(3de) Nonlinear Dynamics of Complex Systems, Both Biological and Rheological

Nonlinear dynamics of complex systems – both biological (e.g. epidemics) and rheological (e.g. wormlike micelles) – are my primary areas of interest as a researcher.

Epidemic models are useful tools in the fight against infectious diseases. However, the need for fine-grained resolution of details related to geography and demographics often constrains the amount of detail that one can include in the underlying model for disease transmission. At the same time, simplifications to the model of disease transmission will propagate through the rest of the epidemic model with effects that are generally difficult to anticipate. In my research, I have developed fast and efficient tools for solving an age-structured epidemic model with time-since-infection disease dynamics. I am eager to continue applying these tools to epidemics and other biological processes (e.g. ecosystems and invasive species) for which transitions between states are well defined and distinctly non-Markov.

From dense suspensions to entangled polymer melts, the processing of complex fluids at industrial scales is often limited by flow instabilities that are poorly understood on a fundamental level. In my research, I seek to address this problem by developing and applying improved constitutive models for complex fluids. For such systems, the coupling between flow and microstructure is generally complex, non-linear, and non-equilibrium, with underlying physics spanning many length-scales and time-scales. In my approach to the subject, I combine new ideas and insights with existing theories to produce a microscopically faithful continuum description of a fluid’s response to deformation and flow. Thereafter, I use insights from physical intuition and numerical simulations to guide well-controlled simplification strategies (asymptotic expansions, pre-averaging approximations, galerkin expansions, moment expansions, etc.) until I reach a suitable balance between fidelity to the full model and computational tractability in complex flows. These simplified models are then applied to study flow instabilities observed in benchmark problems. Initial directions for future work will include polydisperse entangled polymers, dense granular suspensions, and dry fibers.

Selected Research Experience:

Constitutive modelling of wormlike micelles: Department of Applied Maths and Theoretical Physics, University of Cambridge (advised by Mike Cates)

My postdoctoral research focuses on developing and applying simplified constitutive models for the linear and non-linear rheology of wormlike micelles. Wormlike micelles are an important component of many consumer goods (hand-soap, shampoos, detergents, etc.) and industrial processes (e.g. advanced oil recovery). However, the rheology of wormlike micelles is notoriously sensitive to minute changes in formulation, and this creates problems for industrial-scale production. Improved constitutive models could reduce waste and down-time by helping to diagnose and correct problems with the formulation in real-time. By combining existing tube-based constitutive models (e.g. the Rolie Poly model) with a complete population-balance model for scission and reformation kinetics, we have produced a model that directly connects molecular-scale information to linear and non-linear rheological measurements. On some levels, this approach is conceptually similar to what has been done by others – however, by including previously-neglected physics we have upended several well-established results in the existing literature. For example, by incorporating the entire polydisperse molecular weight distribution in our population balances, we show that flow-induced breaking is not generally responsible for shear banding instabilities in wormlike micelles and that a Doi-Edwards type instability is more likely to be at play.

Epidemic Modelling: Department of Applied Maths and Theoretical Physics, University of Cambridge (advised by Mike Cates and Ronojoy Adhikari)

During my postdoctoral work at Cambridge, I was fortunate to be involved in the RAMP (rapid assistance to modelling the pandemic) program. My primary contribution to that effort was to build an age-structured epidemic model with disease dynamics characterized by time-since-infection. Models with time-since-infection in the disease dynamics are often too computationally expensive to be viable for applications such as optimal control of social distancing measures, but I developed a novel Galerkin discretization with spectral accuracy for integrating the equations forwards in time at a cost that is competitive with the standard compartment-based models. I also performed proof-of-concept calculations to optimize non-pharmaceutical interventions and identify desirable herd immunity states.

Teaching Interests and Experience

During my PhD Studies, I was awarded a departmental fellowship to be a co-instructor for an undergraduate chemical engineering course. Because my research is heavily math-oriented, I chose to co-teach the numerical methods class for chemical engineers. This was a great learning experience for me, and I look forward to future opportunities to engage with students in the classroom setting. Although I enjoyed teaching a subject relevant to my own research, I would also welcome the challenge of teaching other subjects as well: my background and training as a chemical engineer makes me qualified to teach any core subject in the discipline (thermo, transport, kinetics, process control, etc).

Publications

1. D. Peterson, J. Guioth, R. Singh, R. Adhikari, “Inference in an age-structured epidemic model with time-since-infection dynamics,” in prep.
2. D. Peterson, M. E. Cates, L. G. Leal, “A full-chain tube-based constitutive model for living linear polymers”, submitted, JOR.
3. D. Peterson, C. Sasmal, V. Boudara, D. J. Read, L. G. Leal, “Nonlinear rheology predictions of bidisperse polymer blends in a complex flow”, submitted, JOR.
4. J. Gillessen, C. Ness, J. D. Peterson, H. J. Wilson, M. E. Cates, “Constitutive model for shear-thickening suspensions: Predictions for steady shear with superposed transverse oscillations.” Journal of Rheology, 2020
5. J. Gillessen, C. Ness, J. D. Peterson, H. J. Wilson, M. E. Cates, “A tensorial constitutive model for dense frictional suspensions”, Physical Review Letters, 2019
6. D. Peterson, G. H. Fredrickson, L. Gary Leal, “Does shear induced demixing resemble a thermodynamically driven instability?” Journal of Rheology, 63(2) pp. 335 – 359
7. Boudara, J. D. Peterson, L. Gary Leal, D. J. Read, “Nonlinear rheology of polydisperse blends of entangled linear polymers: Rolie-Double-Poly models”, Journal of Rheology, 63(1) pp. 71 – 91
8. D. Peterson, M. E. Cromer, G. H. Fredrickson, L. G. Leal, “Shear banding predictions for the two-fluid Rolie-Poly model”, Journal of Rheology 60(5) pp. 927 - 951

Recorded Talks (available online)

“Modelling polymer blends in flow,” Joseph D. Peterson, Glenn H. Fredrickson, L. Gary Leal. Thesis defense (2018). https:/bit.ly/2IRJPmf

“Two-fluid models for polymer melts and solutions,” Joseph D. Peterson, Glenn H. Fredrickson, L. Gary Leal. KITP dense suspensions workshop (2018). (talk begins at 1:00:58) https://bit.ly/2xe7OFl