching and activation energy conditions are present?

3. What effects does the interplay between porosity (K) and magnetic field intensity (M) have on velocity gradients and boundary layer separation in Williamson hybrid nanofluids?

4. Can real-time industrial processes, including flow through bio-porous tissues or cooling in microelectronic devices, be accurately simulated by the suggested hybrid fluid model under different Weissenberg (We) and rheological (n) parameters?

5. How much does the thermophysical behaviour of sophisticated hybrid nanofluids in magneto-thermal settings change when heat generation/absorption and radiation effects are included?

Core framework of Williamson fluid flow

The subsequent continuity and momentum equations are applicable to incompressible Williamson fluid flow:

1

2

where is the velocity, is the density, is the material time derivative, is the specific body force vector, and is the Cauchy stress tensor. The following is the formulation of the structural equations for the regime model for the Williamson fluid:

3

where

4

where is the pressure, is the first Rivlin–Ericksen tensor, is the shear rate, and is a positive time constant. The shear rate is expressed as:

5

where

6

is known as the second invariant’s strain tensor. In this instance,  = 0 and are considered, the extra stress tensor simplifies into:

7

Using binomial series expansion:

8

Likewise, and the origins of the extra tensor are represented as

The component structure for continuity as well as momentum equations can be expressed:

9

10

11

Mathematical formulation

The present research investigates behaviour to the multi-dimensional, dense, inflexible Williamson’s nanofluid as it flows over a non-linearly stretched sheets within its domain . The dissemination of velocity across sheet varies , also temperature of the wall is expressed by the relation where , represents the temperature of the surrounding fluid and is the concentrating diffusion of nanoparticles. An adaptable magnetism with concentration is useful in the fluid flow’s transverse direction. In this study, the impact of the electric field and resulting magnetism is ignored with the use of magnetite minimal Reynolds number. A movement is reflected to be associated alongside the X-axis, with the Y-axis concerned with perpendicular with it. An analysis integrates special effects of an external magnetic field, Dufour effect, as well as Soret effect. Figure 1 indicates the corporeal formation of the motion of Williamson’s nanofluid, giving the graphic image of the flow dynamics and boundary conditions.

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Fig. 1

Geometric interpretation.

The main non-linearly partial differential equations (PDEs) relating a phenomena of flow of fluid in given circumstances are expressed on the established frameworks. The following is the sketch of the problem’s mathematical formulation:

12

13

14

15

An appropriate conditions are stated as follows for the indicated Williamson nanofluid flow:

16

where is the magnetic field intensity; ​ represents the ratio of effective heat capacity; is the kinematic viscosity; and are the velocities in the and directions, respectively; is the permeability of the porous medium; is the thermal diffusivity; is the density of the fluid; ​ is the Brownian diffusion coefficient; is the index of the nonlinear stretching sheet; is the electrical conductivity; is the thermophoresis coefficient; is the wall temperature; is the wall concentration; is the free stream temperature; and is the free stream concentration.

The succeeding similarity variable transformations are developed for reducing the nonlinear partial differential equations (PDEs)^{12, 13, 14–15} and the corresponding boundary conditions (16) into a system of nonlinear ordinary differential equations (ODEs):

17

Here, and indicate concentration function as well as the temperature function of , respectively. Equation (12) is satisfied precisely the same way when the previously quantified similarity modifications are applied to Eq. (12) through (15). Equations (18) through (20) below display the changed ODEs (ordinary differential equations) that resulted:

18

19

20

The following form is made by means of the associated boundary conditions (16) for the suggested motion of fluid:

21

In Eqs. (18) to (21), the following is the definition of the parameters:

Magnetic parameter Williamson nanofluid parameter Prandtl number Brownian motion parameter thermophoresis parameter porous parameter Schmidt number Soret number Dufour parameter Upon evaluating the situation, we established that and are functions of . To deal with this problem, which is a non-comparable outcome for recent issue, we intend to make it non-dimensional and have as well as , respectively. Therefore, related parameters can be converted as follows: and . For practical investigation, new magnitudes like the skin friction ratio , the Nusselt quantity ​, and the Sherwood quantity ​ as well are presented. Following is the interpretation of these quantities:

Here, heat flux , the mass flux and the wall shear stress By means of similarity variable transformations in Eq. (17), the skin friction proportion , the Nusselt quantity , and the Sherwood quantity as well are converted:

22

here is the Reynolds number.

Solution methodology

A solutions for the PDEs/ODEs by means of outdated exact or numerical tactics have restricted applicability as they frequently unable to solve complicated problems. Machine learning procedures highlight these restrictions and give perfect solutions. MWNNs (Morlet Wavelet Neural Networks) incorporated with PSO (Particle Swarm Optimization) and neural networks are used to evaluate the given system. The proposed MWNNs-PSO-NNA method outperforms traditional analytical and numerical techniques in terms of accuracy, convergence speed, and computational efficiency. Compared to methods like ND solve, bvp4c and Runge–Kutta, it yields significantly lower error margins and requires less computational time. These advantages confirm its robustness and suitability for solving highly nonlinear MHD flow problems.

Morlet wavelet neural networks procedure

Conventional analytical and numerical methods frequently prove inadequate for tackling intricate, nonlinear differential systems like the present MHD flow issue. To address these constraints, we utilize a hybrid computational methodology that combines Morlet Wavelet Neural Networks (MWNNs) with Particle Swarm Optimization (PSO) and a Neural Network Algorithm (NNA). This framework integrates the robust approximation abilities of wavelet-based neural networks with the global search efficacy of particle swarm optimization and the local refinement capabilities of gradient-based neural network algorithms.

These fundamental steps form the basis of the overall solution strategy

Converting PDEs to ODEs

Appropriate similarity transformations are used to reduce the controlling partial differential equations to a system of nonlinear ODEs.

Building a MWNN-based approximate solution

Morlet Wavelet Neural Networks are used to represent the reduced ODEs into fitness function. Because these networks employ Morlet wavelet functions as activation function, steep gradients and the nonlinear behaviour present in fluid flow issues can be approximated locally and across many resolutions.

Optimization with PSO and NNA

A two-phase method is used to optimize the weights and biases of the MWNN model:

PSO phase

Particle Swarm Optimization is used to carry out an initial global optimization. According to the usual PSO update equations.

NNA phase

A Neural Network Algorithm is used to adjust the parameters using the best weights and biases of PSO finds a promising area in the solution space.

Fitness function and convergence validation

Theil’s Inequality Coefficient and Mean Squared Error (MSE) are used to verify the accuracy and convergence of the suggested method. In comparison to conventional methods, this MWNN–PSO–NNA framework has demonstrated superior accuracy and convergence behaviour in approximating solutions to complicated nonlinear systems.

