# Dynamics, Modelling and Computation

We have a rich history since 1875, when the group was founded by renown applied mathematician and fluid dynamicist, Sir Horace Lamb. Today, whilst still maintaining our historic strength in fluid mechanics, we are a nationally-leading group with diverse interests in many exciting areas of modern applied mathematics.

Much of our work is highly interdisciplinary, and involves close collaboration with colleagues from fields including nanotechnology, medicine, biology and oceanography, or with industrial partners. Using a combination of modelling, analysis and computation, we can provide new insights into real world problems with important practical applications.

Our researchers have won national awards from the Australian Academy of Science and Australian Mathematical Society, as well as a multitude of Australian Research Council Fellowships. Current and past students have also enjoyed considerable success in winning national prizes and scholarships. Recent graduates are employed in postdoctoral positions in leading universities in Australia and overseas, including the University of Melbourne, University of Auckland and UCLA; others have obtained industry employment, for example at BAE systems, SANTOS, Westpac and the Bureau of Meteorology.

Sanjeeva Balasuriya | Dynamical systems, differential equations, applied analysis, chaotic mixing, geophysical flows, combustion waves, mathematical ecology |

Luke Bennetts | Waves in random and complex media; multiple wave scattering; hydroelasticity; modelling dynamics of the ice-covered ocean. |

Benjamin Binder | Cellular automata and continuum models, free-surface flows, potential flows |

Judith Bunder | Computational multiscale modelling in complex systems, computational algorithms for high performance computing |

David Clements | Anisotropic elasticity, fracture mechanics, seismic wave propagation, heat flow in solids, flow through porous media, fiber-reinforced materials, contact problems |

Barry Cox | Applied mathematical models for nanoscaled systems with applications in electromechanical nanodevices, novel media for gas capture and storage, and advanced materials |

Ed Green | Mathematical modelling of biological problems, Fluid Mechanics, Pattern formation, Reaction-diffusion equations, Free boundary problems, Agent-based models |

Brendan Harding | Particle-laden flows, Computational fluid dynamics, Fractal geometry, High dimensional approximation, High performance computing |

Trent Mattner | Fluid mechanics, including turbulence, direct and large-eddy simulation, bushfires, buoyancy-driven flows, rotating flows, vortical flows and boundary layers |

Tony Roberts | Modelling emergent dynamics in complex systems, simulation and modelling of flood flows and thin fluids, fractal geometry |

Yvonne Stokes | Viscous fluid mechanics, computational fluid dynamics, mathematical biology, industrial mathematics |

Hayden Tronnolone | Viscous Fluid Mechanics, Mathematical Biology |

## Research Areas

**Fluid dynamics**

**Mathematical biology**

**Dynamical systems**

**Numerical modelling**

**Industrial modelling**

## Possible PhD Projects

- Controlling internal flow barriers - Associate Professor Sanjeeva Balasuriya
Fluid flows often have within them regions of fluid moving almost coherently, amidst regions which appear to be of a more random nature. Examples of these coherent objects include the Antarctic circumpolar vortex (‘the ozone hole’), oceanic eddies, oceanic jets such as the Gulf stream, or a blob of one fluid moving within another fluid. The boundaries or edges of the coherent structures can be thought of as ‘internal flow barriers’ which are generally moving with time. These therefore strongly govern the mixing and transport that can occur between different regions of fluid.

This project tackles the issue of whether it would be possible to control the movement of these internal flow barriers, thereby according some level of control on fluid mixing. This is to be achieved by adjusting the fluid velocity in some way. In real life, this might be achieved by applying electrical/magnetic forces on a fluid containing susceptible particles, by flushing fluid into a device using small channels, or by imparting velocities by moving (e.g., vibrating) physical boundaries of a device. The basic question that is to be addressed is: can we find the required protocols in order to achieve a specified movement of internal flow barriers?

This research question turns out to be equivalent to determining how to correct an observational/experimental velocity field in order to ensure that observed flow barriers are in the correct locations. Thus, it also has importance in oceanography.

Required skills: Background in ordinary differential equations, dynamical systems and fluid mechanics, plus coding ability. Experience in either numerical solutions of fluid dynamics equations (CFD), or in data-driven modelling, would be beneficial. The project has the potential of balancing theoretical, computational and applied aspects based on the student’s skills and interests.

- Stochastic impacts on ordinary differential equations - Associate Professor Sanjeeva Balasuriya
Ordinary differential equations (ODEs) generate trajectories in phase space. An example is of a three-dimensional fluid flow, whose particle locations x(t) in the physical 3D phase space are obtained from solutions to the ODE dx/dt = u(x,t), where u(x,t) is the fluid velocity field. Transport may be understood by how the different trajectories move in relation to one another. Some move coherently, while others have a haphazard relationship (‘chaotic’) to their neighbours.

Mixing occurs not just because of transport, but also because of molecular diffusion. This, for example, will cause fluid filaments to break apart and diffuse into the surroundings. One way of modelling diffusion is to have a pollutant density field being subject to a partial differential equation (PDE) in which the pollutant is advected by the velocity field u(x,t), as well as diffused according to a Laplacian operator. Another approach is to consider the stochastic differential equation (SDE) dx/dt = u(x,t) + “noise”. It turns out that these two approaches are intimately linked: the first captures the density evolution of the second.

