Finite difference simulations for Darcy–Forchheimer–Brinkman model of MHD Oldroyd B fluid flow past a permeable sheet containing microorganisms

Abstract This study explores the behavior of gyrotactic microorganisms swimming on a stretching surface in a non‐Newtonian Oldroyd B fluid. The study considers the combined effects of magnetohydrodynamic, Buongiorno and the Darcy–Forchheimer‐Brinkman models. Additionally, the effects of Soret–Dufour, thermophoresis, Brownian motion, viscous dissipation, thermal relaxation time, and thermal generation/absorption with heat flux are investigated. The relevant problems are mathematically formulated, and the ordinary differential equation is obtained by applying appropriate transformations to the governing equations. The finite difference technique is then used to solve the equation. Velocity, temperature, concentration, and microorganism profiles are graphed to analyze the behavior of the flow in the presence of motile gyrotactic microorganisms under various physical and bio‐convection parameters. It is demonstrated that the skin friction of Oldroyd B fluid is affected by the Deborah number and mixed convection parameter. Bio‐convection parameters reduce the local Nusselt number and enhance transfer of motile microorganisms. However, increasing bio‐convection parameters decrease the Sherwood number, resulting in slower mass transfer from the surface.