Codimension-two free boundary problems

Over the past 30 years the study of free boundary problems has stimulated much work. However, there exists a widely occurring, but little studied subclass of free boundary problems in which the free boundary has dimension two fewer than that of the underlying space rather than the more commonly studied case of one less. These problems are called `codimension-two' free boundary problems. In Chapter 1 the typical geometries required for such problems, the main mathematical techniques and the methodology used are discussed. Then, in Chapter 2, the techniques required to solve them are demonstrated using the particular example of the water entry problem. Further results for the water entry problem are then derived including an analysis of the relatively poorly understood water exit problem. In Chapter 3 a review is given of some classical contact and crack problems in solid mechanics. The inclusion of a cohesive zone in a dynamic type-III crack problem is considered. The Muskhelishvili potential method is presented and used to solve both a contact and crack problem. This enables the solution of a type-I crack problem relating to an ink delivery system to be found. In Chapter 4 a problem posed by car windscreen forming is addressed. A local solution near a corner is analysed to explain when and how point forces occur at the corners of the frame on which the simply supported windscreen rests. Then the full problem is solved numerically for different types of boundary condition. Chapters 5 and 6 deal with several sintering problems in viscous flow highlighting the value of the methodology introduced in Chapter 1. It will be shown how the Muskhelishvili potential method also carries over to Stokes flow problems. The difficulties of matching to an inner as opposed to an outer region are investigated. Last two interface problems between immiscible liquids are considered which show how the solution procedure is adapted when the field equation in the thin region is non-trivial. In the final chapter results are summarised, open problems listed and conclusions drawn

Error bounds on block Gauss Seidel solutions of coupled\ud multiphysics problems

Mathematical models in many fields often consist of coupled sub–models, each of which describe a different physical process. For many applications, the quantity of interest from these models may be written as a linear functional of the solution to the governing equations. Mature numerical solution techniques for the individual sub–models often exist. Rather than derive a numerical solution technique for the full coupled model, it is therefore natural to investigate whether these techniques may be used by coupling in a block Gauss–Seidel fashion. In this study, we derive two a posteriori bounds for such linear functionals. These bounds may be used on each Gauss–Seidel iteration to estimate the error in the linear functional computed using the single physics solvers, without actually solving the full, coupled problem. We demonstrate the use of the bound first by using a model problem from linear algebra, and then a linear ordinary differential equation example. We then investigate the effectiveness of the bound using a non–linear coupled fluid–temperature problem. One of the bounds derived is very sharp for most linear functionals considered, allowing us to predict very accurately when to terminate our block Gauss–Seidel iteration.\ud \ud Copyright c 2000 John Wiley & Sons, Ltd

Growth of fat slits and dispersionless KP hierarchy

A "fat slit" is a compact domain in the upper half plane bounded by a curve with endpoints on the real axis and a segment of the real axis between them. We consider conformal maps of the upper half plane to the exterior of a fat slit parameterized by harmonic moments of the latter and show that they obey an infinite set of Lax equations for the dispersionless KP hierarchy. Deformation of a fat slit under changing a particular harmonic moment can be treated as a growth process similar to the Laplacian growth of domains in the whole plane. This construction extends the well known link between solutions to the dispersionless KP hierarchy and conformal maps of slit domains in the upper half plane and provides a new, large family of solutions.Comment: 26 pages, 6 figures, typos correcte

Laplacian Growth, Elliptic Growth, and Singularities of the Schwarz Potential

The Schwarz function has played an elegant role in understanding and in generating new examples of exact solutions to the Laplacian growth (or "Hele- Shaw") problem in the plane. The guiding principle in this connection is the fact that "non-physical" singularities in the "oil domain" of the Schwarz function are stationary, and the "physical" singularities obey simple dynamics. We give an elementary proof that the same holds in any number of dimensions for the Schwarz potential, introduced by D. Khavinson and H. S. Shapiro [17] (1989). A generalization is also given for the so-called "elliptic growth" problem by defining a generalized Schwarz potential. New exact solutions are constructed, and we solve inverse problems of describing the driving singularities of a given flow. We demonstrate, by example, how \mathbb{C}^n - techniques can be used to locate the singularity set of the Schwarz potential. One of our methods is to prolong available local extension theorems by constructing "globalizing families". We make three conjectures in potential theory relating to our investigation

