Scholars@Duke
Thomas P. Witelski

Thomas P. Witelski

Professor in the Department of Mathematics
Mathematics
witelski@math.duke.edu
295 Physics Building, Box 90320, Durham, NC 27708-0320
120 Science Drive, Durham, NC 27708-0320
Office hours Please email me to request a meeting time  

Selected Publications

IMEX methods for thin-film equations and Cahn–Hilliard equations with variable mobility

Journal Article Computational Materials Science · July 1, 2024 We explore a class of splitting schemes employing implicit-explicit (IMEX) time-stepping to achieve accurate and energy-stable solutions for thin-film equations and Cahn–Hilliard models with variable mobility. These splitting methods incorporate a linear, ... Full text Cite

COARSENING OF THIN FILMS WITH WEAK CONDENSATION

Journal Article SIAM Journal on Applied Mathematics · January 1, 2024 A lubrication model can be used to describe the dynamics of a weakly volatile viscous fluid layer on a hydrophobic substrate. Thin layers of the fluid are unstable to perturbations and break up into slowly evolving interacting droplets. A reduced-order dyn ... Full text Cite

The role of exponential asymptotics and complex singularities in self-similarity, transitions, and branch merging of nonlinear dynamics

Journal Article Physica D: Nonlinear Phenomena · November 1, 2023 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 equat ... Full text Cite

CAUCHY-DIRICHLET PROBLEMS FOR THE POROUS MEDIUM EQUATION

Journal Article Discrete and Continuous Dynamical Systems- Series A · March 1, 2023 Featured Publication 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 intermediat ... Full text Cite

Evaporation and deposition in porous media

Conference · April 6, 2022 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 o ... Full text Cite

Uncovering the dynamics of a circadian-dopamine model influenced by the light-dark cycle.

Journal Article Mathematical biosciences · February 2022 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 extracel ... Full text Cite

Acoustohydrodynamic tweezers via spatial arrangement of streaming vortices.

Journal Article Science advances · January 2021 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 ... Full text Cite

Taylor dispersion in osmotically driven laminar flows in phloem

Journal Article Journal of Fluid Mechanics · January 1, 2021 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 thi ... Full text Open Access Cite

Dynamics of spiral waves in the complex Ginzburg–Landau equation in bounded domains

Journal Article Physica D: Nonlinear Phenomena · December 15, 2020 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 twis ... Full text Open Access Cite

Steady states and dynamics of a thin-film-type equation with non-conserved mass

Journal Article European Journal of Applied Mathematics · December 1, 2020 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 vola ... Full text Open Access Cite

Steady states of thin film droplets on chemically heterogeneous substrates

Journal Article IMA Journal of Applied Mathematics · November 25, 2020 AbstractWe 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 s ... Full text Open Access Cite

Erratum: Obtaining self-similar scalings in focusing flows [Phys. Rev. E 92, 043016 (2015)].

Journal Article Physical review. E · May 2020 This corrects the article DOI: 10.1103/PhysRevE.92.043016. ... Full text Cite

Nonlinear dynamics of dewetting thin films

Journal Article AIMS Mathematics · January 1, 2020 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 ... Full text Cite

Thermal Marangoni-driven dynamics of spinning liquid films

Journal Article Physical Review Fluids · August 19, 2019 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 experime ... Full text Cite

Pressure-dipole solutions of the thin-film equation

Journal Article European Journal of Applied Mathematics · April 1, 2019 Featured Publication 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 ... Full text Cite

A vicinal surface model for epitaxial growth with logarithmic free energy

Journal Article Discrete and Continuous Dynamical Systems - Series B · December 1, 2018 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 ... Full text Cite

Principles that govern competition or co-existence in Rho-GTPase driven polarization.

