%% Books
@book{fds323229,
Author = {Witelski, T and Bowen, M},
Title = {Methods of Mathematical Modelling: Continuous Systems and
Differential Equations},
Pages = {1-305},
Publisher = {Springer International Publishing},
Year = {2015},
Month = {September},
ISBN = {9783319230412},
url = {http://dx.doi.org/10.1007/978-3-319-23042-9},
Abstract = {This book presents mathematical modelling and the integrated
process of formulating sets of equations to describe
real-world problems. It describes methods for obtaining
solutions of challenging differential equations stemming
from problems in areas such as chemical reactions,
population dynamics, mechanical systems, and fluid
mechanics. Chapters 1 to 4 cover essential topics in
ordinary differential equations, transport equations and the
calculus of variations that are important for formulating
models. Chapters 5 to 11 then develop more advanced
techniques including similarity solutions, matched
asymptotic expansions, multiple scale analysis, long-wave
models, and fast/slow dynamical systems. Methods of
Mathematical Modelling will be useful for advanced
undergraduate or beginning graduate students in applied
mathematics, engineering and other applied
sciences.},
Doi = {10.1007/978-3-319-23042-9},
Key = {fds323229}
}
%% Papers Published
@article{fds376241,
Author = {Ji, H and Witelski, TP},
Title = {Coarsening of Thin Films with Weak Condensation},
Journal = {SIAM Journal on Applied Mathematics},
Volume = {84},
Number = {2},
Pages = {362-386},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2024},
Month = {April},
url = {http://dx.doi.org/10.1137/23m1559336},
Doi = {10.1137/23m1559336},
Key = {fds376241}
}
@article{fds371622,
Author = {Chapman, SJ and Dallaston, MC and Kalliadasis, S and Trinh, PH and Witelski, TP},
Title = {The role of exponential asymptotics and complex
singularities in self-similarity, transitions, and branch
merging of nonlinear dynamics},
Journal = {Physica D: Nonlinear Phenomena},
Volume = {453},
Year = {2023},
Month = {November},
url = {http://dx.doi.org/10.1016/j.physd.2023.133802},
Abstract = {We study a prototypical example in nonlinear dynamics where
transition to self-similarity in a singular limit is
fundamentally changed as a parameter is varied. Here, we
focus on the complicated dynamics that occur in a
generalised unstable thin-film equation that yields
finite-time rupture. A parameter, n, is introduced to model
more general disjoining pressures. For the standard case of
van der Waals intermolecular forces, n=3, it was previously
established that a countably infinite number of self-similar
solutions exist leading to rupture. Each solution can be
indexed by a parameter, ϵ=ϵ1>ϵ2>⋯>0, and the prediction
of the discrete set of solutions requires examination of
terms beyond-all-orders in ϵ. However, recent numerical
results have demonstrated the surprising complexity that
exists for general values of n. In particular, the
bifurcation structure of self-similar solutions now exhibits
branch merging as n is varied. In this work, we shall
present key ideas of how branch merging can be interpreted
via exponential asymptotics.},
Doi = {10.1016/j.physd.2023.133802},
Key = {fds371622}
}
@article{fds370567,
Author = {Bowen, M and King, JR and Witelski, TP},
Title = {CAUCHY-DIRICHLET PROBLEMS FOR THE POROUS MEDIUM
EQUATION},
Journal = {Discrete and Continuous Dynamical Systems- Series
A},
Volume = {43},
Number = {3-4},
Pages = {1143-1174},
Publisher = {American Institute of Mathematical Sciences
(AIMS)},
Year = {2023},
Month = {March},
url = {http://dx.doi.org/10.3934/dcds.2022182},
Abstract = {We consider the porous medium equation subject to
zero-Dirichlet conditions on a variety of two-dimensional
domains, namely strips, slender domains and sectors,
allowing us to capture a number of different classes of
behaviours. Our focus is on intermediate-asymptotic
descriptions, derived by formal arguments and validated
against numerical computations. While our emphasis is on
non-negative solutions to the slow-diffusion case, we also
derive a number of results for sign-change solutions and for
fast diffusion. Self-similar solutions of various kinds play
a central role, alongside the identification of suitable
conserved quantities. The characterisation of domains
exhibiting infinite-time hole closure is a particular upshot
and we highlight a number of open problems.},
Doi = {10.3934/dcds.2022182},
Key = {fds370567}
}
@article{fds371558,
Author = {Sanaei, P and Breward, C and Ellis, M and Han, S and Holzer, B and Ji, H and El Kahza and H and Smith, SL and Parsa, S and Reynolds, H and Troy, J and Witelski, T and Zhang, N and Zyskin, M},
Title = {Evaporation and deposition in porous media},
Year = {2022},
Month = {April},
url = {http://dx.doi.org/10.33774/miir-2022-wq8fl},
Abstract = {<jats:p>In this work, we consider a porous material that is
filled with a liquid solution containing molecules from
multiple species with known starting concentrations. As the
solvent evaporates, molecules from these species are left
behind and deposited on the internal pore walls within the
porous material. We seek to (i) examine the dependence of
the mass distribution of molecules along the pore walls and
the drying rate/time on the pore diameter, pore length,
fluid wetting properties, and evaporation conditions; (ii)
develop a model a porous structure that has a distribution
of pore sizes and in which. fluid moves due to capillary
force; and (iii) understand how the mass distribution of
molecules change upon cycles of wetting and
drying.</jats:p>},
Doi = {10.33774/miir-2022-wq8fl},
Key = {fds371558}
}
@article{fds361219,
Author = {Kim, R and Witelski, TP},
Title = {Uncovering the dynamics of a circadian-dopamine model
influenced by the light-dark cycle.},
Journal = {Mathematical biosciences},
Volume = {344},
Pages = {108764},
Publisher = {Elsevier},
Year = {2022},
Month = {February},
url = {http://dx.doi.org/10.1016/j.mbs.2021.108764},
Abstract = {The neurotransmitter dopamine (DA) is known to be influenced
by the circadian timekeeping system in the mammalian brain.
We have previously created a single-cell differential
equations model to understand the mechanisms behind
circadian rhythms of extracellular DA. In this paper, we
investigate the dynamics in our model and study different
behaviors such as entrainment to the 24-hour light-dark
cycle and robust periodicity versus decoupling,
quasiperiodicity, and chaos. Imbalances in DA are often
accompanied by disrupted circadian rhythms, such as in
Parkinson's disease, hyperactivity, and mood disorders. Our
model provides new insights into the links between the
circadian clock and DA. We show that the daily rhythmicity
of DA can be disrupted by decoupling between interlocked
loops of the clock circuitry or by quasiperiodic clock
behaviors caused by misalignment with the light-dark cycle.
The model can be used to further study how the circadian
clock affects the dopaminergic system, and to help develop
therapeutic strategies for disrupted DA rhythms.},
Doi = {10.1016/j.mbs.2021.108764},
Key = {fds361219}
}
@article{fds354949,
Author = {Zhu, H and Zhang, P and Zhong, Z and Xia, J and Rich, J and Mai, J and Su, X and Tian, Z and Bachman, H and Rufo, J and Gu, Y and Kang, P and Chakrabarty,
K and Witelski, TP and Huang, TJ},
Title = {Acoustohydrodynamic tweezers via spatial arrangement of
streaming vortices.},
Journal = {Science advances},
Volume = {7},
Number = {2},
Pages = {eabc7885},
Year = {2021},
Month = {January},
url = {http://dx.doi.org/10.1126/sciadv.abc7885},
Abstract = {Acoustics-based tweezers provide a unique toolset for
contactless, label-free, and precise manipulation of
bioparticles and bioanalytes. Most acoustic tweezers rely on
acoustic radiation forces; however, the accompanying
acoustic streaming often generates unpredictable effects due
to its nonlinear nature and high sensitivity to the
three-dimensional boundary conditions. Here, we demonstrate
acoustohydrodynamic tweezers, which generate stable,
symmetric pairs of vortices to create hydrodynamic traps for
object manipulation. These stable vortices enable
predictable control of a flow field, which translates into
controlled motion of droplets or particles on the operating
surface. We built a programmable droplet-handling platform
to demonstrate the basic functions of planar-omnidirectional
droplet transport, merging droplets, and in situ mixing via
a sequential cascade of biochemical reactions. Our
acoustohydrodynamic tweezers enables improved control of
acoustic streaming and demonstrates a previously
unidentified method for contact-free manipulation of
bioanalytes and digitalized liquid handling based on a
compact and scalable functional unit.},
Doi = {10.1126/sciadv.abc7885},
Key = {fds354949}
}
@article{fds355438,
Author = {Nakad, M and Witelski, T and Domec, JC and Sevanto, S and Katul,
G},
Title = {Taylor dispersion in osmotically driven laminar flows in
phloem},
Journal = {Journal of Fluid Mechanics},
Volume = {913},
Publisher = {Cambridge University Press (CUP)},
Year = {2021},
Month = {January},
url = {http://dx.doi.org/10.1017/jfm.2021.56},
Abstract = {Sucrose is among the main products of photosynthesis that
are deemed necessary for plant growth and survival. It is
produced in the mesophyll cells of leaves and translocated
to different parts of the plant through the phloem. Progress
in understanding this transport process remains fraught with
experimental difficulties, thereby prompting interest in
theoretical approaches and laboratory studies. The Münch
pressure and mass flow model is one of the accepted
hypotheses describing the physics of sucrose transport in
the phloem. It is based on osmosis creating an energy
potential difference between the source and the sink. The
flow responding to this energy potential is assumed laminar
and described by the Hagen-Poiseuille equation. This study
revisits such osmotically driven flows in tubes with
membrane walls by including the effects of Taylor dispersion
on mass transport. This effect has been overlooked in phloem
flow studies. Taylor dispersion can increase the effective
transport of solutes by increasing the apparent diffusion
coefficient. It is shown that, in addition to the
conventional diffusive correction derived for impermeable
tube walls, a new adjustment to the mean advective terms
arises because of osmotic effects. Because the molecular
Schmidt number is very large for sucrose in water, the
sucrose front speed and travel times have a direct
dependence on the Péclet number for different ranges of the
Münch number. This study establishes upper limits on
expected Taylor dispersion enhancement of sucrose
transport.},
Doi = {10.1017/jfm.2021.56},
Key = {fds355438}
}
@article{fds352385,
Author = {Aguareles, M and Chapman, SJ and Witelski, T},
Title = {Dynamics of spiral waves in the complex Ginzburg–Landau
equation in bounded domains},
Journal = {Physica D: Nonlinear Phenomena},
Volume = {414},
Year = {2020},
Month = {December},
url = {http://dx.doi.org/10.1016/j.physd.2020.132699},
Abstract = {Multiple-spiral-wave solutions of the general cubic complex
Ginzburg–Landau equation in bounded domains are
considered. We investigate the effect of the boundaries on
spiral motion under homogeneous Neumann boundary conditions,
for small values of the twist parameter q. We derive
explicit laws of motion for rectangular domains and we show
that the motion of spirals becomes exponentially slow when
the twist parameter exceeds a critical value depending on
the size of the domain. The oscillation frequency of
multiple-spiral patterns is also analytically
obtained.},
Doi = {10.1016/j.physd.2020.132699},
Key = {fds352385}
}
@article{fds353091,
Author = {Ji, H and Witelski, T},
Title = {Steady states and dynamics of a thin-film-type equation with
non-conserved mass},
Journal = {European Journal of Applied Mathematics},
Volume = {31},
Number = {6},
Pages = {968-1001},
Publisher = {Cambridge University Press (CUP)},
Year = {2020},
Month = {December},
url = {http://dx.doi.org/10.1017/s0956792519000330},
Abstract = {<jats:p>We study the steady states and dynamics of a
thin-film-type equation with non-conserved mass in one
dimension. The evolution equation is a non-linear
fourth-order degenerate parabolic partial differential
equation (PDE) motivated by a model of volatile viscous
fluid films allowing for condensation or evaporation. We
show that by changing the sign of the non-conserved flux and
breaking from a gradient flow structure, the problem can
exhibit novel behaviours including having two distinct
classes of co-existing steady-state solutions. Detailed
analysis of the bifurcation structure for these steady
states and their stability reveals several possibilities for
the dynamics. For some parameter regimes, solutions can lead
to finite-time rupture singularities. Interestingly, we also
show that a finite-amplitude limit cycle can occur as a
singular perturbation in the nearly conserved
limit.</jats:p>},
Doi = {10.1017/s0956792519000330},
Key = {fds353091}
}
@article{fds353254,
Author = {Liu, W and Witelski, TP},
Title = {Steady states of thin film droplets on chemically
heterogeneous substrates},
Journal = {IMA Journal of Applied Mathematics},
Volume = {85},
Number = {6},
Pages = {980-1020},
Publisher = {Oxford University Press (OUP)},
Year = {2020},
Month = {November},
url = {http://dx.doi.org/10.1093/imamat/hxaa036},
Abstract = {<jats:title>Abstract</jats:title> <jats:p>We 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.</jats:p>},
Doi = {10.1093/imamat/hxaa036},
Key = {fds353254}
}
@article{fds350458,
Author = {Dijksman, JA and Mukhopadhyay, S and Gaebler, C and Witelski, TP and Behringer, RP},
Title = {Erratum: Obtaining self-similar scalings in focusing flows
[Phys. Rev. E 92, 043016 (2015)].},
Journal = {Physical review. E},
Volume = {101},
Number = {5-2},
Pages = {059902},
Year = {2020},
Month = {May},
url = {http://dx.doi.org/10.1103/physreve.101.059902},
Abstract = {This corrects the article DOI: 10.1103/PhysRevE.92.043016.},
Doi = {10.1103/physreve.101.059902},
Key = {fds350458}
}
@article{fds349994,
Author = {Witelski, TP},
Title = {Nonlinear dynamics of dewetting thin films},
Journal = {AIMS Mathematics},
Volume = {5},
Number = {5},
Pages = {4229-4259},
Year = {2020},
Month = {January},
url = {http://dx.doi.org/10.3934/math.2020270},
Abstract = {Fluid films spreading on hydrophobic solid surfaces exhibit
complicated dynamics that describe transitions leading the
films to break up into droplets. For viscous fluids coating
hydrophobic solids this process is called “dewetting”.
These dynamics can be represented by a lubrication model
consisting of a fourth-order nonlinear degenerate parabolic
partial differential equation (PDE) for the evolution of the
film height. Analysis of the PDE model and its regimes of
dynamics have yielded rich and interesting research bringing
together a wide array of different mathematical approaches.
