## Potential Ph.D. topics

The following ideas could serve as Ph.D. projects. Interested students can avail of a UCD research demonstratorship, or can apply to the Irish Research Council for a scholarship. The deadline for the IRC scholarship scheme is usually at the start of November, but please check the website.

### Numerical and analytical modelling of liquid-crystal flows

Liquid crystals are important in established industrial processes (e.g. Liquid Crystal Displays). As liquid crystals are intermediate between a crystalline solid and a fluid, under certain circumstances they flow, and then hydrodynamics are important. This is especially relevant in the very contemporary applications of liquid-crystal theory - where the mathematical theory is used as a model for the flow of active matter, including the flow and alignment of micro-organisms. The mathematics of flowing liquid crystal is extremely complicated due to the large number of degrees of freedom. However, analytical and numerical progress is possible, including by dimensional reduction, lubrication theory, numerical simulation, and asympotics. This project offers an interested and able student the opportunity to work on a choice of these areas.

The goal of this project is to build up theories based on the above methodologies, for instance

- Dimensional Reduction - Explortation of liquid-crystal hydrodynamics in two spatial dimensions, incuding by high-resolution direct numerical simulation.
- Lubrication Theory - Formulation of Navier-Stokes-Landau-de Gennes equations in the longwave limit approporiate for a liquid crystal in a thin film. Analyse the regularity of the resulting solutions using method of a priori bounds.
- Direct numerical simulation of liquid-crystal turbulence in two and three dimensions using methods based on the TPLS computational framework.
- Asyptotics - Explore various limiting cases numerically and analytically, including the case of passive flow where the liquid-crystal orientation is driven by a prescribed external flow.

### Numerical and analytical modelling of suspension flows

Suspensions are everywhere and are one of the great mysteries of Fluid Mechanics. A suspension is a collection of particles embedded in a fluid, dense enough and numerous enough so they can be treated as a continuum, and yet not so dense as to prevent the suspending fluid from moving. This kind of system can be modelled as a kind of complicated Non-Newtonian fluid, with constitutitive modelling for the shear stress and the particle volume fraction. Finding the right constitutive model for the job in hand is a challenging task.

On particular job is to model the flow of a suspension in a rectangular channel. This can be done using a regularized suspension-balance or diffusive flux model. These are standard modelling approaches in the literature and are well established. Indeed, they have been shown to be equivalent, which is good news from the point of view of consistency of all reasonable physical models!

This project is concerned with looking at the linear stability of such parallel flows with respect to small-amplitude perturbations. It is expected that this will lead to a modified Orr-Sommerfeld-type analysis which can be studied so that the parameter range of stable flows can be outlined. It will also be important to look at open-flow instability, which requires the extenstion of the proposed Orr-Sommerfeld analysis to allow for absolute and convective instability.

This modelling approach is challenging from the analytical point of view - especially the theory of absolute instability, which requires good familiarity with Complex Analysis and Laplace transforms. Another possible route of study would be the exploration of the nonlinear instabilty, which would require (in the framework advocated by the Supervisor) numerical simulation of the full nonlinear temporally-evolving equations of motion.