Ira M. Cohen

Melting and solidification of thin wires: a class of phase-change problems with a mobile interface—I. Analysis

Abstract In this paper, a model that describes the transient heating of a thin wire causing the tip to melt, roll-up of the molten mass into a ball due to surface tension forces, and the subsequent solidification of the molten material due to conduction up the wire and convection and radiation from the surface, has been provided. The wire is assumed to be heated at its lower tip to a temperature beyond the melting temperature of the wire material by heat flux from an electrical discharge. The shape of the melt is analytically/numerically determined by solving equations based on minimum energy principles. The departure from sphericity of the melt that is formed is examined by perturbation schemes, based on expansions for small ratio of gravity to surface tension forces and small ratio of surface tension gradient to surface tension forces, both of which are true for the problems considered. Temperature fields in the melt have been obtained by solving the energy equation using a body-fitted coordinate system. Temperature fields in the wire above the melt were calculated as well. Comparisons of those temperatures with experimental measurements described in Part II of this study are excellent.

Variational analysis of the squeezing flow of a yield stress fluid

Abstract The flow produced by slowly squeezing a yield stress fluid between two circular, parallel plates is analyzed using a variational principle. Previous anayses of this problem have been largely unsuccessful because they have relied on direct solution methods which require the complete determination of the flow field. Although a complete determination of the flow field is a desirable objective, it is not necessary for obtaining upper and lower bounds for the rate of squeezing of the yield stress fluid. Here we have developed such bounds for the squeezing flow of a Herschel-Bulkley fluid. The lower bound is found to be similar to the previously reported solutions. We have also carried out some related experiments. These experimental results are in agreement with the form of the lower bound to the solution.

Evaporation and Combustion of a Slowly Moving Liquid Fuel Droplet: Higher-Order Theory

The evaporation and combustion of a single-component fuel droplet which is moving slowly in a hot oxidant atmosphere have been analysed using perturbation methods. Results for the flow field, temperature and species distributions in each phase, interfacial heat and mass transfer, and the enhancement of the mass burning rate due to the presence of convection have all been developed correct to second order in the translational Reynolds number. This represents an advance over a previous study which analysed the problem to first order in the perturbation parameter. The primary motivation for the development of detailed analytical/numerical solutions correct to second order arises from the need for such a higher-order theory in order to investigate fuel droplet ignition and extinction characteristics in the presence of convective flow. Explanations for such a need, based on order of magnitude arguments, are included in this article. With a moving droplet, the shear at the interface causes circulatory motion inside the droplet. Owing to the large evaporation velocities at the droplet surface that usually accompany drop vaporization and burning, the entire flow field is not in the Stokes regime even for low translational Reynolds numbers. In view of this, the formulation for the continuous phase is developed by imposing slow translatory motion of the droplet as a perturbation to uniform radial flow associated with vigorous evaporation at the surface. Combustion is modelled by the inclusion of a fast chemical reaction in a thin reaction zone represented by the Burke-Schumann flame front. The complete solution for the problem correct to second order is obtained by simultaneously solving a coupled formulation for the dispersed and continuous phases. A noteworthy feature of the higher-order formulation is that both the flow field and transport equations require analysis by coupled singular perturbation procedures. The higher-order theory shows that, for identical conditions, compared with the first-order theory both the flame and the front stagnation point are closer to the surface of the drop, the evaporation is more vigorous, the droplet lifetime is shorter, and the internal vortical motion is asymmetric about the drop equatorial plane. These features are significant for ignition/extinction analyses since the prediction of the location of the point of ignition/extinction will depend upon such details. This article is the first of a two-part study ; in the second part, analytical expressions and results obtained here will be incorporated into a detailed investigation of fuel droplet ignition and extinction. In view of the general nature of the formulation considered here, results presented have wider applicability in the general areas of interfacial fluid mechanics and heat/material transport. They are particularly useful in microgravity studies, in atmospheric sciences, in aerosol sciences, and in the prediction of material depletion from spherical particles

Melting and solidification of thin wires: a class of phase-change problems with a mobile interface—II. Experimental confirmation

Abstract In Part I, we formulated the problem of wire heating, melting, roll-up into a ball, cooling and solidification. In this part, we describe experimental observations of the melting and solidification processes using high speed photography in an arc chamber. These observations provide results for comparison with the theoretical model. In the numerical computations for the theoretical model, the heat flux from the arc plasma to the wire is an input parameter. The value of this heat flux is obtained from temperature measurements made by thermocouples embedded in the unmelted wire above the ball, thus enabling comparisons between the predictions of the theoretical model and experimental observations. The heat transfer results indicate that conduction up the wire, for thin wires, is the dominant mechanism; the solidification front in the melt progresses from top downwards. The models described in Parts I and II have many applications apart from demonstrating suitable analytical and numerical procedures for treating phase-change problems with moving interfaces. The most important practical application is in semi-conductor chip assembly and packaging by the ball bonding process.