The early stages of dewetting involve stability analysis and
pattern formation from small perturbations and self-similar
dynamics for finite-time rupture from larger amplitude
perturbations. The intermediate dynamics describes further
instabilities yielding topological transitions in the
solutions producing sets of slowly-evolving near-equilibrium
droplets. The long-time behavior can be reduced to a
finite-dimensional dynamical system for the evolution of the
droplets as interacting quasi-steady localized structures.
This system yields coarsening, the successive re-arrangement
and merging of smaller drops into fewer larger drops. To
describe macro-scale applications, mean-field models can be
constructed for the evolution of the number of droplets and
the distribution of droplet sizes. We present an overview of
the mathematical challenges and open questions that arise
from the stages of dewetting and how they relate to issues
in multi-scale modeling and singularity formation that could
be applied to other problems in PDEs and materials
science.},
Doi = {10.3934/math.2020270},
Key = {fds349994}
}
@article{fds346386,
Author = {Dijksman, JA and Mukhopadhyay, S and Behringer, RP and Witelski,
TP},
Title = {Thermal Marangoni-driven dynamics of spinning liquid
films},
Journal = {Physical Review Fluids},
Volume = {4},
Number = {8},
Year = {2019},
Month = {August},
url = {http://dx.doi.org/10.1103/PhysRevFluids.4.084103},
Abstract = {Thinning dynamics in spin coating of viscous films is
influenced by many physical processes. Temperature gradients
are known to affect thin liquid films through their
influence on the local fluid surface tension as Marangoni
stresses. We show here experimentally and numerically that
adding a static temperature gradient has a significant
effect on the equilibrium film thickness and height profile
reached in spin coating. Most notably, we find that the
thickness of the resulting thin film in spin coating scales
linearly with the strength of the thermal surface tension
gradient. Once equilibrated, the thin film height profile is
controlled by the temperature profile. For small but
nonnegligible Marangoni number (Ma) the surface has a
negative curvature at the center and reaching equilibrium
takes progressively longer with smaller Ma. In this limit,
the steady state reached is set by competition between
Marangoni effects and the disjoining pressure.},
Doi = {10.1103/PhysRevFluids.4.084103},
Key = {fds346386}
}
@article{fds340899,
Author = {Bowen, M and Witelski, TP},
Title = {Pressure-dipole solutions of the thin-film
equation},
Journal = {European Journal of Applied Mathematics},
Volume = {30},
Number = {2},
Pages = {358-399},
Year = {2019},
Month = {April},
url = {http://dx.doi.org/10.1017/S095679251800013X},
Abstract = {We investigate self-similar sign-changing solutions to the
thin-film equation, h t = -(|h| n h xxx ) x , on the
semi-infinite domain x ≥ 0 with zero-pressure-type
boundary conditions h = h xx = 0 imposed at the origin. In
particular, we identify classes of first- and second-kind
compactly supported self-similar solutions (with a
free-boundary x = s(t) = Lt β ) and consider how these
solutions depend on the mobility exponent n; multiple
solutions can exist with the same number of sign changes.
For n = 0, we also construct first-kind self-similar
solutions on the entire half-line x ≥ 0 and show that they
act as limiting cases for sequences of compactly supported
solutions in the limit of infinitely many sign changes. In
addition, at n = 1, we highlight accumulation point-like
behaviour of sign-changes local to the moving interface x =
s(t). We conclude with a numerical investigation of
solutions to the full time-dependent partial differential
equation (based on a non-local regularisation of the
mobility coefficient) and discuss the computational results
in relation to the self-similar solutions.},
Doi = {10.1017/S095679251800013X},
Key = {fds340899}
}
@article{fds338527,
Author = {Gao, Y and Ji, H and Liu, JG and Witelski, TP},
Title = {A vicinal surface model for epitaxial growth with
logarithmic free energy},
Journal = {Discrete and Continuous Dynamical Systems - Series
B},
Volume = {23},
Number = {10},
Pages = {4433-4453},
Publisher = {American Institute of Mathematical Sciences
(AIMS)},
Year = {2018},
Month = {December},
url = {http://dx.doi.org/10.3934/dcdsb.2018170},
Abstract = {We study a continuum model for solid films that arises from
the modeling of one-dimensional step flows on a vicinal
surface in the attachment-detachment-limited regime. The
resulting nonlinear partial differential equation, ut =
-u2(u3 + au)hhhh, gives the evolution for the surface slope
u as a function of the local height h in a monotone step
train. Subject to periodic boundary conditions and positive
initial conditions, we prove the existence, uniqueness and
positivity of global strong solutions to this PDE using two
Lyapunov energy functions. The long time behavior of u
converging to a constant that only depends on the initial
data is also investigated both analytically and
numerically.},
Doi = {10.3934/dcdsb.2018170},
Key = {fds338527}
}
@article{fds336414,
Author = {Chiou, J-G and Ramirez, SA and Elston, TC and Witelski, TP and Schaeffer, DG and Lew, DJ},
Title = {Principles that govern competition or co-existence in
Rho-GTPase driven polarization.},
Journal = {PLoS Comput Biol},
Volume = {14},
Number = {4},
Pages = {e1006095},
Year = {2018},
Month = {April},
url = {http://dx.doi.org/10.1371/journal.pcbi.1006095},
Abstract = {Rho-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.},
Doi = {10.1371/journal.pcbi.1006095},
Key = {fds336414}
}
@article{fds332862,
Author = {Ji, H and Witelski, TP},
Title = {Instability and dynamics of volatile thin
films},
Journal = {Physical Review Fluids},
Volume = {3},
Number = {2},
Publisher = {American Physical Society (APS)},
Year = {2018},
Month = {February},
url = {http://dx.doi.org/10.1103/PhysRevFluids.3.024001},
Abstract = {Volatile viscous fluids on partially wetting solid
substrates can exhibit interesting interfacial instabilities
and pattern formation. We study the dynamics of vapor
condensation and fluid evaporation governed by a one-sided
model in a low-Reynolds-number lubrication approximation
incorporating surface tension, intermolecular effects, and
evaporative fluxes. Parameter ranges for
evaporation-dominated and condensation-dominated regimes and
a critical case are identified. Interfacial instabilities
driven by the competition between the disjoining pressure
and evaporative effects are studied via linear stability
analysis. Transient pattern formation in nearly flat
evolving films in the critical case is investigated. In the
weak evaporation limit unstable modes of finite-amplitude
nonuniform steady states lead to rich droplet dynamics,
including flattening, symmetry breaking, and droplet
merging. Numerical simulations show that long-time behaviors
leading to evaporation or condensation are sensitive to
transitions between filmwise and dropwise
dynamics.},
Doi = {10.1103/PhysRevFluids.3.024001},
Key = {fds332862}
}
@article{fds325294,
Author = {Gao, Y and Ji, H and Liu, JG and Witelski, TP},
Title = {Global existence of solutions to a tear film model with
locally elevated evaporation rates},
Journal = {Physica D: Nonlinear Phenomena},
Volume = {350},
Pages = {13-25},
Publisher = {Elsevier BV},
Year = {2017},
Month = {July},
url = {http://dx.doi.org/10.1016/j.physd.2017.03.005},
Abstract = {Motivated by a model proposed by Peng et al. (2014) for
break-up of tear films on human eyes, we study the dynamics
of a generalized thin film model. The governing equations
form a fourth-order coupled system of nonlinear parabolic
PDEs for the film thickness and salt concentration subject
to non-conservative effects representing evaporation. We
analytically prove the global existence of solutions to this
model with mobility exponents in several different ranges
and present numerical simulations that are in agreement with
the analytic results. We also numerically capture other
interesting dynamics of the model, including finite-time
rupture–shock phenomenon due to the instabilities caused
by locally elevated evaporation rates, convergence to
equilibrium and infinite-time thinning.},
Doi = {10.1016/j.physd.2017.03.005},
Key = {fds325294}
}
@article{fds320453,
Author = {Ji, H and Witelski, TP},
Title = {Finite-time thin film rupture driven by modified evaporative
loss},
Journal = {Physica D: Nonlinear Phenomena},
Volume = {342},
Pages = {1-15},
Publisher = {Elsevier BV},
Year = {2017},
Month = {March},
url = {http://dx.doi.org/10.1016/j.physd.2016.10.002},
Abstract = {Rupture is a nonlinear instability resulting in a
finite-time singularity as a film layer approaches zero
thickness at a point. We study the dynamics of rupture in a
generalized mathematical model of thin films of viscous
fluids with modified evaporative effects. The governing
lubrication model is a fourth-order nonlinear parabolic
partial differential equation with a non-conservative loss
term. Several different types of finite-time singularities
are observed due to balances between conservative and
non-conservative terms. Non-self-similar behavior and two
classes of self-similar rupture solutions are analyzed and
validated against high resolution PDE simulations.},
Doi = {10.1016/j.physd.2016.10.002},
Key = {fds320453}
}
@article{fds320455,
Author = {Sanaei, P and Richardson, GW and Witelski, T and Cummings,
LJ},
Title = {Flow and fouling in a pleated membrane filter},
Journal = {Journal of Fluid Mechanics},
Volume = {795},
Pages = {36-59},
Publisher = {Cambridge University Press (CUP)},
Year = {2016},
Month = {May},
url = {http://dx.doi.org/10.1017/jfm.2016.194},
Abstract = {Pleated membrane filters are widely used in many
applications, and offer significantly better surface area to
volume ratios than equal-area unpleated membrane filters.
However, their filtration characteristics are markedly
inferior to those of equivalent unpleated membrane filters
in dead-end filtration. While several hypotheses have been
advanced for this, one possibility is that the flow field
induced by the pleating leads to spatially non-uniform
fouling of the filter, which in turn degrades performance.
In this paper we investigate this hypothesis by developing a
simplified model for the flow and fouling within a pleated
membrane filter. Our model accounts for the pleated membrane
geometry (which affects the flow), for porous support layers
surrounding the membrane, and for two membrane fouling
mechanisms: (i) adsorption of very small particles within
membrane pores; and (ii) blocking of entire pores by large
particles. We use asymptotic techniques based on the small
pleat aspect ratio to solve the model, and we compare
solutions to those for the closest-equivalent unpleated
filter.},
Doi = {10.1017/jfm.2016.194},
Key = {fds320455}
}
@article{fds317250,
Author = {Smolka, LB and McLaughlin, CK and Witelski, TP},
Title = {Oil capture from a water surface by a falling
sphere},
Journal = {Colloids and Surfaces A: Physicochemical and Engineering
Aspects},
Volume = {497},
Pages = {126-132},
Publisher = {Elsevier BV},
Year = {2016},
Month = {May},
ISSN = {0927-7757},
url = {http://dx.doi.org/10.1016/j.colsurfa.2016.02.026},
Abstract = {Motivated by contaminant remediation, we study the volume of
oil (oleic acid) removed from a liquid lens by a falling
particle. When a spherical particle is dropped from a fixed
height into an oil lens that floats on top of a water
surface, a portion of the oil adheres to the sphere. Once
the sphere comes to rest at the subsurface, the oil forms an
equilibrium pendant drop that remains attached to the
sphere. We find in experiments with spheres of different
sizes and materials, that the pendant drop volume is an
increasing function of sphere mass for each material and a
decreasing function of sphere density. By contrast, the
normalized droplet volume in all of our experiments scales
with sphere mass following Voil/Vsphere ~ M-0.544. Thus, for
a given size, lighter spheres capture more oil relative to
their own volume than do heavier spheres and are more
efficient at removing oil from a water surface in our
experiments. Estimates for the upper bound of the normalized
droplet volume, determined from the continuous family of
solutions of the Young-Laplace equation, show the same
qualitative dependence on the sphere mass.},
Doi = {10.1016/j.colsurfa.2016.02.026},
Key = {fds317250}
}
@article{fds320454,
Author = {George, C and Virgin, LN and Witelski, T},
Title = {Experimental study of regular and chaotic transients in a
non-smooth system},
Journal = {International Journal of Non-Linear Mechanics},
Volume = {81},
Pages = {55-64},
Publisher = {Elsevier BV},
Year = {2016},
Month = {May},
url = {http://dx.doi.org/10.1016/j.ijnonlinmec.2015.12.006},
Abstract = {This paper focuses on thoroughly exploring the finite-time
transient behaviors occurring in a periodically driven
non-smooth dynamical system. Prior to settling down into a
long-term behavior, such as a periodic forced oscillation,
or a chaotic attractor, responses may exhibit a variety of
transient behaviors involving regular dynamics, co-existing
attractors, and super-persistent chaotic transients. A
simple and fundamental impacting mechanical system is used
to demonstrate generic transient behavior in an experimental
setting for a single degree of freedom non-smooth mechanical
oscillator. Specifically, we consider a horizontally driven
rigid-arm pendulum system that impacts an inclined rigid
barrier. The forcing frequency of the horizontal
oscillations is used as a bifurcation parameter. An
important feature of this study is the systematic generation
of generic experimental initial conditions, allowing a more
thorough investigation of basins of attraction when multiple
attractors are present. This approach also yields a
perspective on some sensitive features associated with
grazing bifurcations. In particular, super-persistent
chaotic transients lasting much longer than the conventional
settling time (associated with linear viscous damping) are
characterized and distinguished from regular dynamics for
the first time in an experimental mechanical
system.},
Doi = {10.1016/j.ijnonlinmec.2015.12.006},
Key = {fds320454}
}
@article{fds244178,
Author = {Witelski, TP},
Title = {Preface to the special issue on “Thin films and fluid
interfaces”},
Journal = {Journal of Engineering Mathematics},
Volume = {94},
Number = {1},
Pages = {1-3},
Publisher = {Springer Nature},
Year = {2015},
Month = {October},
ISSN = {0022-0833},
url = {http://dx.doi.org/10.1007/s10665-014-9760-z},
Doi = {10.1007/s10665-014-9760-z},
Key = {fds244178}
}
@article{fds293132,
Author = {Dijksman, JA and Mukhopadhyay, S and Gaebler, C and Witelski, TP and Behringer, RP},
Title = {Obtaining self-similar scalings in focusing
flows.},
Journal = {Physical review. E, Statistical, nonlinear, and soft matter
physics},
Volume = {92},
Number = {4},
Pages = {043016},
Year = {2015},
Month = {October},
ISSN = {1539-3755},
url = {http://dx.doi.org/10.1103/physreve.92.043016},
Abstract = {The surface structure of converging thin fluid films
displays self-similar behavior, as was shown in the work by
Diez et al. [Q. Appl. Math. 210, 155 (1990)]. Extracting the
related similarity scaling exponents from either numerical
or experimental data is nontrivial. Here we provide two such
methods. We apply them to experimental and numerical data on
converging fluid films driven by both surface tension and
gravitational forcing. In the limit of pure gravitational
driving, we recover Diez' semianalytic result, but our
methods also allow us to explore the entire regime of mixed
capillary and gravitational driving, up to entirely
surface-tension-driven flows. We find scaling forms of
smoothly varying exponents up to surprisingly small Bond
numbers. Our experimental results are in reasonable
agreement with our numerical simulations, which confirm
theoretically obtained relations between the scaling
exponents.},
Doi = {10.1103/physreve.92.043016},
Key = {fds293132}
}
@article{fds244182,
Author = {Witelski, T and Virgin, LN and George, C},
Title = {A driven system of impacting pendulums: Experiments and
simulations},
Journal = {Journal of Sound and Vibration},
Volume = {333},
Number = {6},
Pages = {1734-1753},
Year = {2014},
Month = {March},
ISSN = {0022-460X},
url = {http://dx.doi.org/10.1016/j.jsv.2013.11.004},
Abstract = {This paper studies a system composed of two pendulums
attached to a common base that is oscillated horizontally.
