Robert Van Gorder

Bio-sketch: 

I obtained BS (2009) and PhD (2014) degrees in mathematics from the University of Central Florida (UCF), where I held a Trustees Doctoral Fellowship (2009-2011) and then a National Science Foundation Graduate Research Fellowship (2011-2014). After completion of my PhD, I moved to the Mathematical Institute at the University of Oxford, UK, and took up the post of Research Fellow in Nonlinear Dynamics (2014-2015), followed by a Glasstone Research Fellowship in Science (2015-2018). In 2019, I joined the Department of Mathematics and Statistics at the University of Otago, New Zealand as a Senior Lecturer, where I was later "confirmed" (the closest approximation to "tenured" in Otago/NZ parlance) in 2023 and promoted to Associate Professor in 2024. (To see what these ranks signify outside of the four-tiered New Zealand system, click here).


I enjoy traveling and carrying out my work in myriad locales, preferring longer research visits to brief, hectic conferences. I spent the first half of 2023 on leave while a Visiting Scholar at Merton College and the Mathematical Institute, University of Oxford, and a CNRS-funded guest scientist in the Institut de Physique de Nice, France.


Contact Details: 

Mailing address:
Robert A. Van Gorder,
Department of Mathematics and Statistics
University of Otago
P.O. Box 56
 Dunedin 9054 New Zealand

email:
robert dot vangorder at otago dot ac dot nz


Employment:


Education:

Research Interests and Selected Publications:

In my research I seek to better understand how physical phenomena can be described, predicted, and even modified using tools from applied mathematics. I specialize in the trifecta of mathematical modelling (writing down relations between physical quantities within the language of mathematics... this is where all of those equations come from!), analytical and asymptotic solution methods (deriving an exact or approximate solution to a mathematical problem using pen-and-paper approaches), and numerical simulations (using a computer program to solve a mathematical problem). I apply these methods to study physical phenomena primarily falling into one of the following areas:

I. Fluids
If you’re alive, you’re interacting with fluids: The air that you breathe and the water that you drink are examples of fluids. Despite the ubiquitous nature of fluids, there are still many questions we have about their behavior. My interests in fluid mechanics include better understanding the dynamics governing fundamental structures –such as vortices, bubbles, waves, and boundary layers – that can then be used as the building blocks for more complex fluid flows. What forms can these structures take? Do they persist, or break apart over time? What happens if we attempt to manipulate or control these structures? These are the kinds of questions I am interested in.

Selected Publications:
R. A. Van Gorder and P. A. Fisher, Spatial eigenvalue problems for stars in hydrostatic equilibrium: Generalized Lane–Emden equations as boundary value problems, Monthly Notices of the Royal Astronomical Society 523 (2023) 2059-2073
R. A. Van Gorder, Finite time blowup of incompressible flows surrounding compressible bubbles evolving under soft equations of state, Proceedings of the Royal Society A 478 (2022) 20220172
E. K. Luckins, J. M. Oliver, C. P. Please, B. M. Sloman, R, A. Van Gorder, Modelling and analysis of an endothermic reacting counter-current flow, Journal of Fluid Mechanics 949 (2022) A21
B. M. Sloman, C. P. Please, and R. A. Van Gorder, Melting and dripping of a heated material with temperature-dependent viscosity in a thin vertical tube, Journal of Fluid Mechanics 905 (2020) A16
F. Brosa Planella, C. P. Please, and R. A. Van Gorder, Extended Stefan problem for solidification of binary alloys in a finite planar domain, SIAM Journal on Applied Mathematics 79 (2019) 876-913
J. A. Kwiecinski and R. A. Van Gorder, Dynamics of nearly parallel interacting vortex filaments, Journal of Fluid Mechanics 835 (2018) 575-623
D. T. Kennedy and R. A. Van Gorder, Motion of open vortex-current filaments under the Biot–Savart model, Journal of Fluid Mechanics 836 (2018) 532-559.
R. A. Van Gorder, Dynamics of the Rayleigh-Plesset equation modelling a gas-filled bubble immersed in an incompressible fluid, Journal of Fluid Mechanics 807 (2016) 478-508
R. A. Van Gorder, Self-similar vortex filament motion under the non-local Biot–Savart model, Journal of Fluid Mechanics 802 (2016) 760-774.
R. A. Van Gorder, Helical vortex filament motion under the non-local Biot–Savart model, Journal of Fluid Mechanics 762 (2015) 141-155

Cross-sections of the velocity field (top) and scalar pressure (bottom) for an incompressible fluid surrounding a compressible "soft" bubble undergoing collapse and subsequent blowup. For more details, see: R. A. Van Gorder, Proceedings of the Royal Society A 478 (2022) 20220172.

