This page includes summaries of my primary research projects.

## Optimal Mixing

My primary PhD thesis work is on optimal fluid mixing in collaboration with my advisor Prof. Charles Doering. This project investigates the fundamental mixing limitations of the advection-diffusion equation. This single equation describes how a scalar $\theta(\mathbf{x},t)$ such as chemical composition, dye concentration, temperature, or salinity is moved by an incompressible flow $\mathbf{u}(\mathbf{x},t)$ with molecular diffusion coefficient $\kappa$. Mixing is important to industry and science with many open questions. We ask "given mild constraints on the flow $\mathbf{u}$, what is the best mixing rate?" Given these conditions on $\mathbf{u}$, what does this tell us about the transport properties of $\theta$? In particular, we are interested in efficiently mix or homogenize $\theta$ over a domain. To address this problem, we measure 'mixed-ness' of $\theta$ by the $H^{-1}$ Sobolev mix-norm that quantifies mixing with or without diffusion, and utilize techniques from optimal control theory. We have studied optimal mixing within a shell model [1]. A shell model is a reduced model describing the spectral dynamics of the full advection-diffusion equation using a system of ordinary differential equations. Recently, we have explored optimal mixing within the advection-diffusion equation [2] with and without diffusion. The video below shows the optimal stirring stategies for the case of pure advection compared to that of advection and diffusion with a fixed enstrophy constraint in both cases.

[1] C. J. Miles, C. R. Doering, A shell model for optimal mixing, Journal of Nonlinear Science (2017)

[2] C. J. Miles, C. R. Doering, Diffusion-limited mixing by incompressible flows (submitted) arXiv:1712.07187 [physics.flu-dyn]

## Active Matter

In collaboration with Prof. Michael Shelley (NYU) and Prof. Saverio Spagnolie (UW-Madison), we model a bacterial swarm in a continuum model of swimming particles where their interactions are mediated through the surrounding fluid flow. This approach leads to a Smolukowski equation, an equation for the single-particle distribution. We are investigating the dynamics of bacterial clusters within a fluid.

## Acoustic Droplet Vaporization

Acoustic droplet vaporization is the vaporization of micron-sized droplets by ultrasound. Our collaborator Prof. Oliver D. Kripfgans from U. of Michigan's Department of Radiology is investigating how to use this mechanism for a localized drug-delivery method for chemotherapy. The proposed therapy would begin by injecting an emulsion of drug-laden droplets into a cancerous tissue via an IV administration. The droplets will remain neutral to the surrounding tissue until ultrasound is focused on the targeted tissue and cause the the droplets to vaporize. Once the droplets vaporize into a bubble, the embedded chemotherapeutic drug can readily diffuses into the nearby cancerous tissue. The physical mechanism behind acoustic droplet vaporization has been poorly understood until recent developments in the field. In collaboration with Prof. Oliver D. Kripfgans and my advisor Prof. Charles Doering, we have made progress towards this problem by developing a theoretical model of the hydrodynamic interaction between the ultrasound and droplet to determine the induced pressures present within the droplet [3]. Then, by applying classical nucleation theory, we were able to predict which ultrasonic waveforms trigger acoustic droplet vaporization. In addition, we conducted experiments for verification of the model.

[3] C. J. Miles, C. R. Doering, O. D. Kripfgans, Nucleation pressure threshold in acoustic droplet vaporization, Journal of Applied Physics 120, 034903 (2016)