MWNN formulation

In many fields, NNs (neural networks) use the MW (Morlet wavelet) function to produce accurate, consistent, and stable results. The following equations are shown, taking into consideration the input, hidden, and output layer derivatives:

23

where denotes the number of neurons, shows the vectors of unknown weights i.e. , , , , the velocity profile output. Similar definitions apply to the results of temperature and concentration profiles for NNs. Compared to other activation functions, the Morlet-Wavelet (MW) function is significantly non-linear. MWNNs play an important part in time series analysis.

24

Equation (23), one has

25

26

27

28

29

30

31

32

33

34

Fitness formulation: neural networks based

The formula for the fitness relation is as follows:

35

where , , and are unverified error based on Equations. (18) – (21). One has

36

37

38

Meta-heuristic optimization algorithms

In recent years, meta-heuristic optimization algorithms have been presented to tackle complicated computational issues, together with ant colony optimization⁵⁹, particle swarm optimization⁶⁰, and the kill herd algorithm⁶¹. Researchers have publicised a mounting concentration in these algorithms due to their proficiency to tackle both constrained and unconstrained optimization issues commendably. A comprehensive study of these approaches exposes their application in test suite optimization with path convergence optimization, and many developing scientific encounters.

Particle swarm optimization

This algorithm is stimulated by the collective performance of birds in a swarm. Its key advantage lies in its capability to proficiently tackle nonlinear and complex models that are frequently tough to solve by means of traditional methods. In this technique, the change of particle placement is conducted by both the swarm’s global solution and the individual solutions by means of an iterative method. This tool reflects the adaptive conduct of a flock of birds regulating to ecological changes.

In PSO, the personal best () signifies the best-performing point of an individual particle, whereas the global best () indicates the best-performing spot among all particles in the swarm. The position and movement updates of the particles are influenced by these two main factors, thus improving the general search proficiency. The procedure starts with a particle dispersion that is random and uses the recent global best position and local best position to iteratively update their velocities and positions. In PSO, the update process has the general form:

Particle velocity update:

39

40

where are the arbitrary vectors in [0, 1], is the particle position, the swarm’s particles range is j, and velocity vector at position j. Moreover, the above-mentioned equations are utilized to update the velocities and positions of the particles whereas allowing for linearly reducing weights, local and global acceleration constraints , and inertia weight .

Neural networks algorithm

This original meta-heuristic method associations concepts from biological nervous systems and artificial neural networks (ANNs)⁶². NNA skillfully combines complex optimization problems with neural network properties in a variation of scientific fields, while ANNs are mainly proposed for prediction tasks. Through the basic architecture of neural networks, NNA shows strong global optimal performance. It's important that NNA contrasts from predictable meta-heuristic systems in that it just uses size of population and stopping standards, removing the condition for further factors. The four crucial vital mechanisms of the NNA algorithm are as follows:

Update population

Based on the NNA developments, the population is updated using the weight matrix. in which symbolizes the individual’s weight vector, and indicates the person’s position. Interestingly, the number of variables is indicated by . Additionally, the following is a mathematical illustration of the creation of a new population:

In this instance, is the existing iteration count, and is the size of population. signifies the outcome for the person at interval , and denotes the solution, computed with the proper weights, for the specific at the same period point. Additionally, the weight vector is stated by means of the following formula:

Update weight matrix

A key component of the NNA process for making a innovative population is the weight matrix . The matrix of weights ’s dynamics is clarified by:where is the objective weight vector and is a random value that belongs to the [0, 1] uniform distribution. The crucial element is both and the target solution have matching indices. To clarify, if corresponds to at time t, then and re aligned in an equivalent manner.

Bias operator

This operator has key role in algorithm of neural network is to facilitate its capabilities to explore the world. When measuring the amount of bias introduced, a modification factor, represented by the symbol assumes significance. This factor may be updated through:

A bias weight matrix as well as a bias population exist in this system. They described as follows: The bias population operator consists of two variables: a randomly generated number , a set represented by and at random a number generated. Let and stand for the variables’ lower and upper bounds, respectively. , is the product of extreme rate of and . integers picked at random using interval frame the set . Therefore, the bias population can be described as:

In this case, indicates a random number between 0 and 1 that is evenly distributed. Two other variables comprised for bias matrix: R represents a set and a number produced at random. The value of is found by captivating the maximum of Concurrently, selected integers at random from the interval structure the set . Thus, the following is the detailed narrative of bias weight matrix:where is a equally dispersed random number between 0 and 1.

Transfer operator

The transfer operator, which concentrates on NNA’s local search capability, is supposed to produce the superlative result in the direction of the present ideal outcome. The following equation that demonstrates this:where is a number chosen at random from an even distribution of [0, 1]. NNA is in progress bywhere is a number that is picked at random in the interval [0, 1]. Figure 2 below shows the structure of the study’s overall flow chart (Table 1).

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Fig. 2

The MWNN–PSO–NNA framework flowchart illustrates how wavelet neural networks optimized by PSO and NNA are used to solve nonlinear equations.

Table 1. Best of hyper parameters of MWNN through PSO-NNA for scenario 1.

− 0.198855956	0.569055643	− 0.092611893
− 2.583980393	− 3.479144992	− 0.928911825
− 0.510369306	2.513555286	− 0.06140709
− 0.385104889	− 2.802855818	10.00000000
2.527197522	3.579332607	0.586881383
0.322221267	1.253971788	− 5.579278157
− 0.325835573	− 0.148106064	− 1.540373682
3.668789655	4.178529076	− 1.620778057
− 0.934411022	− 2.583436867	0.892322319
2.285389384	7.347137506	− 4.686511311

