### Publications

- Christel Hohenegger & Scott A. McKinley (2018). Reconstructing complex fluid properties from the behavior of fluctuating immersed particles. SIAM. Vol. 78(4), 2200-2226.
Published, 08/2018.

https://doi.org/10.1137/17M1131660 - On equipartion of energy and integrals of generalized Langevin equations with generalized Rouse kernel. We show that if the motion of a particle in a linear viscoelastic liquid is described by a Generalized Langevin Equation with generalized Rouse kernel, then the resulting velocity process satisfies equipartition of energy. In doing so, we present a closed formula for the improper integration along the positive line of the product of a rational polynomial function and of even powers of the sinc function. The only requirements on the rational function are sufficient decay at infinity, no purely real poles and only simple nonzero poles. In such, our results are applicable to a family of exponentially decaying kernels. The proof of the integral result follows from the residue theorem and equipartition of energy is a natural consequence thereof. Furthermore, we apply the integral result to obtain an explicit formulae for the covariance of the position process both in the general case and for the Rouse kernel. We also discuss a numerical algorithm based on residue calculus to evaluate the covariance for the Rouse kernel at arbitrary times. Published, 03/15/2017.
- Mean first passage time in a thermally fluctuating viscoelastic fluid. The motion of a passive spherical particle in a fluid has been widely described via a balance of force equations known as a Generalized Langevin Equation (GLE) where the covariance of the thermal force is related to the time memory function of the fluid. For viscous fluids, this relationship is simply a delta function in time, while for a viscoelastic fluid it depends on the constitutive equation of the fluid memory function. In this paper, we consider a general setting for linear viscoelasticity which includes both solvent and polymeric viscosity contributions, and a family of memory functions known as the generalized Rouse kernel. We present a statistically exact algorithm to generate paths which allows for arbitrary large time steps and which relies on the numerical evaluation of the covariance of the velocity process. As a consequence of the viscoelastic properties of the fluid, the particle exhibits subdiffusive behavior, which we verify as a function of the free parameters in the generalized Rouse kernel. We then numerically compute the mean first passage time of a passive particle through layers of different widths and establish that, for the generalized Rouse kernel, the mean first passage time grows quadratically with the layer’s width independently of the free parameters. Along the way, we also find the linear scaling of the mean first passage time for a layer of fixed width as a function of the particle’s radius. Published, 03/09/2017.
- Spectral measure computations for composite materials. The analytic continuation method of homogenization theory provides Stieltjes integral representations for the effective parameters of composite media. These representations involve the spectral measures of self-adjoint random operators which depend only on the composite geometry. On finite bond lattices, these random operators are represented by random matrices and the spectral measures are given explicitly in terms of their eigenvalues and eigenvectors. Here we provide the mathematical foundation for rigorous computation of spectral measures for such composite media, and develop a numerically efficient projection method to enable such computations. This is accomplished by providing a unified formulation of the analytic continuation method which is equivalent to the original formulation and holds for finite and infinite lattices, as well as in continuum settings. We also introduce a family of bond lattices and directly compute the associated spectral measures and effective parameters. The computed spectral measures are in excellent agreement with known theoretical results. The behavior of the associated effective parameters is consistent with the symmetries and theoretical predictions of models, and the computed values fall within rigorous bounds. Some previous calculations of spectral measures have relied on finding the boundary values of the imaginary part of the effective parameter in the complex plane. Our method instead relies on direct computation of the eigenvalues and eigenvectors which enables, for example, statistical analysis of the spectral data. Published, 03/17/2015.
- Dimensional reduction of a multiscael continuum model of microtubule gliding assays Microtubule gliding assays, in which molecular motors anchored to a plate drive the gliding motion of filaments in a quasi–two-dimensional fluid layer, have been shown to organize into a variety of large-scale patterns. We derive a fully three-dimensional multiscale coarse-grained model of a gliding assay including the evolution of densities of rigid filaments, bound motors, and free motors, coupled to fluid equations. Our model combines continuum theories of polymeric liquids with the force spreading approach of the immersed boundary method. We use dimensional and asymptotic analysis to derive a reduced two-dimensional model and show that, to leading order, the filaments evolve in a plane, similar to what is experimentally observed. We simulate our model numerically with a GPU-based implementation and observe the same qualitative behavior as in experimental work. Published, 10/08/2014.
- Transition in the fractal geometry of Arctic melt ponds. During the Arctic melt season, the sea ice surface undergoes a remarkable transformation from vast expanses of snow covered ice to complex mosaics of ice and melt ponds. Sea ice albedo, a key parameter in climate modeling, is determined by the complex evolution of melt pond configurations. In fact, ice-albedo feedback has played a major role in the recent declines of the summer Arctic sea ice pack. However, understanding melt pond evolution remains a significant challenge to improving climate projections. By analyzing area-perimeter data from hundreds of thousands of melt ponds, we find here an unexpected separation of scales, where pond fractal dimension D transitions from 1 to 2 around a critical length scale of 100 square meters in area. Pond complexity increases rapidly through the transition as smaller ponds coalesce to form large connected regions, and reaches a maximum for ponds larger than 1000 square meters whose boundaries resemble space filling curves with D~2. These universal features of Arctic melt pond evolution are similar to phase transitions in statistical physics. The results impact sea ice albedo, the transmitted radiation fields under melting sea ice, the heat balance of sea ice and the upper ocean, and biological productivity such as under ice phytoplankton blooms. Published, 10/19/2012.

