Modelling cross-linked fibre networks in flow

Summary

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.

Nature and industry abound with examples of highly porous, fibrous materials embedded in a viscous fluid. Examples include the cellular cytoskeleton, scaffolds for tissue engineering, and medical filters, to name just a few. Recent simulations have demonstrated that the coupling between the solid (fibre) phase and the fluid can produce unexpected behaviour that could be exploited for industrial applications, and may have already been exploited by nature; however, the framework employed was mathematically non-rigorous. In this project, the student will develop a more rigorous theoretical framework (analytical, numerical, or both), for assemblies of cross-linked assemblies of flexible fibres in viscous flow. This is an opportunity to work on the interface between two previously separate fields, with the potential for rapid progress and high-impact fundamental research.

The project goals will depend in part on the skills and interest of the student, but initially can be taken to be the following:

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.

Applicants to research degree programmes should normally have at least a first class or an upper second class British Bachelors Honours degree (or equivalent) in an appropriate discipline. The criteria for entry for some research degrees may be higher, for example, several faculties, also require a Masters degree. Applicants are advised to check with the relevant School prior to making an application. Applicants who are uncertain about the requirements for a particular research degree are advised to contact the School or Graduate School prior to making an application.

The minimum English language entry requirement for research postgraduate research study is an IELTS of 6.5 overall with at least 6.5 in writing and at least 6.0 in reading, listening and speaking or equivalent. The test must be dated within two years of the start date of the course in order to be valid. Some schools and faculties have a higher requirement.