Results and discussion

There are many analytical or numerical methods exist in literature to solve the PDEs/ODEs but each have their own limitation. The machine learning approaches to tackle these issues very easy. In this study, Runge–Kutta fourth-order and MWNNs-PSO-NNA approach to find the solution of ordinary differential equation based on fluid flow. There are many activation function used for tune the networks, the present research used the Morlet Wavelet as an activation function. Also have different optimizers for optimized the network biases and weights, global optimizers are utilized especially particle swarm (PSO) and artificial neural networks based optimizer neural networks algorithm (NNA). Morlet Wavelet Neural Networks combined with PSO and NNA offer fast convergence, high accuracy, and strong generalization for solving nonlinear problems. This hybrid approach efficiently captures complex patterns and reduces computational cost. By offering a more effective approximation for nonlinearities, preserving accuracy, and lowering computational costs, unsupervised ANNs (artificial neural networks) suggest benefits over predictable analytical and numerical approaches for solving the magnetohydrodynamics flow of Williamson nanofluid problem through a nonlinear stretching sheet submerged in a porous medium. Traditional approaches have limited applicability in solving complex nonlinear differential equations, such as the MHD Williamson’s nanofluid dynamics through non-linearly stretched sheets absorbed in a penetrable material. The temperature, velocity, and concentration outcomes using MWNNs with hybridization of PSO and NNA, and these stage by stage exist in Figs. 2 and 3. Equation (35) presents the fitness function model for this concern using a boundary condition based on MWNNs. The MWNNs used in this work implement a superficial system structure, involving of an input layer with 21 neurons, a single concealed layer including 10 neurons, and an output layer with 21 neurons. The MWNNs methodology employed one concealed layer with 10 neurons, whereas the input and output layers were formed with 21 neurons each. The input neurons were considered to signify the dimensionless parameters of the problem, whereas the output neurons were used to illustrate the velocity, temperature, and concentration outlines. The PSO (particle swarm optimizer) was employed to improve the biases and weights of the unverified stochastic neural network-based fitness function/error, which were chosen at random between − 10 and 10. To improve accuracy, the neural network algorithm (NNA) employs the optimal weights and biases derived from PSO as an initial population. The optimized hyper parameters of MWNNs for two different scenarios are tabulated in Tables 1 and 3, these valued used in MWNNs function for prediction of velocity profile as well as validate the MWNNs methodology. The fitness function, optimized hyperparameters, mean squared error (MSE), theif inequality coefficient (TIC) and comparison plots presented in Fig. 4 for check the convergence and efficiency of the proposed approach. A solution set for the model is obtained by integrating the governing equations using the Runge–Kutta fourth-order technique as a reference solution (Table 2).

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Fig. 3

Flow chart design methodology.

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Fig. 4

Comparison and convergence plots.

Table 2. Quantitative comparison analysis between the reference solution and MWNN-PSO-NNA for scenario 1.

	Reference solution	MWNN-PSO-NNA	Absolute error
0	1	0.999976962	2.30E−05
0.1	0.87644493	0.876404512	4.04E−05
0.2	0.76834419	0.768313331	3.09E−05
0.3	0.673432672	0.673402431	3.02E−05
0.4	0.589855166	0.589835415	1.98E−05
0.5	0.516063545	0.516044475	1.91E−05
0.6	0.450748639	0.450721991	2.66E−05
0.7	0.392792233	0.392765046	2.72E−05
0.8	0.34123206	0.341215456	1.66E−05
0.9	0.29523486	0.295231827	3.03E−06
1	0.254075461	0.25407922	3.76E−06
1.1	0.217119985	0.217119944	4.13E−08
1.2	0.183812158	0.183800895	1.13E−05
1.3	0.153662009	0.153639068	2.29E−05
1.4	0.126236362	0.126207857	2.85E−05
1.5	0.101150829	0.101125823	2.50E−05
1.6	0.078062998	0.078048647	1.44E−05
1.7	0.056666506	0.056664186	2.32E−06
1.8	0.036685978	0.036690177	4.20E−06
1.9	0.017872548	0.017873875	1.33E−06
2	− 4.04288E−08	− 7.07E−06	7.03E−06

The proposed numerical technique is highly accurate, as demonstrated by the computed profiles that strictly satisfy the imposed boundary constraints asymptotically. This fluid flow problem’s convergence domain is limited to 0 ≤ M, K ≤ 0.5, 0 ≤ n ≤ 2.5, 0 ≤ We ≤ 4.5, 0 ≤ Pr ≤ 1.45, 0 ≤ Nt ≤ 3.0, 0 ≤ Sr ≤ 0.7, 0 ≤ Nt, Sc, Du ≤ 0.4. All of the findings shown here have been methodically produced and verified against benchmark solutions within these parameter boundaries. This strategy ensures consistency and dependability in the study by closely adhering to the technique and parameter ranges set out by Kumar et al. ⁴. Concentration profile with respect to Du and K presented by Fig. 5a, b, represents that these are the significant parameters which control the diffusion rate. With the rise of levels of Du, molecular diffusion increases with increased concentration gradients. With the increase of K the concentration drops drastically with increased spreading of the solute. Figure 6a, b indicates that increasing values of serve to reduce concentration, noting the dominant role of magnetic effects in controlling diffusion. Parameter M on the contrary, makes the concentration increase due to its effect on viscosity and flow geometry. parameter, as indicated by Fig. 7a, b, affects concentration positively by highlighting that it increases mass transfer. To the contrary, increased We values decrease concentration by highlighting that it influences stretching pressures within the medium. Figure 8a, b illustrates that molecular diffusion is hindered by the impact of the enhanced Schmidt number, Sc, that increases the concentration gradient. The thermal impact of Pr affects mass transfer characteristics indirectly by the curve it presents. A decrease with increasing Nt is clear by inspection of Fig. 9a, b, the signature of the thermophoretic effect. Nb enhances the concentration profile by inducing nanoparticle migration.

[See PDF for image]

Fig. 5

(a) Concentration profile with respect to Du obtained using MWNN-PSO-NNA and the reference solution, (b) Concentration profile with respect to K obtained using MWNN-PSO-NNA and the reference solution.

[See PDF for image]

Fig. 6

(a) Concentration profile with respect to M obtained using MWNN-PSO-NNA and the reference solution, (b) Concentration profile with respect to n obtained using MWNN-PSO-NNA and the reference solution.

[See PDF for image]

Fig. 7

(a) Concentration profile with respect to Sr obtained using MWNN-PSO-NNA and the reference solution, (b) Concentration profile with respect to We obtained using MWNN-PSO-NNA and the reference solution.

[See PDF for image]

Fig. 8

(a) Concentration profile with respect to Sc obtained using MWNN-PSO-NNA and the reference solution, (b) Concentration profile with respect to Pr obtained using MWNN-PSO-NNA and the reference solution.

[See PDF for image]

Fig. 9

(a) Concentration profile with respect to Nt obtained using MWNN-PSO-NNA and the reference solution, (b) Concentration profile with respect to Nb obtained using MWNN-PSO-NNA and the reference solution.

In Figs. 10, 11, 12, 13 and 14, Temperature fluctuations for parameters such as Du and We are shown in the profiles. Heat conduction is enhanced by an increase in Du (Fig. 10a) and K (10b). Because of their effects on viscous and nanofluid dynamics, parameters like n (11a) and Nb (11b) exhibit intricate temperature relationships. A higher Nt (12a) and Sc (12b) suppresses temperature because of decreased molecular activity and increased heat dissipation. Temperature changes caused by Sr (13a) and M (13b) indicate interdependent transport mechanisms. Temperature decreases with increasing Pr (14a), indicating less thermal diffusion, whereas stretching impacts with We (14b) intensify thermal effects (Table 3).

[See PDF for image]

Fig. 10

(a) Temperature profile with respect to Du obtained using MWNN-PSO-NNA and the reference solution, (b) Temperature profile with respect to K obtained using MWNN-PSO-NNA and the reference solution.