The pendulums share a common pivot line, but move
independently and are only coupled together through
collisions. Impact dynamics for the collisions of the
pendulums with each other and with fixed barriers yield
complex nonlinear behaviors. Careful numerical simulation of
the equations of motion demonstrates a close correlation
with experimental data collected from the system. There are
many independent parameters in this system, and one of the
motivations for the present study is to establish the extent
to which we can capture observed behavior with a relatively
simple hybrid differential equation model in the face of
several independent energy dissipation mechanisms coming
from friction and impact. Comparison between experiments and
simulations is based on the standard nonlinear dynamical
system analyses of time series, phase projections, time-lag
embedding, Poincaré sections, and frequency content.
Grazing bifurcations and co-existence of
impacting/non-impacting periodic/chaotic states are
observed. © 2013 Elsevier Ltd.},
Doi = {10.1016/j.jsv.2013.11.004},
Key = {fds244182}
}
@article{fds244179,
Author = {Hall Taylor and NS and Hewitt, IJ and Ockendon, JR and Witelski,
TP},
Title = {A new model for disturbance waves},
Journal = {International Journal of Multiphase Flow},
Volume = {66},
Pages = {38-45},
Publisher = {Elsevier BV},
Year = {2014},
Month = {January},
ISSN = {0301-9322},
url = {http://dx.doi.org/10.1016/j.ijmultiphaseflow.2014.06.004},
Abstract = {The first part of this paper surveys the distinctive
features of trains of disturbance waves in high-speed
annular two-phase flow. This data is then used to construct
a mathematical model that predicts relations between the
speed, height, and spacing of the waves, and the net liquid
flow rate. These relations highlight the importance of the
vorticity in the waves, a quantity that has received little
experimental attention. © 2014 Elsevier
Ltd.},
Doi = {10.1016/j.ijmultiphaseflow.2014.06.004},
Key = {fds244179}
}
@article{fds244183,
Author = {Smolka, LB and Witelski, TP},
Title = {Biaxial extensional motion of an inertially driven radially
expanding liquid sheet},
Journal = {Physics of Fluids},
Volume = {25},
Number = {6},
Pages = {062105-062105},
Publisher = {AIP Publishing},
Year = {2013},
Month = {June},
ISSN = {1070-6631},
url = {http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000321272600010&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=47d3190e77e5a3a53558812f597b0b92},
Abstract = {We consider the inertially driven, time-dependent biaxial
extensional motion of inviscid and viscous thinning liquid
sheets. We present an analytic solution describing the base
flow and examine its linear stability to varicose
(symmetric) perturbations within the framework of a
long-wave model where transient growth and long-time
asymptotic stability are considered. The stability of the
system is characterized in terms of the perturbation
wavenumber, Weber number, and Reynolds number. We find that
the isotropic nature of the base flow yields stability
results that are identical for axisymmetric and general
two-dimensional perturbations. Transient growth of
short-wave perturbations at early to moderate times can have
significant and lasting influence on the long-time sheet
thickness. For finite Reynolds numbers, a radially expanding
sheet is weakly unstable with bounded growth of all
perturbations, whereas in the inviscid and Stokes flow
limits sheets are unstable to perturbations in the
short-wave limit. © 2013 AIP Publishing
LLC.},
Doi = {10.1063/1.4811389},
Key = {fds244183}
}
@article{fds244184,
Author = {Chapman, SJ and Trinh, PH and Witelski, TP},
Title = {Exponential Asymptotics for Thin Film Rupture.},
Journal = {SIAM J. Appl. Math.},
Volume = {73},
Number = {1},
Pages = {232-253},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2013},
url = {http://dx.doi.org/10.1137/120872012},
Abstract = {The formation of singularities in models of many physical
systems can be described using self-similar solutions. One
particular example is the finite-time rupture of a thin film
of viscous fluid which coats a solid substrate. Previous
studies have suggested the existence of a discrete,
countably infinite number of distinct solutions of the
nonlinear differential equation which describes the
self-similar behavior. However, no analytical mechanism for
determining these solutions was identified. In this paper,
we use techniques in exponential asymptotics to construct
the analytical selection condition for the infinite sequence
of similarity solutions, confirming the conjectures of
earlier numerical studies. © 2013 Society for Industrial
and Applied Mathematics.},
Doi = {10.1137/120872012},
Key = {fds244184}
}
@article{fds244225,
Author = {Huang, Y and Witelski, TP and Bertozzi, AL},
Title = {Anomalous exponents of self-similar blow-up solutions to an
aggregation equation in odd dimensions},
Journal = {Applied Mathematics Letters},
Volume = {25},
Number = {12},
Pages = {2317-2321},
Publisher = {Elsevier BV},
Year = {2012},
Month = {December},
ISSN = {0893-9659},
url = {http://dx.doi.org/10.1016/j.aml.2012.06.023},
Abstract = {We calculate the scaling behavior of the second-kind
self-similar blow-up solution of an aggregation equation in
odd dimensions. This solution describes the radially
symmetric finite-time blowup phenomena and has been observed
in numerical simulations of the dynamic problem. The
nonlocal equation for the self-similar profile is
transformed into a system of ODEs that is solved using a
shooting method. The anomalous exponents are then retrieved
from this transformed system. © 2012 Elsevier Ltd. All
rights reserved.},
Doi = {10.1016/j.aml.2012.06.023},
Key = {fds244225}
}
@article{fds244180,
Author = {Li, Z and Layton, AT and Bertozzi, A and Ambrose, DM and Witelski, T and Minion, ML and Butters, R},
Title = {Preface},
Journal = {Discrete and Continuous Dynamical Systems - Series
B},
Volume = {17},
Number = {4},
Pages = {i-ii},
Publisher = {American Institute of Mathematical Sciences
(AIMS)},
Year = {2012},
Month = {February},
ISSN = {1531-3492},
url = {http://dx.doi.org/10.3934/dcdsb.2012.17.4i},
Doi = {10.3934/dcdsb.2012.17.4i},
Key = {fds244180}
}
@article{fds244226,
Author = {Wiebe, R and Virgin, LN and Witelski, TP},
Title = {A parametrically forced nonlinear system with reversible
equilibria},
Journal = {International Journal of Bifurcation and
Chaos},
Volume = {22},
Number = {6},
Pages = {1230020-1230020},
Publisher = {World Scientific Pub Co Pte Lt},
Year = {2012},
Month = {January},
ISSN = {0218-1274},
url = {http://dx.doi.org/10.1142/S0218127412300200},
Abstract = {A nonlinear Duffing-type dynamical system, in which the
stability of equilibria is modulated in a time-dependent
manner, is investigated both experimentally and numerically.
This is a low-order dynamical system with some interesting
available choices in the coordinate system. The system is
found to exhibit a variety of interesting nonlinear behavior
including ultrasubharmonic resonance. Frequency content is
used to characterize periodic and chaotic behavior and their
relation to the parameter space. © 2012 World Scientific
Publishing Company.},
Doi = {10.1142/S0218127412300200},
Key = {fds244226}
}
@article{fds244227,
Author = {Aydemir, E and Breward, CJW and Witelski, TP},
Title = {The effect of polar lipids on tear film dynamics.},
Journal = {Bulletin of mathematical biology},
Volume = {73},
Number = {6},
Pages = {1171-1201},
Year = {2011},
Month = {June},
url = {http://www.ncbi.nlm.nih.gov/pubmed/20556530},
Abstract = {In this paper, we present a mathematical model describing
the effect of polar lipids, excreted by glands in the eyelid
and present on the surface of the tear film, on the
evolution of a pre-corneal tear film. We aim to explain the
interesting experimentally observed phenomenon that the tear
film continues to move upward even after the upper eyelid
has become stationary. The polar lipid is an insoluble
surface species that locally alters the surface tension of
the tear film. In the lubrication limit, the model reduces
to two coupled non-linear partial differential equations for
the film thickness and the concentration of lipid. We solve
the system numerically and observe that increasing the
concentration of the lipid increases the flow of liquid up
the eye. We further exploit the size of the parameters in
the problem to explain the initial evolution of the
system.},
Doi = {10.1007/s11538-010-9555-y},
Key = {fds244227}
}
@article{fds244229,
Author = {Aguareles, M and Chapman, SJ and Witelski, T},
Title = {Motion of spiral waves in the complex Ginzburg-Landau
equation},
Journal = {Physica D: Nonlinear Phenomena},
Volume = {239},
Number = {7},
Pages = {348-365},
Publisher = {Elsevier BV},
Year = {2010},
Month = {April},
ISSN = {0167-2789},
url = {http://dx.doi.org/doi:10.1016/j.physd.2009.12.003},
Abstract = {Solutions of the general cubic complex Ginzburg-Landau
equation comprising multiple spiral waves are considered.
For parameters close to the vortex limit, and for a system
of spiral waves with well-separated centres, laws of motion
of the centres are found which vary depending on the order
of magnitude of the separation of the centres. In
particular, the direction of the interaction changes from
along the line of centres to perpendicular to the line of
centres as the separation increases, with the strength of
the interaction algebraic at small separations and
exponentially small at large separations. The corresponding
asymptotic wavenumber and frequency are determined. These
depend on the positions of the centres of the spirals, and
so evolve slowly as the spirals move. © 2009 Elsevier
B.V.},
Doi = {10.1016/j.physd.2009.12.003},
Key = {fds244229}
}
@article{fds244230,
Author = {Bernoff, AJ and Witelski, TP},
Title = {Stability and dynamics of self-similarity in evolution
equations},
Journal = {Journal of Engineering Mathematics},
Volume = {66},
Number = {1},
Pages = {11-31},
Publisher = {Springer Nature},
Year = {2010},
Month = {January},
ISSN = {0022-0833},
url = {http://dx.doi.org/10.1007/s10665-009-9309-8},
Abstract = {A methodology for studying the linear stability of
self-similar solutions is discussed. These fundamental ideas
are illustrated on three prototype problems: a simple ODE
with finite-time blow-up, a second-order semi-linear heat
equation with infinite-time spreading solutions, and the
fourth-order Sivashinsky equation with finite-time
self-similar blow-up. These examples are used to show that
self-similar dynamics can be studied using many of the ideas
arising in the study of dynamical systems. In particular,
the use of dimensional analysis to derive scaling invariant
similarity variables is discussed, as well as the role of
symmetries in the context of stability of self-similar
dynamics. The spectrum of the linear stability problem
determines the rate at which the solution will approach a
self-similar profile. For blow-up solutions it is
demonstrated that the symmetries give rise to positive
eigenvalues associated with the symmetries, and it is shown
how this stability analysis can identify a unique stable
(and observable) attracting solution from a countable
infinity of similarity solutions. © Springer
Science+Business Media B.V. 2009.},
Doi = {10.1007/s10665-009-9309-8},
Key = {fds244230}
}
@article{fds244241,
Author = {Witelski, TP},
Title = {The subtle art of blowing bubbles (News and Views: Fluid
Dynamics)},
Journal = {Nature Physics},
Volume = {5},
Number = {5},
Pages = {315-316},
Year = {2009},
Month = {May},
ISSN = {1745-2473},
url = {http://links.ealert.nature.com/ctt?kn=65&m=32736150&r=MTc2NjI2MDg2NwS2&b=2&j},
Doi = {10.1038/nphys1265},
Key = {fds244241}
}
@article{fds244243,
Author = {Witelski, TP and Bowen, M},
Title = {Singular perturbation theory.},
Journal = {Scholarpedia},
Volume = {4},
Number = {4},
Pages = {3951-3951},
Publisher = {Scholarpedia},
Year = {2009},
Month = {April},
url = {http://www.scholarpedia.org/article/Singular_perturbation_theory},
Doi = {10.4249/scholarpedia.3951},
Key = {fds244243}
}
@article{fds244244,
Author = {Hwang, HJ and Witelski, TP},
Title = {Short-time pattern formation in thin film
equations},
Journal = {Discrete and Continuous Dynamical Systems},
Volume = {23},
Number = {3},
Pages = {867-885},
Publisher = {American Institute of Mathematical Sciences
(AIMS)},
Year = {2009},
Month = {March},
ISSN = {1078-0947},
url = {http://aimsciences.org/journals/displayArticles.jsp?paperID=3829},
Abstract = {We study the early stages of the nonlinear dynamics of
pattern formation for unstable generalized thin film
equations. For unstable constant steady states, we obtain
rigorous estimates for the short- to intermediate-time
nonlinear evolution which extends the mathematical
characterization for pattern formation derived from linear
analysis: formation of patterns can be bounded by the
finitely many dominant growing eigenmodes from the initial
perturbation.},
Doi = {10.3934/dcds.2009.23.867},
Key = {fds244244}
}
@article{fds244188,
Author = {Peterson, E and Shearer, M and Witelski, TP and Levy,
R},
Title = {Stability of traveling waves in thin liquid films driven by
gravity and surfactant},
Journal = {HYPERBOLIC PROBLEMS: THEORY, NUMERICS AND APPLICATIONS, PART
2},
Volume = {67},
Number = {2},
Pages = {855-+},
Publisher = {AMER MATHEMATICAL SOC},
Editor = {Tadmor, E and Liu, J and Tzavaras, A},
Year = {2009},
Month = {January},
ISBN = {978-0-8218-4730-5},
url = {http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000282769400053&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=47d3190e77e5a3a53558812f597b0b92},
Key = {fds244188}
}
@article{fds244236,
Author = {Gratton, MB and Witelski, TP},
Title = {Transient and self-similar dynamics in thin film
coarsening},
Journal = {Physica D: Nonlinear Phenomena},
Volume = {238},
Number = {23-24},
Pages = {2380-2394},
Publisher = {Elsevier BV},
Year = {2009},
Month = {January},
ISSN = {0167-2789},
url = {http://dx.doi.org/10.1016/j.physd.2009.09.015},
Abstract = {We study coarsening in a simplified model of one-dimensional
thin films of viscous fluids on hydrophobic substrates.
