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Todd F. Dupont

Bio: Todd F. Dupont is an academic researcher from University of Chicago. The author has contributed to research in topics: Galerkin method & Parabolic partial differential equation. The author has an hindex of 42, co-authored 92 publications receiving 13542 citations. Previous affiliations of Todd F. Dupont include University of Minnesota & Rice University.


Papers
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Journal ArticleDOI
23 Oct 1997-Nature
TL;DR: In this article, the authors ascribe the characteristic pattern of the deposition to a form of capillary flow in which pinning of the contact line of the drying drop ensures that liquid evaporating from the edge is replenished by liquid from the interior.
Abstract: When a spilled drop of coffee dries on a solid surface, it leaves a dense, ring-like deposit along the perimeter (Fig 1a) The coffee—initially dispersed over the entire drop—becomes concentrated into a tiny fraction of it Such ring deposits are common wherever drops containing dispersed solids evaporate on a surface, and they influence processes such as printing, washing and coating1,2,3,4,5 Ring deposits also provide a potential means to write or deposit a fine pattern onto a surface Here we ascribe the characteristic pattern of the deposition to a form of capillary flow in which pinning of the contact line of the drying drop ensures that liquid evaporating from the edge is replenished by liquid from the interior The resulting outward flow can carry virtually all the dispersed material to the edge This mechanism predicts a distinctive power-law growth of the ring mass with time—a law independent of the particular substrate, carrier fluid or deposited solids We have verified this law by microscopic observations of colloidal fluids

5,553 citations

Journal ArticleDOI
TL;DR: A theory is described that predicts the flow velocity, the rate of growth of the ring, and the distribution of solute within the drop that is driven by the loss of solvent by evaporation and the geometrical constraint that the drop maintain an equilibrium droplet shape with a fixed boundary.
Abstract: Solids dispersed in a drying drop will migrate to the edge of the drop and form a solid ring. This phenomenon produces ringlike stains and occurs for a wide range of surfaces, solvents, and solutes. Here we show that the migration is caused by an outward flow within the drop that is driven by the loss of solvent by evaporation and geometrical constraint that the drop maintain an equilibrium droplet shape with a fixed boundary. We describe a theory that predicts the flow velocity, the rate of growth of the ring, and the distribution of solute within the drop. These predictions are compared with our experimental results.

2,051 citations

Book ChapterDOI
TL;DR: In this paper, a Galerkin method based on C0-piecewise-polynomial spaces was proposed and analyzed for elliptic and parabolic equations, where a penalty on the jump in the normal derivative across the interior edges of elements can produce an apparent stiffness intermediate between C0 and C1.
Abstract: Standard Galerkin methods based on C0-piecewise-polynomial spaces often can lead to unsatisfactory approximations of solutions of problems having dominant transport terms. A penalty on the jump in the normal derivative across the interior edges of elements can produce an apparent stiffness intermediate between C0 and C1, and such a method is proposed and analyzed for elliptic and parabolic equations.

611 citations

Journal ArticleDOI
TL;DR: In this paper, the velocity and radius of a column of axisymmetric fluid with a free surface were derived from the Navier-Strokes equation, where the equations form singularities as the fluid neck is pinching off.
Abstract: We consider the viscous motion of a thin axisymmetric column of fluid with a free surface. A one-dimensional equation of motion for the velocity and the radius is derived from the Navier–Strokes equation. We compare our results with recent experiments on the breakup of a liquid jet and on the bifurcation of a drop suspended from an orifice. The equations form singularities as the fluid neck is pinching off. The nature of the singularities is investigated in detail.

550 citations

Journal ArticleDOI
TL;DR: In this paper, a polynomial plus a remainder is represented as a Taylor series and the remainder can be manipulated in many ways to give different types of bounds, including integer order and nonstandard Sobolev-like spaces.
Abstract: Constructive proofs and several generalizations of approximation results of J. H. Bramble and S. R. Hilbert are presented. Using an averaged Taylor series, we represent a function as a polynomial plus a remainder. The remainder can be manipulated in many ways to give different types of bounds. Approximation of functions in fractional order Sobolev spaces is treated as well as the usual integer order spaces and several nonstandard Sobolev-like spaces.

447 citations


Cited by
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Journal ArticleDOI
TL;DR: Modules for Experiments in Stellar Astrophysics (MESA) as mentioned in this paper is a suite of open source, robust, efficient, thread-safe libraries for a wide range of applications in computational stellar astrophysics.
Abstract: Stellar physics and evolution calculations enable a broad range of research in astrophysics. Modules for Experiments in Stellar Astrophysics (MESA) is a suite of open source, robust, efficient, thread-safe libraries for a wide range of applications in computational stellar astrophysics. A one-dimensional stellar evolution module, MESAstar, combines many of the numerical and physics modules for simulations of a wide range of stellar evolution scenarios ranging from very low mass to massive stars, including advanced evolutionary phases. MESAstar solves the fully coupled structure and composition equations simultaneously. It uses adaptive mesh refinement and sophisticated timestep controls, and supports shared memory parallelism based on OpenMP. State-of-the-art modules provide equation of state, opacity, nuclear reaction rates, element diffusion data, and atmosphere boundary conditions. Each module is constructed as a separate Fortran 95 library with its own explicitly defined public interface to facilitate independent development. Several detailed examples indicate the extensive verification and testing that is continuously performed and demonstrate the wide range of capabilities that MESA possesses. These examples include evolutionary tracks of very low mass stars, brown dwarfs, and gas giant planets to very old ages; the complete evolutionary track of a 1 M ☉ star from the pre-main sequence (PMS) to a cooling white dwarf; the solar sound speed profile; the evolution of intermediate-mass stars through the He-core burning phase and thermal pulses on the He-shell burning asymptotic giant branch phase; the interior structure of slowly pulsating B Stars and Beta Cepheids; the complete evolutionary tracks of massive stars from the PMS to the onset of core collapse; mass transfer from stars undergoing Roche lobe overflow; and the evolution of helium accretion onto a neutron star. MESA can be downloaded from the project Web site (http://mesa.sourceforge.net/).

