Mathematical modeling of the isothermal impingement of liquid droplets in spraying processes
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INTRODUCTION
FLUIDflow and heat-transfer phenomena associated with the impingement, spreading, and solidification of liquid droplets on solid surfaces are of broad general importance in a range of materials processing operations. Typical examples include plasma spraying (PS), spray forming (SF), and the in situ production of composite materials. This is a very complex problem, because it involves substantial deformation of a free surface over a very short time scale. A comprehensive understanding of these phenomena would be highly desirable from the materials processing standpoint, in order to analyze the needed relationships between process parameters such as droplet size, material properties and droplet velocity, and the structure and properties of the deposits formed. This article will address the fluid flow aspects (spreading/splatting) of the problem only; this is the first necessary condition of tackling the most complete heat transfer-solidification problem, which will be examined in a subsequent publication. In reviewing previous work, the classical fluid mechanics approach to these problems, exemplified by References 1 through 5 has used oversimplifying assumptions and emphasized rather low impact velocit i e s - l a r g e l y outside of the range of the materials processing interest. The first numerical effort in this field was done by Harlow and Shannon, t6] who investigated the flow dynamics of the splash of a liquid droplet onto a flat plate and into a pool of the same liquid, using the then developed "marker and cell" technique, t7] They attempted to explain experimental observations of the splashing process obtained by high-speed photography, t8,91 However, in their calculations, surface tension and viscous GERARDO TRAPAGA, formerly Graduate Student, Postdoctoral Associate, and JULIAN SZEKELY, Professor of Materials Engineering, are with the Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139. Manuscript submitted March 13, 1990. METALLURGICAL TRANSACTIONS B
effects were totally neglected, and a close representation was obtained only for the very initial stages of the splashing process. Nevertheless, their work was a pioneering approach in fluid dynamics, because the nature of the problem involves solving for the free surface of the fluid. The high-speed impact between a liquid droplet and a solid surface has been extensively studied due to its importance regarding damage and erosion of materials, such as in turbine applications, tl~ It has been common in this field to consider an entirely compressible fluid, and the elastic impact theory has been widely used to predict the maximum impact pressures generated during the critical first instant of liquid-solid impact. A similar approach, using the shock theory, has been proposed to treat the problem of collision of liquid droplets in thermal spraying processes, t~41Based on this approach, tentative explanations have been suggested for morphological characteristics of the resulting splats. Thi
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