Designed fluids: ferrofluids and beyond

Objective:

Some of the most useful technology in our daily lives is based on advanced, novel materials, that have spectacular properties. Most of these materials have been discovered experimentally, but some of them have been designed with a specific purpose in mind. This is the case of ferrofluids, the study of which is the main focus of this project. An in-depth understanding of this material will allow to propose redesigned versions of it, by combining it with liquid crystals materials. The most obvious mathematical feature of ferrofluid models is the remarkable complexity of the equations involved, a coupling of the incompressible Navier-Stokes equations, with two parabolic-type systems and one elliptic type system. Due to this complexity only a handful of results are available, mostly concerning basic well-posedness issues. Our objectives in this grants are to provide an in-depth study of the system by focusing on qualitative aspects, on those features of ferrofluids that exhibit the intrinsic properties that only this material has. Furthermore we will extend this study to related materials that can be obtained by combining ferrofluids and liquid crystals. Ferrofluids are a type of complex non-Newtonian fluid that is intrinsically related to liquid crystals, presenting nevertheless significant characteristic features. Such materials are studied both from fluid mechanics and from materials science points of view, using significantly different mathematical methods. The studies concern various versions of the models, the fluid mechanics perspective keeping the complicated Navier-Stokes part, and using usually simplified versions of the other equations, while the materials science perspective usually ignores the effect of the flow (thus dropping the Navier-Stokes part) and focuses on the remaining equations. It is a fundamental issue to understand the relationships between the various reduced models and their predictions. We propose to study problems relevant to both perspectives, namely problems concerning well-posedness and model reduction from the fluids perspective, respectively for the material science perspective we will look into the pattern formation and study stability issues. Moreover we will study a more complex version of ferrofluids, namely combinations between ferrofluids and liquid crystals, aiming to solve a number of design problems for this mixture of two types of materials, in particular related to the homogenization of the mixture.