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where Bx and By are the characteristic width along the x and y axis, respectively.

The modeling results of the spray stage provide input data and initial conditions for the modeling of droplet-substrate interactions in the ensuing deposition stage of the spray forming process.

5.4.0MODELING OF DROPLET DEFORMATION ON A SURFACE

Several research groups have conducted numerical analyses and simulations of the transient flow of a single liquid droplet impinging onto a flat surface, into a shallow or deep pool,[397] and related phenomena such as liquid breakup and dispersion,[422] heat transfer,[334][368] and solidification.[50][157][371][401][515] Recently, important progress in the modeling of droplet impingement processes has

been accomplished by Liu et al.[18][51][52][144][338][339][378][379][388][389]

The modeling studies and their applications will be reviewed in this section.

5.4.1Modeling of Deformation of a Single Droplet on a Flat Surface

Numerous 2-D models have been developed to simulate droplet deformation processes during impact on a smooth surface. Most of these models assumed axisymmetric deformation of a spherical or cylindrical droplet. The models may be conveniently divided into two groups, i.e., compressible and incompressible.

The compressible models resolve a very early stage of droplet deformation when the compressible wave generated by impact has not yet traveled throughout the droplet. In this stage, the shock wave separates the compressed liquid from the undisturbed liquid that is above the compressed liquid and at the initial impact velocity. The intersection of the compressed and undisturbed liquids constitutes a contact ring. As long as the velocity of the contact ring is

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greater than the sonic speed in the liquid, the shock wave will be attached to the contact ring and the shape of the droplet is a truncated sphere. This stage is referred to as subcritical flow period and the stage following it is termed postcritical flow period. Radial spreading of the droplet starts after the subcritical flow period and is driven by the high pressures in the compressed liquid. A simple scaling argument shows that the time scale for this early stage is on the order of:

	Eq. (14)	tcom	=	D0
				U s

where Us is the speed of sound in the liquid. Due to the complexity, the compressible models have only been used to study the initial stage of droplet impact on a solid substrate to obtain the peak pressure. No results have been available for final splat size.

The incompressible models offer a simplified examination of the radial flow during droplet impact. The time scale of the impact process may be estimated using the following equation:[515]

Eq. (15)	tincom	=	2D0	Re0.2
			3u0

which is at least an order of magnitude longer than the time scale for the compressible flow. Thus, the incompressible models should yield reasonably accurate results. It should be indicated, however, that the extension of the numerical results from 2-D models to actual spherical droplets is not readily achievable. Further, instabilities that may cause breakup upon impact (for example, the classical crown splash geometries presented in Ref. 337) cannot be resolved in any 2-D models.

The first numerical study on the transient flow of a single liquid droplet impinging onto a flat surface, into a shallow or deep pool was performed by Harlow and Shannon.[397] In their work, the full Navier-Stokes equations were solved numerically in cylindrical

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coordinates using the Marker-and-Cell (MAC) technique.[633] The axisymmetric calculations showed that the liquid sheet jet created on a flat surface by an impinging droplet spreads radially at a velocity 1.6 times the droplet impact velocity, independent of any other parameters. Splashing was resolved for the cases where a droplet impinges into a shallow or deep pool. However, the effects of the surface tension and viscosity, that are important to the deformation of molten metal droplets on a surface, were not taken into consideration. In addition, the numerical simulations did not yield a final splat diameter because all the mechanisms that would limit droplet spreading were ignored. Tsurutani et al.[334] extended the method of Harlow and Shannon by considering the effects of the surface tension and viscosity, and applied the method to a single droplet impinging onto a hot flat surface. These pioneering studies provided valuable insight into the problem of droplet impingement. In both studies, however, interactions between multiple droplets during spreading were not taken into account. In addition, the impact speed, density, viscosity and surface tension of the liquid droplets considered in these studies were significantly lower than those present normally in spray forming and thermal spray processes.

Montavon et al.[634] investigated the transient contact pressure at the interface between an impinging droplet and a flat substrate surface under thermal spray conditions. These researchers divided a droplet impingement process into two stages: (a) impact and (b) flattening and solidification. To address the transient dynamic phenomena in the impact stage, they developed a finite element model to solve the transient nonlinear Navier-Stokes equations in primitive variables on an Eulerian rectangular mesh with the Euler implicit and NewtonRaphson methods. The numerical results showed that the impact of a droplet on a substrate surface may induce a high pressure at the dropletsubstrate interface, and this contact pressure is strongly influenced by spray process parameters, especially droplet diameter, density and impact velocity. The contact pressure increases almost linearly with increasing Reynolds number of droplet at impact. As the interaction time increases, the contact pressure diminishes.

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Samsonov and Shcherbakov[635] used the Gibbs energy approach based on non-equilibrium thermodynamics to study the evolution of a spherical droplet after point contact with a solid surface. The study was restricted to the isothermal spreading of an incompressible and non-volatile liquid droplet on a flat, chemically inert, rigid solid surface in an atmosphere of inert gas that is not soluble in the liquid. A two-cell model for the droplet bulk volume and the region of wetting perimeter was used to derive an integral freeenergy balance equation for the droplet-substrate-gas system and to determine the thermodynamic forces that govern the spreading kinetics. They also analyzed the role of the linear free energy of the wetting perimeter and the energy of point contact in the primary wetting action.

For droplet impingement processes at very low or zero impact velocities, the driving force for droplet spreading is the difference between two competing factors, i.e., the attractive substrate potential that promotes the spreading, and the surface tension of the liquid that attempts to keep the droplet in spherical shape in order to minimize its surface area.[342] Direct numerical solution of the Langevin equation of motion, Monte Carlo method, and molecular dynamics method have been used to study the spreading dynamics.[48][342][636] Heiniö et al.[48] investigated the spreading of a viscous fluid droplet on a solid surface using a horizontal solid-on-solid model with Langevin theory for 2-D case and the Monte Carlo method for both 2-D and 3-D cases. The dropletsubstrate interaction was described first with one-monolayer-range chemical potential and second with a long-range van der Waals type potential. Dynamic droplet profiles, spreading rates, contact angles, and their characteristic time behaviors were examined for both partial and complete wetting regimes. The transition from partial to complete wetting occurs when the total droplet-substrate interaction exceeds the interatomic interaction between fluid particles. Thus, both the thickness and speed of the liquid film are determined by the value and range of the substrate potential. Nieminen et al.[636] conducted molecular dynamic simulations of Lennard-Jones systems to analyze the spreading of a