In this study, we proposed a numerical technique for solving time-dependent partial differential equations that arise in the electro-osmotic flow of Carreau fluid across a stationary plate based on a modified exponential inte... | Find, read and cite all the research you need on Tech Science Press

This study develops a high-order computational scheme for analyzing unsteady tangent hyperbolic fluid flow with variable thermal conductivity, thermal radiation, and coupled heat and mass transfer effects. A modified two-stage Exponential Time Integrator is introduced for temporal discretization, providing second-order accuracy in time. A compact finite difference method is employed for spatial discretization, yielding sixth-order accuracy at most grid points. The proposed framework ensures numerical stability and convergence when solving stiff, nonlinear parabolic systems arising in fluid flow and heat transfer problems. The novelty of the work lies in combining exponential integrator schemes with compact high-order spatial discretization, enabling accurate and efficient simulations of tangent hyperbolic fluids under complex boundary conditions, such as oscillatory plates and varying thermal conductivity. This approach addresses limitations of classical Euler, Runge–Kutta, and spectral methods by significantly reducing numerical errors (up to 45%) and computational cost. Comprehensive parametric studies demonstrate how viscous dissipation, chemical reactions, the Weissenberg number, and the Hartmann number influence flow behaviour, heat transfer, and mass transfer. Notably, heat transfer rates increase by 18.6% with stronger viscous dissipation, while mass transfer rates rise by 21.3% with more intense chemical reactions. The real-world relevance of the study is underscored by its direct applications in polymer processing, heat exchanger design, radiative thermal management in aerospace, and biofluid transport in biomedical systems. The proposed scheme thus provides a robust numerical framework that not only advances the mathematical modelling of non-Newtonian fluid flows but also offers practical insights for engineering systems involving tangent hyperbolic fluids.

In this paper, we examine aspects of general variational inequalities (GVIs) that are similar to fixed-point problems. Special cases of the proposed unique iterative methods are introduced, including the implicit, explicit, and extra-gradient methods of the proximal point approach. The convergence of the derived expression is analyzed under appropriate assumptions to demonstrate the effectiveness of the proposed method. Numerical results are also presented to evaluate the algorithms’ performance in practice. In these experiments, matrix-scaling tests were conducted, and parameter sensitivity analyses were performed; the convergence speed, computational efficiency, robustness, and stability of the experiments were evaluated. From the results, the implicit proximal method seems to give very good results, and convergence is achieved after fewer iterations than other algorithms. Additionally, we examine the non-convergent behavior of other algorithms under varying parameters. Overall, our study validates the theoretical findings and highlights the effectiveness of advanced proximal methods for large-scale GVI problems, while also suggesting directions for future research in this area.

This study introduces an efficient and accurate two-stage explicit computational scheme for solving partial differential equations (PDEs) containing first-order time derivatives. The suggested method is a modification of the classical Runge–Kutta scheme that introduces a new first-stage formulation. This minimizes numerical error with moderate step sizes while preserving the stability region of the classical method. Spatial discretization is performed using a sixth-order compact finite-difference scheme to obtain high-resolution solutions. The analysis of stability and convergence is strictly determined for both scalar and system forms of convection–diffusion-type equations. To illustrate the suitability of the method, a dimensionless mathematical model of the unsteady, incompressible, laminar flow of a Prandtl-type non-Newtonian nanofluid over a Riga plate is considered, accounting for viscous dissipation, thermophoresis, Brownian motion, and a magnetic field. Here, the Prandtl ternary nanofluid is defined as a non-Newtonian nanofluid that follows the Prandtl rheological model, and it exhibits three critical transport phenomena: heat conduction, viscous dissipation, and nanoparticle diffusion. Representative values of the Prandtl number $P_r=3$ and Reynolds number $R_e=5$ are used to perform the simulation, and other parameters, including but not limited to the Hartmann number $H_a$, Williamson number $W_e$, thermophoresis $N_t$ and Brownian motion $N_b$, are varied to evaluate the flow behavior. Moreover, an artificial neural network (ANN)-developed surrogate model is used to calculate the skin friction coefficient and the local Sherwood number, using five input parameters: the Reynolds number, Prandtl number, Schmidt number, Brownian motion parameter, and thermophoresis parameter. The governing partial differential equations yield high-fidelity numerical data used to train the surrogate model. The data is split into 80% for training, 10% for validation, and 10% for testing. The ANN is tested using regression analysis and error histograms, which demonstrate high accuracy and generalization capacity. Numerical simulation combined with AI-based prediction is a cost-efficient method for real-time estimation of complex non-Newtonian nanofluid systems.