# Browsing by Author "Witelski, Thomas P"

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Item Open Access Dynamics and Steady-states of Thin Film Droplets on Homogeneous and Heterogeneous Substrates(2019) Liu, WeifanIn this dissertation, we study the dynamics and steady-states of thin liquid films on solid substrates using lubrication equations. Steady-states and bifurcation of thin films on chemically patterned substrates have been previously studied for thin films on infinite domains with periodic boundary conditions. Inspired by previous work, we study the steady-state thin film on a chemically heterogeneous 1-D domain of finite length, subject to no-flux boundary conditions. Based on the structure of the bifurcation diagram, we classify the 1-D steady-state solutions that could exist on such substrates into six different branches and develop asymptotic approximation of steady-states on each branch. We show that using perturbation expansions, the leading order solutions provide a good prediction of steady-state thin film on a stepwise-patterned substrate. We also show that all of the analysis in 1-D can be easily extended to axisymmetric solutions in 2-D, which leads to qualitatively the same results.

Subject to long-wave instability, thin films break up and form droplets. In presence of small fluxes, these droplets move and exchange mass. In 2002, Glasner and Witelski proposed a simplified model that predicts the pressure and position evolution of droplets in 1-D on homogeneous substrates when fluxes are small. While the model is capable of giving accurate prediction of the dynamics of droplets in presence of small fluxes, the model becomes less accurate as fluxes increase. We present a refined model that computes the pressure and position of a single droplet on a finite domain. Through numerical simulations, we show that the refined model captures single-droplet dynamics with higher accuracy than the previous model.

Item Open Access Modeling Temperature Dependence in Marangoni-driven Thin Films(2015) Potter, Harrison DavidThin liquid films are often studied by reducing the Navier-Stokes equations

using Reynolds lubrication theory, which leverages a small aspect ratio

to yield simplified governing equations. In this dissertation a plate

coating application, in which polydimethylsiloxane coats a silicon substrate,

is studied using this approach. Thermal Marangoni stress

drives fluid motion against the resistance of gravity, with the parameter

regime being chosen such that these stresses lead to a stable advancing front.

Additional localized thermal Marangoni stress is used to control the thin film;

in particular, coating thickness is modulated through the intensity of such

localized forcing. As thermal effects are central to film dynamics, the dissertation

focuses specifically on the effect that incorporating temperature dependence

into viscosity, surface tension, and density has on film dynamics and control.

Incorporating temperature dependence into viscosity, in particular,

leads to qualitative changes in film dynamics.

A mathematical model is developed in which the temperature dependence

of viscosity and surface tension is carefully taken into account.

This model is then

studied through numerical computation of solutions, qualitative analysis,

and asymptotic analysis. A thorough comparison is made between the

behavior of solutions to the temperature-independent and

temperature-dependent models. It is shown that using

localized thermal Marangoni stress as a control mechanism is feasible

in both models. Among constant steady-state solutions

there is a unique such solution in the temperature-dependent model,

but not in the temperature-independent model, a feature that

better reflects the known dynamics of the physical system.

The interaction of boundary conditions with finite domain size is shown

to generate both periodic and finite-time blow-up solutions, with

qualitative differences in solution behavior between models.

This interaction also accounts for the fact that locally perturbed solutions,

which arise when localized thermal Marangoni forcing is too weak

to effectively control thin film thickness, exist only for a discrete

set of boundary heights.

Modulating the intensity of localized thermal Marangoni forcing is

an effective means of modulating the thickness of a thin film

for a plate coating application; however, such control must be initiated before

the film reaches the full thickness it would reach in the absence of

such localized forcing. This conclusion holds for both the temperature-independent

and temperature-dependent mathematical models; furthermore, incorporating

temperature dependence into viscosity causes qualitative changes in solution

behavior that better align with known features of the underlying physical system.