Journal Article PLoS Comput Biol · April 2018 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 ... Full text Open Access Link to item Cite

Instability and dynamics of volatile thin films

Journal Article Physical Review Fluids · February 1, 2018 Featured Publication 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 ... Full text Cite

Global existence of solutions to a tear film model with locally elevated evaporation rates

Journal Article Physica D: Nonlinear Phenomena · July 1, 2017 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 thickn ... Full text Cite

Finite-time thin film rupture driven by modified evaporative loss

Journal Article Physica D: Nonlinear Phenomena · March 1, 2017 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 e ... Full text Cite

Flow and fouling in a pleated membrane filter

Journal Article Journal of Fluid Mechanics · May 25, 2016 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 u ... Full text Cite

Oil capture from a water surface by a falling sphere

Journal Article Colloids and Surfaces A: Physicochemical and Engineering Aspects · May 20, 2016 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 th ... Full text Cite

Experimental study of regular and chaotic transients in a non-smooth system

Journal Article International Journal of Non-Linear Mechanics · May 1, 2016 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, ... Full text Cite

Preface to the special issue on “Thin films and fluid interfaces”

Journal Article Journal of Engineering Mathematics · October 29, 2015 Full text Cite

Obtaining self-similar scalings in focusing flows.

Journal Article Physical review. E, Statistical, nonlinear, and soft matter physics · October 2015 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 ... Full text Cite

Methods of Mathematical Modelling: Continuous Systems and Differential Equations

Book · September 18, 2015 Featured Publication 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 a ... Full text Cite

A driven system of impacting pendulums: Experiments and simulations

Journal Article Journal of Sound and Vibration · March 17, 2014 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 colli ... Full text Cite

A new model for disturbance waves

Journal Article International Journal of Multiphase Flow · January 1, 2014 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 th ... Full text Cite

Biaxial extensional motion of an inertially driven radially expanding liquid sheet

Journal Article Physics of Fluids · June 1, 2013 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 wit ... Full text Cite

Exponential Asymptotics for Thin Film Rupture.

Journal Article SIAM J. Appl. Math. · 2013 Featured Publication Full text Cite

Anomalous exponents of self-similar blow-up solutions to an aggregation equation in odd dimensions

Journal Article Applied Mathematics Letters · December 1, 2012 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 t ... Full text Cite

Preface

Journal Article Discrete and Continuous Dynamical Systems - Series B · February 2012 Full text Cite

A parametrically forced nonlinear system with reversible equilibria

Journal Article International Journal of Bifurcation and Chaos · January 1, 2012 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 th ... Full text Cite

The effect of polar lipids on tear film dynamics.

Journal Article Bulletin of mathematical biology · June 2011 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 experimentall ... Full text Cite

Motion of spiral waves in the complex Ginzburg-Landau equation

Journal Article Physica D: Nonlinear Phenomena · April 1, 2010 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 ... Full text Cite

Stability and dynamics of self-similarity in evolution equations

Journal Article Journal of Engineering Mathematics · January 1, 2010 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 ... Full text Cite

Singular perturbation theory.

Journal Article Scholarpedia · April 7, 2009 Full text Cite

Short-time pattern formation in thin film equations

Journal Article Discrete and Continuous Dynamical Systems · March 1, 2009 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 t ... Full text Cite

On the planar extensional motion of an inertially driven liquid sheet

Journal Article Physics of Fluids · January 1, 2009 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 ... Full text Cite

Transient and self-similar dynamics in thin film coarsening

Journal Article Physica D: Nonlinear Phenomena · January 1, 2009 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 p ... Full text Cite

Fluid dynamics: The subtle art of blowing bubbles

Journal Article Nature Physics · January 1, 2009 Full text Cite

Stability of traveling waves in thin liquid films driven by gravity and surfactant

Conference HYPERBOLIC PROBLEMS: THEORY, NUMERICS AND APPLICATIONS, PART 2 · January 1, 2009 Link to item Cite

Interaction of spiral waves in the complex Ginzburg-Landau equation.

Journal Article Physical review letters · November 2008 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 ... Full text Cite

Large oscillations of beams and columns including self-weight

Journal Article International Journal of Non-Linear Mechanics · October 1, 2008 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 ... Full text Cite

On spiking models for synaptic activity and impulsive differential equations

Journal Article SIAM Review · September 1, 2008 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 distribution ... Full text Cite

Nonmonotonic traveling wave solutions of infiltration into porous media

Journal Article Water Resources Research · February 1, 2008 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 ... Full text Cite

Coarsening of unstable thin films subject to gravity.

