Rheology of crosslinked assemblies of flexible fibres in viscous flow

许多工业和生物医学材料由浸入粘性流体中的纤维固体相组成，例如细胞骨架、组织工程支架、医疗过滤器、纤维增强材料等。对这种复合材料的流动进行建模，可以合理地设计新产品，增强我们对自然系统的理解——因此，支架的时间依赖性机械（粘弹性）反应控制着干细胞的分化，而细胞骨架在运动的细胞中传播应力。我们最近的模拟显示了浸入式弹簧网络中纤维网络黏弹性响应的意想不到的和潜在的可开发模式。然而，该模型中流体和光纤网络之间的耦合在数学上并不严谨。虽然完整的数值模拟会非常昂贵，但存在更严格的框架（例如浸入式边界，细长体和正则Stokeslet方法）来计算浸入式纤维的流动。但是，它们不能立即应用于网络，因为(a)没有考虑到光纤的交联，(b)网络可以跨越整个系统。在这个项目中，学生将开发一个严格的理论框架（通过分析和/或数值方法的结合）交联弹性纤维在粘性流动中的装配，生成线性粘弹性和更复杂的非线性现象的预测。这个项目的成果将是：1。模拟振荡剪切流中交联弹性细纤维内流体动力相互作用的理论框架。代表化学或物理键合的交联可以通过约束每个纤维弧长上的预定义点来实现，以始终占据空间中的相同位置。框架可以是解析的、数值的或两者的结合，采用流固耦合文献中的现有思想，包括细长体理论（局部或非局部）和/或浸入边界法。虽然公式应该是完全非线性的，但在第一个实例中将产生线性粘弹性响应的预测。2. 将框架扩展到系统跨越网络，即弹性阶段在概念上是无限的，或跨越周期系统。这里的一个问题是在计算光纤网络的遥远部分之间的流体动力相互作用时实现数值效率。一种方法是实现Stokes方程的快速求解方法，如快速多极方法。对于浸入边界法，可以考虑现有的IBAMR自适应网格软件包。

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.