Lubrication theory shows that such films are unstable and
dewet to form droplets that then aggregate over long
timescales. The masses and positions of the droplets can be
described by a coarsening dynamical system (CDS) consisting
of ODEs and deletion rules. We develop discrete and
continuous mean-field models that reproduce the well-known N
(t) = O (t- 2 / 5) long-time statistical power law for the
number of drops. A Lifshitz-Slyozov-Wagner-type (LSW)
continuous model predicts the self-similar distribution of
drop masses matching with histograms produced by CDS
simulations and the discrete mean-field model. We also
describe the distribution of drops following homogeneous
versus heterogeneous dewetting and use these as initial
conditions for the CDS simulations that yield characteristic
"staircasing" transients. Transients can also include
recurring spike formation behavior in the mass distribution.
For idealized initial conditions, we show that the transient
dynamics can span the full coarsening process, bypassing the
power law regime entirely. © 2009 Elsevier B.V. All rights
reserved.},
Doi = {10.1016/j.physd.2009.09.015},
Key = {fds244236}
}
@article{fds244242,
Author = {Smolka, LB and Witelski, TP},
Title = {On the planar extensional motion of an inertially driven
liquid sheet},
Journal = {Physics of Fluids},
Volume = {21},
Number = {4},
Pages = {042101-042101},
Publisher = {AIP Publishing},
Year = {2009},
Month = {January},
ISSN = {1070-6631},
url = {http://link.aip.org/link/?PHF/21/042101},
Abstract = {We derive a time-dependent exact solution of the free
surface problem for the Navier-Stokes equations that
describes the planar extensional motion of a viscous sheet
driven by inertia. The linear stability of the exact
solution to one- and two-dimensional symmetric perturbations
is examined in the inviscid and viscous limits within the
framework of the long-wave or slender body approximation.
Both transient growth and long-time asymptotic stability are
considered. For one-dimensional perturbations in the axial
direction, viscous and inviscid sheets are asymptotically
marginally stable, though depending on the Reynolds and
Weber numbers transient growth can have an important effect.
For one-dimensional perturbations in the transverse
direction, inviscid sheets are asymptotically unstable to
perturbations of all wavelengths. For two-dimensional
perturbations, inviscid sheets are unstable to perturbations
of all wavelengths with the transient dynamics controlled by
axial perturbations and the long-time dynamics controlled by
transverse perturbations. The asymptotic stability of
viscous sheets to one-dimensional transverse perturbations
and to two-dimensional perturbations depends on the
capillary number (Ca); in both cases, the sheet is unstable
to long-wave transverse perturbations for any finite Ca. ©
2009 American Institute of Physics.},
Doi = {10.1063/1.3094026},
Key = {fds244242}
}
@article{fds304502,
Author = {Witelski, TP},
Title = {Fluid dynamics: The subtle art of blowing
bubbles},
Journal = {Nature Physics},
Volume = {5},
Number = {5},
Pages = {315-316},
Publisher = {Springer Nature},
Year = {2009},
Month = {January},
ISSN = {1745-2473},
url = {http://dx.doi.org/10.1038/nphys1265},
Doi = {10.1038/nphys1265},
Key = {fds304502}
}
@article{fds304501,
Author = {Aguareles, M and Chapman, SJ and Witelski, T},
Title = {Interaction of spiral waves in the complex Ginzburg-Landau
equation.},
Journal = {Physical review letters},
Volume = {101},
Number = {22},
Pages = {224101},
Year = {2008},
Month = {November},
ISSN = {0031-9007},
url = {http://dx.doi.org/10.1103/physrevlett.101.224101},
Abstract = {Solutions of the general cubic complex Ginzburg-Landau
equation comprising multiple spiral waves are considered,
and laws of motion for the centers are derived. The
direction of the motion changes from along the line of
centers to perpendicular to the line of centers as the
separation increases, with the strength of the interaction
algebraic at small separations and exponentially small at
large separations. The corresponding asymptotic wave number
and frequency are also determined, which evolve slowly as
the spirals move.},
Doi = {10.1103/physrevlett.101.224101},
Key = {fds304501}
}
@article{fds244232,
Author = {Santillan, ST and Plaut, RH and Witelski, TP and Virgin,
LN},
Title = {Large oscillations of beams and columns including
self-weight},
Journal = {International Journal of Non-Linear Mechanics},
Volume = {43},
Number = {8},
Pages = {761-771},
Publisher = {Elsevier BV},
Year = {2008},
Month = {October},
ISSN = {0020-7462},
url = {http://dx.doi.org/10.1016/j.ijnonlinmec.2008.04.007},
Abstract = {Large-amplitude, in-plane beam vibration is investigated
using numerical simulations and a perturbation analysis
applied to the dynamic elastica model. The governing
non-linear boundary value problem is described in terms of
the arclength, and the beam is treated as inextensible. The
self-weight of the beam is included in the equations. First,
a finite difference numerical method is introduced. The
system is discretized along the arclength, and
second-order-accurate finite difference formulas are used to
generate time series of large-amplitude motion of an upright
cantilever. Secondly, a perturbation method (the method of
multiple scales) is applied to obtain approximate solutions.
An analytical backbone curve is generated, and the results
are compared with those in the literature for various
boundary conditions where the self-weight of the beam is
neglected. The method is also used to characterize
large-amplitude first-mode vibration of a cantilever with
non-zero self-weight. The perturbation and finite difference
results are compared for these cases and are seen to agree
for a large range of vibration amplitudes. Finally,
large-amplitude motion of a postbuckled, clamped-clamped
beam is simulated for varying degrees of buckling and
self-weight using the finite difference method, and backbone
curves are obtained. © 2008 Elsevier Ltd.},
Doi = {10.1016/j.ijnonlinmec.2008.04.007},
Key = {fds244232}
}
@article{fds244245,
Author = {Catllá, AJ and Schaeffer, DG and Witelski, TP and Monson, EE and Lin,
AL},
Title = {On spiking models for synaptic activity and impulsive
differential equations},
Journal = {SIAM Review},
Volume = {50},
Number = {3},
Pages = {553-569},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2008},
Month = {September},
ISSN = {0036-1445},
url = {http://link.aip.org/link/?SIR/50/553},
Abstract = {We illustrate the problems that can arise in writing
differential equations that include Dirac delta functions to
model equations with state-dependent impulsive forcing.
Specifically, difficulties arise in the interpretation of
the products of distributions with discontinuous functions.
We suggest several methods to resolve these ambiguities,
such as using limiting sequences and asymptotic analysis,
with applications of the results given for discrete maps.
These suggestions are applied to a popular model describing
synaptic connections in the brain. © 2008 Society for
Industrial and Applied Mathematics.},
Doi = {10.1137/060667980},
Key = {fds244245}
}
@article{fds244234,
Author = {DiCarlo, DA and Juanes, R and LaForce, T and Witelski,
TP},
Title = {Nonmonotonic traveling wave solutions of infiltration into
porous media},
Journal = {Water Resources Research},
Volume = {44},
Number = {2},
Pages = {W02406},
Publisher = {American Geophysical Union (AGU)},
Year = {2008},
Month = {February},
ISSN = {0043-1397},
url = {http://dx.doi.org/10.1029/2007WR005975},
Abstract = {In uniform soils that are susceptible to unstable
preferential flow, the water saturation may exhibit a
nonmonotonic profile upon continuous infiltration. As this
nonmonotonicity (also known as saturation overshoot) cannot
be described by the conventional Richards equation, there
have been proposed possible extensions to the unsaturated
flow equations, including a nonmonotonic capillary
pressure-saturation curve and a second-order hyperbolic
term. Here, we present analytic traveling wave solutions to
the extended Richards equation. These new solutions indeed
display a nonmonotonic saturation profile, similar to
previous simulation results. We show that these extensions
need a regularization term to produce a unique solution. We
develop complete analytic solutions using a relaxation
regularization term, and we discuss the results in terms of
recent measurements of saturation overshoot. Copyright 2008
by the American Geophysical Union.},
Doi = {10.1029/2007WR005975},
Key = {fds244234}
}
@article{fds244222,
Author = {Gratton, MB and Witelski, TP},
Title = {Coarsening of unstable thin films subject to
gravity.},
Journal = {Physical review. E, Statistical, nonlinear, and soft matter
physics},
Volume = {77},
Number = {1 Pt 2},
Pages = {016301},
Year = {2008},
Month = {January},
ISSN = {1539-3755},
url = {http://dx.doi.org/10.1103/physreve.77.016301},
Abstract = {Thin films of viscous fluids coating hydrophobic substrates
are unstable to dewetting instabilities, and long-time
evolution leads to the formation of an array of
near-equilibrium droplets connected by ultrathin fluid
layers. In the absence of gravity, previous use of
lubrication theory has shown that coarsening dynamics will
ensue-the system will evolve by successively eliminating
small drops to yield fewer larger drops. While gravity has
only a weak influence on the initial thin film, we show that
it has a significant influence on the later stages of the
coarsening dynamics, dramatically slowing the rate of
coarsening for large drops. Small drops are relatively
unaffected, but as coarsening progresses, these aggregate
into larger drops whose shape and dynamics are dominated by
gravity. The change in the mean drop shape causes a
corresponding gradual transition from power-law coarsening
to a logarithmic behavior.},
Doi = {10.1103/physreve.77.016301},
Key = {fds244222}
}
@article{fds244231,
Author = {Aguareles, M and Chapman, SJ and Witelski, TP},
Title = {Interaction of spiral waves in the Complex Ginzburg-Landau
equation},
Journal = {Physical Review Letters},
Volume = {101},
Number = {224101},
Year = {2008},
ISSN = {0031-9007},
url = {http://link.aps.org/abstract/PRL/v101/e224101},
Abstract = {Solutions of the general cubic complex Ginzburg-Landau
equation comprising multiple spiral waves are considered,
and laws of motion for the centers are derived. The
direction of the motion changes from along the line of
centers to perpendicular to the line of centers as the
separation increases, with the strength of the interaction
algebraic at small separations and exponentially small at
large separations. The corresponding asymptotic wave number
and frequency are also determined, which evolve slowly as
the spirals move. © 2008 The American Physical
Society.},
Doi = {10.1103/PhysRevLett.101.224101},
Key = {fds244231}
}
@article{fds244246,
Author = {Gratton, MB and Witelski, TP},
Title = {Coarsening of dewetting thin films subject to
gravity},
Journal = {Physical Review E},
Volume = {77},
Number = {016301},
Pages = {1-11},
Year = {2008},
url = {http://link.aps.org/abstract/PRE/v77/e016301},
Key = {fds244246}
}
@article{fds244233,
Author = {Levy, R and Shearer, M and Witelski, TP},
Title = {Gravity-driven thin liquid films with insoluble surfactant:
Smooth traveling waves},
Journal = {European Journal of Applied Mathematics},
Volume = {18},
Number = {6},
Pages = {679-708},
Publisher = {Cambridge University Press (CUP)},
Year = {2007},
Month = {December},
ISSN = {0956-7925},
url = {http://dx.doi.org/10.1017/S0956792507007218},
Abstract = {The flow of a thin layer of fluid down an inclined plane is
modified by the presence of insoluble surfactant. For any
finite surfactant mass, traveling waves are constructed for
a system of lubrication equations describing the evolution
of the free-surface fluid height and the surfactant
concentration. The one-parameter family of solutions is
investigated using perturbation theory with three small
parameters: the coefficient of surface tension, the
surfactant diffusivity, and the coefficient of the
gravity-driven diffusive spreading of the fluid. When all
three parameters are zero, the nonlinear PDE system is
hyperbolic/degenerate-parabolic, and admits traveling wave
solutions in which the free-surface height is piecewise
constant, and the surfactant concentration is piecewise
linear and continuous. The jumps and corners in the
traveling waves are regularized when the small parameters
are nonzero; their structure is revealed through a
combination of analysis and numerical simulation. © 2007
Cambridge University Press.},
Doi = {10.1017/S0956792507007218},
Key = {fds244233}
}
@article{fds244237,
Author = {Schaeffer, DG and Shearer, M and Witelski, TP},
Title = {Boundary-value problems for hyperbolic equations related to
steady granular flow},
Journal = {Mathematics and Mechanics of Solids},
Volume = {12},
Number = {6},
Pages = {665-699},
Publisher = {SAGE Publications},
Year = {2007},
Month = {December},
ISSN = {1081-2865},
url = {http://dx.doi.org/10.1177/1081286506067325},
Abstract = {Boundary value problems for steady-state flow in
elastoplasticity are a topic of mathematical and physical
interest. In particular, the underlying PDE may be
hyperbolic, and uncertainties surround the choice of
physically appropriate stress and velocity boundary
conditions. The analysis and numerical simulations of this
paper address this issue for a model problem, a system of
equations describing antiplane shearing of an elastoplastic
material. This system retains the relevant mathematical
structure of elastoplastic planar flow. Even if the flow
rule is associative, two significant phenomena appear: (i)
For boundary conditions suggestive of granular flow in a
hopper, in which it seems physically natural to specify the
velocity everywhere along a portion of the boundary, no such
solutions of the equations exist; rather, we construct a
solution with a shear band (velocity jump) along part of the
boundary, and an appropriate relaxed boundary condition is
satisfied there. (ii) Rigid zones appear inside deforming
regions of the flow, and the stress field in such a zone is
not uniquely determined. For a nonassociative flow rule, an
extreme form of nonuniqueness-both velocity and stress-is
encountered. © SAGE Publications 2007.},
Doi = {10.1177/1081286506067325},
Key = {fds244237}
}
@article{fds244235,
Author = {Witelski, TP and Shearer, M and Levy, R},
Title = {Growing surfactant waves in thin liquid films driven by
gravity},
Journal = {Applied Mathematics Research eXpress},
Volume = {2006},
Number = {15487},
Pages = {1-21},
Publisher = {Oxford University Press (OUP)},
Year = {2006},
Month = {December},
ISSN = {1687-1200},
url = {http://dx.doi.org/10.1155/AMRX/2006/15487},
Abstract = {The dynamics of a gravity-driven thin film flow with
insoluble surfactant are described in the lubrication
approximation by a coupled system of nonlinear PDEs. When
the total quantity of surfactant is fixed, a traveling wave
solution exists. For the case of constantflux of surfactant
from an upstream reservoir, global traveling waves no longer
exist as the surfactant accumulates at the leading edge of
the thin film profile. The dynamics can be described using
matched asymptotic expansions for t→∞ . The solution is
constructed from quasistatically evolving traveling waves.
