# Browsing by Author "Witelski, Thomas P"

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Item Open Access An efficient finite element method for embedded interface problems(2013) Annavarapu, ChandrasekharWe focus on developing a computationally efficient finite element method for interface problems. Finite element methods are severely constrained in their ability to resolve interfaces. Many of these limitations stem from their inability in independently representing interface geometry from the underlying discretization. We propose an approach that facilitates such an independent representation by embedding interfaces in the underlying finite element mesh. This embedding, however, raises stability concerns for existing algorithms used to enforce interfacial kinematic constraints. To address these stability concerns, we develop robust methods to enforce interfacial kinematics over embedded interfaces. We begin by examining embedded Dirichlet problems – a simpler class of embedded constraints. We develop both stable methods, based on Lagrange multipliers,and stabilized methods, based on Nitsche’s approach, for enforcing Dirichlet constraints over three-dimensional embedded surfaces and compare and contrast their performance. We then extend these methods to enforce perfectly-tied kinematics for elastodynamics with explicit time integration. In particular, we examine the coupled aspects of spatial and temporal stability for Nitsche’s approach.We address the incompatibility of Nitsche’s method for explicit time integration by (a) proposing a modified weighted stress variational form, and (b) proposing a novel mass-lumpingprocedure.We revisit Nitsche’s method and inspect the effect of this modified variational form on the interfacial quantities of interest. We establish that the performance of this method, with respect to recovery of interfacial quantities, is governed significantly by the choice for the various method parameters viz.stabilization and weighting. We establish a relationship between these parameters and propose an optimal choice for the weighting. We further extend this approach to handle non-linear,frictional sliding constraints at the interface. The naturally non-symmetric nature of these problems motivates us to omit the symmetry term arising in Nitsche’s method.We contrast the performance of the proposed approach with the more commonly used penalty method. Through several numerical examples, we show that with the pro-posed choice of weighting and stabilization parameters, Nitsche’s method achieves the right balance between accurate constraint enforcement and flux recovery - a balance hard to achieve with existing methods. Finally, we extend the proposed approach to intersecting interfaces and conduct numerical studies on problems with junctions and complex topologies.Item Open Access Analysis of the Elastica with Applications to Vibration Isolation(2007-05-02T17:38:28Z) Santillan, Sophia TeresaLinear theory is useful in determining small static and dynamic deflections. However, to characterize large static and dynamic deflections, it is no longer useful or accurate, and more sophisticated analysis methods are necessary. In the case of beam deflections, linear beam theory makes use of an approximate curvature expression. Here, the exact curvature expression is used to derive the governing partial differential equations that describe the in-plane equilibrium and dynamics of a long, thin, inextensible beam, where the self-weight of the beam is included in the analysis. These beam equations are expressed in terms of arclength, and the resulting equilibrium shape is called the elastica. The analysis gives solutions that are accurate for any deflection size, and the method can be used to characterize the behavior of many structural systems. Numerical and analytical methods are used to solve or to approximate solutions to the governing equations. Both a shooting method and a finite difference, time-stepping algorithm are developed and implemented to find numerical solutions and these solutions are compared with some analytical approximation method results. The elastica equations are first used to determine both linear and nonlinear equilibrium configurations for a number of boundary conditions and loading types. In the case of a beam with a significant self-weight, the system can exhibit nonlinear static behavior even in the absence of external loading, and the elastica equations are used to determine the weight corresponding to the onset of instability (or self-weight buckling). The equations are also used to characterize linear and nonlinear vibrations of some structural systems, and experimental tests are conducted to verify the numerical results. The linear vibration analysis is applied to a vibration isolator system, where a postbuckled clamped-clamped beam or otherwise highly-deformed structure is used (in place of a conventional spring) to reduce system motion. The method is also used to characterize nonlinear dynamic behavior, and the resulting frequency-response curves are compared with those in the literature. Finally, the method is used to investigate the dynamics of subsea risers, where the effects of gravity, buoyancy, and the current velocity are considered.Item Open Access Coarsening of Thin Fluid Films(2008-04-15) Gratton, Michael B.Observed in many physical systems, coarsening is an orderly decrease in the number of localized structures, such as particles, drops, shear bands, solitons, or point defects. Coarsening is a type of pattern formation in which the characteristic length scale between features grows while the total number of features decreases. These phenomena have been studied in many problems and several mathematical techniques for modeling these phenomena have been developed. This dissertation examines the aggregation of drops in the thin film equation, where drops may coarsen through two general mechanisms: collision and collapse. A series of simplifications to model this process is developed. Slender-body asymptotics is applied to the Navier-Stokes equations for fluid motion in order to derive the Reynolds lubrication equation. The lubrication equation is in turn simplified to a coarsening dynamical system (CDS) model for interacting drops through solvability conditions for a perturbation about a drop-type steady state. Lastly, the dynamical system is averaged into an ensemble model to describe the dynamics of the distribution of drop sizes. The ensemble model takes the form of an integro-differential equation for the distribution function, much like the model of Ostwald ripening proposed by Lifshitz and Slyozov. A convenient choice of scaling yields an intermediate asymptotic self-similar solution. This solution is compared to numerical simulations of the ensemble model and histograms of drop masses from the CDS model. The early-time dynamics before similarity are explored by varying the initial distribution of drop sizes. Interesting far-from-similarity ``stairstep'' behavior is observed in the coarsening rate when the initial distribution has a very small variance. A well-chosen initial condition with a fractal-like structure is shown to replicate the stairstep behavior. At very long times, the mean drop size grows large, requiring the inclusion of gravity in the model. The CDS model parameters are modified as a result of the dependence of drop shapes on both size and gravity. The new dynamical system predicts the coarsening rate slowing from a power law to an inverse logarithmic rate. The energy liberated by each coarsening event is shown to approach a gravity-dependent constant as the mean drop mass increases. This suggests a reason for the coarsening slow-down.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 Optical Precursor Behavior(2007-05-07T19:07:14Z) LeFew, William R.Controlling and understanding the propagation of optical pulses through dispersive media forms the basis for optical communication, medical imaging, and other modern technological advances. Integral to this control and understanding is the ability to describe the transients which occur immediately after the onset of a signal. This thesis examines the transients of such a system when a unit step function is applied. The electromagnetic field is described by an integral resulting from Maxwell’s Equations. It was previously believed that optical precursors, a specific transient effect, existed only for only a few optical cycles and contributed only small magnitudes to the field. The main results of this thesis show that the transients arising from this integral are entirely precursors and that they may exist on longer time scales and contribute larger magnitudes to the field. The experimental detection of precursors has previously been recognized only through success comparison to the transient field resulting from an application of the method of steepest descent to that field integral. For any parameter regime where steepest descents may be applied, this work gives iterative methods to determine saddle points which are both more accurate than the accepted results and to extend into regimes where the current theory has failed. Furthermore, asymptotic formulae have been derived for regions where previous attempts at steepest descent have failed. Theory is also presented which evaluates the applicability of steepest descents in the represention of precursor behavior for any set of parameters. Lastly, the existence of other theoretical models for precursor behavior who may operate beyond the reach of steepest descent is validated through successful comparisons of the transient prediction of those methods to the steepest descent based results of this work.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.