Rheology of crosslinked assemblies of flexible fibres in viscous flow

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Many industrial and biomedical materials consist of a fibrous solid phase immersed in a viscous fluid, e.g. the cellular cytoskeleton, tissue engineering scaffolds, medical filters, fibre reinforced materials etc. Modelling the flow of such composites provides the capability to rationally design new products and enhance our understanding of natural systems - thus, the time-dependent mechanical (viscoelastic) response of scaffolds controls stem cell differentiation, and the cytoskeleton propagates stress in moving cells. Our recent simulations demonstrated unexpected and potentially exploitable modalities in the viscoelastic response of fibre networks modelled as immersed spring networks. However, the coupling between the fluid and the fibre network in this model is not mathematically rigorous. Whilst a full numerical simulation would be prohibitively expensive, more rigorous frameworks exist (e.g. immersed boundary, slender body and regularised Stokeslet methods) for calculating the flow through immersed fibres. However, they cannot be immediately applied to networks as (a) the crosslinking of fibres has not been considered, and (b) networks can span the system. In this project, the student will develop a rigorous theoretical framework (through a combination of analytical and/or numerical approaches) for assemblies of crosslinked elastic fibres in viscous flow, generating predictions for linear viscoelasticity and more complex non-linear phenomena.The outcomes of this project will be:1. A theoretical framework for modelling hydrodynamic interactions within crosslinked elastic slender fibres in oscillatory shear flow. The crosslinking, representing chemical or physical bonding, can be implemented as constraining predefined points along each fibre's arc length to occupy the same position in space at all times. The framework may be analytical, numerical, or a combination of both, adapting existing ideas from the fluid-structure coupling literature, including slender body theory (local or non-local), and/or the immersed boundary method. Although the formulation should be fully non-linear, predictions will be generated for the linear viscoelastic response in the first instance. 2. Expand the framework to system-spanning networks, i.e. where the elastic phase is conceptually infinite, or spans a periodic system. One issue here is to achieve numerical efficiency in computing the hydrodynamic interactions between distant parts of the fibre network. One approach would be to implement fast solution methodologies for the Stokes' equations such as the Fast Multipole Method. For the immersed boundary method, IBAMR, an existing software package with adaptive meshing, could be considered.