Item Open Access Principles that govern competition or co-existence in Rho-GTPase driven polarization.(PLoS computational biology, 2018-04-12) Chiou, Jian-Geng; Ramirez, Samuel A; Elston, Timothy C; Witelski, Thomas P; Schaeffer, David G; Lew, Daniel JRho-GTPases are master regulators of polarity establishment and cell morphology. Positive feedback enables concentration of Rho-GTPases into clusters at the cell cortex, from where they regulate the cytoskeleton. Different cell types reproducibly generate either one (e.g. the front of a migrating cell) or several clusters (e.g. the multiple dendrites of a neuron), but the mechanistic basis for unipolar or multipolar outcomes is unclear. The design principles of Rho-GTPase circuits are captured by two-component reaction-diffusion models based on conserved aspects of Rho-GTPase biochemistry. Some such models display rapid winner-takes-all competition between clusters, yielding a unipolar outcome. Other models allow prolonged co-existence of clusters. We investigate the behavior of a simple class of models and show that while the timescale of competition varies enormously depending on model parameters, a single factor explains a large majority of this variation. The dominant factor concerns the degree to which the maximal active GTPase concentration in a cluster approaches a "saturation point" determined by model parameters. We suggest that both saturation and the effect of saturation on competition reflect fundamental properties of the Rho-GTPase polarity machinery, regardless of the specific feedback mechanism, which predict whether the system will generate unipolar or multipolar outcomes.Item Open Access Steady states of thin film droplets on chemically heterogeneous substrates(IMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications), 2020-12-01) Liu, Weifan; Witelski, Thomas PWe study steady-state thin films on chemically heterogeneous substrates of finite size, subject to no-flux boundary conditions. Based on the structure of the bifurcation diagram, we classify the 1D steady-state solutions that exist on such substrates into six different branches and develop asymptotic estimates for the steady states on each branch. Using perturbation expansions, we show that leading-order solutions provide good predictions of the steady-state thin films on stepwise-patterned substrates. We show how the analysis in one dimension can be extended to axisymmetric solutions. We also examine the influence of the wettability contrast of the substrate pattern on the linear stability of droplets and the time evolution for dewetting on small domains. Results are also applied to describe 2D droplets on hydrophilic square patches and striped regions used in microfluidic applications.Item Open Access Thin Films with Non-conservative Effects(2017) Ji, HangjieThin viscous fluids films (or “thin films”) spreading over a solid domain have appli- cations in many physical and biological systems. The dynamics of these thin films can be subject to fluid evaporation and vapor condensation with applications to pre- corneal tear film and thermal management. In this thesis, we study three models that arise from applications of thin fluid films with non-conservative effects. First, inspired by an evaporating thin film model studied by Ajaev in 2005, we investigate the pattern formation of a thin film equation with evaporation or condensation effects. Unlike the conservative thin film models where the steady states usually satisfy a second-order ODE, our model has a rich family of steady state solutions that satisfy a fourth-order ODE. Using bifurcation theory and stability analysis, we show that the coexistence of these different types of steady states yields interesting bifurcation structures and dynamics. Second, an interesting finite-time singularity phenomenon in this model motivates us to investigate various types of rupture solutions of a family of generalized thin film equations with non-conservative effects. Third, we consider a tear film model proposed by Peng in 2014 that characterizes the tear film rupture driven by locally-elevated evaporation effects. In addition to analytical results to a related generalized model, we show an interesting rupture-shock dynamics in the film thickness and osmolarity.

Item Open Access Two Coating Problems: Thin Film Rupture and Spin Coating(2009) Froehlich, MihaelaIn this work we study two fluid dynamics problems which are of particular interest in industries where thin film coating is a part of a production process, like optical coating, insulating layers in micro-circuitry, adhesives and painting. Experiments show that for very thin films of viscous liquids van der Waals intermolecular forces can produce instabilities leading to film ruptures. We consider this problem of thin film rupture driven by van der Waals forces and look for axisymmetric steady state solutions. Small perturbations from these solutions will lead to finite-time rupture. Using different numerical approaches we look for the solutions close to rupture. We obtain more solutions via shooting method and show that finite difference schemes on uniform grids are inferior to shooting for this problem. The second problem comes from a process called spin coating, one of the methods used to coat uniform thin films in variety of industrial applications such as manufacturing magnetic and optical discs. In spin coating a drop of liquid spreads radially due to centrifugal effects from spinning and eventually yields a thin film of uniform thickness formed on the solid surface. In experiments fingering instabilities at the expanding front of the fluid layer have been observed. This has renewed interest in the study of the nonlinear dynamics of such problems. We derive the evolution equation for thin films in rotating polar coordinates. We solve the axisymmetric problem and give numerical and analytical results for steady states. For large fluid volumes in rotating cylinders, we show that when centrifugal and gravity forces dominate, steady state solution free-surface profiles are parabolas for the angular velocity less then the critical velocity, and for higher velocities film ruptures, producing truncated parabolic profiles with circular contact lines. We find the similar results for the steady state case when centrifugal and surface tension forces are dominant. FInally, we study the dynamics of the spin coating again considering three cases, as weill as including the influence of van der Waals and Marangoni forces.