Data-driven discovery of equations for laminar dispersion

Traditionally, mathematical models are developed using basic conservations laws (for mass, momentum, energy, etc) combined with constitutive equations (e.g., Fick's law of diffusion). Well known examples of such models include the energy equation for heat conduction and the Navier-Stokes equations for fluid flow. However, there exist many complex and multi-scale systems, of great practical significance, for which we do not know the governing equations, and furthermore, are unable to derive the equations using conventional methods. Prominent examples of such systems include the flow of complex Non-newtonian fluids like magma and polymer melts, the collective flocking of birds, and the dispersion of non-simple solutes in pipe flows.

The last example may seem relatively mundane, but in fact it is an intriguing, non-intuitive phenomena that has important implications for pollutant transport in rivers, drug transport in arteries, the functioning of chromatographs, and the conversion achieved in tubular reactors. Using traditional methods, we can derive effective equations for this emergent phenomena only under restrictions of small Peclet numbers, and that too for simple solutes. 

The goal of this project is to use recently developed, data-driven techniques from the Machine Learning community to discover the dispersion equation for high Peclet numbers, as well as for complex solutes like microswimmers, active particles, etc.

Resources:

SINDY data-driven method to discover equations:

https://www.youtube.com/watch?v=gSCa78TIldg

https://www.youtube.com/watch?v=NxAn0oglMVw

https://www.youtube.com/watch?v=8-hoWTJwmrE

Dispersion phenomena:

https://www.youtube.com/watch?v=RNPHl0ZyhwY&list=PLmfNoDzkx8K3e1zDbAGSC5pHNJPmk9vSZ&index=44

https://en.wikipedia.org/wiki/Taylor_dispersion

 

Name of Faculty