[See PDF for image]

Fig. 11

(a) Temperature profile with respect to n obtained using MWNN-PSO-NNA and the reference solution, (b) Temperature profile with respect to Nb obtained using MWNN-PSO-NNA and the reference solution.

[See PDF for image]

Fig. 12

(a) Temperature profile with respect to Nt obtained using MWNN-PSO-NNA and the reference solution, (b) Temperature profile with respect to Sc obtained using MWNN-PSO-NNA and the reference solution.

[See PDF for image]

Fig. 13

(a) Temperature profile with respect to Sr obtained using MWNN-PSO-NNA and the reference solution, (b) Temperature profile with respect to M obtained using MWNN-PSO-NNA and the reference solution.

[See PDF for image]

Fig. 14

(a) Temperature profile with respect to Pr obtained using MWNN-PSO-NNA and the reference solution, (b) Temperature profile with respect to We obtained using MWNN-PSO-NNA and the reference solution,

Table 3. Best of hyper parameters of MWNN through PSO-NNA for scenario 2.

0.725423391	− 5.112661839	4.32757544
− 3.511127203	− 5.045653576	10
0.445141973	0.59424218	1.337980396
− 0.353871468	− 0.821661439	− 0.722141419
− 0.136401352	− 2.035333456	5.749435967
− 0.292139056	− 0.944195423	− 3.704389771
− 9.999933263	− 10.000000	− 5.279155917
− 5.024447708	− 4.130751454	− 1.235112127
2.271738052	3.834279071	− 8.195852174
1.321099882	3.43721917	− 9.999991541

All things considered, the velocity profiles react predictably to each governing parameter: In Fig. 16a raising the Hartmann number M, introduces magnetic drag that slows the fluid down, while in Fig. 15a raising the power-law index n, increases, the velocity profile decreases further slowly, representative a gentler decline as well as denser momentum boundary layer used for greater n values. Localized velocity boosts are produced via enhanced Brownian motion Nb represents in Fig. 16b and greater radiation parameter Sr, showed in Fig. 18a, which both encourage particle migration and layer stretching. On the other hand, increasing the Prandtl Pr in represents Fig. 17a or Schmidt Sc showed in Fig. 17b numbers increases the resistance of the medium, damping the momentum boundary layer and decreasing velocity. The fluid’s elasticity is captured by the Weissenberg number We showed in Fig. 18b; a greater We denotes dominating elastic stresses that more efficiently stretch fluid layers, lower internal friction, and further enhance velocity in accordance with typical viscoelastic behavior. Likewise, increased permeability K in Fig. 15b increases velocity uniformly throughout the domain by reducing drag via the porous matrix (Table 4).

[See PDF for image]

Fig. 15

(a) Velocity profile with respect to n obtained using MWNN-PSO-NNA and the reference solution, (b) Velocity profile with respect to K obtained using MWNN-PSO-NNA and the reference solution.

[See PDF for image]

Fig. 16

(a) Velocity profile with respect to M obtained using MWNN-PSO-NNA and the reference solution, (b) Velocity profile with respect to Nb obtained using MWNN-PSO-NNA and the reference solution.

[See PDF for image]

Fig. 17

(a) Velocity profile with respect to Pr obtained using MWNN-PSO-NNA and the reference solution. (b) Velocity profile with respect to Sc obtained using MWNN-PSO-NNA and the reference solution.

[See PDF for image]

Fig. 18

(a) Velocity profile with respect to Sr obtained using MWNN-PSO-NNA and the reference solution, (b) Velocity profile with respect to We obtained using MWNN-PSO-NNA and the reference solution.

Table 4. Quantitative comparison analysis between the reference solution and MWNN-PSO-NNA for scenario 2.

	Reference solution	MWNN-PSO-NNA	Absolute error
0	1	0.999999586	4.14E−07
0.1	0.851130053	0.850974645	1.55E−04
0.2	0.72938801	0.729282164	1.06E−04
0.3	0.627380857	0.62728772	9.31E−05
0.4	0.540738755	0.540674927	6.38E−05
0.5	0.466471788	0.466410732	6.11E−05
0.6	0.402374343	0.402312121	6.22E−05
0.7	0.346743151	0.346696798	4.64E−05
0.8	0.298222145	0.29819804	2.41E−05
0.9	0.25570786	0.255694489	1.34E−05
1	0.218287661	0.218269049	1.86E−05
1.1	0.185197041	0.185166379	3.07E−05
1.2	0.155788885	0.155751661	3.72E−05
1.3	0.129510584	0.129478523	3.21E−05
1.4	0.105886526	0.105868808	1.77E−05
1.5	0.084504388	0.084502024	2.36E−06
1.6	0.065004194	0.065010249	6.06E−06
1.7	0.047069437	0.047074404	4.97E−06
1.8	0.03041978	0.030418736	1.04E−06
1.9	0.014804932	0.014801542	3.39E−06
2	− 4.25E−07	1.07E−06	1.50E−06

Furthermore, there are noticeable synergies produced by the interaction of elastic and magnetic forces: at moderate M values and large We, elastic stretching can partially offset magnetic damping, resulting in a non-monotonic velocity distribution close to the wall. Optimizing applications such as porous-media reactors or MHD-driven viscoelastic pumps requires this delicate balancing. Furthermore, the velocity field’s sensitivity to thermophoretic effects and Brownian motion (Nb) implies that temperature gradients and nanoparticle loading can be adjusted to target particular flow increases. For example, raising particle concentration to raise Nb increases mixing but may also increase viscous dissipation; our ANN model captures this trade-off with less than 1% inaccuracy compared to benchmark solutions. The ANN-based solution employing the MWNN-PSO-NNA technique demonstrated outstanding convergence, with the mean square error (MSE) falling below 10⁻⁶. This demonstrates the effectiveness and dependability of the model in precisely resolving the nonlinear differential equations controlling the flow of nanofluids. Similar to this, changes in Cr and Sc have distinct effects on thermal and species boundary layers, underscoring the necessity of customized mass-transfer or cooling techniques in hybrid nanofluid systems. Lastly, parametric experiments demonstrate the MWNN solver’s practical usefulness for engineering design and real-time control of complex MHD hybrid nanofluid flows by confirming that it stays stable and convergent throughout extreme parameter ranges.

Key findings

• The damping by the magnetic field reduces velocity and enhances the thickness of the momentum layer.

• Greater K mitigates drag and uniformly accelerates flow.

• Nb increases nanoparticle mixing; Nt shifts peak thermal gradients.

• More shear-thinning (n > 1.5) implies a 25% increase in the near-wall velocity.

• Heat-transfer rates (30%) and skin-friction coefficients (18%) are increased.

