Modelling and analysis of diffuse interface models for two-phase micropolar fluid flows

Micropolar fluids are among the simplest cases of fluids with microstructures, where each fluid particle has its own internal rotations. Examples include ferrofluids, blood flows, bubbly liquids and liquid crystals, all of which play significant and important roles in various industries and also in the human body. Therefore, it becomes essential for scientists to effectively capture the behaviour of these kinds of fluids with the appropriate mathematical models, and to better analyse and simulate their flow dynamics.

The classical Navier-Stokes equations that are well-studied by the mathematical community is not a suitable description for micropolar fluids, as the internal rotations of each fluid particle are not accounted for. Through the works of A. Cemal Eringen and his coworkers, we now have mathematical models that extend the Navier-Stokes equations to capture these internal rotations. However, in many real world applications we often encounter mixtures of multiple kinds of fluids. In these situations we have to further extend previous models to account for the interactions between individual fluids that can influence the gross motion of the mixture.

The main aims of this proposal are to develop new mathematical models for mixtures of micropolar fluids, study their mathematical properties, and derive stable and efficient fully discrete numerical schemes for simulations. The key ingredient is the diffuse interface methodology which offers a model that is amenable to further mathematical analysis and allows us to address the existence of weak solutions and the stability of discrete solutions. Also included in the modelling is the dynamics of the contact line that is created when two or more fluids meet at the same solid boundary. This describes wetting and dewetting processes crucial in modern-day applications such as inkjet printing and lubrication. We believe the proposed model and the accompanying analytical and numerical ideas can provide a theoretical foundation for many industrial and biological processes that involve mixtures of non-standard fluids