The Physical Mathematics Lab
Where our theoretical and experimental developments both inspire and constrain one another.
PHILOSOPHY
At PML, we have assembled a multidisciplinary research group that in-tegrates theory and simulation with in-house experiments to uncover new physical phenomena and rationalize them me-chanistically. We believe that creative ideas and genuine understanding emerge from the dialogue between mathematics and experiment, from the analysis of equations and simulations to direct observation in the laboratory. To that end, we maintain a fully equipped wet lab within the Department of Mathematics, where we design and perform controlled experiments that both inform and challenge our theoretical models.
EXPERIMENTS
We conduct table-top experiments using the equipment available in our wet lab, located in Chapman Hall. Typical techniques include high-speed imaging, laser-sheet and particle image velocimetry (PIV) for flow visualization, and Schlieren methods to capture wave patterns and interfacial dynamics. Our studies usually rely on custom-built apparatuses, such as vibrating tables for exploring parametrically forced systems. Experiments serve as both a source of discovery and a benchmark for our models, revealing new phenomena, validating theoretical predictions, and guiding the development of reduced descriptions.
A few snapshots of the lab
These videos provide a sample of the footage we capture
Walking Droplets
Faraday Waves
Galloping Bubbles
Active Matter
SIMULATIONS
We complement our studies with numerical simulations, working with a hierarchy of models that span different levels of complexity — from partial differential equation (PDE) formulations, including direct numerical simulations (DNS) that resolve full flow fields and intermediate reduced wave models that emerge in appropriate asymptotic limits, to minimal ordinary differential equation (ODE) systems that result from the reduction of these models to their essential degrees of freedom. We take advantage of UNC’s significant high-performance computing (HPC) facilities, which enable large-scale simulations and extensive parameter sweeps. Simulations allow us to isolate physical mechanisms and explore parameter regimes that are difficult to access experimentally, thereby bridging theory and experiment through quantitative comparison and mechanistic insight.
UNC's extensive HPC resources are managed by ITS Research Computing and RENCI
Direct Numerical Simulations
Reduced PDE Model
ODE Model
THEORY
Our theoretical work is grounded in the Applied Mathematics graduate program at UNC, which has long traditions in asymptotic and perturbation theory, analysis of PDEs, nonlinear dynamics, and fluid mechanics. Theory provides the framework for mechanistic understanding — enabling us to identify key balances, uncover organizing principles, and reduce complex systems to their minimal mathematical essence. Mechanistic insights allow us to recognize universal behaviors across systems that appear entirely different, extending the reach of our discoveries beyond their original context. Together, theory, computation, and experiment form a self-consistent cycle of discovery that defines the lab’s approach.
Asymptotic and perturbative approximations, for instance, help us rationalize certain behaviors of walking droplets...
... or the galloping locomotion of vibrating bubbles.