Conclusion

In this research, MHD (magnetohydrodynamic) motion of Williamson’s nanofluid over the non-linear elongating sheets in a permeable intermediate was studied, incorporating Dufour and Soret cross-diffusion special effects. A key challenge was accurately modelling the nonlinear interactions between velocity, temperature, and concentration fields under magnetic and porous medium influences. To address this, similarity transformations converted the main partial differential equations into ordinary differential equations (ODEs). To find the solutions of the ODEs artificial intelligence technique were employed especially, MWNNs-PSO-NNA framework, yielding results that closely matched RK4 findings.

• The current research checks that MWNNs-PSO-NNA expressively improve prediction precision and computational productivity in solving complex, nonlinear non-Newtonian nanofluid models with MHD as well as porous media effects.

• Cross-diffusion phenomena, mainly Brownian motion and thermophoresis, were established to increase heat and mass transfer, particularly under nonlinear stretching and activation energy situations.

• The interaction between magnetic field strength (M) and porosity (K) strongly affects velocity gradients and boundary layer performance, by means of higher M and lower K endorsing flow stabilization.

• The suggested hybrid Williamson nanofluid model established high consistency in simulating practical scenarios for instance microelectronic cooling as well as bio-porous flow through changing Weissenberg number along with rheological index.

• Additionally, the presence of heat generation/absorption as well as radiation influences pointedly modified thermal distributions, emphasising the model’s potential in magneto-thermal applications.

• The MWNNs-PSO-NNA model requires high computational resources due to the complexity of hybrid optimization. Additionally, it may face slower convergence in highly noisy or unstable systems. Future research can focus on optimizing the computational efficiency of MWNNs-PSO-NNA by incorporating adaptive learning rates and parallel computing.

Acknowledgements

The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through research groups program under Grant No. R.G.P2/127/46.

Author contributions

Khalid Arif: Conceptualization, Investigation, Software, Writing—original draft, Writing—reviews & editing. Syed Tauseef Saeed: Investigation, Methodology, Project Administration, Supervision Muhammad Naeem: Software, Validation, Writing—original draft. Jihad Younis: Supervision, Validation, Writing—original draft. Arshad Zia: Conceptualization, Data Curation, Formal analysis, Writing—reviews & editing. Salman Saleem: Writing—review & editing, Supervision, Funding acquisition.

Data availability

The datasets analysed during the current study available from the corresponding author on reasonable request.

Competing interests

The authors declare no competing interests.

List of symbols

Concentration of the nanoparticles

Concentration of the wall

Stretching velocity

Free stream temperature

Temperature of the nanofluid

Wall temperature distribution

Free nanoparticle concentration

Strength of the uniform magnetic field

G

Acceleration due to gravity

Coefficient of mass diffusivity

Coefficient of Thermophoresis

Brownian diffusion coefficient

Dimensionless stream function

Permeability of porous medium

Ratio of thermal diffusion

Concentration susceptibility

Specific heat at constant pressure

Williamson parameter

Prandtl number

Dufour parameter

Schmidt number

Soret parameter

Porous parameter

Stretching sheet index

Magnetic field parameter

Positive constants

Intensity of the variable magnetic field

Local skin friction coefficient

Brownian motion parameter

Thermophoresis parameter

Local Nusselt number

Local Sherwood parameter

Wall shear stress

Heat flux

Mass flux

Cartesian coordinates along the plate and normal to it, respectively

Velocity along x and y axes

Reference velocity

Dimensionless temperature

Kinematic viscosity

Positive time constant

Coefficient of viscosity

Density of the fluid

Electrical conductivity

Reynolds number

Thermal conductivity

Dimensionless concentration

Ratio of effective heat capacity

Similarity variable

Thermal diffusivity

Wall condition

Free stream condition

Publisher's note

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References

1. Raju, RS; Reddy, GJ; Kumar, MA; Gorla, RSR. Unsteady magnetohydrodynamic chemically reacting fluid flow past an inclined vertical permeable moving plate. Int. J. Fluid Mech. Res.; 2020; 47, 3 pp. 191-215. [DOI: https://dx.doi.org/10.1615/InterJFluidMechRes.2020028808]

2. Rallabandi, SR. Analytical and numerical investigation of heat and mass transfer effects on magnetohydrodynamic natural convective flow past a vertical porous plate. J. Heat Mass Transf. Res.; 2017; 4, 2 pp. 117-133. [DOI: https://dx.doi.org/10.22075/JHMTR.2017.1854.1142]

3. Raju, RS. Application of finite element method to MHD mixed convection chemically reacting flow past a vertical porous plate with cross diffusion and biot number effects. Am. J. Heat Mass Transfer; 2017; 4, 3 pp. 53-74.1:CAS:528:DC%2BC1cXls1eiu7Y%3D [DOI: https://dx.doi.org/10.7726/ajhmt.2017.1009]

4. Kumar, RM; Raju, RS; Mebarek-Oudina, F; Kumar, MA; Narla, VK. Cross-diffusion effects on an MHD Williamson nanofluid flow past a nonlinear stretching sheet immersed in a permeable medium. Front. Heat Mass Transfer; 2024; 22, 1 pp. 15-34. [DOI: https://dx.doi.org/10.32604/fhmt.2024.048045]

5. Khader, MM; Ahmad, H; Adel, M; Megahed, AM. Numerical analysis of the MHD Williamson nanofluid flow over a nonlinear stretching sheet through a Darcy porous medium: Modeling and simulation. Open Phys.; 2024; 22, 1 20240016.2024OPhy..2240016K1:CAS:528:DC%2BB2cXhslOksr3I [DOI: https://dx.doi.org/10.1515/phys-2024-0016]

6. Faizan, M; Ajithkumar, M; Reddy, MV; Jamal, MA; Almutairi, B; Shah, NA; Chung, JD. A theoretical analysis of the ternary hybrid nano-fluid with Williamson fluid model. Ain Shams Eng. J.; 2024; 15, 8 [DOI: https://dx.doi.org/10.1016/j.asej.2024.102839] 102839.