The rate of growth of the surfactant profile is shown to be
O(√t) and is supported by numerical simulations.},
Doi = {10.1155/AMRX/2006/15487},
Key = {fds244235}
}
@article{fds244240,
Author = {Bowen, M and Witelski, TP},
Title = {The linear limit of the dipole problem for the thin film
equation},
Journal = {SIAM Journal on Applied Mathematics},
Volume = {66},
Number = {5},
Pages = {1727-1748},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2006},
Month = {October},
ISSN = {0036-1399},
url = {http://dx.doi.org/10.1137/050637832},
Abstract = {We investigate self-similar solutions of the dipole problem
for the one-dimensional thin film equation on the half-line
{x ≥ 0}. We study compactly supported solutions of the
linear moving boundary problem and show how they relate to
solutions of the nonlinear problem. The similarity solutions
are generally of the second kind, given by the solution of a
nonlinear eigenvalue problem, although there are some
notable cases where first-kind solutions also arise. We
examine the conserved quantities connected to these
first-kind solutions. Difficulties associated with the lack
of a maximum principle and the non-self-adjointness of the
fundamental linear problem are also considered. Seeking
similarity solutions that include sign changes yields a
surprisingly rich set of (coexisting) stable solutions for
the intermediate asymptotics of this problem. Our results
include analysis of limiting cases and comparisons with
numerical computations. © 2006 Society for Industrial and
Applied Mathematics.},
Doi = {10.1137/050637832},
Key = {fds244240}
}
@article{fds244247,
Author = {Munch, A and Wagner, B and Witelski, TP},
Title = {Lubrication models with small to large slip
lengths},
Journal = {Journal of Engineering Mathematics},
Volume = {53},
Number = {3-4},
Pages = {259-283},
Publisher = {Springer Nature},
Year = {2005},
Month = {December},
ISSN = {0022-0833},
url = {http://www.springerlink.com/(gpximmrigyvfihbxa2cbda45)/app/home/contribution.asp?referrer=parent&backto=issue,11,12;journal,3,197;linkingpublicationresults,1:100287,1},
Abstract = {A set of lubrication models for the thin film flow of
incompressible fluids on solid substrates is derived and
studied. The models are obtained as asymptotic limits of the
Navier-Stokes equations with the Navier-slip boundary
condition for different orders of magnitude for the
slip-length parameter. Specifically, the influence of slip
on the dewetting behavior of fluids on hydrophobic
substrates is investigated here. Matched asymptotics are
used to describe the dynamic profiles for dewetting films
and comparison is given with computational simulations. The
motion of the dewetting front shows transitions from being
nearly linear in time for no-slip to t2/3 as the slip is
increased. For much larger slip lengths the front motion
appears to become linear again. Correspondingly, the
dewetting profiles undergo a transition from oscillatory to
monotone decay into the uniform film layer for large slip.
Increasing the slip further, to very large values, is
associated with an increasing degree of asymmetry in the
structure of the dewetting ridge profile. © Springer
2005.},
Doi = {10.1007/s10665-005-9020-3},
Key = {fds244247}
}
@article{fds244248,
Author = {Witelski, TP and Rienstra, SW},
Title = {Introduction to practical asymptotics III},
Journal = {Journal of Engineering Mathematics},
Volume = {53},
Number = {3-4},
Pages = {199},
Publisher = {Springer Nature},
Year = {2005},
Month = {December},
ISSN = {0022-0833},
url = {http://www.springerlink.com/(gpximmrigyvfihbxa2cbda45)/app/home/contribution.asp?referrer=parent&backto=issue,1,12;journal,3,197;linkingpublicationresults,1:100287,1},
Abstract = {Introduction to special issue in the journal, TPW and SWR
guest co-editors.},
Doi = {10.1007/s10665-005-9027-9},
Key = {fds244248}
}
@article{fds244249,
Author = {Glasner, KB and Witelski, TP},
Title = {Collision versus collapse of droplets in coarsening of
dewetting thin films},
Journal = {Physica D: Nonlinear Phenomena},
Volume = {209},
Number = {1-4 SPEC. ISS.},
Pages = {80-104},
Publisher = {Elsevier BV},
Year = {2005},
Month = {September},
url = {http://dx.doi.org/10.1016/j.physd.2005.06.010},
Abstract = {Thin films of viscous fluids coating solid surfaces can
become unstable due to intermolecular forces, leading to
break-up of the film into arrays of droplets. The long-time
dynamics of the system can be represented in terms of
coupled equations for the masses and positions of the
droplets. Analysis of the decrease of energy of the system
shows that coarsening, decreasing the number of droplets
with increasing time, is favored. Here we describe the two
coarsening mechanisms present in dewetting films: (i) mass
exchange leading to collapse of individual drops, and (ii)
spatial motion leading to droplet collisions and merging
events. Regimes where each of mechanisms are dominant are
identified, and the statistics of the coarsening process are
explained. © 2005 Elsevier B.V. All rights
reserved.},
Doi = {10.1016/j.physd.2005.06.010},
Key = {fds244249}
}
@article{fds244250,
Author = {Haskett, RP and Witelski, TP and Sur, J},
Title = {Localized Marangoni forcing in driven thin
films},
Journal = {Physica D: Nonlinear Phenomena},
Volume = {209},
Number = {1-4 SPEC. ISS.},
Pages = {117-134},
Publisher = {Elsevier BV},
Year = {2005},
Month = {September},
url = {http://dx.doi.org/10.1016/j.physd.2005.06.019},
Abstract = {We consider the use of localized Marangoni forcing to
produce a thermocapillary "microfluidic valve" that allows
us to control the downstream flow of a thin film of viscous
fluid. To this end, we analyze the influence of this
localized forcing on a flow driven by a combination of
uniform Marangoni stresses and gravity in a one-dimensional
model. Long-time solutions approach states that can be
categorized in two classes, where the film thickness
downstream of the forcing is: (I) determined by the upstream
thickness, or (II) controlled by the forcing amplitude. The
type II solutions are stable stationary hydraulic jumps for
thin films. We give careful attention to the relation
between the forcing and the downstream film flow for the
resulting bi-stable solutions. We include a comparison of
the one-dimensional theory with two-dimensional computations
and experimental results. © 2005 Elsevier B.V. All rights
reserved.},
Doi = {10.1016/j.physd.2005.06.019},
Key = {fds244250}
}
@article{fds244238,
Author = {Fetzer, R and Jacobs, K and Münch, A and Wagner, B and Witelski,
TP},
Title = {New slip regimes and the shape of dewetting thin liquid
films.},
Journal = {Physical review letters},
Volume = {95},
Number = {12},
Pages = {127801},
Year = {2005},
Month = {September},
ISSN = {0031-9007},
url = {http://www.ncbi.nlm.nih.gov/pubmed/16197111},
Abstract = {We compare the flow behavior of liquid polymer films on
silicon wafers coated with either octadecyl-(OTS) or
dodecyltrichlorosilane (DTS). Our experiments show that
dewetting on DTS is significantly faster than on OTS. We
argue that this is tied to the difference in the
solid/liquid friction. As the film dewets, the profile of
the rim advancing into the undisturbed film is monotonically
decaying on DTS but has an oscillatory structure on OTS. For
the first time we can describe this transition in terms of a
lubrication model with a Navier-slip condition for the flow
of a viscous Newtonian liquid.},
Doi = {10.1103/physrevlett.95.127801},
Key = {fds244238}
}
@article{fds244239,
Author = {Witelski, TP},
Title = {Motion of wetting fronts moving into partially pre-wet
soil},
Journal = {Advances in Water Resources},
Volume = {28},
Number = {10 SPEC. ISS.},
Pages = {1133-1141},
Publisher = {Elsevier BV},
Year = {2005},
Month = {January},
url = {http://dx.doi.org/10.1016/j.advwatres.2004.06.006},
Abstract = {We study the motion of wetting fronts for vertical
infiltration problems as modeled by Richards' equation.
Parlange and others have shown that wetting fronts in
infiltration flows can be described by traveling wave
solutions. If the soil layer is not initially dry, but has
an initial distribution of water content then the motion of
the wetting front will change due to the interaction of the
infiltrating flow with the pre-existing soil conditions.
Using traveling wave profiles, we construct simple
approximate solutions of initial-boundary value problems for
Richards' equation that accurately describe the position and
moisture distribution of the wetting front. We show that the
influences of surface boundary conditions and initial
conditions produce shifts to the position of the wetting
front. The shifts can be calculated by examining the
cumulative infiltration, and are validated numerically for
several problems for Richards' equation and the linear
advection-diffusion equation. © 2005 Elsevier Ltd. All
rights reserved.},
Doi = {10.1016/j.advwatres.2004.06.006},
Key = {fds244239}
}
@article{fds244251,
Author = {Smolka, LB and Belmonte, A and Henderson, DM and Witelski,
TP},
Title = {Exact solution for the extensional flow of a viscoelastic
filament},
Journal = {European Journal of Applied Mathematics},
Volume = {15},
Number = {6},
Pages = {679-712},
Publisher = {Cambridge University Press (CUP)},
Year = {2004},
Month = {December},
url = {http://dx.doi.org/10.1017/S0956792504005789},
Abstract = {We solve the free boundary problem for the dynamics of a
cylindrical, axisymmetric viscoelastic filament stretching
in a gravity-driven extensional flow for the Upper Convected
Maxwell and Oldroyd-B constitutive models. Assuming the
axial stress in the filament has a spatial dependence
provides the simplest coupling of viscoelastic effects to
the motion of the filament, and yields a closed system of
ODEs with an exact solution for the stretch rate and
filament thickness satisfied by both constitutive models.
This viscoelastic solution, which is a generalization of the
exact solution for Newtonian filaments, converges to the
Newtonian power-law scaling as t → ∞. Based on the exact
solution, we identify two regimes of dynamical behavior
called the weakly- and strongly-viscoelastic limits. We
compare the viscoelastic solution to measurements of the
thinning filament that forms behind a falling drop for
several semi-dilute (strongly-viscoelastic) polymer
solutions. We find the exact solution correctly predicts the
time-dependence of the filament diameter in all of the
experiments. As t → ∞, observations of the filament
thickness follow the Newtonian scaling 1/√t. The
transition from viscoelastic to Newtonian scaling in the
filament thickness is coupled to a stretch-to-coil
transition of the polymer molecules. © 2004 Cambridge
University Press.},
Doi = {10.1017/S0956792504005789},
Key = {fds244251}
}
@article{fds244252,
Author = {Sur, J and Witelski, TP and Behringer, RP},
Title = {Steady-profile fingering flows in Marangoni driven thin
films.},
Journal = {Physical review letters},
Volume = {93},
Number = {24},
Pages = {247803},
Year = {2004},
Month = {December},
ISSN = {0031-9007},
url = {http://www.ncbi.nlm.nih.gov/pubmed/15697861},
Abstract = {We present experimental and computational results indicating
the existence of finite-amplitude fingering solutions in a
flow of a thin film of a viscous fluid driven by thermally
induced Marangoni stresses. Using carefully controlled
experiments, spatially periodic perturbations to the contact
line of an initially uniform thin film flow are shown to
lead to the development of steady-profile two-dimensional
traveling wave fingers. Using an infrared laser and scanning
mirror, we impose thermal perturbations with a known
wavelength to an initially uniform advancing fluid front. As
the front advances in the experiment, we observe convergence
to fingers with the initially prescribed wavelength.
Experiments and numerical computations show that this family
of solutions arises from a subcritical bifurcation.},
Doi = {10.1103/physrevlett.93.247803},
Key = {fds244252}
}
@article{fds244253,
Author = {Borucki, LJ and Witelski, T and Please, C and Kramer, PR and Schwendeman, D},
Title = {A theory of pad conditioning for chemical-mechanical
polishing},
Journal = {Journal of Engineering Mathematics},
Volume = {50},
Number = {1},
Pages = {1-24},
Publisher = {Springer Nature},
Year = {2004},
Month = {December},
url = {http://ipsapp007.kluweronline.com/ips/frames/issues.aspx?J=4678&N=JournalContents&ADS=0},
Abstract = {Statistical models are presented to describe the evolution
of the surface roughness of polishing pads during the
pad-conditioning process in chemical-mechanical polishing.
The models describe the evolution of the surface-height
probability-density function of solid pads during fixed
height or fixed cut-rate conditioning. An integral equation
is derived for the effect of conditioning on a foamed pad in
terms of a model for a solid pad. The models that combine
wear and conditioning are then discussed for both solid and
foamed pads. Models include the dependence of the surface
roughness on the shape and density of the cutting tips used
in the conditioner and on other operating parameters. Good
agreement is found between the model, Monte Carlo
simulations and with experimental data. © 2004 Kluwer
Academic Publishers.},
Doi = {10.1023/B:ENGI.0000042116.09084.00},
Key = {fds244253}
}
@article{fds244255,
Author = {Witelski, TP and Bernoff, AJ and Bertozzi, AL},
Title = {Blowup and dissipation in a critical-case unstable thin film
equation},
Journal = {European Journal of Applied Mathematics},
Volume = {15},
Number = {2},
Pages = {223-256},
Publisher = {Cambridge University Press (CUP)},
Year = {2004},
Month = {April},
url = {http://dx.doi.org/10.1017/S0956792504005418},
Abstract = {We study the dynamics of dissipation and blow-up in a
critical-case unstable thin film equation. The governing
equation is a nonlinear fourth-order degenerate parabolic
PDE derived from a generalized model for lubrication flows
of thin viscous fluid layers on solid surfaces. There is a
critical mass for blow-up and a rich set of dynamics
including families of similarity solutions for finite-time
blow-up and infinite-time spreading. The structure and
stability of the steady-states and the compactly-supported
similarity solutions is studied.},
Doi = {10.1017/S0956792504005418},
Key = {fds244255}
}
@article{fds244220,
Author = {Witelski, TP},
Title = {Nonlinear Differential Equations, Mechanics and
Bifurcation},
Journal = {Discrete and Continuous Dynamical Systems - Series
B},
Volume = {3},
Number = {4},
Pages = {i},
Year = {2003},
Month = {November},
url = {http://aimsciences.org/journals/dcdsB/B3_4.htm},
Key = {fds244220}
}
@article{fds304499,
Author = {Shearer, M and Schaeffer, DG and Witelski, TP},
Title = {Stability of shear bands in an elastoplastic model for
granular flow: The role of discreteness},
Journal = {Mathematical Models and Methods in Applied
Sciences},
Volume = {13},
Number = {11},
Pages = {1629-1671},
Publisher = {World Scientific Pub Co Pte Lt},
Year = {2003},
Month = {November},
url = {http://dx.doi.org/10.1142/S0218202503003069},
Abstract = {Continuum models for granular flow generally give rise to
systems of nonlinear partial differential equations that are
linearly ill-posed. In this paper we introduce discreteness
into an elastoplasticity model for granular flow by
approximating spatial derivatives with finite differences.
