I am a lecturer in the School of Mathematics and Statistics at the University of Glasgow. I am a member of the Solid Mechanics group.

# Research

My research concerns aspects of materials science, including fracture, solidification and pattern formation. I have particular interests in applications of finite elasticity, free boundary problems and techniques for the analysis and simulation of nonlinear partial differential equations, particularly in the reduction of partial differential equations to systems of coupled ordinary differential equations via perturbation methods. Particular systems of interest include foams, metal alloys and porelastic materials. I am generally interested in continuum mechanics, especially thermodynamics, solid mechanics and fluid mechanics and problems that involve differential geometry. I have a general interest in numerical methods, especially for partial differential equations, but also for optimal location and transport problems and in computational geometry (generalised voronoi diagrams).## Solidification

When a multicomponent alloy is cooled compositional and thermal effects can alter the form of the solidified material, producing microsegregation. With the addition of gravity there is also the possibility of large scale (compared to the micro scales) convective fluid motion, which can redistribute heat and solute and lead to interesting forms of macrosegregation. In particular the formation of mushy layers during the solidification of binary alloys can lead to chimney formation and the presence of freckles in the sample. Interesting interactions between fluid flow, thermal and solutal diffusion and surface energetics occur.

## Nanowire growth

Nanowires are rods of semi-conductor (such as silicon or germanium) which are 10-100 nm in diameter and can micrometres in length. They have desirable properties for the construction of nanoscale structures and in electronic applications. They can be grown via a technique known as Vapour-Liquid-Solid deposition, in which a catalyst particle (usually gold) is melted on a substrate, forming an alloy. The atmosphere surrounding the droplet contains the desired semi-conductor in a molecular compound (silicon, for example as silane SiH4), the decomposition of which is enhanced by the catalyst droplet. The droplet absorbs the semi-conductor until a certain super-saturation is achieved and the nanowire column precipitates at the base of the droplet. A variety of morphologies exist depending the precise nature of the component elements and the growth conditions. Straight wires, branched wires and exotic helical wires have all been observed.

Droplet on plane summary## Fluid Driven Fracture

When a fluid fills a crack, the elastic-fluid interaction can lead to an extension of the crack. This phenomenon is called fluid-driven fracture. A geophysical example is the motion of lava through a vertical crack, called a dyke. The dyke may solidify and the surrounding country rock eroded to reveal the dyke's structure, or the dyke may reach the surface and give rise to curtains of fire, as seen in effusive eruptions in Hawaii. Fluid driven fracture is also a concern of the oil industry, where fractures are propagated in order to access reservoirs of oil. Laboratory experiments to model geophysical fluid fracture use gelatin and and, typically, water. In all situations the governing equations combine elasticity, fracture mechanics and fluid dynamics and can the situation can be further complicated by solidification of the fluid, including the effects of porosity on the solid, allowing for a variety of rheologies for the fluid and exsolution of vapour.

## Optimal transportation

## Numerical methods for fourth order PDEs

The Cahn-Hilliard equation is an excellent example of a fourth-order partial differential equation arising in materials science. It describes the phase separation of a binary mixture. I am interested in numerical methods for the simulation of solutions to the Cahn-Hilliard equation.

# Teaching

I have taught at all undergraduate levels and supervised undergraduate and postgraduate projects. Here are some resources: Click here to validate this page.## Dr. Steven Roper

School of Mathematics and Statistics

University of Glasgow

University Avenue

Glasgow, G12 8QQ

United Kingdom

T: +44 (0)141 330 6535

F: +44 (0)131 330 4111

## Links

## Mathematical/Computational

- ESAM.
- DAMTP, University of Cambridge, UK.
- The Journal of Fluid Mechanics.
- The American Physical Society.
- A useful FORTRAN 77 page.
- The Numerical Recipes homepage.
- The LaTeX project site and the TeX users group.
- Perl, Bash, Awk, Python

## Funding links

- The EPSRC
- The NERC
- The Royal Society.

## Visualisation

- The Gnuplot homepage
- Gnuplot tricks