A two-stage computational approach for stochastic Darcy-forchheimer non-newtonian flows

The study of stochastic non-Newtonian fluid flows in porous media has significant applications in engineering and scientific fields, particularly in geophysical transport, biomedical flows, and industrial filtration systems. This research develops a high-order numerical scheme to solve deterministic and stochastic partial differential equations governing the Darcy–Forchheimer flow of Williamson fluid over a stationary sheet. This study aims to formulate and validate a computationally efficient two-stage method that accurately captures the effects of non-Newtonian behavior, porous media resistance, and stochastic perturbations. The proposed two-stage numerical method integrates a modified time integrator with a second-stage Runge-Kutta scheme, ensuring second-order accuracy in time for deterministic problems. The Euler-Maruyama approach handles Wiener processes for stochastic models, providing robust performance under random fluctuations. A compact sixth-order spatial discretization scheme enhances solution accuracy while maintaining computational efficiency. Numerical experiments, including Stokes’ first problem, demonstrate the superior accuracy and reliability of the proposed method compared to existing second-order Runge-Kutta schemes. The results confirm that the technique effectively captures complex interactions between deterministic and stochastic effects while significantly improving computational efficiency. This study advances numerical techniques for stochastic fluid dynamics, providing a practical framework for modeling and analyzing non-Newtonian fluid flows in porous media with real-world applications in engineering, geophysics, and industrial systems.