This project addresses the stochastic impacts on mixing using both the above approaches, in the situation where the noise is considered small. Several research questions will be pursued, using both models for the velocity field as well as experimental/observational data. Are there signature features that are more susceptible to mixing when stochasticity is included? What features of the ODE are robust under noise? Are there connections to the theory of Lagrangian coherent structures (a well-established theory for transport barriers in the absence of noise)?

Required skills: Theoretical background in ODEs and PDEs, and the ability to numerically solve them. Exposure to any of SDEs, Ito calculus, or advanced statistical methods is advantageous.

- Quantifying and modelling yeast growth - Associate Professor Ben Binder
Yeasts are single cell fungi organisms, familiar to most for their use in the baking of bread and brewing of alcoholic drinks. One of the most significant issues related to yeasts today is their causation of pathogenic infections in humans (e.g. oral and vaginal thrush). Moreover, a persistent problem is in their formation of biofilms on medical devices such as catheters, placing patient populations at risk of infection in our major hospitals. The real and varied impacts that yeast’s have on our everyday lives is one of the reasons why it is one of the most studied organisms in modern biology today.

The aim of this project is to work on developing both spatial statistics and mathematical models (continuum and discrete) to gain a better understanding of the yeast growth process.

The project will be supervised by Associate Professor Ben Binder at the University of Adelaide, and provides an opportunity to collaborate directly with biologists at the Australian Wine Research Institute, University of Adelaide (Waite Campus).

- Three-dimensional water waves over topography (exchange with the University of East Anglia, UK) - Associate Professor Ben Binder
The purpose of this project is to study water waves and how they interact with bottom topography. One reason for doing this is to provide a means for using surface observations (of the ocean surface, for example) to infer the shape and structure of the water-bed. Of particular interest to this project is the potential formation of localised three-dimensional waves, which decay in the far-field away from the topographic forcing. Progress will be made on this by studying solutions to the Kadomtsev-Petviashvili equation. More challenging will be the study of such structures for the fully nonlinear equations, and this will be tackled using boundary integral methods.

The project will be supervised by Associate Professor Ben Binder at the University of Adelaide.

A key feature of this project is the opportunity for one or more research visits to the School of Mathematics at the University of East Anglia, UK (UEA). At the UEA you will work on the project with Dr Blyth.

- Geometry of Nanostructures - Dr Barry Cox
It is clear from the various structures seen at the nanoscale such as the cylindrical shapes of nanotubes and spherical and spheroidal shapes of fullerenes that the complex interactions of these structures often lead to symmetric conformations. So in satisfying a minimum energy constraint the system often (although not always) adopts a symmetric structure which shares the energetic costs of bending and stretching covalent bonds equally to all components in the structure. By assuming a symmetric conformation up front, it is possible to reduce fundamentally complex problems of molecular structure to problems with are more mathematically tractable and thereby derive results which can be confirmed by experiment and simulation and can also be used to predict ideal systems and novel structures in certain extreme cases.

- Predicting Properties of Nanomaterials - Dr Barry Cox
The design of many novel electronic devices will hinge on our understanding of the joining of certain nanostructures. For example, the joining of carbon nanotubes to graphene sheets, fullerenes and other carbon nanotubes applies to the situation of constructing nano-electro-mechanical devices including nanoscale FETs. Connecting carbon nanostructures essentially involves a discrete geometric procedure and we have attempted to solve such problems by invoking the principle that the bond lengths and bond angles at the join are determined in such a manner that the total squared deviation from some ideal configuration is a minimum. We have also applied the calculus of variations to compute various join configurations in such a way that the total curvature squared is minimised, subject to other constraints regarding the interatomic bond lengths.

- Modelling growth and morphogenesis of colonic organoids - Dr Edward Green
Organoids are three-dimensional in vitro tissue cultures which mimic (to some degree) the distinctive in vivo structure of the organ from which they derive. In this project, we will focus on organoids grown from intestinal tissue, which are used for research into colorectal cancer, one of the most common cancer types. Although colonic organoids are being grown successfully by various research groups, at present their growth and development are not well understood. They can vary in morphology e.g. `budded' and `cystic' types are observed, but the reason for this, and its possible significance for the usefulness of the organoids in research is unknown. Furthermore, for potential applications in personalised medicine, there is a need to optimise the culture process to grow larger quantities of tissue more rapidly.

Budded and cystic organoids (picture courtesy of Dr Tamsin Lannagan, SAHMRI)

This project will develop new mathematical models for organoid growth and development, based on the principles of morphoelasticity. We will investigate the role of growth induced buckling in determining their form (cystic or budded), taking into account the potential roles of different cell populations, and chemical signals in this process. Our models will be validated against experiments being undertaken by Dr Daniel Worthley's group at the South Australian Health and Medical Research Institute (SAHMRI).

Note that this project is available as part of the joint Adelaide-Nottingham PhD programme.

