Kara L. Maki
Assistant Professor
Rochester Institute of Technology
School of Mathematical Sciences
kmaki@rit.edu

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Overview

My research relies on the art of model formulation, mathematical analysis, and scientific computation to predict, reconstruct, and describe physical and industrial processes. The systems I study pertain to the flows of biological and complex fluids.

Settling Dynamics of the Contact Lens:

The U.S. contact lens industry is a multi-billion dollar industry with 34 million contact lenses wearers nationwide. It has been reported that the main reasons for contact lens discontinuation are discomfort and dryness. The current state-of-the-art contact lens design process is primarily empirical. A contact lens is manufactured; a clinical trial is run where measurements are collected; and then the lens design is altered. In collaboration with engineers at Bausch + Lomb, we are streamlining the design process with the aid of mathematical modeling. Our initial efforts have been on modeling the tear film under the contact lens with the immediate goal of predicting the suction pressure in response to applied pressure by the eyelid.

For more information,

See this poster created by RIT student Emily (Molly) Holz.

Evaporative Self-Assembly of Particles:

When a coffee droplet dries on a countertop, a dark ring of coffee solute is left behind, a phenomenon often referred to as "the coffee-ring effect''. A closely related yet less-well-explored phenomenon is the formation of a layer of particles, or skin, at the surface of the droplet during drying. The figure below shows an image of skin formation from an experiment in which a droplet of 1 micron-diameter polystyrene spheres suspended in water evaporates on a microscope slide under ambient conditions. Experimentally, the skin only forms in suspensions with larger-sized particles. We are working to identify the mechanisms causing skin formation. Our initial modeling efforts suggest that skin formation is due to a competition of evaporative (collecting particles at the interface) and diffusive (smoothing out particle concentration gradients) time scales.

An image of a drying droplet of 1 micron-diameter polystyrene spheres (Thermo Scientific) suspended in water evaporating on a microscope slide under ambient conditions. The scale bar represents 1 mm, and the volume fraction of particles is approximately 0.001. The light circular-like line is the edge of the droplet and the dark cloud in the center is the skin of particles. The image is taken at a relatively early time; at later times (not shown) the skin breaks up and the particles collect at the droplet edge to produce a coffee-ring. Image taken in the Coating Process and Visualization Lab at the University of Minnesota.

Modeling Thixotropic Yield Stress Fluids: A Mathematical Perspective of Ketchup?

Rheology is the study of the deformation and flow of matter such as pastes, suspensions, slurries, or foods. One rheological effect that distinguishes a rheologically interesting fluid, i.e., a non-Newtonain fluid, from a conventional fluid, i.e., Newtonian fluid, is yield stress. Yield is the tendency for a complex fluid to flow as a liquid only above a nonzero critical shear stress. Put another way, a yield stress fluid can support its own weight to a certain extent. In thixotropic fluids, the yield stress is time dependent. These characteristics are exploited in both nature and engineering. For example, snails can climb on walls and ceilings because they excrete and crawl on a yield stress fluid. We investigate the yielding and unyielding dynamics of a simple model that captures features of a thixotropic yield stress fluid, ketchup.

LEFT: Temporal evolution of the compliance for ketchup; experimental data, reproduced from Figure 2 of [1]. Each line is a single experiment, at a prescribed shear stress, which varies from the highest for the top-most curve, to half that value for the lowest curve. We have superimposed our interpretation based on our analysis. RIGHT: Numerical simulations of transient dynamics the of our of simplified model from equilibrium to steady shear flow. [1] F. Caton and C. Baravian. Plastic behavior of some yield stress fluids: from creep to long-time yield. Rheol. Acta 47 (2008), 601-607.

Dynamics of the Human Tear Film:

Each time someone blinks, a thin multilayered film of fluid must reestablish itself, within a second or so, on the front of the eye. This thin film is essential for both the health and optical quality of the human eye. An important first step towards effectively managing eye syndromes, like dry eye, is understanding the fluid dynamics of the tear film. In close collaboration with the University of Delaware tear film group, optometrist P. Ewen King-Smith, and national laboratory computational scientist Bill Henshaw, we focus on understanding the movement of the tear film on the eye. To do so, mathematical models for the tear film thickness are derived from the Navier-Stokes equations using lubrication theory. The highly nonlinear governing evolution equations are simulated with overset grid based computational methods in the Overture framework. We have found that the shape of the eye itself helps to steer fluid into the canthi regions.

       

LEFT: A contour plot of the tear film thickness, where blue represents thickness below 5 microns and maroon thickness above 15 microns, from our mathematical model. When the eye opens, the tear film thins (dark blue) near the edge of the lids; this has been seen experimentally and is referred to as the black lines. We found the tear fluid to collect in the canthi regions (maroon corners) as the shape of the eye steers fluid there. During the simulation, fluid enters above the temporal canthus (right corner) to mimic tear supply from the lacrimal gland. The fluid was able to penetrate the black line and move down the front of the eye. In addition, fluid is removed in the nasal canthus (left corner) to model drainage. RIGHT: A piece of the overlapping grid created in Overture to simulate the tear film dynamics.

Rarity of Large Growth Factors:

I have worked on simulating rare events. In particular, on reconstructing the probability distribution functions for growth factors of random matrices. The growth factor of a matrix, denoted by rho, quantifies potential error growth when a linear system is solved by Gaussian elimination with partial pivoting. While the growth factor has a maximum of 2^(n-1) for an nxn matrix, the occurrences of matrices with exponentially large growth factors is extremely rare. We implemented a multicanonical Monte Carlo method to explore the tails of growth factor probability distributions for random matrices. Our results attain a probability level of 10^(-12) and suggests the occurrence of an 8x8 matrix with a growth factor of 40 is on the order of a once-in-the-age-of-the-universe event.