BNL Citric Acid Technology: Pilot Scale Demonstration

The objective of this project is to remove toxic metals such as lead and cadmium from incinerator ash using the Citric Acid Process developed at Brookhaven National Laboratory. In this process toxic metals in bottom ash from the incineration of municipal solid waste were first extracted with citric acid followed by biodegradation of the citric acid-metal extract by the bacterium Pseudomonas fluorescens for metals recovery. The ash contained the following metals: Al, As, Ba, Ca, Cd, Cr, Cu, Fe, Mg, Mn, Ni, Pb, Se, Sr, Ti, and Zn. Optimization of the Citric Acid Process parameters which included citric acid molarity, contact time, the impact of mixing aggressiveness during extraction and pretreatment showed lead and cadmium removal from incinerator ash of >90%. Seeding the treated ash with P. fluorescens resulted in the removal of residual citric acid and biostabilization of any leachable lead, thus allowing it to pass EPA?s Toxicity Characteristic Leaching Procedure. Biodegradation of the citric acid extract removed >99% of the lead from the extract as well as other metals such as Al, Ca, Cu, Fe, Mg, Mn, Ti, and Zn. Speciation of the bioprecipitated lead by Extended X-ray Absorption Fine Structure at the National Synchrotron Light Source showed that the lead is predominantly associated with the phosphate and carboxyl functional groups in a stable form. Citric acid was completely recovered (>99%) from the extract by sulfide precipitation technique and the extraction efficiency of recovered citric acid is similar to that of the fresh citric acid. Recycling of the citric acid should result in considerable savings in the overall treatment cost. We have shown the potential application of this technology to remove and recover the metal contaminants from incinerator ash as well as from other heavy metal bearing wastes (i.e., electric arc furnace dust from steel industry) or soils. Information developed from this project is being applied to demonstrate the remediation of lead paint contaminated soils on Long Island

Random Matrices in 2D, Laplacian Growth and Operator Theory

Since it was first applied to the study of nuclear interactions by Wigner and Dyson, almost 60 years ago, Random Matrix Theory (RMT) has developed into a field of its own within applied mathematics, and is now essential to many parts of theoretical physics, from condensed matter to high energy. The fundamental results obtained so far rely mostly on the theory of random matrices in one dimension (the dimensionality of the spectrum, or equilibrium probability density). In the last few years, this theory has been extended to the case where the spectrum is two-dimensional, or even fractal, with dimensions between 1 and 2. In this article, we review these recent developments and indicate some physical problems where the theory can be applied.Comment: 88 pages, 8 figure

Multilevel interpolation of divergence-free vector fields

We introduce a multilevel technique for interpolating scattered data of divergence-free vector fields with the help of matrix-valued compactly supported kernels. The support radius at a given level is linked to the mesh norm of the data set at that level. There are at least three advantages of this method: no grid structure is necessary for the implementation, the multilevel approach is computationally cheaper than solving a large one-shot system and the interpolant is guaranteed to be analytically divergence-free. Furthermore, though we will not pursue this here, our multilevel approach is able to represent multiple scales in the data if present. We will prove convergence of the scheme, stability estimates and give a numerical example. For the first time, we will also prove error estimates for derivatives and give approximation orders in terms of the fill distance of the finest data set

Computation of currents at microelectrodes using hp-DGFEM

We describe how a discontinuous Galerkin finite element method with interior penalty can be used to compute the current flowing at a microdisc electrode for the steady-state E and EC′ reaction mechanisms and at a channel microband electrode for the steady-state E, ECE, EC2E, DISP1 and DISP2 reaction mechanisms. We then state an a posteriori error bound for the computed current and use this as the basis of an hp-adaptive discontinuous Galerkin finite element algorithm to deliver the current to within a prescribed error tolerance. For all problems we demonstrate excellent agreement with previous results using comparatively coarse meshes. © 2005 Elsevier B.V. All rights reserved

Multilevel interpolation of divergence-free vector fields

We introduce a multilevel technique for interpolating scattered data of divergence-free vector fields with the help of matrix-valued compactly supported kernels. The support radius at a given level is linked to the mesh norm of the data set at that level. There are at least three advantages of this method: no grid structure is necessary for the implementation, the multilevel approach is computationally cheaper than solving a large one-shot system and the interpolant is guaranteed to be analytically divergence-free. Furthermore, though we will not pursue this here, our multilevel approach is able to represent multiple scales in the data if present. We will prove convergence of the scheme, stability estimates and give a numerical example. For the first time, we will also prove error estimates for derivatives and give approximation orders in terms of the fill distance of the finest data set