Publications of Thomas P Witelski :chronological combined listing:
%% Papers Published
@article{fds158293,
Author = {T.P. Witelski},
Title = {The subtle art of blowing bubbles (News and Views: Fluid
Dynamics)},
Journal = {Nature Physics},
Volume = {5},
Pages = {315-316},
Year = {2009},
Month = {May},
url = {http://links.ealert.nature.com/ctt?kn=65&m=32736150&r=MTc2NjI2MDg2NwS2&b=2&j},
Key = {fds158293}
}
@article{fds154256,
Author = {H.-J. Hwang and T.P. Witelski},
Title = {Short-time Pattern formation in thin film
equations},
Journal = {Discrete and Continuous Dynamical Systems
A},
Volume = {23},
Number = {3},
Pages = {867-885},
Year = {2009},
Month = {March},
ISSN = {1078-0947},
url = {http://aimsciences.org/journals/displayArticles.jsp?paperID=3829},
Key = {fds154256}
}
@article{fds165136,
Author = {M. B. Gratton and T.P. Witelski},
Title = {Transient and self-similar dynamics in thin film
coarsening},
Journal = {Physica D},
Volume = {238},
Number = {23-24},
Pages = {2380-2394},
Year = {2009},
Key = {fds165136}
}
@article{fds157908,
Author = {L. B. Smolka and T. P. Witelski},
Title = {On the planar extensional motion of an inertially driven
liquid sheet},
Journal = {Physics of Fluids},
Volume = {21},
Number = {4},
Pages = {042101},
Year = {2009},
url = {http://link.aip.org/link/?PHF/21/042101},
Key = {fds157908}
}
@article{fds157864,
Author = {T.P. Witelski and M. Bowen},
Title = {Singular perturbation theory},
Journal = {Scholarpedia},
Volume = {4},
Number = {4},
Pages = {3951},
Year = {2009},
url = {http://www.scholarpedia.org/article/Singular_perturbation_theory},
Key = {fds157864}
}
@article{fds159830,
Author = {A.J.Bernoff and T.P. Witelski},
Title = {Stability and dynamics of self-similarity in evolution
equations},
Journal = {Journal of Engineering Mathematics},
Year = {2009},
ISSN = {1573-2703},
url = {http://dx.doi.org/10.1007/s10665-009-9309-8},
Key = {fds159830}
}
@article{fds145237,
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},
Year = {2008},
Month = {February},
ISSN = {0043-1397},
url = {DOI: 10.1029/2007WR005975},
Key = {fds145237}
}
@article{fds151811,
Author = {M. Aguareles and S. J. Chapman and T.P. Witelski},
Title = {Interaction of spiral waves in the Complex Ginzburg-Landau
equation},
Journal = {PRL},
Volume = {101},
Number = {224101},
Year = {2008},
url = {http://link.aps.org/abstract/PRL/v101/e224101},
Key = {fds151811}
}
@article{fds148596,
Author = {S. T. Santillian and R. H. Plaut and T. P. Witelski and L. N.
Virgin},
Title = {Large oscillations of beams and columns including
self-weight},
Journal = {International Journal of Nonlinear mechanics},
Volume = {43},
Pages = {761-771},
Year = {2008},
ISSN = {0020-7462},
url = {doi:10.1016/j.ijnonlinmec.2008.04.007},
Key = {fds148596}
}
@article{fds141438,
Author = {M.B. Gratton and T.P. Witelski},
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 = {fds141438}
}
@article{fds148304,
Author = {A.J. Catlla and D.G. Schaeffer and T.P. Witelski and E. E. Monson and A.
L. Lin},
Title = {On spiking models for synaptic activity and impulsive
differential equations},
Journal = {SIAM Review},
Volume = {50},
Number = {3},
Pages = {553-569},
Year = {2008},
url = {http://link.aip.org/link/?SIR/50/553},
Key = {fds148304}
}
@article{fds49715,
Author = {David G. Schaeffer and Michael Shearer and T.P.
