Dr. Philippe H. Trinh
Departmental Lecturer in Mathematical Modelling
University of Oxford
Oxford, Oxfordshire, UK
This paper, now published in the Proceedings of the Royal Society A (PRSA) has a few interesting distinctions. It's the first paper I've published in PRSA—but hopefully not the last as it's certainly a strong journal with an illustrious history. It's the first solo paper I've published. And it has the longest title of any other paper I've worked on.
In any case, it's a paper where I explore exponential asymptotic techniques for free-surface flows (now well known) from a slightly different viewpoint. It turns out that the situation of gravity waves permits the governing equations to be re-formulated in a particularly simple way: that of a first-order nonlinear differential equation. In this paper, I show how the differential equation is studied using steepest descents. What results is a visual and beautiful way of understanding wave-structure interactions through a correspondence with the topology of certain Riemann surfaces (seen above).
I'm happy to announce the publication of a paper in collaboration with Samuel Crew (Lincoln College, Oxford). In this paper, we revisit the classic problem of a periodic wave travelling without change of form, which was first considered by Stokes in the late 19th century.
Interestingly, there are still several unanswered questions regarding this canonical example of a water wave. Some of these questions concern the existence of the singularities in the complex plane, along with their behaviour as the wave steepens. You can read the paper here.
A late congratulations to second-year student Tom Chandler (Lincoln College, Oxford) for successfully presenting his poster Jet flows from angled nozzles at the British Applied Mathematics Colloquium 2016 that took place last month in Oxford. Tom's work studies the situation of fluid ejected from an angled nozzle and driven by gravitational forces. It turns out that there are some rather non-trivial issues concerning the angle that a liquid separates from a solid, and this deals with the interplay between geometry, surface tension, and gravity. Tom's poster forms part of his third year dissertation project and will continue to a summer research project at the Mathematical Institute.
I'm happy to announce the publication of my paper in the journal Nonlinearity with Michael J. Ward (University of British Columbia) on localized spot patterns on the surface of the sphere. In this paper, we develop the detailed asymptotics that describe the slow dynamics of spot patterns modelled by a system of reaction-diffusion equations. The interesting twist is that when the patterns occur on a surface of non-zero curvature, the methodology must account for higher-order terms due to the changing geometry. It is a wonderful paper with some beautiful mathematics and beautiful pictures.
I'll be attending the Fluid and Elasticity 2015 conference, from June 22-24 in Biarritz, France, and presenting some joint work with Stephen K. Wilson (Strathclyde University) and Howard A. Stone (Princeton University).
I'm happy to announce the publication of two new papers. The first paper is published in Nonlinearity and is on the topic of developing exponential asymptotics for problems with coalescing singularities (motivated by the study of ship waves, above left) with Jonathan Chapman (Oxford). The second paper, published in the Journal of Fluid Mechanics, and collaboration with Weiqing Ren (National University of Singapore) and Weinan E (Princeton) seeks to explain the importance of distinguished limits in the classical contact line problem. Both papers can be downloaded from the research section.