### Research Keywords

- Suspensions
- Stochastic Processes
- Solid - Liquid Interfaces
- Partial Differential Equations
- Fluid Dynamics
- Computational Fluid Dynamics
- Applied Mathematics

### Presentations

- Numerical simulations of tracers in a thermally fluctuating viscoelastic fluid, SIAM Louisiana-Texas Sectional Meeting. Contributed Talk, Presented, 11/02/2019.
- Towards simulations of tracer motion in a thermally fluctuating viscoelastic fluid, University of California Merced, Applied Mathematics Seminar. Presentation, Presented, 10/11/2019.
- A First step towards simulations of particles in a thermally fluctuating viscoelastic fluid, {Mathematical Fluids, Materials, and Biology Conference, Ann Arbor, MI.
Invited Talk/Keynote,
Presented, 06/13/2019.

https://indico.flatironinstitute.org/event/30/ - A First step towards simulations of tracer motion in a thermally fluctuating viscoelastic fluid, Women in Numerical PDEs, BIRS, Canada.
Invited Talk/Keynote,
Presented, 05/14/2019.

https://www.birs.ca/events/2019/5-day-workshops/19... - Uncertainty quantification in passive microrheology, 71st Annual Meeting of the Division of Fluid Dynamics.
Contributed Talk,
Presented, 11/19/2018.

http://meetings.aps.org/Meeting/DFD18/Session/F37.... - Uncertainty propagation in complex fluids, Complex Fluids in Biological Systems, BIRS, Canada.
Invited Talk/Keynote,
Presented, 07/2018.

https://www.birs.ca/events/2018/5-day-workshops/18... - Undergraduate Colloquium, Department of Mathematics: The Scallop Theorem. Presentation, Presented, 04/2018.
- Mathematical inference in one point microrheology, 69th Annual Meeting of the Division of Fluid Dynamics.
Contributed Talk,
Presented, 11/21/2016.

http://meetings.aps.org/Meeting/DFD16/Session/L26.... - Diffusion in fluids with memory, Colloquium, Department of Mathematics, Utah State University. Presentation, Presented, 11/10/2016.
- Calculus of bubbles and drops, Undergraduate Colloquium, Department of Mathematics, University of Utah. Presentation, Presented, 11/02/2016.
- Immersed Particle Dynamics in Fluctuating Fluids with Memory, SIAM Conference on Mathematical Aspects of Material Science.
Invited Talk/Keynote,
Presented, 05/11/2016.

http://meetings.siam.org/sess/dsp_talk.cfm?p=75423 - Immersed particle dynamics in fluids with memory, Applied Mathematics Seminar, Applied Mathematics, University of California, Merced. Presentation, Presented, 04/15/2016.
- Immersed particle dynamics in fluids with memory, Applied Mathematics Seminar, Department of Mathematics, Brigham Young University. Presentation, Presented, 03/31/2016.
- Fluid coupling in continuum modeling of microtubule motility assays, 65th Annual Meeting of the Division of Fluid Dynamics. Contributed Talk, Presented, 11/2012.
- Fluctuating hydrodynamics of immersed particles in a Maxwellian fluid, AMS Southeastern Sectional Meeting. Invited Talk/Keynote, Presented, 10/2012.
- Dynamics of Active Suspensions near Boundaries, SIAM Life Sciences (LS2012). Invited Talk/Keynote, Presented, 08/2012.
- Dynamics of Active Suspensions near Boundaries, Frontiers in Applied Mathematics (FACM'12). Invited Talk/Keynote, Presented, 06/2012.
- The Scallop Theorem: How to Swim in Honey, Undergraduate Colloquium, Department of Mathematics, University of Utah. Presentation, Presented, 04/2012.
- Dynamics of Active Suspensions near Boundaries, Applied Mathematics Seminar, Department of Mathematics, University of North Carolina, Chapel Hill, NC. Presentation, Presented, 03/2012.
- Swimming near a Wall, Early Research Directions Seminar, Department of Mathematics, University of Utah. Presentation, Presented, 02/2012.

### Languages

- English, fluent.
- French, fluent.
- German, fluent.

### Geographical Regions of Interest

- Western Europe