Bifurcation and stability analysis of exact solution of couple stress fluid creeping flow in a Darcy porous slit

In this paper, the investigation examines the problem of creeping flow of a non-Newtonian couple-stress fluid through a linear porous-walled slit within a Darcy porous material. A method uses similar shapes made by changing coordinates and making complex equations simpler to find clear formulas for the flow variables, leading to an exact solution to the nonlinear field equations. We use dynamical system theory and nonlinear stability analysis to establish a systematic framework for analyzing and controlling creeping flows. This framework allows for a more in-depth understanding of their stability and bifurcations, as well as a thorough exploration of the whole phase space while taking into account the interactions between various flow modes. As a novel result, it shows that the identification of homoclinic and heteroclinic orbits involving several distinct saddle stagnation points causes a qualitative change within the attraction basin, resulting in a trapping zone. The analytical results indicated the absence of bubbles or trapping in the wall boundaries and clarified the determination of the maximum and minimum retention limits. The results also showed that changes and improvements from earlier studies can be used in microfiltration devices, biological porous membranes, and energy transfer systems that deal with non-Newtonian fluids.