Journal Article Physical review. E, Statistical, nonlinear, and soft matter physics · January 2008 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, pre ... Full text Cite

Boundary-value problems for hyperbolic equations related to steady granular flow

Journal Article Mathematics and Mechanics of Solids · December 1, 2007 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 bo ... Full text Cite

Gravity-driven thin liquid films with insoluble surfactant: Smooth traveling waves

Journal Article European Journal of Applied Mathematics · December 1, 2007 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 ... Full text Cite

Growing surfactant waves in thin liquid films driven by gravity

Journal Article Applied Mathematics Research eXpress · December 1, 2006 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 o ... Full text Cite

The linear limit of the dipole problem for the thin film equation

Journal Article SIAM Journal on Applied Mathematics · October 25, 2006 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 nonli ... Full text Cite

Introduction to practical asymptotics III

Journal Article Journal of Engineering Mathematics · December 1, 2005 Full text Cite

Lubrication models with small to large slip lengths

Journal Article Journal of Engineering Mathematics · December 2005 Link to item Cite

Lubrication models with small to large slip lengths

Journal Article Journal of Engineering Mathematics · December 1, 2005 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 ... Full text Cite

Localized Marangoni forcing in driven thin films

Journal Article Physica D: Nonlinear Phenomena · September 15, 2005 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 dr ... Full text Cite

Collision versus collapse of droplets in coarsening of dewetting thin films

Journal Article Physica D: Nonlinear Phenomena · September 15, 2005 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 mas ... Full text Cite

New slip regimes and the shape of dewetting thin liquid films.

Journal Article Physical review letters · September 2005 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 dif ... Full text Cite

Motion of wetting fronts moving into partially pre-wet soil

Journal Article Advances in Water Resources · January 1, 2005 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 initi ... Full text Cite

A theory of pad conditioning for chemical-mechanical polishing

Journal Article Journal of Engineering Mathematics · December 1, 2004 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 ... Full text Cite

Steady-profile fingering flows in Marangoni driven thin films.

Journal Article Physical review letters · December 2004 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 ... Full text Cite

Exact solution for the extensional flow of a viscoelastic filament

Journal Article European Journal of Applied Mathematics · December 1, 2004 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 fil ... Full text Cite

Blowup and dissipation in a critical-case unstable thin film equation

Journal Article European Journal of Applied Mathematics · April 1, 2004 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 ... Full text Cite

Nonlinear Differential Equations, Mechanics and Bifurcation

Journal Article Discrete and Continuous Dynamical Systems - Series B · November 1, 2003 Cite

Stability of shear bands in an elastoplastic model for granular flow: The role of discreteness

Journal Article Mathematical Models and Methods in Applied Sciences · November 1, 2003 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 der ... Full text Cite

ADI schemes for higher-order nonlinear diffusion equations

Journal Article Applied Numerical Mathematics · May 1, 2003 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 ... Full text Cite

Intermediate asymptotics for Richards' equation in a finite layer

Journal Article Journal of Engineering Mathematics · April 1, 2003 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/Robi ... Full text Cite

Coarsening dynamics of dewetting films.

Journal Article Physical review. E, Statistical, nonlinear, and soft matter physics · January 2003 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 ... Full text Cite

One-dimensional solutions of an elastoplasticity model of granular material

Journal Article Math. Models and Methods in Appl. Sciences · 2003 Cite

Coarsening dynamics of dewetting films

Journal Article Physical Review E - Statistical, Nonlinear, and Soft Matter Physics · January 1, 2003 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 d ... Cite

Linear stability of source-type similarity solutions of the thin film equation

Journal Article Applied Mathematics Letters · January 1, 2002 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 ... Full text Cite

Computing finite-time singularities in interfacial flows

Conference MODERN METHODS IN SCIENTIFIC COMPUTING AND APPLICATIONS · January 1, 2002 Link to item Cite

A discrete model for an ill-posed nonlinear parabolic PDE

Journal Article Physica D: Nonlinear Phenomena · December 15, 2001 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 ... Full text Cite

Symmetry and self-similarity in rupture and pinchoff: A geometric bifurcation

Journal Article European Journal of Applied Mathematics · December 1, 2001 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 independen ... Full text Cite

Dewetting films: Bifurcations and concentrations

Journal Article Nonlinearity · November 1, 2001 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 paraboli ... Full text Cite

