Understanding non-Newtonian fluids using the XY model

Rheology is the study of the mechanical properties (viscosity, viscoelasticity etc.) of flowing fluids. Every large manufacturer that develops new foods, cosmetics, surgical implants, paints, glues, oils, drilling muds or other engineering fluids, has a rheology lab where they measure “constitutive relations” (shear stress as a function of strain rate) and other quantities. [20] Non-Newtonian fluids in continuous shear flow are routinely developed in commercial rheology laboratories. They are an example of non-equilibrium steady states, for which there is no rigorous microscopically-based mathematical theory [1] and no non-trivial, microscopic-physics-based models have been solved analytically. Rheology is a huge commercial and academic discipline. Unlike equilibrium thermodynamics, where the static properties of substances are calculated and understood via the rigorous unifying formalism of statistical mechanics, rheology has no equivalent rigorous formalism. Instead, it relies on ad hoc approximations and intuition to develop a separate mathematical model of each distinct material of commercial interest. This situation is problematic, as it hinders the development of new medicinal and commercial fluids. As an example, equilibrium statistical mechanics has proven that the critical point of water (or other fluids), where its liquid and vapour states merge, must share its properties (its “critical exponents”) with the Curie point of magnetic materials, where ferromagnetism merges with paramagnetism. Hence, the exactly solved magnetic models that are the cornerstone of statistical mechanics (the Ising model and the XY model [34–37]) inform our wider understanding of all substances at equilibrium. No equivalently profound insights are attainable for rheological properties because statistical mechanics does not apply to flowing systems and we currently have no equivalent framework for them. [1] We propose an intensive period of research, working on a potentially tractable, physically valid microscopic model exhibiting non-equilibrium phase transitions in continuous shear flow, which could establish the standard model for statistical mechanics of rheology. We aim to analyse the sheared state of the simple model fluid and derive large-scale principles from the micro-scale dynamics. It will be the first microscopically-based analytical calculation for a physically valid system exhibiting non-equilibrium phase transitions in continuous shear flow (the standard situation for commercial labs developing non-equilibrium fluids). It will be a springboard for further research.