7. Kumar, R; Mehta, R; Bhatnagar, A; Ozsahin, I; Uzun, B; Almohsen, B; Ahmad, H. Numerical study of Williamson fluid flow over a stretching sheet with newtonian heating embedded in a porous medium in presence of thermal radiation and heat source/sink. J. Nonlinear Math. Phys.; 2025; 32, 1 pp. 1-12.48475841:CAS:528:DC%2BB2MXhsVSjtb%2FP [DOI: https://dx.doi.org/10.1007/s44198-024-00258-1]

8. Nadeem, S; Hussain, ST; Lee, C. Flow of a Williamson fluid over a stretching sheet. Braz. J. Chem. Eng.; 2013; 30, pp. 619-625.1:CAS:528:DC%2BC3sXhslyhsLnI [DOI: https://dx.doi.org/10.1590/S0104-66322013000300019]

9. M. Megahed, A. Williamson fluid flow due to a nonlinearly stretching sheet with viscous dissipation and thermal radiation. J. Egypt. Math. Soc.; 2019; 27, 1 12.4092035 [DOI: https://dx.doi.org/10.1186/s42787-019-0016-y]

10. Bibi, M; Malik, MY; Tahir, M. Numerical study of unsteady Williamson fluid flow and heat transfer in the presence of MHD through a permeable stretching surface. Eur. Phys. J. Plus; 2018; 133, pp. 1-15. [DOI: https://dx.doi.org/10.1140/epjp/i2018-11991-2]

11. Salahuddin, T; Iqbal, MA; Bano, A; Awais, M; Muhammad, S. Cattaneo-Christov heat and mass transmission of dissipated Williamson fluid with double stratification. Alex. Eng. J.; 2023; 80, pp. 553-558. [DOI: https://dx.doi.org/10.1016/j.aej.2023.09.012]

12. Nadeem, S; Hussain, ST. Flow and heat transfer analysis of Williamson nanofluid. Appl. Nanosci.; 2014; 4, pp. 1005-1012.2014ApNan..4.1005N1:CAS:528:DC%2BC3sXhvVeisLzM [DOI: https://dx.doi.org/10.1007/s13204-013-0282-1]

13. Bhatti, MM; Rashidi, MM. Effects of thermo-diffusion and thermal radiation on Williamson nanofluid over a porous shrinking/stretching sheet. J. Mol. Liq.; 2016; 221, pp. 567-573.1:CAS:528:DC%2BC28XhtValtbjN [DOI: https://dx.doi.org/10.1016/J.MOLLIQ.2016.05.049]

14. Reddy, S; Naikoti, K; Rashidi, MM. MHD flow and heat transfer characteristics of Williamson nanofluid over a stretching sheet with variable thickness and variable thermal conductivity. Trans. A. Razmadze Math. Inst.; 2017; 171, 2 pp. 195-211.3660934 [DOI: https://dx.doi.org/10.1016/J.TRMI.2017.02.004]

15. Mabood, F; Ibrahim, SM; Lorenzini, G; Lorenzini, E. Radiation effects on Williamson nanofluid flow over a heated surface with magnetohydrodynamics. Int. J. Heat Technol.; 2017; 35, 1 pp. 196-204. [DOI: https://dx.doi.org/10.18280/IJHT.350126]

16. Mebarek-Oudina, FP; Preeti,; Sabu, AS; Vaidya, H; Lewis, RW; Areekara, S; Mathew, A; Ismail, AI. Hydromagnetic flow of magnetite–water nanofluid utilizing adapted Buongiorno model. Int. J. Mod. Phys. B; 2024; 38, 01 2450003.2024IJMPB.3850003M1:CAS:528:DC%2BB3sXksFamu74%3D [DOI: https://dx.doi.org/10.1142/S0217979224500036]

17. Ramesh, K; Mebarek-Oudina, F; Ismail, AI; Jaiswal, BR; Warke, AS; Lodhi, RK; Sharma, T. Computational analysis on radiative non-Newtonian Carreau nanofluid flow in a microchannel under the magnetic properties. Sci. Iran.; 2023; 30, 2 pp. 376-390. [DOI: https://dx.doi.org/10.24200/SCI.2022.58629.5822]

18. Abbas, A; Jeelani, MB; Alnahdi, AS; Ilyas, A. MHD Williamson nanofluid fluid flow and heat transfer past a non-linear stretching sheet implanted in a porous medium: Effects of heat generation and viscous dissipation. Processes; 2022; 10, 6 1221.1:CAS:528:DC%2BB38XhslKhs77J [DOI: https://dx.doi.org/10.3390/pr10061221]

19. Bouselsal, M; Mebarek-Oudina, F; Biswas, N; Ismail, AAI. Heat transfer enhancement using Al₂O₃-MWCNT hybrid-nanofluid inside a tube/shell heat exchanger with different tube shapes. Micromachines; 2023; 14, 5 1072. [DOI: https://dx.doi.org/10.3390/mi14051072]

20. Mebarek-Oudina, F; Chabani, I. Review on nano enhanced PCMs: Insight on nePCM application in thermal management/storage systems. Energies; 2023; 2023, 16 1066.1:CAS:528:DC%2BB3sXjtFCmsLY%3D [DOI: https://dx.doi.org/10.3390/en16031066]

21. Loganathan, P; Sangeetha, S. Effect of Williamson parameter on Cu-water Williamson nanofluid over a vertical plate. Int. Commun. Heat Mass Transfer; 2022; 137, 1:CAS:528:DC%2BB38Xit1WrtLrE [DOI: https://dx.doi.org/10.1016/J.ICHEATMASSTRANSFER.2022.106273] 106273.

22. Das, K. Nanofluid flow over a non-linear permeable stretching sheet with partial slip. J. Egypt. Math. Soc.; 2015; 23, 2 pp. 451-456.3355443 [DOI: https://dx.doi.org/10.1016/j.joems.2014.06.014]

23. Bilal, S; Malik, MY; Hussain, A; Khan, M. Effects of temperature dependent conductivity and absorptive/generative heat transfer on MHD three dimensional flow of Williamson fluid due to bidirectional non-linear stretching surface. Results Phys.; 2017; 7, pp. 204-212.2017ResPh..7.204B [DOI: https://dx.doi.org/10.1016/j.rinp.2016.11.063]

24. Abo-Dahab, SM; Abdelhafez, MA; Mebarek-Oudina, F; Bilal, SM. MHD Casson nanofluid flow over nonlinearly heated porous medium in presence of extending surface effect with suction/injection. Indian J. Phys.; 2021; 95, 12 pp. 2703-2717.2021InJPh.95.2703A1:CAS:528:DC%2BB3MXjtFartg%3D%3D [DOI: https://dx.doi.org/10.1007/s12648-020-01923-z]

25. Hayat, T; Ullah, I; Alsaedi, A; Farooq, M. MHD flow of Powell-Eyring nanofluid over a non-linear stretching sheet with variable thickness. Results Phys.; 2017; 7, pp. 189-196.2017ResPh..7.189H [DOI: https://dx.doi.org/10.1016/j.rinp.2016.12.008]

26. Qayyum, S; Hayat, T; Alsaedi, A. Chemical reaction and heat generation/absorption aspects in MHD nonlinear convective flow of third grade nanofluid over a nonlinear stretching sheet with variable thickness. Results Phys.; 2017; 7, pp. 2752-2761.2017ResPh..7.2752Q [DOI: https://dx.doi.org/10.1016/j.rinp.2017.07.043]

27. Ahmed, K; McCash, LB; Akbar, T; Nadeem, S. Effective similarity variables for the computations of MHD flow of Williamson nanofluid over a non-linear stretching surface. Processes; 2022; 10, 6 1119. [DOI: https://dx.doi.org/10.3390/pr10061119]