Rupture of thin viscous films by van der waals forces: Evolution and self-similarity

Journal Article Physics of Fluids · January 1, 2001 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, ex ... Full text Cite

Critical wave speeds for a family of scalar reaction-diffusion equations

Journal Article Applied Mathematics Letters · January 1, 2001 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 an ... Full text Cite

Dynamics of three-dimensional thin film rupture

Journal Article Physica D: Nonlinear Phenomena · December 1, 2000 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 ... Full text Cite

On axisymmetric traveling waves and radial solutions of semi-linear elliptic equations

Journal Article Natural Resource Modeling · January 1, 2000 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-aut ... Full text Cite

Large bearing number stability analysis for tango class gas bearing sliders

Journal Article Tribology Transactions · January 1, 1999 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 de ... Full text Cite

Stability of gas bearing sliders for large bearing number: Convective instability of the tapered slider©

Journal Article Tribology Transactions · January 1, 1999 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 l ... Full text Cite

Stability of self-similar solutions for van der Waals driven thin film rupture

Journal Article Physics of Fluids · January 1, 1999 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 singularitie ... Full text Cite

On the properties of polymer globules in the high density limit

Journal Article Journal of Chemical Physics · June 1, 1998 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 ... Full text Cite

On spherically symmetric gravitational collapse

Journal Article Journal of Statistical Physics · January 1, 1998 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-simil ... Full text Cite

Axisymmetric surface diffusion: Dynamics and stability of self-similar pinchoff

Journal Article Journal of Statistical Physics · January 1, 1998 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 a ... Full text Cite

Equilibrium interface solutions of a degenerate singular Cahn-Hilliard equation

Journal Article Applied Mathematics Letters · January 1, 1998 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 de ... Full text Cite

Horizontal infiltration into wet soil

Journal Article Water Resources Research · January 1, 1998 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 in ... Full text Cite

Dynamics of air bearing sliders

Journal Article Physics of Fluids · January 1, 1998 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 slider ... Full text Cite

Self-similar asymptotics for linear and nonlinear diffusion equations

Journal Article Studies in Applied Mathematics · January 1, 1998 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 gi ... Full text Cite

Similarity solutions of the lubrication equation

Journal Article Applied Mathematics Letters · September 12, 1997 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 coef ... Full text Cite

Perturbation Analysis for Wetting Fronts in Richards' Equation

Journal Article Transport in Porous Media · January 1, 1997 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 co ... Full text Cite

Segregation and mixing in degenerate diffusion in population dynamics

Journal Article Journal of Mathematical Biology · January 1, 1997 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 r ... Full text Cite

Traveling wave solutions for case II diffusion in polymers

Journal Article Journal of Polymer Science, Part B: Polymer Physics · January 15, 1996 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 travel ... Full text Cite

Inaccessible states in time-dependent reaction diffusion

Journal Article Studies in Applied Mathematics · January 1, 1996 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 ... Full text Cite

The structure of internal layers for unstable nonlinear diffusion equations

Journal Article Studies in Applied Mathematics · January 1, 1996 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- ... Full text Cite

Stopping and merging problems for the porous media equation

Journal Article IMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications) · December 1, 1995 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 ... Full text Cite

Perturbed reversible systems

Journal Article Physics Letters A · October 23, 1995 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 particu ... Full text Cite

Forbidden Regions for Shock Formation in Diffusive Systems

Journal Article Studies in Applied Mathematics · October 1, 1995 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 le ... Full text Cite

Shocks in nonlinear diffusion

Journal Article Applied Mathematics Letters · January 1, 1995 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 nonmo ... Full text Cite

Merging traveling waves for the porous-Fisher's equation

Journal Article Applied Mathematics Letters · January 1, 1995 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 ... Full text Cite

Shock formation in a multidimensional viscoelastic diffusive system

Journal Article SIAM Journal on Applied Mathematics · January 1, 1995 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 ... Full text Cite

An asymptotic solution for traveling waves of a nonlinear-diffusion Fisher's equation

Journal Article Journal of Mathematical Biology · November 1, 1994 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 disper ... Full text Cite

An application of pattern recognition and infrared spectroscopy to water analysis

Journal Article International Journal of Environmental Analytical Chemistry · 1991 Cite