Most of my research is focused on the theory of the lattice Boltzmann method (LBM) and its application to complex and multiphase flow.  Another area of particular interest to me is non-Newtonian fluid dynamics.

The LBM is a powerful alternative to the traditional methods of computational fluid dynamics (CFD). Rather than discretising hydrodynamic equations directly, the LBM is derived from a velocity-space truncation of the Boltzmann equation of classical kinetic theory. This discrete formulation yields a linear, constant coefficient hyperbolic system where all nonlinearities are confined to algebraic source terms. The linear differential operators may be discretised exactly by integrating along their characteristics, while the hydrodynamic equations with their nonlinear convection terms are recovered by seeking slowly varying solutions to the kinetic equations.

Rayleigh-Benard convection simulated with LBM and moment-based boundary conditions. Research with Rebecca Allen

An overview of the LBM is given in these slides. They are from a seminar I gave as part of the Leslie Comrie Series.  Dr Paul Dellar is acknowledged for his contribution – the slides are greatly inspired by his excellent introductory lectures on the LBM. Much of the work came from my former PhD students and collaborators: Andreas Hantsch, Rebecca Allen, and Seemaa Mohammed. It builds on work I started with Paul Dellar and Sam Bennett.

Some recent research of mine has concerned the development of moment-based boundary conditions for the LBM. This was inspired, in part, by applications of the LBM to microfluidics, where we had the goal of imposing Navier-Maxwell slip in pressure-driven flows in microchannels. The methodology can be extended quite naturally to other flows and allowed us to investigate wetting conditions for multiphase LBMs. I am also interested in developing models similar in spirit to “colour gradient” multiphase LBMs, and trying to capture shape interfaces in particular.

Some other early work on the LBM included revising axisymmetric models (see here and here, and the erratum), and analysing the generalised lattice Boltzmann equation.

I have also collaborated with other applied mathematicians to understand industrial problems through the “study group” format. Problems I have worked on include reacting flows and vortices (with Rolls Royce), reaction-diffusion models for decontamination (with DSTL), and football dynamics (with Trabzonspor Football Club).

Please visit the Research Menu at the top of the page for more details.