The resulting ordinary differential equations have bounded
solutions for all time, a consequence of both discreteness
and nonlinearity. We study how the large-time behavior of
solutions in this model depends on an elastic shear modulus
ε. For large and moderate values of ε, the model has
stable steady-state solutions with uniform shearing except
for one shear band; almost all solutions tend to one of
these as t → ∞. However, when ε becomes sufficiently
small, the single-shear-band solutions lose stability
through a Hopf bifurcation. The value of ε at the
bifurcation point is proportional to the ratio of the mesh
size to the macroscopic length scale. These conclusions are
established analytically through a careful estimation of the
eigenvalues. In numerical simulations we find that: (i)
after stability is lost, time-periodic solutions appear,
containing both elastic and plastic waves, and (ii) the
bifurcation diagram representing these solutions exhibits
bi-stability.},
Doi = {10.1142/S0218202503003069},
Key = {fds304499}
}
@article{fds244256,
Author = {Witelski, TP and Bowen, M},
Title = {ADI schemes for higher-order nonlinear diffusion
equations},
Journal = {Applied Numerical Mathematics},
Volume = {45},
Number = {2-3},
Pages = {331-351},
Publisher = {Elsevier BV},
Year = {2003},
Month = {May},
url = {http://dx.doi.org/10.1016/S0168-9274(02)00194-0},
Abstract = {Alternating Direction Implicit (ADI) schemes are constructed
for the solution of two-dimensional higher-order linear and
nonlinear diffusion equations, particularly including the
fourth-order thin film equation for surface tension driven
fluid flows. First and second-order accurate schemes are
derived via approximate factorization of the evolution
equations. This approach is combined with iterative methods
to solve nonlinear problems. Problems in the fluid dynamics
of thin films are solved to demonstrate the effectiveness of
the ADI schemes. © 2002 IMACS. Published by Elsevier
Science B.V. All rights reserved.},
Doi = {10.1016/S0168-9274(02)00194-0},
Key = {fds244256}
}
@article{fds244257,
Author = {Witelski, TP},
Title = {Intermediate asymptotics for Richards' equation in a finite
layer},
Journal = {Journal of Engineering Mathematics},
Volume = {45},
Number = {3-4},
Pages = {379-399},
Year = {2003},
Month = {April},
url = {http://dx.doi.org/10.1023/A:1022609020200},
Abstract = {Perturbation methods are applied to study an
initial-boundary-value problem for Richards' equation,
describing vertical infiltration of water into a finite
layer of soil. This problem for the degenerate diffusion
equation with convection and Dirichlet/Robin boundary
conditions exhibits several different regimes of behavior.
Boundary-layer analysis and short-time asymptotics are used
to describe the structure of similarity solutions, traveling
waves, and other solution states and the transitions
connecting these different intermediate asymptotic
regimes.},
Doi = {10.1023/A:1022609020200},
Key = {fds244257}
}
@article{fds244258,
Author = {Glasner, KB and Witelski, TP},
Title = {Coarsening dynamics of dewetting films.},
Journal = {Physical review. E, Statistical, nonlinear, and soft matter
physics},
Volume = {67},
Number = {1 Pt 2},
Pages = {016302},
Year = {2003},
Month = {January},
ISSN = {1539-3755},
url = {http://www.ncbi.nlm.nih.gov/pubmed/12636597},
Abstract = {Lubrication theory for unstable thin liquid films on solid
substrates is used to model the coarsening dynamics in the
long-time behavior of dewetting films. The dominant physical
effects that drive the fluid dynamics in dewetting films are
surface tension and intermolecular interactions with the
solid substrate. Instabilities in these films lead to
rupture and other morphological changes that promote
nonuniformity in the films. Following the initial
instabilities, the films break up into near-equilibrium
droplets connected by an ultrathin film. For longer times,
the fluid will undergo a coarsening process in which
droplets both move and exchange mass on slow time scales.
The dynamics of this coarsening process will be obtained
through the asymptotic reduction of the long-wave PDE
governing the thin film to a set of ODEs for the evolution
of the droplets. From this, a scaling law that governs the
coarsening rate is derived.},
Doi = {10.1103/physreve.67.016302},
Key = {fds244258}
}
@article{fds318344,
Author = {Glasner, KB and Witelski, TP},
Title = {Coarsening dynamics of dewetting films},
Journal = {Physical Review E - Statistical, Nonlinear, and Soft Matter
Physics},
Volume = {67},
Number = {1 2},
Pages = {163021-1630212},
Year = {2003},
Month = {January},
Abstract = {The modelling of coarsening dynamics of dewetting films
using lubrication theory for unstable thin liquid films on
solid substrates was discussed. Surface tension and
intermolecular interactions with the solid substrate were
the dominant physical effects driving the fluid dynamics.
The fluid underwent a coarsening process in which droplets
moved and exchanged mass on slow time scales.},
Key = {fds318344}
}
@article{fds325967,
Author = {Schaeffer, DG and Shearer, M and Witelski, T},
Title = {One-dimensional solutions of an elastoplasticity model of
granular material},
Journal = {Math. Models and Methods in Appl. Sciences},
Volume = {13},
Pages = {1629-1671},
Year = {2003},
Key = {fds325967}
}
@article{fds244187,
Author = {Witelski, TP},
Title = {Computing finite-time singularities in interfacial
flows},
Journal = {MODERN METHODS IN SCIENTIFIC COMPUTING AND
APPLICATIONS},
Volume = {75},
Pages = {451-487},
Publisher = {SPRINGER},
Editor = {Bourlioux, A and Gander, MJ and Sabidussi, G},
Year = {2002},
Month = {January},
ISBN = {1-4020-0782-5},
ISSN = {1568-2609},
url = {http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000180113600012&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=47d3190e77e5a3a53558812f597b0b92},
Abstract = {Finite-time singularities occuring in mathematical models of
free-surface flows indicate that important qualitative
changes are taking place; for problems in solid and fluid
mechanics this includes topological transitions -- blow-up,
and pinch-off. For many problems, the dynamics leading to
the formation of such singularities are described by
self-similar solutions of the governing nonlinear partial
differential equations. We present an analytical and
numerical study of these similarity solutions and discuss
their stability.},
Key = {fds244187}
}
@article{fds244217,
Author = {Bernoff, AJ and Witelski, TP},
Title = {Linear stability of source-type similarity solutions of the
thin film equation},
Journal = {Applied Mathematics Letters},
Volume = {15},
Number = {5},
Pages = {599-606},
Publisher = {Elsevier BV},
Year = {2002},
Month = {January},
ISSN = {0893-9659},
url = {http://dx.doi.org/10.1016/S0893-9659(02)80012-X},
Abstract = {We study the stability of compactly-supported source-type
self-similar solutions of the generalized thin film equation
ht = -(hnhxxx)x. Using linear stability analysis, applied to
the problem in similarity variables, we show that the
source-type solutions are stable. These results are related
to the underlying symmetries of the PDE. For the special
case of n = 1, analytical results are obtained for the
spectrum, and the eigenfunctions are given in terms of
classical orthogonal polynomials. © 2002 Elsevier Science
Ltd. All rights reserved.},
Doi = {10.1016/S0893-9659(02)80012-X},
Key = {fds244217}
}
@article{fds244214,
Author = {Witelski, TP and Schaeffer, DG and Shearer, M},
Title = {A discrete model for an ill-posed nonlinear parabolic
PDE},
Journal = {Physica D: Nonlinear Phenomena},
Volume = {160},
Number = {3-4},
Pages = {189-221},
Publisher = {Elsevier BV},
Year = {2001},
Month = {December},
ISSN = {0167-2789},
url = {http://dx.doi.org/10.1016/S0167-2789(01)00350-5},
Abstract = {We study a finite-difference discretization of an ill-posed
nonlinear parabolic partial differential equation. The PDE
is the one-dimensional version of a simplified
two-dimensional model for the formation of shear bands via
anti-plane shear of a granular medium. For the discretized
initial value problem, we derive analytically, and observed
numerically, a two-stage evolution leading to a
steady-state: (i) an initial growth of grid-scale
instabilities, and (ii) coarsening dynamics. Elaborating the
second phase, at any fixed time the solution has a piecewise
linear profile with a finite number of shear bands. In this
coarsening phase, one shear band after another collapses
until a steady-state with just one jump discontinuity is
achieved. The amplitude of this steady-state shear band is
derived analytically, but due to the ill-posedness of the
underlying problem, its position exhibits sensitive
dependence. Analyzing data from the simulations, we observe
that the number of shear bands at time t decays like t-1/3.
From this scaling law, we show that the time-scale of the
coarsening phase in the evolution of this model for granular
media critically depends on the discreteness of the model.
Our analysis also has implications to related ill-posed
nonlinear PDEs for the one-dimensional Perona-Malik equation
in image processing and to models for clustering
instabilities in granular materials. © 2001 Elsevier
Science B.V. All rights reserved.},
Doi = {10.1016/S0167-2789(01)00350-5},
Key = {fds244214}
}
@article{fds244213,
Author = {Vaynblat, D and Lister, JR and Witelski, TP},
Title = {Symmetry and self-similarity in rupture and pinchoff: A
geometric bifurcation},
Journal = {European Journal of Applied Mathematics},
Volume = {12},
Number = {3},
Pages = {209-232},
Publisher = {Cambridge University Press (CUP)},
Year = {2001},
Month = {December},
url = {http://dx.doi.org/10.1017/S0956792501004375},
Abstract = {Long-wavelength models for van der Waals driven rupture of a
free thin viscous sheet and for capillary pinchoff of a
viscous fluid thread both give rise to families of
first-type similarity solutions. The scaling exponents in
these solutions are independent of the dimensionality of
problem. However, the structure of the similarity solutions
exhibits an intriguing geometric dependence on the
dimensionality of the system: van der Waals driven sheet
rupture proceeds symmetrically, whereas thread rupture is
inherently asymmetric. To study the bifurcation of rupture
from symmetric to asymmetric forms, we generalize the
governing equations with the dimension serving as a control
parameter. The bifurcation is governed by leading-order
inviscid dynamics in which viscous effects are
asymptotically small but nevertheless provide the selection
mechanism.},
Doi = {10.1017/S0956792501004375},
Key = {fds244213}
}
@article{fds244216,
Author = {Bertozzi, AL and Grün, G and Witelski, TP},
Title = {Dewetting films: Bifurcations and concentrations},
Journal = {Nonlinearity},
Volume = {14},
Number = {6},
Pages = {1569-1592},
Publisher = {IOP Publishing},
Year = {2001},
Month = {November},
url = {http://dx.doi.org/10.1088/0951-7715/14/6/309},
Abstract = {Under the influence of long-range attractive and short-range
repulsive forces, thin liquid films rupture and form complex
dewetting patterns. This paper studies this phenomenon in
one space dimension within the framework of fourth-order
degenerate parabolic equations of lubrication type. We
derive the global structure of the bifurcation diagram for
steady-state solutions. A stability analysis of the solution
branches and numerical simulations suggest coarsening
occurs. Furthermore, we study the behaviour of solutions in
the limit that short-range repulsive forces are neglected.
Both asymptotic analysis and numerical experiments show that
this limit can concentrate mass in δ-distributions.},
Doi = {10.1088/0951-7715/14/6/309},
Key = {fds244216}
}
@article{fds244212,
Author = {Witelski, TP and Ono, K and Kaper, TJ},
Title = {Critical wave speeds for a family of scalar
reaction-diffusion equations},
Journal = {Applied Mathematics Letters},
Volume = {14},
Number = {1},
Pages = {65-73},
Publisher = {Elsevier BV},
Year = {2001},
Month = {January},
url = {http://dx.doi.org/10.1016/S0893-9659(00)00114-2},
Abstract = {We study the set of traveling waves in a class of
reaction-diffusion equations for the family of potentials
fm(U) = 2Um(1 - U). We use perturbation methods and matched
asymptotics to derive expansions for the critical wave speed
that separates algebraic and exponential traveling wave
front solutions for m → 2 and m → ∞. Also, an integral
formulation of the problem shows that nonuniform convergence
of the generalized equal area rule occurs at the critical
wave speed. © 2000 Elsevier Science Ltd. All rights
reserved.},
Doi = {10.1016/S0893-9659(00)00114-2},
Key = {fds244212}
}
@article{fds244215,
Author = {Vaynblat, D and Lister, JR and Witelski, TP},
Title = {Rupture of thin viscous films by van der waals forces:
Evolution and self-similarity},
Journal = {Physics of Fluids},
Volume = {13},
Number = {5},
Pages = {1130-1141},
Publisher = {AIP Publishing},
Year = {2001},
Month = {January},
url = {http://ojps.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=PHFLE6000013000005001130000001&idtype=cvips&gifs=Yes},
Abstract = {The van der Waals driven rupture of a freely suspended thin
viscous sheet is examined using a long-wavelength model.
Dimensional analysis shows the possibility of first-type
similarity solutions in which the dominant physical balance
is between inertia, extensional viscous stresses and the van
der Waals disjoining pressure, while surface tension is
negligible. For both line rupture and point rupture the film
thickness decreases like (t* - t)1/3 and the lateral length
scale like (t* - t)1/2, where t* - t is the time remaining
until rupture. In each geometry these scalings are confirmed
by numerical simulations of the time-dependent behavior, and
a discrete family of similarity solutions is found. The
"lowest-order" mode in the family is the one selected by the
time-dependent dynamics. © 2001 American Institute of
Physics.},
Doi = {10.1063/1.1359749},
Key = {fds244215}
}
@article{fds244211,
Author = {Witelski, TP and Bernoff, AJ},
Title = {Dynamics of three-dimensional thin film rupture},
Journal = {Physica D: Nonlinear Phenomena},
Volume = {147},
Number = {1-2},
Pages = {155-176},
Publisher = {Elsevier BV},
Year = {2000},
Month = {December},
url = {http://dx.doi.org/10.1016/S0167-2789(00)00165-2},
Abstract = {We consider the problem of thin film rupture driven by van
der Waals forces. A fourth-order nonlinear PDE governs the
low Reynolds number lubrication model for a viscous liquid
on a solid substrate. Finite-time singularities in this
equation model rupture leading to formation of dry spots in
the film. Our study addresses the problem of rupture in the
full three-dimensional geometry. We focus on stability and
selection of the dynamics determined by the initial
conditions on small finite domains with planar and
axisymmetric geometries. We also address the final stages of
the dynamics - self-similar dynamics for point, line, and
ring rupture. We will demonstrate that line and ring rupture
are unstable and will generically destabilize to produce
axisymmetric rupture at isolated points.},
Doi = {10.1016/S0167-2789(00)00165-2},
Key = {fds244211}
}
@article{fds342143,
Author = {Witelski, TP and Ono, K and Kaper, TJ},
Title = {On axisymmetric traveling waves and radial solutions of
semi-linear elliptic equations},
Journal = {Natural Resource Modeling},
Volume = {13},
Number = {3},
Pages = {339-388},
Year = {2000},
Month = {January},
url = {http://www.math.duke.edu/~witelski/local/radial.ps},
Abstract = {Combining analytical techniques from perturbation methods
and dynamical systems theory, we present an
elementaryapproach to the detailed construction of
axisymmetric diffusive interfaces in semi-linear elliptic
equations. Solutions of the resulting non-autonomous radial
differential equations can be expressed in terms of a
slowlyvarying phase plane system. Special analytical results
for the phase plane system are used to produce closed-form
solutions for the asymptotic forms of the curved front
solutions. These axisym-metric solutions are fundamental
examples of more general curved fronts that arise in a wide
variety of scientific fields, and we extensivelydiscuss a
number of them, with a particular emphasis on connections to
geometric models for the motion of interfaces. Related
classical results for traveling waves in one-dimensional
problems are also reviewed briefly. Manyof the results
contained in this article are known, and in presenting known
results, it is intended that this article be expositoryin
nature, providing elementarydemonstrations of some of the
central dynamical phenomena and mathematical techniques. It
is hoped that the article serves as one possible avenue of
entree to the literature on radiallysymmetric solutions of
semilinear elliptic problems, especiallyto those articles in
which more advanced mathematical theoryis developed. © 2000
Rocky Mountain Mathematics Consortium.},
Doi = {10.1111/j.1939-7445.2000.tb00039.x},
Key = {fds342143}
}
@article{fds244186,
Author = {Witelski, TP and Bernoff, AJ},
Title = {Stability of self-similar solutions for van der Waals driven
thin film rupture},
Journal = {Physics of Fluids},
Volume = {11},
Number = {9},
Pages = {2443-2445},
Publisher = {AIP Publishing},
Year = {1999},
Month = {January},
ISSN = {1070-6631},
url = {http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000081906000002&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=47d3190e77e5a3a53558812f597b0b92},
Abstract = {Recent studies of pinch-off of filaments and rupture in thin
films have found infinite sets of first-type similarity
solutions. Of these, the dynamically stable similarity
solutions produce observable rupture behavior as localized,
finite-time singularities in the models of the flow. In this
letter we describe a systematic technique for calculating
such solutions and determining their linear stability. For
the problem of axisymmetric van der Waals driven rupture
(recently studied by Zhang and Lister), we identify the
unique stable similarity solution for point rupture of a
thin film and an alternative mode of singularity formation
corresponding to annular “ring rupture.”. © 1999,
American Institute of Physics. All rights
reserved.},
Doi = {10.1063/1.870138},
Key = {fds244186}
}
@article{fds244209,
Author = {Witelski, TP and Hendriks, F},
Title = {Stability of gas bearing sliders for large bearing number:
Convective instability of the tapered slider©},
Journal = {Tribology Transactions},
Volume = {42},
Number = {1},
Pages = {216-222},
Publisher = {Informa UK Limited},
Year = {1999},
Month = {January},
ISSN = {1040-2004},
url = {http://www.math.duke.edu/~witelski/local/stle97.ps},
Abstract = {The dynamics and stability of tapered air bearing sliders
used for computer hard disk drive magnetic recording heads
is examined using analytical methods. Lubrication theory is
applied to determine the lift on the slider from the
Reynolds equation in the limit of large bearing number. The
dynamics of the slider are given by a nonlinear
integro-differential equation. Linear stability analysis of
this model yields valuable insight into the behavior of the
slider. Most significantly, it is found that convective
effects can not be neglected and yield either damping or
instability depending on the slider configuration. This
analysis is also applied to determine the response of the
slider motion due to deviations in the disk surface. © 1999
Taylor & Francis Group, LLC.},
Doi = {10.1080/10402009908982211},
Key = {fds244209}
}
@article{fds244210,
Author = {Witelski, TP and Hendriks, F},
Title = {Large bearing number stability analysis for tango class gas
bearing sliders},
Journal = {Tribology Transactions},
Volume = {42},
Number = {3},
Pages = {668-674},
Publisher = {Informa UK Limited},
Year = {1999},
Month = {January},
ISSN = {1040-2004},
url = {http://www.math.duke.edu/~witelski/local/gas2.ps},
Abstract = {Air bearing sliders in the Tango class use load bearing pads
with inlet-throttled leading edges to control the mass flux
and lift. The influence of leakage or diffusion effects is
always present in real sliders. In some designs such as
railed taper flat designs leakage is dominant. The behavior
of such sliders must be determined with numerical methods
that obscure deeper understanding. Many aspects of the
behavior of Tango class sliders can be understood with the
vast simplification allowed by inlet throttling. In this
paper such a simplified analysis is applied to describe a
complete air bearing slider composed of two pads. The
conditions for static level flight are determined, as well
as the linear stability of heaving and pitching
oscillations. Both stable and unstable modes are identified.
Either damping or amplification can result from convective
effects in the absence of mechanical damping. In real
implementations of Tango class sliders instability has not
been observed thanks to diffusion. The present analysis can
serve as a guide to select initial choices for the operating
parameters that correspond to maximum convective damping. ©
1999 Taylor and Francis Group, LLC.},
Doi = {10.1080/10402009908982268},
Key = {fds244210}
}
@article{fds244204,
Author = {Witelski, TP and Grosberg, AY and Tanaka, T},
Title = {On the properties of polymer globules in the high density
limit},
Journal = {Journal of Chemical Physics},
Volume = {108},
Number = {21},
Pages = {9144-9149},
Publisher = {AIP Publishing},
Year = {1998},
Month = {June},
url = {http://ojps.aip.org/journal_cgi/getabs?KEY=JCPSA6&cvips=JCPSA6000108000021009144000001&gifs=Yes},
Abstract = {We re-examine quantitative mean-field theory for the
collapsed globule phase of a polymer chain taking full
account of its finite compressibility. The mathematical
properties of the nonlinear mean-field equations describing
the structure of the globule are formulated. Our results
explain findings and observations of the recent computer
simulation and experimental studies. While the effects are
due to the restricted compressibility, they are seen well
before the globule reaches its dry maximal density. © 1998
American Institute of Physics.},
Doi = {10.1063/1.476361},
Key = {fds244204}
}
@article{fds244202,
Author = {Witelski, TP and Bernoff, AJ},
Title = {Self-similar asymptotics for linear and nonlinear diffusion
equations},
Journal = {Studies in Applied Mathematics},
Volume = {100},
Number = {2},
Pages = {153-193},
Publisher = {WILEY},
Year = {1998},
Month = {January},
url = {http://www.math.duke.edu/~witelski/articles/selfsim.ps.gz},
Abstract = {The long-time asymptotic solutions of initial value problems
for the heat equation and the nonlinear porous medium
equation are self-similar spreading solutions. The
symmetries of the governing equations yield three-parameter
families of these solutions given in terms of their mass,
center of mass, and variance. Unlike the mass and center of
mass, the variance, or "time-shift," of a solution is not a
conserved quantity for the nonlinear problem. We derive an
optimal linear estimate of the long-time variance. Newman's
Lyapunov functional is used to produce a maximum entropy
time-shift estimate. Results are applied to nonlinear
merging and time-dependent, inhomogeneously forced diffusion
problems.},
Doi = {10.1111/1467-9590.00074},
Key = {fds244202}
}
@article{fds244203,
Author = {Witelski, TP},
Title = {Dynamics of air bearing sliders},
Journal = {Physics of Fluids},
Volume = {10},
Number = {3},
Pages = {698-708},
Publisher = {AIP Publishing},
Year = {1998},
Month = {January},
url = {http://ojps.aip.org/journal_cgi/getabs?KEY=PHFLE6&cvips=PHFLE6000010000003000698000001&gifs=Yes},
Abstract = {In this paper we present new results for the dynamics of a
problem tor the interaction of a compressible gas flow with
a movable rigid surface. Compressible lubrication theory is
applied to describe the dynamics of the vertical motion of
air bearing sliders used in computer hard disk drives. In
the limit of large bearing number we show this problem can
be reduced to a nonlinear integrodifferential equation.
Linear stability analysis and perturbation methods show that
over the range of possible slider positions there is an
infinite sequence of Hopf bifurcations yielding stable large
amplitude periodic orbits. The dynamics of near-crash
behavior and interaction with a moving disk surface are also
addressed. © 1998 American Institute of
Physics.},
Doi = {10.1063/1.869595},
Key = {fds244203}
}
@article{fds244205,
Author = {Witelski, TP},
Title = {Horizontal infiltration into wet soil},
Journal = {Water Resources Research},
Volume = {34},
Number = {7},
Pages = {1859-1863},
Publisher = {American Geophysical Union (AGU)},
Year = {1998},
Month = {January},
url = {http://www.math.duke.edu/~witelski/articles/wetsoil.ps.gz},
Abstract = {We obtain the long-time asymptotic similarity solution for
the wetting front for water absorption from a constant
source into a homogenous layer of soil with a preexisting
moisture distribution. The presence of the initial water
distribution in the soil introduces a time shift that
advances the position of the wetting front. The time shift
be explicitly calculated for any form of diffusivity. A
dynamic time shift is derived to yield a very efficient
means for estimating the water content distribution and
front position for all times in Brooks-Corey-type soil
models.},
Doi = {10.1029/98WR00775},
Key = {fds244205}
}
@article{fds244206,
Author = {Witelski, TP},
Title = {Equilibrium interface solutions of a degenerate singular
Cahn-Hilliard equation},
Journal = {Applied Mathematics Letters},
Volume = {11},
Number = {5},
Pages = {127-133},
Publisher = {Elsevier BV},
Year = {1998},
Month = {January},
url = {http://dx.doi.org/10.1016/S0893-9659(98)00092-5},
Abstract = {We present an analysis of the equilibrium diffusive
interfaces in a model for the interaction of layers of pure
polymers. The discussion focuses on the important
qualitative features of the solutions of the nonlinear
singular Cahn-Hilliard equation with degenerate mobility for
the Flory-Huggins-deGennes free energy model. The spatial
structure of possible equilibrium phase separated solutions
are found. Using phase plane analysis, we obtain
heteroclinic and homoclinic degenerate weak compact-support
solutions that are relevant to finite domain boundary value
problems and localized impurities in infinite layers. ©
1998 Elsevier Science Ltd. AU rights reserved.},
Doi = {10.1016/S0893-9659(98)00092-5},
Key = {fds244206}
}
@article{fds244207,
Author = {Bernoff, AJ and Bertozzi, AL and Witelski, TP},
Title = {Axisymmetric surface diffusion: Dynamics and stability of
self-similar pinchoff},
Journal = {Journal of Statistical Physics},
Volume = {93},
Number = {3-4},
Pages = {725-776},
Publisher = {Springer Nature},
Year = {1998},
Month = {January},
url = {http://www.math.duke.edu/~witelski/articles/surfdiff.ps.gz},
Abstract = {The dynamics of surface diffusion describes the motion of a
surface with its normal velocity given by the surface
Laplacian of its mean curvature. This flow conserves the
volume enclosed inside the surface while minimizing its
surface area. We review the axisymmetric equilibria: the
cylinder, sphere, and the Delaunay unduloid. The sphere is
stable, while the cylinder is long-wave unstable. A
subcritical bifurcation from the cylinder produces a
continuous family of unduloid solutions. We present
computations that suggest that the stable manifold of the
unduloid forms a separatrix between states that relax to the
cylinder in infinite time and those that tend toward
finite-time pinchoff. We examine the structure of the
pinchoff, showing it has self-similar structure, using
asymptotic, numerical, and analytical methods. In addition
to a previously known similarity solution, we find a
countable set of similarity solutions, each with a different
asymptotic cone angle. We develop a stability theory in
similarity variables that selects the original similarity
solution as the only linearly stable one and consequently
the only observable solution. We also consider similarity
solutions describing the dynamics after the topological
transition.},
Doi = {10.1023/b:joss.0000033251.81126.af},
Key = {fds244207}
}
@article{fds244208,
Author = {Brenner, MP and Witelski, TP},
Title = {On spherically symmetric gravitational collapse},
Journal = {Journal of Statistical Physics},
Volume = {93},
Number = {3-4},
Pages = {863-899},
Publisher = {Springer Nature},
Year = {1998},
Month = {January},
url = {http://www.math.duke.edu/~witelski/articles/grav.ps.gz},
Abstract = {This paper considers the dynamics of a classical problem in
astrophysics, the behavior of spherically symmetric
gravitational collapse starting from a uniform, density
cloud of interstellar gas. Previous work on this problem
proposed a universal self-similar solution for the collapse
yielding a collapsed mass much smaller than the mass
contained in the initial cloud. This paper demonstrates the
existence of a second threshold not far above the marginal
collapse threshold - above which the asymptotic collapse is
not universal. In this regime, small changes in the initial
data or weak stochastic forcing leads to qualitatively
different collapse dynamics. In the absence of
instabilities, a progressing wave solution yields a
collapsed uniform core with infinite density. Under some
conditions the instabilities ultimately lead to the
well-known self-similar dynamics. However, other
instabilities can cause the density profile to become
non-monotone and produce a shock in the velocity. In
presenting these results, we outline pitfalls of numerical
schemes that can arise when computing collapse.},
Doi = {10.1023/b:joss.0000033167.19114.b8},
Key = {fds244208}
}
@article{fds244201,
Author = {Witelski, TP},
Title = {Similarity solutions of the lubrication equation},
Journal = {Applied Mathematics Letters},
Volume = {10},
Number = {5},
Pages = {107-113},
Publisher = {Elsevier BV},
Year = {1997},
Month = {September},
url = {http://www.math.duke.edu/~witelski/articles/amslube.ps.gz},
Abstract = {We present a method for constructing closed-form similarity
solutions of the fourth-order nonlinear lubrication
equation. By extending a technique used to study
second-order degenerate diffusion problems, corresponding
interface profiles and diffusion coefficient functions can
be derived in exact form. Different classes of spreading and
shrinking solutions are obtained using this
approach.},
Doi = {10.1016/S0893-9659(97)00092-X},
Key = {fds244201}
}
@article{fds244199,
Author = {Witelski, TP},
Title = {Segregation and mixing in degenerate diffusion in population
dynamics},
Journal = {Journal of Mathematical Biology},
Volume = {35},
Number = {6},
Pages = {695-712},
Publisher = {Springer Nature},
Year = {1997},
Month = {January},
ISSN = {0303-6812},
url = {http://www.math.duke.edu/~witelski/articles/jmbseg.ps.gz},
Abstract = {We study the qualitative properties of degenerate diffusion
equations used to describe dispersal processes in population
dynamics. For systems of interacting populations, the forms
of the diffusion models used determine if the population
will intermix or remain disjoint (segregated). The dynamics
and stability of segregation boundaries between competing
populations is analyzed. General population models with
segregation and mixing interactions are derived and
connections to behavior in fluid mechanical systems are
addressed. © Springer-Verlag 1997.},
Doi = {10.1007/s002850050072},
Key = {fds244199}
}
@article{fds244200,
Author = {Witelski, TP},
Title = {Perturbation Analysis for Wetting Fronts in Richards'
Equation},
Journal = {Transport in Porous Media},
Volume = {27},
Number = {2},
Pages = {121-134},
Year = {1997},
Month = {January},
url = {http://www.math.duke.edu/~witelski/articles/richards.ps.gz},
Abstract = {Perturbation methods are used to study the interaction of
wetting fronts with impervious boundaries in layered soils.
