Rheology of crosslinked assemblies of flexible fibres in viscous flow

许多工业和生物医学材料由浸没在粘性流体中的纤维状固相组成，例如细胞骨架、组织工程支架、医疗过滤器、纤维增强材料等。对此类复合材料的流动进行建模提供了合理设计新产品的能力，并增强了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）响应控制干细胞分化，并且细胞骨架在运动细胞中传播应力。我们最近的模拟表明，意外的和潜在的可利用的方式在粘弹性响应的纤维网络建模为浸入弹簧网络。然而，该模型中流体和纤维网络之间的耦合在数学上并不严格。虽然一个完整的数值模拟将是昂贵的，更严格的框架存在（例如浸入边界，细长体和正则化Stokeslet方法）计算通过浸入纤维的流量。然而，它们不能立即应用于网络，因为（a）纤维的交联尚未被考虑，并且（B）网络可以跨越系统。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/或数值方法的组合）在粘性流中的交联弹性纤维的组件，产生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长纤维内流体动力相互作用的理论框架。代表化学或物理结合的交联可以被实施为沿着沿着每个纤维的弧长约束预定义的点，以始终占据空间中的相同位置。该框架可以是分析的，数值的，或两者的组合，适应现有的想法，从流体-结构耦合文献，包括细长体理论（本地或非本地），和/或浸没边界法。虽然该公式应该是完全非线性的，但首先将生成线性粘弹性响应的预测。2.将框架扩展到系统跨越网络，即弹性相在概念上是无限的，或跨越周期性系统。这里的一个问题是在计算纤维网络的远距离部分之间的流体动力学相互作用时实现数值效率。一种方法将是实施用于斯托克斯方程的快速求解方法，例如快速多极方法。对于浸入边界法，可以考虑IBAMR，一个现有的软件包与自适应网格，。

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.