Witelski},
Title = {Boundary-value problems for hyperbolic PDE related to steady
granular flow},
Journal = {Mathematics and Mechanics of Solids},
Volume = {12},
Number = {6},
Pages = {665-699},
Year = {2007},
Key = {fds49715}
}
@article{fds139614,
Author = {R. Levy,M. Shearer and T.P. Witelski},
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},
Year = {2006},
Key = {fds139614}
}
@article{fds53982,
Author = {T.P. Witelski and R. Levy and M. Shearer},
Title = {Growing surfactant waves in thin liquid films driven by
gravity},
Journal = {Applied Mathematics Research Express},
Volume = {2006},
Number = {15487},
Pages = {1-21},
Year = {2006},
Key = {fds53982}
}
@article{fds48367,
Author = {Mark Bowen and Thomas P. Witelski},
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},
Year = {2006},
Key = {fds48367}
}
@article{fds47572,
Author = {A. Munch and B. Wagner and T.P. Witelski},
Title = {Lubrication models with small to large slip
lengths},
Journal = {Journal of Engineering Mathematics},
Volume = {53},
Number = {3-4},
Pages = {259-283},
Year = {2005},
Month = {December},
url = {http://www.springerlink.com/(gpximmrigyvfihbxa2cbda45)/app/home/contribution.asp?referrer=parent&backto=issue,11,12;journal,3,197;linkingpublicationresults,1:100287,1},
Key = {fds47572}
}
@article{fds47573,
Author = {T.P. Witelski and S.W. Rienstra},
Title = {Introduction to Practical Asymptotics III},
Journal = {Journal of Engineering Mathematics},
Volume = {53},
Number = {3-4},
Pages = {199-199},
Year = {2005},
Month = {December},
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.},
Key = {fds47573}
}
@article{fds42870,
Author = {R. Fetzer and K. Jacobs and A.Munch, B. Wagner and T.P.
Witelski},
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},
Key = {fds42870}
}
@article{fds40649,
Author = {R. P. Haskett and T. P. Witelski and J. Sur},
Title = {Localized Marangoni forcing in driven thin
films},
Journal = {Physica D},
Volume = {209},
Number = {1-4},
Pages = {117-134},
Year = {2005},
Month = {September},
url = {http://dx.doi.org/10.1016/j.physd.2005.06.019},
Key = {fds40649}
}
@article{fds40650,
Author = {K. B. Glasner and T.P. Witelski},
Title = {Collision vs. collapse of droplets in coarsening of
dewetting thin films},
Journal = {Physica D},
Volume = {209},
Number = {1-4},
Pages = {80-104},
Year = {2005},
Month = {September},
url = {http://dx.doi.org/10.1016/j.physd.2005.06.010},
Key = {fds40650}
}
@article{fds37311,
Author = {L. B. Smolka and A. Belmonte and D. M. Henderson and T.P.
Witelski},
Title = {Exact solution for the extensional flow of a viscoelastic
filament},
Journal = {European Journal of Applied Mathematics},
Volume = {15},
Number = {6},
Pages = {679-712},
Year = {2005},
Abstract = {We derive an exact solution for a purely
extensionalcylindrical filament of viscoelastic fluid that
satisfies both the upper convected Maxwell and Oldroyd-B
equations. The resulting prediction of decreasing filament
thickness agrees with our experimental measurements for
semi-dilute polymer solutions. In the limit of time to
infinity, the exact solution approaches that for a Newtonian
fluid.},
Key = {fds37311}
}
@article{fds41205,
Author = {T.P. Witelski},
Title = {Motion of wetting fronts moving into partially pre-wet
soil},
Journal = {Advances in Water Resources},
Volume = {28},
Number = {10},
Pages = {1133-1141},
Year = {2005},
Key = {fds41205}
}
@article{fds28645,
Author = {Jeanman Sur and T. P. Witelski and R. P.