II. Patterns
Diffusive instabilities – such as the Turing and Benjamin-Feir instabilities – have been proposed as mechanisms for the formation of patterns in many real-world systems, ranging from spot formation on the coats of big cats to optical turbulence in lasers. I am interested in understanding how these instability mechanisms extended to more generic non-autonomous or spatially heterogeneous systems, where they result in messier yet perhaps more realistic patterning. These heterogeneous systems arise from models of reaction-diffusion processes such as chemical reactions in the presence of thermal forcing, fluid flows, or evolving space domains, and are found in applications ranging from physics and chemistry to biology and epidemiology. Under what conditions can spatial or spatiotemporal patterns emerge from such systems? 

Selected Publications:
Y. Önder and R. A. Van Gorder, Modification of Turing patterns through the use of time-varying anisotropic diffusion, Proceedings of the Royal Society A 479 (2023) 20230487
R. A. Van Gorder, Pattern formation from spatially heterogeneous reaction-diffusion systems, Philosophical Transactions of the Royal Society A 379 (2021) 20210001
R. A. Van Gorder, A theory of pattern formation for reaction-diffusion systems on temporal networks, Proceedings of the Royal Society A 477 (2021) 20200753
R. A. Van Gorder, V. Klika, and A. L. Krause, Turing conditions for pattern forming systems on evolving manifolds, Journal of Mathematical Biology 82 (2021) 4
R. A. Van Gorder, Influence of temperature on Turing pattern formation, Proceedings of the Royal Society A 476 (2020) 20200356
R. A. Van Gorder, Turing and Benjamin-Feir instability mechanisms in non-autonomous systems, Proceedings of the Royal Society A 476 (2020) 20200003
R. A. Van Gorder, H. Kim, and A. L. Krause, Diffusive instabilities and spatial patterning from the coupling of reaction-diffusion processes with Stokes flow in complex domains, Journal of Fluid Mechanics 877 (2019) 759-823

Control of spatial patterns within a disc using a sharp diffusion and reaction rate differential. For more details, see R. A. Van Gorder, Philosophical Transactions of the Royal Society A 379 (2021) 20210001.

III. Waves
My interests in theoretical physics include quantum mechanics and quantum field theory, with a particular focus on modelling problems in low-temperature physics and condensed matter physics as quantum fluids. Some of my work in this area has involved understanding the dynamics Bose-Einstein condensates, quantized vortex filaments in superfluid helium, and confined quantum systems. While these topics are farther removed from our daily experiences, they exhibit exotic and interesting behaviors, providing a number of interesting scientific challenges. The study of waves is pertinent to these and other applications, and I am interested in understanding how nonlinear waves evolve under various conditions relevant to realistic experimental configurations, such as confined space domains or imposed heterogeneity.  

Selected Publications:
R. A. Van Gorder, Quantum mechanics on time-varying space domains, Proceedings of the Royal Society A 479 (2023) 20220759
R. A. Van Gorder, Compressed hydrogen atoms confined within generic boxes, Proceedings of the Royal Society A 478 (2022) 20220467
R. A. Van Gorder, Adiabatic soliton management: Controlling solitary wave motion while keeping the wave envelope unchanged, Physics Letters A 446 (2022) 128284
R. A. Van Gorder, Time-varying Bose-Einstein condensates, Proceedings of the Royal Society A 477 (2021) 20210443
R. A. Van Gorder, Perturbation theory for Bose-Einstein condensates on bounded space domains, Proceedings of the Royal Society A 476 (2020) 20200674
L. W. S. Baines and R. A. Van Gorder, Soliton wave-speed management: slowing, stopping, or reversing a solitary wave, Physical Review A 97 (2018) 063814
R. A. Van Gorder, Breathers and nonlinear waves on open vortex filaments in the nonrelativistic Abelian Higgs model, Physical Review D 95 (2017) 096007
R. A. Van Gorder, Solitons and nonlinear waves along quantum vortex filaments under the low-temperature two-dimensional local induction approximation, Physical Review E 93 (2016) 052208
R. A. Van Gorder, Quantum Hasimoto transformation and nonlinear waves on a superfluid vortex filament under the quantum local induction approximation, Physical Review E 91 (2015) 053201
R. A. Van Gorder, The Biot-Savart description of Kelvin waves on a quantum vortex filament in the presence of mutual friction and a driving fluid, Proceedings of the Royal Society A 471 (2015) 20150149
R. A. Van Gorder, Decay of helical Kelvin waves on a quantum vortex filament, Physics of Fluids 26 (2014) 075101

Density plots for the ground state of an attractive Bose-Einstein condensate on space domains of different shapes. For more details, see R. A. Van Gorder, Proceedings of the Royal Society A 476 (2020) 20200674.

Other research interests have included:

These days, my papers appear on Google Scholar quickly after they are published, and you can find a publication list sorted in reverse chronological order here:

Feel free to get in touch if you'd like a copy of a published paper emailed to you.  I am happy to supervise mathematics or physics PhD and honours projects in any of the three areas mentioned above.

Research student supervision:

I've supervised the following doctoral students:

I've also supervised a number of MSc, BSc, and other short-term research projects.

Teaching


Miscellaneous: