The drag force on a particle approaching a solidifying interface
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dance of natural and industrial processes that involve the freezing of a liquid suspension.[1–5] Of these processes, the melt processing of metal matrix composites has attracted the most attention from researchers. A central theme dominates these processes, namely, the suspended particles end up interacting with the crystallization front. If the liquid suspension is highly dilute, the particles usually interact with the interface but not with each other. Although considerable theoretical, numerical, and experimental efforts have been laid out to understand the interaction of a single particle with a solidification front, many relevant questions remain unanswered.[6,7,8] This article is restricted to the investigation of the drag force on a particle as it approaches steadily a solidifying interface, with the particular intention of isolating the influence of the thermal effects. In the absence of shrinkage flow, the fluid motion is caused solely by the movement of the particle. The influence of the thermal factors on the particle-interface interaction has been investigated by numerous scientists. The first ever experimental inquiry into the role of the thermal effects on the capture of particles by growing crystals was conducted by Zubko et al.,[9] who examined the solidification, by the Bridgman method, of various liquid substances that have been mixed with particles of different material. The interactions between the particle and interface were detected using a microscope. They discovered that the nature of the interaction depended primarily on the relative magnitudes of the coefficients of thermal conductivity of the particle (kp) and the melt (kl), namely, the particle is captured by the solid front if kp . kl and rejected if kp , kl regardless of the wetting properties of the particles. These authors accounted for the strong dependence of the coefficients of thermal conductivity on temperature, LAYACHI HADJI, Associate Professor, is with the Department of Mathematics, The University of Alabama, Tuscaloosa, AL 35487. Contact e-mail: [email protected] Manuscript submitted January 22, 2006. METALLURGICAL AND MATERIALS TRANSACTIONS A
but did not offer a physical explanation for the behavior of the particles at the interface. Subsequent theoretical and experimental studies found that the crystal behind the particle bulges into the melt if kp , kl and forms a trough otherwise, and went on to explain the rejection and the engulfment of the particle in terms of the interface profile; i.e., the formation of a trough is conducive to engulfment while the formation of a bump is conducive to pushing. Several studies have been carried out since to shed light on this thermal puzzle. These studies investigated the effect of the thermal factors on the forces that the particle is subjected to during its interaction with the interface. The drag force on a particle of radius a approaching steadily with velocity v a planar rigid wall of infinite extent is given by[10,11] F ; Cpmyan =dm
[1]
where m is the dynamic viscosity of the fluid, d is
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