Behringer},
Title = {Steady-Profile Fingering Flows in Marangoni Driven Thin
Films},
Journal = {Physical Review Letters},
Volume = {93},
Number = {24},
Pages = {7803},
Year = {2004},
Month = {December},
url = {http://link.aps.org/abstract/PRL/v93/e247803},
Key = {fds28645}
}
@article{fds26415,
Author = {L. J. Borucki and Thomas P. Witelski and C. P. Please and P. Kramer and D. Schwendeman},
Title = {A theory of pad conditioning for chemical-mechanical
polishing},
Journal = {Journal of Engineering Mathematics},
Volume = {50},
Number = {1},
Pages = {1-24},
Year = {2004},
Month = {September},
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 time-dependent development 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. 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
diamonds used in the conditioner and on tool operating
parameters. Good agreement is found with Monte Carlo
simulations and with experimental data.},
Key = {fds26415}
}
@article{fds22790,
Author = {Thomas P. Witelski and Andrew J. Bernoff and Andrea L.
Bertozzi},
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},
Year = {2004},
Month = {April},
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.},
Key = {fds22790}
}
@article{fds13478,
Author = {T.P. Witelski (guest},
Title = {Nonlinear Differential Equations, Mechanics and
Bifurcation},
Journal = {Discrete and Continuous Dynamical Systems: Series
B},
Volume = {3},
Number = {4},
Year = {2003},
Month = {November},
url = {http://aimsciences.org/journals/dcdsB/B3_4.htm},
Key = {fds13478}
}
@article{fds14008,
Author = {D. G. Schaeffer and M. Shearer and Thomas P.
Witelski},
Title = {One-dimensional solutions in an elastoplasticity model of
granular materials},
Journal = {Mathematical Models and Methods in Applied
Sciences},
Volume = {13},
Number = {11},
Pages = {1629-1671},
Year = {2003},
Month = {November},
url = {http://www.worldscinet.com/m3as/13/1311/S0218202503003069.html},
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
${\calE}$. For large and moderate values of ${\calE}$, the
model has stable steady-state solutions with uniform
shearing except for one shear band; indeed, almost all
solutions tend to one of these as $t \to \infty$. However,
when ${\calE}$ becomes sufficiently small, the
single-shear-band solutions lose stability through a Hopf
bifurcation. The value of ${\calE}$ 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,
solutions containing both elastic and plastic waves, and
(ii)~the bifurcation diagram representing these solutions
exhibits bi-stability.},
Key = {fds14008}
}
@article{fds10426,
Author = {Karl B. Glasner and Thomas P. Witelski},
Title = {Coarsening dynamics of dewetting films},
Journal = {Physical Review E},
Volume = {67},
Pages = {016302},
Year = {2003},
url = {http://ojps.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=PLEEE8000067000001016302000001&idtype=cvips&gifs=Yes},
Abstract = {Many thin fluid films are subject to instabilities caused by
a competition of short and long range intermolecular forces.
After breaking into droplets connected by an ultra-thin
film, the fluid will undergo a coarsening process in which
droplets both move and exchange mass on slow timescales. In
the context of a one-dimensional lubrication model, the slow
dynamics can be characterized in terms of a finite
dimensional set of evolution equations. From this, a scaling
law which governs the coarsening rate is
derived.},
Key = {fds10426}
}
@article{fds10442,
Author = {Thomas P. Witelski},
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},
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
states.},
Key = {fds10442}
}
@article{fds10446,
Author = {Thomas P. Witelski and M. Bowen},
Title = {ADI schemes for fourth-order nonlinear diffusion
equations},
Journal = {Applied Numerical Mathematics},
Volume = {45},
Number = {2-3},
Pages = {331-351},
Year = {2003},
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
for the solution of nonlinear problems. Problems in the
fluid dynamics of thin films are solved to demonstrate the
effectiveness of the ADI schemes.},
Key = {fds10446}
}
@article{fds10321,
Author = {Thomas P. Witelski},
Title = {Computing finite-time singularities in interfacial
flows},
Journal = {Modern Methods in Scientific Computing and Applications
(NATO ASI series II proceedings, volume 75), 2002, pp.
451-487.},
url = {http://www.wkap.nl/prod/b/1-4020-0782-5},
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 = {fds10321}
}
@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{fds10221,
Author = {A. J. Bernoff and Thomas P. Witelski},
Title = {Linear stability of source-type similarity solutions of the
thin film equation},
Journal = {Applied Mathematics Letters 18 (2002) pp.
599--606.},
Abstract = {We study the stability of compactly-supported source-type
self-similar solutions of the generalized lubrication
equation $h_t=-(h^nh_{xxx})_x$. 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.},
Key = {fds10221}
}
@article{fds10135,
Author = {Thomas P. Witelski and D. G. Schaeffer and M.
Shearer},
Title = {A Discrete Model for an Ill-posed Nonlinear Parabolic
PDE},
Journal = {Physica D, 160 (2001) pp. 189--221.},
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.},
Key = {fds10135}
}
@article{fds9864,
Author = {A. L. Bertozzi and G. Grun and Thomas P. Witelski},
Title = {Dewetting films: bifurcations and concentrations},
Journal = {Nonlinearity, 14 (2001) pp 1569--1592},
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 in 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 behavior 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 $\delta$-distributions.},
Key = {fds9864}
}
@article{fds9832,
Author = {D. Vaynblat and J. R. Lister and Thomas P. Witelski},
Title = {Symmetry and self-similarity in rupture and pinchoff:A
geometric bifurcation},
Journal = {European Journal of Applied Mathematics (2001) 12, 3, pp.
209-232.},
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.},
Key = {fds9832}
}
@article{fds9802,
Author = {D. Vaynblat and J. R. Lister and Thomas P. Witelski},
Title = {Rupture of Thin Viscous Films by van der Waals Forces:
Evolution and Self-similarity},
Journal = {Physics of Fluids 13, 5 (2001) 1130-1140.},
url = {http://ojps.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=PHFLE6000013000005001130000001&idtype=cvips&gifs=Yes},
Abstract = {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 $(\ts-t)^{1/3}$ and the lateral
lengthscale like $(\ts-t)^{1/2}$, where $\ts-t$ is the time
remaining until rupture. In each geometry these scalings are
confirmed by numerical simulations of the time-dependent
behaviour, 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.},
Key = {fds9802}
}
@article{fds9673,
Author = {Thomas P. Witelski and K. Ono and T. J. Kaper},
Title = {Analysis of the critical wave speeds of scalar
reaction-diffusion equations},
Journal = {Applied Math Letters, 14/1 (2000) pp. 65-73.},
url = {http://www.math.duke.edu/~witelski/local/woki.ps},
Abstract = {We study the set of traveling waves in a class of
reaction-diffusion equations for the family of potentials
$f_m(U)=2U^m(1-U)$. 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 $m\to 2$ and $m\to\infty$. Also, an
integral formulation of the problem shows that nonuniform
convergence of the generalized equal area rule occurs at the
critical wave speed.},
Key = {fds9673}
}
@article{fds9674,
Author = {Thomas P. Witelski and A. J. Bernoff},
Title = {Three-dimensional van der Waals driven thin film
rupture},
Journal = {Physica D, 147 (1-2), pp. 155--176, (2000).},
url = {http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TVK-41F63R0-11&_user=38557&_coverDate=12%2F01%2F2000&_rdoc=9&_fmt=summary&_orig=browse&_srch=%23toc%235537%232000%23998529998%23214216!&_cdi=5537&_sort=d&_docanchor=&_acct=C000004358&_version=1&_urlVersion=0&_userid=38557&md5=9a20a3a8161050a3b8c3368a20a75fc2},
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 which lead 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 from the initial conditions in
planar and axisymmetric geometries as well as the final
stages of 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.},
Key = {fds9674}
}
@article{fds9634,
Author = {Thomas P. Witelski and K. Ono and T. J. Kaper},
Title = {On Axi-symmetric Traveling waves and Radial solutions of
semi-linear elliptic equations},
Journal = {Natural Resource Modeling 13, 3, 2000, pp.
339-387},
url = {http://www.math.duke.edu/~witelski/local/radial.ps},
Abstract = {Combining analytical techniques from perturbation methods
and dynamical systems theory, we present an elementary
approach to the detailed construction of axi-symmetric
diffusive interfaces in semi-linear elliptic equations.
Solutions of the resulting non-autonomous radial
differential equations can be expressed in terms of a slowly
varying 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 fronts
solutions. These axi-symmetric problems are fundamental
examples of more general curved fronts that arise in a wide
variety of scientific fields, and we extensively discuss 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. Many of the results
contained in this article are known, and in presenting known
results, it is intended that this article be expository in
nature, providing elementary demonstrations 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 radially symmetric solutions of
semilinear elliptic problems, especially to those articles
in which more advanced mathematical theory is
developed.},
Key = {fds9634}
}
@article{fds9121,
Author = {Thomas P. Witelski and Andrew J. Bernoff},
Title = {Stability of self-similar solutions for van der Waals driven
thin film rupture},
Journal = {Physics of Fluids, 11,9, (1999), pp. 2443-2445},
url = {http://ojps.aip.org/journal_cgi/getabs?KEY=PHFLE6&cvips=PHFLE6000011000009002443000001&gifs=Yes},
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 & 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.''},
Key = {fds9121}
}
@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{fds9162,
Author = {Thomas P Witelski and F. Hendriks},
Title = {Stability of Gas Bearing Sliders for Large Bearing Number:
Convective Instability of the Tapered slider},
Journal = {Tribology Transactions, 42, 1, pp. 216-222,
(1999)},
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.},
Key = {fds9162}
}
@article{fds9253,
Author = {Thomas P. Witelski and F. Hendriks},
Title = {Large Bearing Number Stability Analysis for Tango Class Gas
Bearing Sliders},
Journal = {Tribology Transactions, 42 (3), pp. 668-674
(1999)},
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.},
Key = {fds9253}
}
@article{fds8723,
Author = {Thomas P Witelski},
Title = {Dynamics of air bearing sliders},
Journal = {Physics of Fluids, 10, 3, (1998), pp. 698-708},
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 for 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 integro-differential 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.},
Key = {fds8723}
}
@article{fds8724,
Author = {Thomas P Witelski},
Title = {Horizontal infiltration into wet soil},
Journal = {Water Resources Research, 34, 7, pp. 216-222
(1998)},
url = {http://www.math.duke.edu/~witelski/articles/wetsoil.ps.gz},
Key = {fds8724}
}
@article{fds8725,
Author = {Thomas P. Witelski and A. J. Bernoff},
Title = {Self-similar asymptotics for linear and nonlinear diffusion
equations},
Journal = {Studies in Applied Mathematics, 100, pp. 153-193,
(1998)},
url = {http://www.math.duke.edu/~witelski/articles/selfsim.ps.gz},
Key = {fds8725}
}
@article{fds8726,
Author = {A. Y. Grosberg and T. Tanaka and Thomas P Witelski},
Title = {On the properties of polymer globules in the high-density
limit},
Journal = {Journal of Chemical Physics, 108, 21, pp. 9144-9149,
(1998)},
url = {http://ojps.aip.org/journal_cgi/getabs?KEY=JCPSA6&cvips=JCPSA6000108000021009144000001&gifs=Yes},
Key = {fds8726}
}
@article{fds8727,
Author = {A. J. Bernoff and A. L. Bertozzi and Thomas P Witelski},
Title = {Axisymmetric surface diffusion: dynamics and stability of
self-similar pinch-off},
Journal = {Journal of Statistical Physics, 93, 3/4, pp. 725-776,
(1998)},
url = {http://www.math.duke.edu/~witelski/articles/surfdiff.ps.gz},
Key = {fds8727}
}
@article{fds8728,
Author = {M. P. Brenner and Thomas P Witelski},
Title = {On spherically symmetric gravitational collapse},
Journal = {Journal of Statistical Physics, 93, 3/4, pp. 863-900,
(1998)},
url = {http://www.math.duke.edu/~witelski/articles/grav.ps.gz},
Key = {fds8728}
}
@article{fds8729,
Author = {Thomas P Witelski},
Title = {Equilibrium solutions of a degenerate singular Cahn-Hilliard
