Possible Areas for PhD Projects

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I am happy to consider applications from students wishing to study for a PhD with me on any of my research interests. In addition, there may be specific projects for which I am keen to attract students. Details of these projects can be found below.

I am more than happy to talk to potential applicants, so please get in touch with me for more information. For instructions on how to make a formal application, please see the UEA Maths Research Degrees page.

There is no specific funding allocated to these project. However, some funded studentships are available from the University for well-qualified students. Self-funded students can be considered too.


Oscillations in Flow Through an Elastic-Walled Tube

Project Description

Flow-induced oscillations of fluid-conveying elastic vessels arise in many engineering and biomechanical systems. Examples include pipe flutter, wheezing during forced expiration from the pulmonary airways, and the development of Korotkoff sounds during blood pressure measurement by sphygmomanometry.

Experimental studies of flow in collapsible tubes are typically performed with a Starling resistor. A finite-length elastic tube is mounted between two rigid tubes and flow is driven through the system. The collapsible segment is contained inside a pressure chamber which allows the external pressure acting on the elastic tube to be controlled. If the pressure outside the tube becomes sufficiently large, it will buckle non-axisymmetrically. Once buckled, the tube is very flexible leading to strong fluid-structure interaction. Experiments show that in this buckled state, the elastic tube segment has a propensity to develop large-amplitude self-excited oscillations of great complexity when the flow rate is increased beyond a certain value.

This project aims to further our understanding of some of the mechanisms that can lead to this instability of flow through an elastic-walled tube. The fluid flow will be described by the Navier–Stokes equations, and an appropriate elastic model will be used for the tube wall. Whittaker et al (2010) developed a relatively simple model for small amplitude, long-wavelength, high-frequency oscillations. Work is currently underway on extensions to include shear effects, wall inertia and axial bending. This project will start by working to relax some of the remaining assumptions in the previous model, e.g. adding nonlinear effects and allowing for different cross-sectional shapes. The project will likely focus on developing reduced analytic models (which may need to be solved analytically or numerically) though there is also scope for conducting full-scale numerical simulations.

References

High-Frequency Self-Excited Oscillations in a Collapsible-Channel Flow
Oliver E. Jensen & Matthias Heil, 2003.
Annual Review of Fluid Mechanics 481, 235.
Biofluid Mechanics in Flexible Tubes
James B. Grotberg & Oliver E. Jensen, 2004.
Ann. Rev. Fluid Mechanics 36, 121.
Predicting the Onset of High-Frequency Self-Excited Oscillations in Elastic-Walled Tubes
Robert J. Whittaker, Matthias Heil, Oliver E. Jensen & Sarah L. Waters, 2010.
Proceedings of the Royal Society A 466 (2124), 3635–3657.
Fluid–Structure Interaction in Internal Physiological Flows
Matthias Heil & Andrew L. Hazel, 2011.
Annual Review of Fluid Mechanics 43, 141–162.

Further details of my work in this area can be found on my collapsible tubes research page.


Stability of Soap Films and Foams

This project is suitable for a graduate of applied mathematics, engineering or physics with a strong background in theoretical continuum mechanics and mathematical modelling.

Project Description

Foams, comprising thin liquid films surrounding many small gas bubbles, have many domestic and industrial applications. These often rely on properties such as a high interfacial area, suface activity and yield stress. Foams can also be used as proxies for the structure of more complicated biological systems. The structure of a foam us largely determined by the minimization of the interfacial surface area. For relatively dry foams, this is dominated by the surface films, but there is also a contribution from the joins — known as Plateau borders — between the surfaces.

In any application, the stability of the foam will be important. We may want to promote either a stable long-lived foam, or an unstable foam that quickly breaks down. Previous work by the supervisor and a collaborator has investigated the stability of a single twisted Plateau border. Initially, just the effects of the connected surfaces were considered, and then some simple mechanics of the Plateau border was added too. Theoretical results were compared with simulations conducted using the 'Surface Evolver' software.

In this project you will continue the investigation of the stability of foams using theoretical mathematical modelling techniques. The starting point will be to develop a model for the mechanics of the Plateau border, to correctly account for restoring forces in response to extension, bending and twisting. Further work will look at integrating models for a single Plateau border to determine the bulk properties of a whole foam.

References

Instability of Stretched and Twisted Soap Films in a Cylinder
Simon J. Cox & Siân A. Jones, 2014.
Journal of Engineering Mathematics, 86, 1–7.
Stability of a Helicoidal Surface inside a Cylinder with Pinned Diameters
Robert J. Whittaker & Simon J. Cox, 2015.
Quarterly Journal of Mechanics and Applied Mathematics, 68 (1), 23–52.
Stability of Twisted Plateau Border with Line Tension and Bending Stiffness
Robert J. Whittaker & Simon J. Cox, 2019.
IMA Journal of Applied Mathematics, 84(2), 385–415.
The Surface Evolver
Ken Brakke, 1992.
Experimental Mathematics, 1(2), 141–165.

Further details of my work in this area can be found on my soap films research page.

See the project advert for details of how to apply.