Journal articleCommunications on Applied Mathematics and Computation · June 1, 2025
In this paper, we investigate the instability of growing tumors by employing both analytical and numerical techniques to validate previous results and extend the analytical findings presented in a prior study by Feng et al. (Z Angew Math Phys 74:107, 2023) ...
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Journal articleComputational 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, ...
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Journal articleSIAM 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 ...
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Journal articlePhysica 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 ...
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Journal articleDiscrete and Continuous Dynamical Systems Series A · March 1, 2023
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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 ...
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Journal articleMathematical 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 ...
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Journal articleScience 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 ...
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Journal articleJournal 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 ...
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Journal articlePhysica 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 ...
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Journal articleEuropean 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 ...
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Journal articleIMA 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 ...
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Journal articleAims 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 ...
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Journal articlePhysical 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 ...
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Journal articleEuropean Journal of Applied Mathematics · April 1, 2019
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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 impos ...
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Journal articleDiscrete 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
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Journal articlePLoS 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 ...
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Journal articlePhysical Review Fluids · February 1, 2018
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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 ...
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Journal articlePhysica 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 ...
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Journal articlePhysica 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 ...
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Journal articleJournal 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 ...
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Journal articleColloids 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 ...
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Journal articleInternational 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, ...
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Journal articlePhysical 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 ...
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Journal articleJournal 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 ...
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Journal articleInternational 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 ...
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Journal articlePhysics 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 ...
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Journal articleApplied 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 ...
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Journal articleInternational 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 ...
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Journal articleBulletin 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 ...
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Journal articlePhysica 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 ...
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Journal articleJournal 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 ...
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Journal articleDiscrete 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 ...
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Journal articlePhysics 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 ...
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Journal articlePhysica 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 ...
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Journal articlePhysical 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 ...
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Journal articleInternational 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 ...
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Journal articleSIAM 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 ...
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Journal articleWater 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 ...
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Journal articlePhysical 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 ...
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Journal articleMathematics 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 ...
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Journal articleEuropean 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 ...
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Journal articleApplied 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 ...
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Journal articleSIAM 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 ...
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Journal articleJournal 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 ...
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Journal articlePhysica 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 ...
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Journal articlePhysica 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 ...
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Journal articlePhysical 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 ...
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Journal articleAdvances 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 ...
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Journal articleJournal 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 ...
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Journal articlePhysical 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 ...
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Journal articleEuropean 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 ...
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Journal articleEuropean 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 ...
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Journal articleMathematical 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 ...
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Journal articleApplied 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 ...
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Journal articleJournal 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 ...
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Journal articlePhysical 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 ...
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Journal articlePhysical 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 ...
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Journal articleApplied 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 variab ...
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Journal articlePhysica 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 ...
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Journal articleEuropean 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 ...
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Journal articleNonlinearity · 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 ...
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Journal articlePhysics 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 ...
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Journal articleApplied 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 ...
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Journal articlePhysica 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 ...
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Journal articleNatural 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 ...
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Journal articleTribology 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 ...
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Journal articleTribology 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 ...
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Journal articlePhysics 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 ...
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Journal articleJournal 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 ...
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Journal articleJournal 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 ...
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Journal articleJournal 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 ...
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Journal articleApplied 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 ...
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Journal articleWater 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 ...
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Journal articlePhysics 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 ...
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Journal articleStudies 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 ...
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Journal articleApplied 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 ...
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Journal articleTransport 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 ...
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Journal articleJournal 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 ...
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Journal articleJournal 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 ...
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Journal articleStudies 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 ...
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Journal articleStudies 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- ...
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Journal articleIMA 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 ...
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Journal articlePhysics 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 ...
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Journal articleStudies 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 ...
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Journal articleApplied 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 ...
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Journal articleApplied 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 ...
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Journal articleSIAM 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 ...
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Journal articleJournal 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 ...
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