equation},
Journal = {Applied Mathematics Letters, 11, 5, pp. 127-133,
(1998)},
url = {http://www.math.duke.edu/~witelski/articles/dch.ps.gz},
Key = {fds8729}
}
@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}
}
@article{fds8731,
Author = {Thomas P Witelski},
Title = {Segregation and Mixing in degenerate diffusion in population
dynamics},
Journal = {Journal of Mathematical Biology, 35, pp. 695-712,
(1997)},
url = {http://www.math.duke.edu/~witelski/articles/jmbseg.ps.gz},
Key = {fds8731}
}
@article{fds8732,
Author = {Thomas P Witelski},
Title = {Similarity solutions of the lubrication equation},
Journal = {Applied Mathematics Letters, 10, 5, pp. 107-113,
(1997)},
url = {http://www.math.duke.edu/~witelski/articles/amslube.ps.gz},
Key = {fds8732}
}
@article{fds8733,
Author = {Thomas P Witelski},
Title = {Perturbation analysis for wetting fronts in Richards'
equation},
Journal = {Transport in Porous Media, 27, pp. 121-134,
(1997)},
url = {http://www.math.duke.edu/~witelski/articles/richards.ps.gz},
Key = {fds8733}
}
@article{fds8734,
Author = {D. S. Cohen and Thomas P Witelski},
Title = {Inaccessible states in time-dependent reaction-diffusive
systems},
Journal = {Studies in Applied Mathematics, 97, pp. 301-319,
(1996)},
url = {http://www.math.duke.edu/~witelski/articles/inaccess.ps.gz},
Key = {fds8734}
}
@article{fds8735,
Author = {Thomas P Witelski},
Title = {The structure of internal layers for unstable nonlinear
diffusion equations},
Journal = {Studies in Applied Mathematics, 96, pp. 277-300,
(1996)},
url = {http://www.math.duke.edu/~witelski/articles/vch.ps.gz},
Key = {fds8735}
}
@article{fds8737,
Author = {Thomas P Witelski},
Title = {Traveling wave solutions for Case II diffusion in
polymers},
Journal = {Journal of Polymer Science B: Polymer Physics, 34, pp.
141-150, (1996)},
url = {http://www.math.duke.edu/~witelski/articles/case2.ps.gz},
Key = {fds8737}
}
@article{fds8736,
Author = {Thomas P Witelski},
Title = {Shocks in nonlinear diffusion},
Journal = {Applied Mathematics Letters, 8, 5, pp. 27-32,
(1995)},
url = {http://www.math.duke.edu/~witelski/articles/shock.ps.gz},
Key = {fds8736}
}
@article{fds8738,
Author = {D. S. Cohen and Thomas P Witelski},
Title = {Perturbed Reversible Systems},
Journal = {Physics Letters A, 207, pp. 83-86, (1995)},
Key = {fds8738}
}
@article{fds8739,
Author = {D. S. Cohen and Thomas P Witelski},
Title = {Forbidden regions for shock formation in diffusive
systems},
Journal = {Studies in Applied Mathematics, 95, pp. 297-317,
(1995)},
url = {http://www.math.duke.edu/~witelski/articles/forbidden.ps.gz},
Key = {fds8739}
}
@article{fds8740,
Author = {Thomas P Witelski},
Title = {Merging traveling waves for the pouous-Fisher's
equation},
Journal = {Applied Mathematics Letters, 8, 4, pp. 57-62,
(1995)},
url = {http://www.math.duke.edu/~witelski/articles/amspfish.ps.gz},
Key = {fds8740}
}
@article{fds8741,
Author = {Thomas P Witelski},
Title = {Stopping and merging problems for the porous media
equation},
Journal = {IMA Journal of Applied Mathematics, 54, pp. 227-243,
(1995)},
url = {http://www.math.duke.edu/~witelski/articles/porous.ps.gz},
Key = {fds8741}
}
@article{fds8742,
Author = {D. S. Cohen and A. B. White Jr and Thomas P Witelski},
Title = {Shock formation in a multidimensional viscoelastic diffusive
system},
Journal = {SIAM Journal on Applied Mathematics, 55, 2, pp. 348-368,
(1995)},
url = {http://www.math.duke.edu/~witelski/articles/vemshock.ps.gz},
Key = {fds8742}
}
@article{fds8743,
Author = {Thomas P Witelski},
Title = {An asymptotic solution for traveling waves of a
nonlinear-diffusion Fisher's equation},
Journal = {Journal of Mathematical Biology, 33, pp. 1-16,
(1994)},
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., {\bf 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.},
Key = {fds8743}
}
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Mathematics Department