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<p>This study investigates the slow, steady rotation of a permeable sphere in an incompressible Jeffrey fluid under the influence of a magnetic field using an analytical approach and an artificial neural network (ANN) framework. In order to train and validate the ANN, high-quality reference data is generated by using exact analytical solutions to the governing equations. The suggested ANN reliably forecasts the swirl and external couple for a variety of controlling parameters, such as the magnetic and Jeffrey fluid characteristics. The ANN model converges quickly, maintains high accuracy, and greatly reduces the amount of computation that is required. These findings support the ANN’s efficacy as a trustworthy surrogate model for challenging non-Newtonian MHD flow issues. The governing dynamics of the flow are represented through nonlinear partial differential equations (PDEs) articulated in the form of swirl velocity, subject to the relevant boundary conditions. The conventional no-slip condition is imposed at the spherical surface, while the regularity condition for the fluid velocity is applied outside of the sphere in the far-field area. Azimuthal direction is used to apply a constant magnetic field. For both the inside and outside of the sphere, the velocity field is represented by the swirl function. A new method for estimating Navier–Stokes equation solutions has been created, based on artificial neural networks. The scheme’s performance and result correctness are assessed by comparison with established analytical methods and solutions in the literature. To create and analyze the solution for controlled fluid flow, Feed-Forward Neural Network (FFNN) approaches are used. A multilayer perceptron (MLP) neural network is used to construct the sample functions, and the adaptive moment estimation (ADAM) algorithm is applied to optimize the adjustable parameters. The mathematical calculations for both ANN and precise solutions are shown in tables and graphs for different values of physical parameters. The numerical results obtained from both the ANN and analytical solutions are presented in tabular form and further illustrated graphically for a range of physical parameter values. The accuracy of the ANN-based solution is found to improve with rise in both the amount of neurons and the volume of training data used in the network. The swirl velocity predicted by the ANN approach achieved a high coefficient of determination, with an R-Squared value of 0.99951. Additionally, the suggested ANN model shows better adaptation to more complicated mathematical frameworks than the analytical method, and it drastically reduces the amount of computation and resources needed to solve problems.</p>

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Artificial neural network modeling of rotary flow around a permeable sphere in Jeffrey fluid with magnetic field effects

  • 期刊论文

  • 发表日期：

    2026年2月12日

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  • P. Aparna,
  • M. Pavan Kumar Reddy,
  • Rajani Indrakanti,
  • MD. Shamshuddin,
  • V. Ganesh Kumar

摘要

This study investigates the slow, steady rotation of a permeable sphere in an incompressible Jeffrey fluid under the influence of a magnetic field using an analytical approach and an artificial neural network (ANN) framework. In order to train and validate the ANN, high-quality reference data is generated by using exact analytical solutions to the governing equations. The suggested ANN reliably forecasts the swirl and external couple for a variety of controlling parameters, such as the magnetic and Jeffrey fluid characteristics. The ANN model converges quickly, maintains high accuracy, and greatly reduces the amount of computation that is required. These findings support the ANN’s efficacy as a trustworthy surrogate model for challenging non-Newtonian MHD flow issues. The governing dynamics of the flow are represented through nonlinear partial differential equations (PDEs) articulated in the form of swirl velocity, subject to the relevant boundary conditions. The conventional no-slip condition is imposed at the spherical surface, while the regularity condition for the fluid velocity is applied outside of the sphere in the far-field area. Azimuthal direction is used to apply a constant magnetic field. For both the inside and outside of the sphere, the velocity field is represented by the swirl function. A new method for estimating Navier–Stokes equation solutions has been created, based on artificial neural networks. The scheme’s performance and result correctness are assessed by comparison with established analytical methods and solutions in the literature. To create and analyze the solution for controlled fluid flow, Feed-Forward Neural Network (FFNN) approaches are used. A multilayer perceptron (MLP) neural network is used to construct the sample functions, and the adaptive moment estimation (ADAM) algorithm is applied to optimize the adjustable parameters. The mathematical calculations for both ANN and precise solutions are shown in tables and graphs for different values of physical parameters. The numerical results obtained from both the ANN and analytical solutions are presented in tabular form and further illustrated graphically for a range of physical parameter values. The accuracy of the ANN-based solution is found to improve with rise in both the amount of neurons and the volume of training data used in the network. The swirl velocity predicted by the ANN approach achieved a high coefficient of determination, with an R-Squared value of 0.99951. Additionally, the suggested ANN model shows better adaptation to more complicated mathematical frameworks than the analytical method, and it drastically reduces the amount of computation and resources needed to solve problems.