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In this paper, the influence of velocity slip and the Newtonian heating parameters on Williamson Nanofluid across a variable stretching sheet is studied. The basic fluid flow equations are turned into a system of coupled ordinary differential equations with appropriate conditions, which are solved numerically with the Keller-box approach. The outcomes show that, increasing Williamson parameter, due to the viscosity velocity of the fluid slowdown. As a result, it is noticed that the velocity slip parameter reduces Williamson Nanofluid velocity, while increasing shear stress, according to the results. Furthermore, it has been discovered that unlike velocity profiles, the slip influence is more pronounced on temperature and concentration profiles. Furthermore, the Newtonian heating parameter may be shown to have rising functions for velocity and dimensionless temperature. Newtonian heating has uses in the petroleum industry, heat exchangers, conjugate heat transfer around fins and solar radiation. It is discovered that the Joule heating parameter improves the thermal boundary layer thickness and slows down the velocity of the fluid, which aids in maintaining system temperature. Additionally, it has been found that the temperature profile exhibits a stronger slip effect compared with the velocity profile. To prepare toffee, chocolate and other delicacies as efficiently as possible, it is crucial to investigate the flow of Williamson fluids (such as drilling muds, clay coatings, various suspensions and some lubricating oils, thermoplastic melts, and a variety of colloids) in the incidence of heat transfer. The novelty of this paper is the effect of joule heating, Newtonian heating parameter and velocity slip on Williamson fluid velocity and temperature gradient. Furthermore, the heat and mass transfer rates are calculated for various parameters numerically and shown in tabular form. [ABSTRACT FROM AUTHOR]

In the present article, the intelligence based numerical investigation by Levenberg–Marquardt algorithm with Back‐propagated Artificial Neural Networks (LMA–BANN) is exploited to analyze the nonlinear radiative stagnation point flow of cross nanofluid system (NRS‐CNFS) past a stretching surface. The designed NRS‐CNFS, initially represented by system of partial differential equations (PDEs), is converted into system of non‐linear ODEs through the applicability of mathematical conversion analysis. The desired reference solution in the form of a dataset for LMA–BANN is achieved from the Adam method for NRS‐CNFS's different scenarios by varying the magnetic parameter (Ha), Prandlt number (Pr), thermal radiation parameter(R), Eckert number (Ec), thermophoresis diffusion coefficients (Nt), and non‐dimensional activation energy (E). The obtained results are construed for the NRS‐CNFS through the performances of the testing, validation, and training. In addition, the comparison is provided using the LMA ‐BANN analysis, which is validated in terms of regression, fitness estimations, and histograms along with MSE performances. Hence, the article provides an impeccable innovative approach to problem‐solving by implementing a soft computation paradigm. It promotes the use of an effective and dependable alternative framework that is based on soft computing environments and emphasizes descriptive analysis to successfully handle the problems brought on by varied physical features. [ABSTRACT FROM AUTHOR]

The fluid flow between the rotating disk has wide range of applications and are used in medical equipment's, car brake system, gas turbines, manufacturing of glass and plastic films. Inspired from these applications, we scrutinized the behavior of an axisymmetric expansion–contraction rotating disk on three‐dimensional flow of hybrid nanoparticles in the presence of microorganisms. Here, we considered copper and silica nanoparticles suspended in base fluid water. It is known that, the inequality of concentration in chemically reactive species affecting mass transfer mechanism and activation energy plays an important role in increasing the production speed of chemical reactions. Hence, the Arrhenius activation energy and chemical reaction effects are added in the mass equation. Autorelated transformations were adopted to reduce PDE's to ODE's and then, we utilized the RKF‐45 by adopting shooting skill to solve reduced ODE's. The analysis of rate of heat and mass transfer is done through graphs. Furthermore, change in velocity, thermal and concentration profiles for various nondimensional parameters are deliberated briefly by using suitable plots. Result reveals that rise in wall expansion/contraction parameter supports up the velocity gradients, higher the values of bioconvection Peclet number and concentration difference parameter declines the microorganism profile and Nusselt number decreases for rise in values of thermophoresis parameter. [ABSTRACT FROM AUTHOR]

This study aims to develop a supervised learning artificial recurrent neural network algorithm supported by Bayesian regularization called (ARNN‐BR) to analyze the impact of physical parameters, including radius of curvature (κ$\kappa $), Casson parameter (β$\beta $), heat generation parameter (λ$\lambda $) and radiation parameter (Rd${{R}_d}$) on velocity fʹ(η), and temperature profiles θ(η) in Casson nanofluid consisting of carbon nanotubes (CNTs‐CNF) model for single and multiwalled CNTs across a curved stretched surface. The numerical dataset of the proposed model has been constructed by varying various parameters for five scenarios that are used in a Bayesian regularization‐based intelligent computing method to build networks for approximating the numerical solutions of CNTs‐CNF model. It is observed that increment in the dimensionless radius of curvature (κ$\kappa $) causes to rise an increase in the velocity profile fʹ(η) for both SWCNTs and MWCNTs. However, a contrasting trend is observed when the Casson parameter (β$\beta $) is increased to higher values. The temperature θ(η) of fluid increases as the heat generation parameter (λ$\lambda $) and radiation parameter (Rd${{R}_d}$) increase. However, an opposite behavior is noticed when the dimensionless radius of curvature (κ$\kappa $) varies. The effectiveness and significance of designed Bayesian regularization based artificial recurrent neural networks (ARNN‐BR) is demonstrated through regression index measurements, error histogram studies, auto‐correlation analysis and convergence curves showing a minimal level of mean square error (E‐11 to E‐04) for the comprehensive simulations of CNTs‐CNF model. The designed ARNN‐BR algorithm is employed in many domains such as voice recognition, machine translation, identification of neurological brain illnesses as well as for automated translation of texts across different languages. [ABSTRACT FROM AUTHOR]