Ahmad Deeb

In this article, we present a finite element method for studying the dynamic behavior of deformable vesicles, which mimic red blood cells, in a non-Newtonian Casson fluid. The fluid membrane, represented by an implicit level-set function, adheres to the Canham–Helfrich model and maintains surface inextensibility constraint through penalty. We propose a two-step time integration scheme that incorporates higher-order accuracy by using an asymmetric composition of discrete flow based on the second-order backward difference formula, followed by a projection onto the real axis. Our framework incorporates variable time steps generated by an appropriate adaptation criterion. We validate our model through numerical simulations against existing experimental and numerical results in the case of purely Newtonian flow. Furthermore, we provide preliminary results demonstrating the influence of the non-Newtonian fluid model on membrane regimes.