Mathematical Sciences: Computations in Fluids and Materials

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9707494 Michael Shelley These projects concern dynamics and pattern formation in complex fluids, and singularity formation and topological transitions in Newtonian fluids. The first project considers the hydrodynamics of slender elastic filaments, such as arise in liquid crystal flows, the dynamics of phospho-lipid bilayer tubes, and in the dynamics of biological polymers. Building tractable computational models, that account for hydrodynamic interactions of the filament with itself, relies on discriminating exploitation of slenderness. This still gives a computationally intensive problem with high-order time-step constraints from elasticity, interaction integrals with singular kernels, and integral equations to be solved at every time-step. The second project continues towards an understanding of topological transitions of fluid/fluid interfaces between immiscible liquids. The fundamental questions are: How does surface tension provoke or mediate transitions? What characterizes the singularity? What physics needs to be added to follow the transition? And what is left of the singularity in its aftermath? Building upon previous work on such singularities in the Kelvin-Helmholtz instability between immiscible fluids, it is proposed to study, computationally and analytically, singularities and transitions in jets that separate immiscible fluids, both by using sharp interface models, and fluid models that have viscosity and allow some miscibility. The third project studies the effect of shear-thinning, a property shared with many non-Newtonian fluids and liquid crystal flows, on the development of the Saffman-Taylor instability. Some of the modelling work has already been done, yielding a natural non-Newtonian version of Darcy's law, relating the fluid velocity to the solution of a nonlinear elliptic problem. It is proposed to now simulate the full nonlinear dynamics of such a bubble expanding into a shear-thinning liquid. This is a very challenging comp utational problem as it involves the solution of nonlinear elliptic problems on an evolving domain. Much of the fundamental dynamics of fluids and materials -- singularity and pattern formation are two central examples -- will be understood by a progression from mathematical modelling, to developing computational methods and relevant mathematical understanding, and thence to large-scale simulation and data analysis through high-performance computing. The three projects to be pursued here all lie at the intersection of fluid dynamics and materials science, and all illustrate the above statement. In the first project, it is proposed to understand and simulate the dynamics of filamentary structures, as arise in phase transitions of liquid crystalline fluids, the dynamics of phospho-lipid bilayer tubes, and in the dynamics of biological polymers. In first example, such filaments are of potential technological importance in the manufacture of high-strength filaments. The second project continues towards a theoretical understanding of what drives the break-up into droplets of a jet of fluid into a second, different fluid (say, oil and water). While easy and common to observe, such behavior is strongly associated with surface tension, an effect that is still ill-understood, and yet lies at the heart of much basic fluid phenomena. This will be studied by a combination of modelling, analysis, and large-scale computation. The final project concerns the dynamics of shear-thinning liquids flowing in thin gaps. Such flows are important to display device design, and to injection molding. Of particular interest is the instability and pattern formation associated with a gas/liquid interface which is driven but mediated by surface tension. This is an extremely challenging computational problem, requiring the development of new simulational methods.