Models for Solidification and Splashing in Laser Percussion Drilling

This paper studies systems of partial differential equations modelling laser percussion drilling. The particular phenomenon considered in detail is the ejection of the thin layer of molten material. This thin layer is modelled as an inviscid flow between the fluid surface and fluid/solid interface, both of which are unknown moving boundaries. Through a regular asymptotic expansion, the governing equations are reduced to a combination of the shallow water equations in the zero gravity limit and a two-phase Stefan problem; the key small parameter is the square of the aspect ratio. These leading-order problems exhibit shocks which represent a possible mechanism for the previously unexplained fluid clumping. Approximate formulas and a parameter grouping are derived to predict the rate of melt solidification during ejection. Finally, weak formulations of the convection-diffusion equation for energy conservation are presented. These weak formulations are novel because the fluid is moving across a solid surface. An appropriate extension to the enthalpy method is suggested as a first stage toward numerical calculations.