28. Ragulkumar, E; Palani, G; Sambath, P; Chamkha, AJ. Dissipative MHD free convective nanofluid flow past a vertical cone under radiative chemical reaction with mass flux. Sci. Rep.; 2023; 13, 1 2878.2023NatSR.13.2878R1:CAS:528:DC%2BB3sXjsFCmtLw%3D [DOI: https://dx.doi.org/10.1038/s41598-023-28702-0]

29. Ragulkumar, E; Sambath, P; Chamkha, AJ. Free convection nanofluid flow past a vertical isothermal cone surface in the presence of viscous dissipation and MHD with heat flux. Eur. Phys. J. Plus; 2022; 137, 8 894.1:CAS:528:DC%2BB38XitVyms7rO [DOI: https://dx.doi.org/10.1140/epjp/s13360-022-03115-6]

30. Choudhary, S; Kumar Jarwal, V; Choudhary, P; Loganathan, K; Pattanaik, B. Mass-based hybrid nanofluid model for thermal radiation analysis of MHD Flow over a wedge embedded in porous medium. J. Eng.; 2024; 2024, 1 9528362. [DOI: https://dx.doi.org/10.1155/2024/9528362]

31. Choudhary, P; Choudhary, S; Jat, K; Loganathan, K; Eswaramoorthi, S. Impacts of unsteady MHD hybrid nanofluid over a non-linear stretchable porous sheet with thermal radiation and gyrotatic microorganisms. Int. J. Thermofluids; 2024; 23, 1:CAS:528:DC%2BB2cXhslKrur%2FK [DOI: https://dx.doi.org/10.1016/j.ijft.2024.100788] 100788.

32. Loganathan, K; Thamaraikannan, N; Eswaramoorthi, S; Jain, R. Entropy framework of the bioconvective Williamson nanofluid flow over a Riga plate with radiation, triple stratification and swimming microorganisms. Int. J. Thermofluids; 2025; 25, 1:CAS:528:DC%2BB2cXivVejsbfN [DOI: https://dx.doi.org/10.1016/j.ijft.2024.101000] 101000.

33. Eswaramoorthi, S; Sivasankaran, S. Entropy optimization of MHD Casson-Williamson fluid flow over a convectively heated stretchy sheet with Cattaneo-Christov dual flux. Sci. Iran.; 2022; 29, 5 pp. 2317-2331. [DOI: https://dx.doi.org/10.24200/SCI.2022.58291.5654]

34. Farayola, PI; Aselebe, LO; Salaudeen, KA; Ogundiran, SD; Asiru, TM. Influence of temperature-dependent thermophysical properties of TiO2-SiO2-ZnO-Fe2O3/PAO tetra-hybrid nanofluid along a vertical porous surface with suction. Open J. Fluid Dyn.; 2024; 14, 3 pp. 123-146. [DOI: https://dx.doi.org/10.1080/16583655.2024.2336327]

35. Haq, I; Bilal, M; Ahammad, NA; Ghoneim, ME; Ali, A; Weera, W. Mixed convection nanofluid flow with heat source and chemical reaction over an inclined irregular surface. ACS Omega; 2022; 7, 34 pp. 30477-30485.1:CAS:528:DC%2BB38XitFeitLnN [DOI: https://dx.doi.org/10.1021/acsomega.2c03919]

36. Patgiri, B; Paul, A; Sarma, N. Numerical assessment of viscoelastic tetra hybrid nanofluid flow across a stretchable rotatory disk under the Soret and Dufour aspects. Multidiscip. Model. Mater. Struct.; 2024; 20, 4 pp. 688-706.1:CAS:528:DC%2BB2cXisFKltr%2FK [DOI: https://dx.doi.org/10.1108/MMMS-12-2023-0416]

37. Paul, A; Patgiri, B; Sarma, N. Transformer oil-based Casson ternary hybrid nanofluid flow configured by a porous rotating disk with hall current. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik; 2024; 104, 44741905 [DOI: https://dx.doi.org/10.1002/zamm.202300704] e202300704.

38. Paul, A; Patgiri, B; Sarma, N. Darcy-Forchheimer flow of Ag–ZnO–CoFe2O4/H2O Casson ternary hybrid nanofluid induced by a rotatory disk with EMHD. Int. J. Ambient Energy; 2024; 45, 1 2313697.1:CAS:528:DC%2BB2cXjsl2msL8%3D [DOI: https://dx.doi.org/10.1080/01430750.2024.2313697]

39. Paul, A; Patgiri, B; Sarma, N. Mixed convective flow of engine oil-based non-Newtonian tri-hybrid nanofluid across a porous rotating disk. World Journal of Engineering; 2024; [DOI: https://dx.doi.org/10.1108/WJE-01-2024-0025]

40. Abhijith, P; Areekara, S; Sabu, AS; Mathew, A. MHD Reiner-Rivlin nanofluid flow considering Soret-Dufour and nonlinear chemical reaction. Multiscale and Multidisciplinary Modeling, Experiments and Design; 2025; 8, 1 33. [DOI: https://dx.doi.org/10.1007/s41939-024-00616-y]

41. Tak, P; Poonia, H; Areekara, S; Sabu, AS; Mathew, A. Non-similar approach on the MHD Carreau nanofluid flow with quadratic radiation and Soret-Dufour effects. Phys. Scr.; 2023; 98, 102023PhyS..98j5225T1:CAS:528:DC%2BB2cXitVehu7rL [DOI: https://dx.doi.org/10.1088/1402-4896/acf4cf] 105225.

42. Areekara, S; Sabu, AS; Mathew, A; Parvathy, KS; Rana, P. Significance of nanoparticle radius on EMHD Casson blood-gold nanomaterial flow with non-uniform heat source and Arrhenius kinetics. J. Therm. Anal. Calorim.; 2023; 148, 17 pp. 8945-8968.1:CAS:528:DC%2BB3sXhsVens7%2FP [DOI: https://dx.doi.org/10.1007/s10973-023-12288-w]

43. Areekara, S; Sabu, AS; Mathew, A; Parvathy, KS; Wakif, A. Significance of nanoparticle radius on EMHD Casson nanomaterial flow with non-uniform heat source and second-order velocity slip. Numer. Heat Transf. Part B Fundam.; 2024; 85, 5 pp. 604-621.2024NHTB..85.604A1:CAS:528:DC%2BB3sXhvFSksrnJ [DOI: https://dx.doi.org/10.1080/10407790.2023.2252598]

44. Seid, E; Haile, E; Walelign, T. Multiple slip, Soret and Dufour effects in fluid flow near a vertical stretching sheet in the presence of magnetic nanoparticles. Int. J. Thermofluids; 2022; 13, 1:CAS:528:DC%2BB38XisFKnt7bK [DOI: https://dx.doi.org/10.1016/J.IJFT.2022.100136] 100136.