3,474 citations

Journal ArticleDOI
TL;DR: In this paper, a framework for the analysis of a large class of discontinuous Galerkin methods for second-order elliptic problems is provided, which allows for the understanding and comparison of most of the discontinuous methods that have been proposed over the past three decades.
Abstract: We provide a framework for the analysis of a large class of discontinuous methods for second-order elliptic problems. It allows for the understanding and comparison of most of the discontinuous Galerkin methods that have been proposed over the past three decades for the numerical treatment of elliptic problems.

3,293 citations

Book
01 Jan 1987
TL;DR: In this book, which focuses on the use of iterative methods for solving large sparse systems of linear equations, templates are introduced to meet the needs of both the traditional user and the high-performance specialist.
Abstract: In this book, which focuses on the use of iterative methods for solving large sparse systems of linear equations, templates are introduced to meet the needs of both the traditional user and the high-performance specialist Templates, a description of a general algorithm rather than the executable object or source code more commonly found in a conventional software library, offer whatever degree of customization the user may desire

2,976 citations

Journal ArticleDOI
TL;DR: In this article, a unified mathematical theory is presented that takes advantage of the disparity of the length scales and is based on the asymptotic procedure of reduction of the full set of governing equations and boundary conditions to a simplified, highly nonlinear, evolution equation or to a set of equations.
Abstract: Macroscopic thin liquid films are entities that are important in biophysics, physics, and engineering, as well as in natural settings. They can be composed of common liquids such as water or oil, rheologically complex materials such as polymers solutions or melts, or complex mixtures of phases or components. When the films are subjected to the action of various mechanical, thermal, or structural factors, they display interesting dynamic phenomena such as wave propagation, wave steepening, and development of chaotic responses. Such films can display rupture phenomena creating holes, spreading of fronts, and the development of fingers. In this review a unified mathematical theory is presented that takes advantage of the disparity of the length scales and is based on the asymptotic procedure of reduction of the full set of governing equations and boundary conditions to a simplified, highly nonlinear, evolution equation or to a set of equations. As a result of this long-wave theory, a mathematical system is obtained that does not have the mathematical complexity of the original free-boundary problem but does preserve many of the important features of its physics. The basics of the long-wave theory are explained. If, in addition, the Reynolds number of the flow is not too large, the analogy with Reynolds's theory of lubrication can be drawn. A general nonlinear evolution equation or equations are then derived and various particular cases are considered. Each case contains a discussion of the linear stability properties of the base-state solutions and of the nonlinear spatiotemporal evolution of the interface (and other scalar variables, such as temperature or solute concentration). The cases reducing to a single highly nonlinear evolution equation are first examined. These include: (a) films with constant interfacial shear stress and constant surface tension, (b) films with constant surface tension and gravity only, (c) films with van der Waals (long-range molecular) forces and constant surface tension only, (d) films with thermocapillarity, surface tension, and body force only, (e) films with temperature-dependent physical properties, (f) evaporating/condensing films, (g) films on a thick substrate, (h) films on a horizontal cylinder, and (i) films on a rotating disc. The dynamics of the films with a spatial dependence of the base-state solution are then studied. These include the examples of nonuniform temperature or heat flux at liquid-solid boundaries. Problems which reduce to a set of nonlinear evolution equations are considered next. Those include (a) the dynamics of free liquid films, (b) bounded films with interfacial viscosity, and (c) dynamics of soluble and insoluble surfactants in bounded and free films. The spreading of drops on a solid surface and moving contact lines, including effects of heat and mass transport and van der Waals attractions, are then addressed. Several related topics such as falling films and sheets and Hele-Shaw flows are also briefly discussed. The results discussed give motivation for the development of careful experiments which can be used to test the theories and exhibit new phenomena.

2,689 citations

Book
01 Jan 2000
TL;DR: In this paper, a summary account of the subject of a posteriori error estimation for finite element approximations of problems in mechanics is presented, focusing on methods for linear elliptic boundary value problems.
Abstract: This monograph presents a summary account of the subject of a posteriori error estimation for finite element approximations of problems in mechanics. The study primarily focuses on methods for linear elliptic boundary value problems. However, error estimation for unsymmetrical systems, nonlinear problems, including the Navier-Stokes equations, and indefinite problems, such as represented by the Stokes problem are included. The main thrust is to obtain error estimators for the error measured in the energy norm, but techniques for other norms are also discussed.

2,607 citations