Unified positive column theory of gas discharges

The effects of electron impact ionization, three‐body recombination, and a thermal energy balance are considered on the positive column of a gas discharge confined between plane parallel walls. The diffusion flux equations (with temperature‐dependent ionization and recombination) for conservation of ions and electrons, Poissons equation for the self‐consistent electric field, and the Elenbaas‐Heller energy equation for the temperature distribution, are used. Thermal and electrical conductivities are given by mixture rules that are linear interpolations of collision cross sections in fraction ionized. This formulation allows the description of the entire range of gas discharges from dark discharges through arcs. The emphasis here is devoted to the range from the normal glow through the transition to an arc. The end result is the calculation of semi‐internal characteristics, ionization collision rate versus current (per unit depth), and electric field versus, current (per unit depth). It is found that the ...

Oscillatory enhancement of the squeezing flow of yield stress fluids: A novel experimental result

The extrusion of a yield stress fluid from the space between two parallel plates is investigated experimentally. Oscillating the magnitude of the squeezing force about a mean value ( F = f [1+αcos(ω t )]) was observed to significantly enhance the flow rate of yield stress fluids, while having no effect on the flow rate of Newtonian fluids. This is a novel result. The enhancement depends on the magnitude of the force, the oscillatory frequency and amplitude, the fluid being squeezed, and the thickness of the fluid layer. Non-dimensional results for the various flow quantities have been presented by using the flow predicted for the constant-force squeezing of a Herschel–Bulkley yield stress fluid as the reference. In the limit of constant-force squeezing, the present experimental results compare very well with those of our earlier theoretical model for this situation (Zwick, Ayyaswamy & Cohen 1996). The results presented in this paper have significance, among many applications, for injection moulding, in the adhesive bonding of microelectronic chips, and in surgical procedures employed in health care.

Continuum analysis of the photoionization chamber in the transition from low to high rates of ionization

The photoionization chamber consists of a test gas which is bounded by parallel plates and ionized to a small extent by incident radiation. The charged particles formed by the ionization process diffuse to the walls of the chamber where they recombine. The particle densities and electric field distributions as well as the current‐voltage characteristic are studied. At low levels of ionization the electron density is neglected with respect to the ion density. By coupling this simplification with correct scaling, it is possible to obtain analytic solutions. The resulting current‐voltage characteristic reflects the rate of ionization as well as the ratio of the ion to the electron temperature. When this ratio is equal to unity, the ion and electron current fluxes saturate at the same rate. If this ratio is less than unity, the electron current flux saturates more slowly than the ion current flux. At higher levels of ionization the electron density can no longer be neglected with respect to the ion density an...

Electrode Heating in a Wire-to-Plane Arc

A steady wire‐to‐plane electric discharge has been modeled in a prolate spheroidal coordinate system with the wire shape taken as a hyperboloid of revolution. A set of continuum conservation equations for the charged particle densities and temperatures together with Poisson’s equation for the self‐consistent electric potential describe the steady electric discharge process. These equations have been solved numerically to obtain ion and electron densities, temperature distribution, and electrode heat fluxes. Particle densities show the main body of the arc is quasineutral bounded by space charge sheaths at both electrodes. The temperature is greatest in a region around the discharge axis about one‐third of the distance from the wire to the plane. Strong electric fields are concentrated in the electrode sheaths. The heat flux to the wire is sharply peaked near the tip but on the plane it decays slowly away from the discharge axis. The knowledge of heat transfer from the arc to the electrodes is useful in de...

Continuum theory of spherical electrostatic probes in a stationary, moderately ionized plasma

A theory is developed for spherical electrostatic probes in a stationary, collision‐dominated plasma in which the ratio of charge‐charge to charge‐neutral collision frequencies may range between zero and unity. The former limit corresponds to the usual “weakly ionized” condition of earlier probe analyses, while for non‐zero values of this ratio, transport coefficients must be assumed to vary with position in the plasma. Electron temperature variation is allowed for through the consideration of electron energy conservation and transport. Quasilinearization is used to solve the equations numerically, and an orthonormalization technique is developed to overcome the “ill‐conditioning” problem which often occurs in the solution of linear two‐point boundary value problems. Plasma property profiles in the vicinity of the probe, for several values of collision frequency ratio and a wide range of probe potentials, are investigated. A few representative current‐voltage characteristics are calculated. The effects of...

Melting of a Wire Anode Followed by Solidification: A Three-Phase Moving Interface Problem

A fine metallic wire electrode is heated from below (by an electric discharge) causing melting and roll-up into a ball by surface tension. After the heating is terminated, a solidification front progresses through the melt until a solid ball is formed and cooled to ambient conditions. In this paper we numerically simulate the heating, melt motion and roll up and subsequent cooling and solidification. This is a three-phase problem (solid, liquid, and the ambient medium-plasma/gas) with two simultaneously moving phase interfaces, the outer one tracked by orthogonal grid generation conformal with the evolving boundary surface at each time interval A novel observation in this study is that the wire end first drops until the melt radius equals the wire radius and then it begins to roll up into a ball consuming the wire. In other words, the inter-electrode gap first reduces and subsequently increases during an electronic flame off (EFO) discharge heating/phase-change process.