James Graham-Eagle

Monotone methods for semilinear elliptic equations in unbounded domains

Abstract Monotone methods in conjuction with upper and lower solutions have proved to be extremely powerful tools in the study of nonlinear elliptic boundary value problems in bounded domains. In this paper the theory is extended to unbounded (not necessarily exterior) domains and general linear boundary conditions.

Contact angle calculations from the contact/maximum diameter of sessile drops

We present two algorithms for computing the contact angle of sessile liquid drops given data about the drops. The first yields the contact angle given the volume, surface tension and maximum diameter (or contact diameter) of a single drop. This algorithm is an extension of existing algorithms based on knowledge of the maximum diameter or of the contact diameter of a drop. A sensitivity analysis is included for this algorithm, allowing estimates to be made of the error in computed contact angle caused by errors in the measurement of the volume and/or diameter. The second algorithm requires only the volume and maximum or contact diameter of two different drops as input, and it produces both the contact angle and surface tension as output. Both algorithms are based on Newtons method applied to a function whose value is computed by solving a system of ordinary differential equations obtained from the Laplace equation of capillarity. The techniques are applicable to both hydrophobic and hydrophilic surfaces

ON THE EXISTENCE OF TWO-DIMENSIONAL, LOCALIZED, ROTATING, SELF-SIMILAR VORTICAL STRUCTURES*

We prove that a Gaussian monopole, also known as the Lamb-Oseen vortex, is the only localized, rotating, self-similar solution to the two-dimensional, incompressible Navier-Stokes equations where level sets of vorticity and corotating streamfunction coincide. Our definition of self-similarity is restricted to the natural linear combination of space, time, and viscous diffusion. We arrive at this conclusion by analytically determining the azimuthal Fourier modes for all possible solutions to this problem and then proving that the amplitude of all but the first (axisymmetric) is zero. Since coherent vortex multipoles are observed to be in a state in which lines of vorticity and corotating streamfunction correspond, this casts doubt on the existence of any localized, rotating, self-similar asymptotic structure other than the monopole.

Implementation and examination of a new drop shape analysis algorithm to measure contact angle and surface tension from the diameters of two sessile drops

The performance of a new algorithm developed to measure contact angle and surface tension of sessile drops is examined. To calculate the contact angle and surface tension, the new algorithm (ADSA-TD) requires the radius (contact or equatorial) and volume of two sessile drops of different sizes that are placed on the same surface. Initially, the algorithm was tested using synthetic drops (synthetic or theoretical drops are produced by numerical integration of the Laplace equation). The radii and volumes of synthetic drops were used as ADSA-TD inputs. The calculated contact angle (θ) and surface tension (γ) by ADSA-TD matched perfectly the assumed values of θ and γ used to produce the synthetic drops, confirming theoretically the validity of the new algorithm. In the next step, the sensitivity of the algorithm to input errors was examined. It was shown experimentally that both calculated contact angle and surface tension are affected by the errors in volume and radius. Besides the error in input values, it was shown that the size difference between the paired drops and the differences in their contact angles would affect the output of ADSA-TD. As it turns out, the calculated surface tension is so sensitive to the above factors that ADSA-TD does not appear to be promising as a surface tension measurement technique. However, ADSA-TD produced acceptable contact angle values as compared to measurements made by other proven techniques such as axisymmetric drop shape analysis-profile. Thus, ADSA-TD may be of interest as a contact angle measurement technique which does not require the liquid surface tension as input.

SOME NEW FINDINGS ON THE ZEROS OF BAND-LIMITED FUNCTIONS

We investigate relationships between the structure of a real object of compact support and the realness of the zeros of its Fourier transform. We introduce a necessary condition, and some sufficient conditions are presented for one-dimensional functions. A very useful property of the sinc function is also presented.

Strain-Rate Effects in Shear Highlighted by a Dynamical Systems Model

AbstractSoil shear can be described as a dynamical system in which particles move at random shear strains into the steady-state flow structure in a process governed by simple friction. This paper studies strain-rate effects in soil shear in the context of dynamical systems soil-shear theory. The theory highlights the fact that conditions at the start of plastic deformation are strain-rate dependent and that much of the initial linear variation of stresses with strain is not attributable to the elasticity behavior of the soil but rather is the expected small-strain behavior of a nonlinear process of plastic deformation. The static coefficient of friction dominates at very small strains, after which the friction coefficient reduces to its dynamic value. Variations with strain rate in the stress-strain and void ratio–strain curves are small because of the correspondingly small dependence of the friction coefficients on strain rate.

The effect of wind on combustion waves with reactant depletion

A simple model for the propagation of an exothermic reaction is proposed accounting for reactant depletion and for heat transport via advection (wind). To study travelling-wave solutions, we assume that the reactant ignites at a specified temperature. The speed of propagation of the resulting combustion wave depends on ambient wind conditions. This dependence on wind conditions is determined explicitly for simple reaction dependence on temperature, and stability is examined. An existence criterion is determined for general reaction dependence on temperature. It is shown that no travelling wave can exist if there is too great a headwind and that tailwinds do not cause such extinction. We also determine conditions on the reaction rate term for which only a single temperature maximum is allowed. Numerically obtained temperature profiles are included to demonstrate these results.

Simpson's rule for estimating n! (and proving Stirling's formula, almost)

Using a tool familiar to first-year calculus students (Simpsons rule) surprisingly good estimates are deduced for values of n!—or more precisely ln (n!)—along with error bounds. These estimates can be implemented on a simple hand-held calculator or computer. Moreover, it is demonstrated how to arrive at analogous, improved estimates (with error bounds) for all higher-order Newton-Cotes integration methods. Along the way, the error bounding naturally, and in short order, leads to the conclusion that n! ∼ C(n/e)n √n. While these methods cannot show the entirety of Stirlings formula (namely that C = √2φ), they do show how C can be approximated to any desired accuracy.

The Draining Cylinder

H^^^^^H James Graham-Eagle (james-grahameagle@uml.edu) ^^H^^M^^H received a D.Phil, from Oxford University in England in J^^H 1984. Before arriving at the University of Massachusetts ^H^^^^^H Lowell he held positions at Victoria University of Wellington ^^Bk 11 i^H and the University ?f Auckland in New Zealand and the ^^B?|;?i^i^^H University of Delaware. His mathematical interests include ^^^F^^^H nonlinear partial differential equations and the calculus of I^^S^mmHBB variations. Despite failing knees he still enjoys chasing after a soccer ball with other old sweaty guys.

Modelling smectite illitization in earthen barriers of buried radioactive wastes

The textural characteristics of engineered clay barriers of deeply buried radioactive waste repositories are likely to change in response to hydro-thermo-mechanical stresses during the 10 000-year service and risk analysis period. Expected textural changes stem from a combination of processes, the most significant of which is smectite illitization. To pose significant risks of radionuclide release from buried waste canisters, illitization must, as a minimum, produce barrier textural changes that are significant within the timeframe in which radioactive activity is still significant. Besides, the canister must be damaged, principally through corrosion by substance-laden infiltrating moisture. In the current paper, simulations of three scenarios of illitization of a clay barrier that is initially 100% smectite under a temperature regime that does not exceed 150°C indicate that the accumulation of potassium ions ( ) borne by infiltrating water close to the waste canister is likely to be insignificant within the first 1000 years of waste entombment. Some are extracted from the infiltrating moisture to feed the illitization reaction which is herein assumed to occur with an activation energy of 1.17 × 10−5 J/mole.