45. Zafar, SS; Zaib, A; Ali, F; Faizan, M; Khan, U; Sherif, ESM; Pop, I. Dynamics of active and passive control on bioconvection Carreau nanofluid with thermal radiation and Cattaneo-Christov double-diffusion effects. J. Radiat. Res. Appl. Sci.; 2025; 18, 21:CAS:528:DC%2BB2MXktlCru7k%3D [DOI: https://dx.doi.org/10.1016/j.jrras.2025.101336] 101336.

46. Kamal, M; Ali, F; Khan, N; Faizan, M; Gul, N; Muhammad, T; Kolsi, L. Second law analysis of two phase Maxwell mixed convective nanofluid using Marangoni flow and gyrotactic microorganism framed by rotating disk. J. Therm. Anal. Calorim.; 2025; [DOI: https://dx.doi.org/10.1007/s10973-024-13792-3]

47. Ali, F; Kamal, M; Faizan, M; Zafar, SS. Mixed convective magneto flow of nanofluid for the Sutterby fluid containing micro-sized selfpropelled microorganisms with chemical reaction: Keller box analysis. Phys. Scr.; 2024; 99, 91:CAS:528:DC%2BB2cXisVCjtbnJ [DOI: https://dx.doi.org/10.1088/1402-4896/ad6642] 095202.

48. Shah, NA; Ali, F; Yook, SJ; Faizan, M; Zafar, SS; Sidi, MO. Dynamics of chemical reactive on magneto hybrid nanomaterial with heat radiation due to porous exponential plate: Laplace transform technique for the heat and mass. J. Radiat. Res. Appl. Sci.; 2025; 18, 11:CAS:528:DC%2BB2MXisVSqs74%3D [DOI: https://dx.doi.org/10.1016/j.jrras.2025.101295] 101295.

49. Ahmed, MF; Zaib, A; Ali, F; Khan, U; Zafar, SS. Thermal radiation of Walter-B magneto bioconvection nanofluid due to the stretching surface under convective condition and heat source/sink: A semi-analytical technique for the stagnation point. J. Radiat. Res. Appl. Sci.; 2025; 18, 11:CAS:528:DC%2BB2MXisVarurc%3D [DOI: https://dx.doi.org/10.1016/j.jrras.2025.101291] 101291.

50. Imran, M; Naveed, M; Abbas, Z. Dynamics of Soret and Dufour effects on oscillatory flow of couple stress fluid due to stretchable curved surface. Adv. Mech. Eng.; 2023; 15, 2 16878132231156742. [DOI: https://dx.doi.org/10.1177/16878132231156742]

51. Dharmaiah, G; Mebarek-Oudina, F; Balamurugan, KS; Vedavathi, N. Numerical analysis of the magnetic dipole effect on a radiative ferromagnetic liquid flowing over a porous stretched sheet. Fluid Dyn. Mater. Process.; 2024; 20, 2 pp. 293-310. [DOI: https://dx.doi.org/10.32604/fdmp.2023.030325]

52. Ali, AH; Amir, M; Rahman, JU; Raza, A; Arif, GE. Design of Morlet wavelet neural networks for solving the nonlinear Van der Pol–Mathieu–Duffing oscillator model. Computers; 2025; 14, 1 14. [DOI: https://dx.doi.org/10.3390/computers14010014]

53. Gad, AG. Particle swarm optimization algorithm and its applications: A systematic review. Arch. Comput. Methods Eng.; 2022; 29, 5 pp. 2531-2561.4453187 [DOI: https://dx.doi.org/10.1007/s11831-021-09694-4]

54. Kumar, MD; Dharmaiah, G; Yook, SJ; Raju, CSK; Shah, NA. Deep learning approach for predicting heat transfer in water-based hybrid nanofluid thin film flow and optimization via response surface methodology. Case Stud. Therm. Eng.; 2025; 68, 105930. [DOI: https://dx.doi.org/10.1016/j.csite.2025.105930]

55. Kumar, MD; Raju, CSK; Shah, NA; Yook, SJ; Gurram, D. Support vector machine learning classification of heat transfer rate in tri-hybrid nanofluid over a 3D stretching surface with suction effects for water at 10° C and 50° C. Alex. Eng. J.; 2025; 118, pp. 556-578. [DOI: https://dx.doi.org/10.1016/j.aej.2025.01.061]

56. Kumar, MD; Jawad, M; Ramanuja, M; Ghodhbani, R; Yook, SJ; Abdallah, SAO. Forecasting heat and mass transfer enhancement in magnetized non-Newtonian nanofluids using Levenberg-Marquardt algorithm: influence of activation energy and bioconvection. Mech. Time-Depend. Mater.; 2025; 29, 1 14.1:CAS:528:DC%2BB2cXisl2ru7fI [DOI: https://dx.doi.org/10.1007/s11043-024-09739-8]

57. Kumar, MD; Gurram, D; Yook, SJ; Raju, CSK; Shah, NA. Optimizing thermal performance of water-based hybrid nanofluids with magnetic and radiative effects over a spinning disc. Chemom. Intell. Lab. Syst.; 2025; [DOI: https://dx.doi.org/10.1016/j.chemolab.2025.105336]

58. Kumar, MD; Raju, CSK; Sajjan, K; Dharmaiah, G; Shah, NA; Yook, SJ. Deep neural network-based prediction and computational fluid dynamics analysis of convective heat transfer in dusty fluid flow over heated surface. Phys. Fluids; 2025; [DOI: https://dx.doi.org/10.1063/5.0250396]

59. Dorigo, M; Birattari, M; Stutzle, T. Ant colony optimization. IEEE Comput. Intell. Mag.; 2007; 1, 4 pp. 28-39. [DOI: https://dx.doi.org/10.1109/MCI.2006.329691]

60. Azadeh, A; Sangari, MS; Amiri, AS. A particle swarm algorithm for inspection optimization in serial multi-stage processes. Appl. Math. Model.; 2012; 36, 4 pp. 1455-1464.2878121 [DOI: https://dx.doi.org/10.1016/j.apm.2011.09.037]

61. Gandomi, AH; Alavi, AH. Krill herd: A new bio-inspired optimization algorithm. Commun. Nonlinear Sci. Numer. Simul.; 2012; 17, 12 pp. 4831-4845.2012CNSNS.17.4831G2960279 [DOI: https://dx.doi.org/10.1016/j.cnsns.2012.05.010]

62. Sadollah, A; Sayyaadi, H; Yadav, A. A dynamic metaheuristic optimization model inspired by biological nervous systems: Neural network algorithm. Appl. Soft Comput.; 2018; 71, pp. 747-782. [DOI: https://dx.doi.org/10.1016/j.asoc.2018.07.039]