- Large-eddy simulation of turbulent flows - Dr Trent Mattner
Turbulent flows are characterised by irregular unsteady three-dimensional fluid motion over a wide range of spatial and temporal scales. They occur in a wide variety of situations, such as super-novae, atmospheric and oceanic circulation, combustion, breaking waves, flows past ships and aircraft, and so on. In these applications, the range of scales is so enormous that is impossible to simulate them all, even using the largest supercomputers. In a large-eddy simulation (LES), only the largest scales are simulated. While this is computationally expedient, it is necessary to model the effects of the smaller scales. The aim of this project is to refine the stretched-vortex subgrid model und use it to simulate turbulent mixing in various flows.

- Harvesting ocean wave energy - Dr Luke Bennetts
The Ocean Wave Energy Research Group at the University of Adelaide is leading development of devices to harvest the vast natural and renewable resource of wave energy along Australia’s southern coastline. The key challenges are to design the devices so that they will capture significant proportions of energy from the range of possible sea states, and to arrange devices into arrays that capture greater energy than the same number of devices in isolation. Advances are based on PDE models of the devices and the waves, and efficient mathematical methods need to be developed for these models to provide insights into optimal energy harvesting. Available HDR projects focus on low-cost nonlinear models of individual devices, and techniques for trapping wave energy in arrays. Applicants interested in working in multidisciplinary teams, involving numerical modellers, experimentalists, engineers and computer scientists, are particularly welcome.

- Designing acoustic metamaterials - Dr Luke Bennetts
Acoustic metamaterials are artificial, composite materials, designed to control acoustic vibrations, and often used to prevent sound waves travelling through the material, so that it can insulate a body from sound. They are constructed from a large number of repeating microstructural elements, with carefully tuned resonance properties. Developing advanced mathematical techniques to calculate the overall material properties from the properties of an individual element, is the key to designing the most effective metamaterials. HDR projects are available to explore how local resonances can be combined with nonlinearities and different modes of motion, potentially creating new metamaterials with control over much wider acoustic ranges than currently possible.

Requirements: Undergrad training in differential equations, mechanics and coding (e.g. MATLAB).

- Mathematical Modelling of Biochemical Sensor Fabrication - Associate Professor Yvonne Stokes
Microstructured optical fibres (MOFs) have revolutionised optical fibre technology, with a virtually limitless range of designs for a wide range of applications, including as whispering gallery resonator biochemical sensors with the limit of detection potentially down to a single molecule. These are fabricated from commercially available capillaries (preforms) by first drawing or tapering the capillary down to the required diameter (typically < 100m), and then heating a small section while, at the same time, pressurising the air inside to form a micro-bottle or micro-bubble in the heated region.

PhD projects are available on the mathematical modelling of the fabrication of tapered capillaries and of micro-bottles and micro-bubbles. We will develop coupled flow and temperature models using asymptotic methods exploiting the slender/thin geometry, and use these models to determine and investigate the key parameters in the fabrication process and how to achieve the desired sensor design. In addition to modelling, there will be opportunity to be involved in experiments with skilled technicians at the world-renowned Institute for Photonics and Advanced Sensing (IPAS) at the University of Adelaide, for validation of your models.

Supervisors: Assoc. Prof. Yvonne Stokes (School of Mathematical Sciences), Prof. Heike Ebendorff-Heidepriem (IPAS), Dr Yinlan Ruan (IPAS).

- Calcium signalling and travelling wave response to oocyte fertilisation - Associate Professor Yvonne Stokes
The successful fertilisation of oocytes (eggs) is a major event in the reproduction of many species including humans. There is now compelling evidence that so-called cumulus cells, which surround the mature oocyte, play a crucial role in this process. Recent experimental work by Thompson and others in the Robinson Institute, The University of Adelaide, showed that upon fertilisation cumulus cells move away from the oocyte in a travelling wave-like fashion. These travelling waves have been observed in bovine and amphibian embryos and are known to be a result of cellular calcium signalling. Importantly, these calcium signals are triggered in the first instance by oocyte fertilisation and thus provide a causal link between fertilisation and the wave-like motion of cumulus cells. In this project, we will investigate the connection between cell movement and cellular calcium signalling in the cumulus-oocyte complex (COC).

The new COC model that we will develop will amalgamate ordinary and partial differential equations with an agent-based framework. Using tools from applied nonlinear dynamical systems and scientific computation we will explore how this model integrates intra- and intercellular signals to create movement. A main question that we wish to investigate is how the information that the oocyte is fertilised is transmitted through the COC. Is the oocyte solely responsible for the calcium signal that then diffuses through the surrounding cells and into the medium surrounding the COC, or do the cumulus cells contribute to the signalling? By answering this question, we will gain a deeper understanding of the cellular calcium signalling cascades involved in fertilisation and the physical organisation of the cumulus cell network. Existing experimental data as well as new experiments will be used to determine model parameters and for model validation.

Supervisors: Assoc. Prof. Yvonne Stokes (School of Mathematical Sciences, The University of Adelaide), Dr Ruediger Thul and Prof. Stephen Coombes (School of Mathematical Sciences, University of Nottingham), and Assoc. Prof. Jeremy Thompson (School of Medicine, The University of Adelaide).