Modelling and simulation of biofluid mechanics with multiphysics and multiple scales

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I propose to explore problems in biofluid mechanics where the fluid flow interacts with other biological or physical processes. My aim is to develop a deeper understanding of these complex flows and uncover new insights through a combination of analytical and computational methods. The problems of interest fall under two themes: (I) Fluid-structure interaction involving deformable, elastic structures immersed in incompressible fluids. Applications include suspensions of active swimmers such as marine worms and jellyfish; and biomechanics of the fluid-filled cochlea (or inner ear). (II) Multiphase flow and transport in micro-structured porous media. The primary focus is on the study of sap flow in trees and the coupling with nutrient transport and other bio-physical processes, more specifically the freeze-thaw process that drives sap exudation in sugar maple. Another related problem that impacts a much wider range of tree species is freeze-induced embolism.  These two themes comprise a seemingly diverse collection of physical and biological phenomena, but a few common features unify our modelling and computational approaches: * dynamics are governed by nonlinear systems of partial differential equations with strongly-coupled solution components; * solutions are characterized by interior and boundary layers and a clear separation of spatial scales, which naturally benefit from the application of multiscale analysis and adaptive numerical methods; and * widely varying time scales that give rise to numerical stiffness, and which require the use of implicit time-stepping algorithms. I apply a similar solution methodology to all of these problems: (1) deriving a detailed mathematical model based on careful biological and physical reasoning; (2) employing analytical techniques such as asymptotics, linear stability, and periodic homogenization to simplify the governing equations and obtain insight into solution behavior; and (3) developing accurate and efficient numerical schemes that enable extensive parameter studies to validate the model against analytical solutions and experimental data. A defining characteristic of this proposal is the tight interplay between mathematical analysis and algorithm development, where insight into the mathematical structure of solutions is exploited to make informed decisions on numerical methods. This work will have significant impact in terms of novel algorithms for solving complex problems in biofluid dynamics, as well as advancing knowledge of multi-physics and multiscale flow phenomena. For example, studies of interactions between active cell structures and cochlear fluid in the inner ear should lead to improved understanding of mammalian hearing mechanisms. Moreover, new insights into the multiscale nature of cellular processes governing sap flow in maple trees will lead to advances in tree physiology as well as improving sap yields in the maple syrup industry.