Kara L. Maki 

Overview
Settling Dynamics of the Contact Lens:
The U.S. contact lens industry is a multibillion 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 stateoftheart 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.
Evaporative SelfAssembly 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 coffeering effect''.
A closely related yet lesswellexplored 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 microndiameter polystyrene spheres suspended in water evaporates
on a microscope slide under ambient conditions. Experimentally, the skin only forms in suspensions with largersized 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.
For more information, K.L. Maki and S. Kumar. Fast evaporation of spreading droplets of colloidal suspensions. Langmuir 27 (2011), 1134711363. 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 nonNewtonain 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.
For more information, K.L. Maki and Y. Renardy. The dynamics of a viscoelastic liquid which displays thixotropic yield stress behavior. Journal of NonNewtonian Fluid Mechanics 181 (2012), 3050. K.L. Maki and Y. Renardy. The dynamics of a simple model for a unified treatment of thixotropic yield stress fluids. Journal of NonNewtonian Fluid Mechanics 165 (2010), 13731385. 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 KingSmith, 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 NavierStokes 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.
For more information, K.L. Maki, R.J. Braun, P. Ucciferro, W.D. Henshaw, and P.E. KingSmith. Tear film dynamics on an eyeshaped domain II: Flux Boundary Conditions. Journal of Fluid Mechanics 165 (2010), 13731385. K.L. Maki, R.J. Braun, W.D. Henshaw, and P.E. KingSmith. Tear film dynamics on an eyeshaped domain I: Pressure Boundary Conditions. Mathematical Medicine and Biology 27 (2010), 227254. K.L. Maki, R.J. Braun, T.A. Driscoll, and P.E. KingSmith. An overset grid method for the study of reflex tearing. Mathematical Medicine and Biology 25 (2008), 187214. A. Heryudono, R.J. Braun, T.A. Driscoll, L.P. Cook, K.L. Maki and P.E. KingSmith. Singleequation models for the tear film in a blink cycle: Realistic lid motion. Mathematical Medicine and Biology 24 (2007), 347377. 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^(n1) 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
onceintheageoftheuniverse event.
For more information, T.A. Driscoll and K.L. Maki. Searching for rare growth factors using multicanonical Monte Carlo methods. SIAM Review 49 (2007), 673692.