Solutions of Richards' equation for horizontal and vertical
infiltration problems are considered. Asymptotically
accurate solutions are constructed from outer solutions and
boundary-layer corrections. Results are compared with
numerical simulations to demonstrate a high level of
accuracy.},
Doi = {10.1023/A:1006513009125},
Key = {fds244200}
}
@article{fds244197,
Author = {Witelski, TP},
Title = {Traveling wave solutions for case II diffusion in
polymers},
Journal = {Journal of Polymer Science, Part B: Polymer
Physics},
Volume = {34},
Number = {1},
Pages = {141-150},
Publisher = {WILEY},
Year = {1996},
Month = {January},
url = {http://www.math.duke.edu/~witelski/articles/case2.ps.gz},
Abstract = {Case II diffusion of penetrant liquids in polymer films is
characterized by constant-velocity propagation of a phase
interface. We review the development of viscoelastic models
describing case II diffusion and then present a phase plane
analysis for traveling wave solutions. For simplified,
piecewise-constant coefficient models we give closed-form
analytic solutions showing the dependence on various
physical parameters in both viscous and viscoelastic
diffusive systems. We will also compare the results of our
analysis with results from numerical simulations of more
general models. © 1996 John Wiley & Sons,
Inc.},
Doi = {10.1002/(SICI)1099-0488(19960115)34:1<141::AID-POLB12>3.0.CO},
Key = {fds244197}
}
@article{fds244196,
Author = {Witelski, TP},
Title = {The structure of internal layers for unstable nonlinear
diffusion equations},
Journal = {Studies in Applied Mathematics},
Volume = {97},
Number = {3},
Pages = {277-300},
Publisher = {WILEY},
Year = {1996},
Month = {January},
url = {http://www.math.duke.edu/~witelski/articles/vch.ps.gz},
Abstract = {We study the structure of diffusive layers in solutions of
unstable nonlinear diffusion equations. These equations are
regularizations of the forward-backward heat equation and
have diffusion coefficients that become negative. Such
models include the Cahn-Hilliard equation and the
pseudoparabolic viscous diffusion equation. Using singular
perturbation methods we show that the balance between
diffusion and higher-order regularization terms uniquely
determines the interface structure in these equations. It is
shown that the well-known "equal area" rule for the
Cahn-Hilliard equation is a special case of a more general
rule for shock construction in the viscous Cahn-Hilliard
equation.},
Doi = {10.1002/sapm1996973277},
Key = {fds244196}
}
@article{fds244198,
Author = {Cohen, DS and Witelski, TP},
Title = {Inaccessible states in time-dependent reaction
diffusion},
Journal = {Studies in Applied Mathematics},
Volume = {97},
Number = {4},
Pages = {301-319},
Publisher = {WILEY},
Year = {1996},
Month = {January},
url = {http://dx.doi.org/10.1002/sapm1996974301},
Abstract = {Using asymptotic methods we show that the long-time dynamic
behavior in certain systems of nonlinear parabolic
differential equations is described by a time-dependent,
spatially inhomogeneous nonlinear evolution equation. For
problems with multiple stable states, the solution develops
sharp fronts separating slowly varying regions. By studying
the basins of attraction of Abel's nonlinear differential
equation, we demonstrate that the presence of explicit time
dependence in the asymptotic evolution equation creates
"forbidden regions" where the existence of interfaces is
excluded. Consequently, certain configurations of stable
states in the nonlinear system become inaccessible and
cannot be achieved from any set of real initial
conditions.},
Doi = {10.1002/sapm1996974301},
Key = {fds244198}
}
@article{fds244191,
Author = {Witelski, TP},
Title = {Stopping and merging problems for the porous media
equation},
Journal = {IMA Journal of Applied Mathematics (Institute of Mathematics
and Its Applications)},
Volume = {54},
Number = {3},
Pages = {227-243},
Publisher = {Oxford University Press (OUP)},
Year = {1995},
Month = {December},
ISSN = {0272-4960},
url = {http://www.math.duke.edu/~witelski/articles/porous.ps.gz},
Abstract = {A class of boundary value problems for nonlinear diffusion
equations is studied. Using singular perturbation theory and
matched asymptotic expansions, the author analyses the
interactions of compact-support solutions of the porous
media equation with fixed boundaries and with other
solutions. The boundary layer analysis yields results on how
'stopping' and 'merging' disturbances at the interface
propagate back into the solution. Analysis is also extended
to cover merging problems for the fourth-order lubrication
equation. © 1995 Oxford University Press.},
Doi = {10.1093/imamat/54.3.227},
Key = {fds244191}
}
@article{fds244195,
Author = {Witelski, TP and Cohen, DS},
Title = {Perturbed reversible systems},
Journal = {Physics Letters A},
Volume = {207},
Number = {1-2},
Pages = {83-86},
Publisher = {Elsevier BV},
Year = {1995},
Month = {October},
ISSN = {0375-9601},
url = {http://dx.doi.org/10.1016/0375-9601(95)00662-M},
Abstract = {For a class of nonlinear evolution equations describing
reversible processes with several equilibrium solutions, we
will demonstrate that the addition of time-dependent
disturbances can significantly change the stability
properties of the model. In particular, we will show that
the introduction of bounded time-dependent forcing can cause
singular changes in the basins of attraction for Abel's
nonlinear ordinary differential equation. ©
1995.},
Doi = {10.1016/0375-9601(95)00662-M},
Key = {fds244195}
}
@article{fds244185,
Author = {Witelski, TP and Cohen, DS},
Title = {Forbidden Regions for Shock Formation in Diffusive
Systems},
Journal = {Studies in Applied Mathematics},
Volume = {95},
Number = {3},
Pages = {297-317},
Publisher = {WILEY},
Year = {1995},
Month = {October},
ISSN = {0022-2526},
url = {http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:A1995RY07600004&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=47d3190e77e5a3a53558812f597b0b92},
Abstract = {We consider an initial-boundary value problem for a
nonlinear parabolic system. Using perturbation methods, this
problem is reduced to one of considering an evolution
equation for the long-time asymptotics of the system. This
model can be related to the leading order approximation for
a spatially inhomogeneous reaction-diffusion system with
time-dependent forcing. The evolution equation yields
solutions with steady state shocks. We study some of the
subtle effects introduced by time-dependent forcing. Most
significant among these effects is the creation of
"forbidden regions" where stationary shocks cannot form.
Results are presented for bi- and tri-stable one-dimensional
models as well as multidimensional systems.},
Doi = {10.1002/sapm1995953297},
Key = {fds244185}
}
@article{fds244192,
Author = {Cohen, DS and White, AB and Witelski, TP},
Title = {Shock formation in a multidimensional viscoelastic diffusive
system},
Journal = {SIAM Journal on Applied Mathematics},
Volume = {55},
Number = {2},
Pages = {348-368},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {1995},
Month = {January},
url = {http://www.math.duke.edu/~witelski/articles/vemshock.ps.gz},
Abstract = {We examine a model for non-Fickian 'sorption overshoot'
behavior in diffusive polymer-penetrant systems. The
equations of motion proposed by Cohen and White [SIAM J.
Appl. Math., 51 (1991), pp. 472-483] are solved for
two-dimensional problems using matched asymptotic
expansions. The phenomenon of shock formation predicted by
the model is examined and contrasted with similar behavior
in classical reaction-diffusion systems. Mass uptake curves
produced by the model are examined and shown to compare
favorably with experimental observations.},
Doi = {10.1137/S0036139993269333},
Key = {fds244192}
}
@article{fds244193,
Author = {Witelski, TP},
Title = {Merging traveling waves for the porous-Fisher's
equation},
Journal = {Applied Mathematics Letters},
Volume = {8},
Number = {4},
Pages = {57-62},
Publisher = {Elsevier BV},
Year = {1995},
Month = {January},
ISSN = {0893-9659},
url = {http://dx.doi.org/10.1016/0893-9659(95)00047-T},
Abstract = {We study a reaction-diffusion equation model for population
dynamics. By focusing on the diffusive behavior expected in
a population that seeks to avoid over-crowding, we derive a
nonlinear-diffusion porous-Fisher's equation. Using explicit
traveling wave solutions, initially-separated, expanding
populations are studied as they first coalesce. The
nonlinear interactions of the merging populations are
examined using perturbation theory and the method of matched
asymptotic expansions. Results are also extended to the
axisymmetric case. © 1995.},
Doi = {10.1016/0893-9659(95)00047-T},
Key = {fds244193}
}
@article{fds244194,
Author = {Witelski, TP},
Title = {Shocks in nonlinear diffusion},
Journal = {Applied Mathematics Letters},
Volume = {8},
Number = {5},
Pages = {27-32},
Publisher = {Elsevier BV},
Year = {1995},
Month = {January},
ISSN = {0893-9659},
url = {http://www.math.duke.edu/~witelski/articles/shock.ps.gz},
Abstract = {Using two models that incorporate a nonlinear
forward-backward heat equation, we demonstrate the existence
of well-defined weak solutions containing shocks for
diffusive problems. Occurrence of shocks is connected to
multivalued inverse solutions and nonmonotone potential
functions. Unique viscous solutions are determined from
perturbation theory by matching to a shock layer condition.
Results of direct numerical simulations are also discussed.
© 1995.},
Doi = {10.1016/0893-9659(95)00062-U},
Key = {fds244194}
}
@article{fds244190,
Author = {Witelski, TP},
Title = {An asymptotic solution for traveling waves of a
nonlinear-diffusion Fisher's equation},
Journal = {Journal of Mathematical Biology},
Volume = {33},
Number = {1},
Pages = {1-16},
Publisher = {Springer Nature},
Year = {1994},
Month = {November},
ISSN = {0303-6812},
url = {http://www.math.duke.edu/~witelski/articles/jmbfish.ps.gz},
Abstract = {We examine traveling-wave solutions for a generalized
nonlinear-diffusion Fisher equation studied by Hayes [J.
Math. Biol. 29, 531-537 (1991)]. The density-dependent
diffusion coefficient used is motivated by certain polymer
diffusion and population dispersal problems. Approximate
solutions are constructed using asymptotic expansions. We
find that the solution will have a corner layer (a shock in
the derivative) as the diffusion coefficient approaches a
step function. The corner layer at z = 0 is matched to an
outer solution for z < 0 and a boundary layer for z > 0 to
produce a complete solution. We show that this model also
admits a new class of nonphysical solutions and obtain
conditions that restrict the set of valid traveling-wave
solutions. © 1994 Springer-Verlag.},
Doi = {10.1007/BF00160171},
Key = {fds244190}
}
@article{fds244228,
Author = {Witelski, T and Ng, P and Ying, J and Jundy, J and Bove,
J},
Title = {An application of pattern recognition and infrared
spectroscopy to water analysis},
Journal = {International Journal of Environmental Analytical
Chemistry},
Volume = {44},
Number = {2},
Pages = {127-136},
Publisher = {Informa UK Limited},
Year = {1991},
ISSN = {0306-7319},
url = {http://dx.doi.org/10.1080/03067319108027542},
Abstract = {A mathematical methodology is presented that shows potential
for the interpretation of infrared spectra through a
technique of pattern recognition. A number of water samples
and simple alkanes were studied to examine the sensitivity
and discriminating qualities of the methodology. With 2500
comparisons each of tap water versus tap water or distilled
water versus tap water, the technique performed well in
selecting the targeted reagent. The same is true for the
comparisons of n-pentane, as the window compound, versus
n-heptane, n-octane, n-nonane and n-decane. Comparisons of
these n-alkanes to n-pentane gave fitting tolerances of
15.3, 21.2, 20.9 and 24.7%, respectively. When n-pentane was
compared to itself, the tolerance fit was 2.7 %, showing the
ease of discrimination. These results suggest that this
computer-aided phase space transformation method is
sensitive, offers good analytical precision, and is capable
of detecting small differences in the infrared spectra of
compounds and mixture studied. Preliminary data also suggest
that the method has potential for monitoring the quality of
water. © 1991, Taylor & Francis Group, LLC. All rights
reserved.},
Doi = {10.1080/03067319108027542},
Key = {fds244228}
}
@article{fds10239,
Author = {Ferdinand Hendriks and Thomas P. Witelski and et
al},
Title = {Shape optimization of pressurized air bearings},
Journal = {Proceedings 2001 Workshop on Mathematical Problems in
Industry, RPI},
Abstract = {Group project report edited and compiled by
TPW.},
Key = {fds10239}
}
@article{fds9561,
Author = {F. Hendriks and Thomas P. Witelski and et al},
Title = {Design of planar coils of minimum resistance for magnetic
recording devices},
Journal = {Proceedings of the Workshop on Mathematical Problems in
Industry, University of Delaware, 1999},
url = {http://www.math.duke.edu/~witelski/local/spiral.ps.gz},
Key = {fds9561}
}
@article{fds8730,
Author = {W. L. Hogarth and J. Y. Parlange and Thomas P Witelski},
Title = {The superposition principle for infiltration with power law
diffusivity},
Journal = {Hydrology Days - Proceedings of the 17th Annual American
Geophysical Union, pp. 365-374 (1997)},
Key = {fds8730}
}
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