Laser energy absorption and melt-vapor front

At the melt-vapor front, the Stefan boundary condition is normally applied to describe the laser energy absorption (Kar and Mazumda, 1990; Yao, et al., 2001).

${\displaystyle I_{abs}+k\left({\frac {\partial T}{\partial z}}+r{\frac {\partial T}{\partial r}}\right)+\rho _{l}\nu _{i}L_{v}-\rho _{v}\nu _{v}(c_{p}T_{i}+E_{v})=0}$ (1)

where ${\displaystyle I_{abs}=I(t)^{-\beta z}}$ is the absorbed laser intensity, β is the laser absorption coefficient depending on laser wavelength and target material, and I(t) describes temporal input laser intensity including pulse width, repetition rate, and pulse temporal shape. k is the heat conductivity, T is the temperature, z and r are distances along axial and radial directions, p is density, v the velocity, L_v the latent heat of vaporization. The subscripts l, v and i denote liquid phase, vapor phase and vapor-liquid interface, respectively.

If the laser intensity is high and pulse duration is short, the so-called Knudsen layer is assumed to exist at the melt-vapor front where the state variables undergo discontinuous changes across the layer. By considering the discontinuity across the Knudsen layer, Yao, et al. (2001) simulated the surface recess velocity V_v distribution, along the radial direction at different times, which indicates the material ablation rate is changing significantly across the Knudsen layer.

Melt expulsion

After obtaining the vapor pressure p_v, the melt layer flow and melt expulsion can be modeled using hydrodynamic equations (Ganesh et al.,1997). Melt expulsion occurs when the vapor pressure is applied on the liquid free surface which in turn pushes the melt away in the radial direction. In order to achieve fine melt expulsion, the melt flow pattern needs to be predicted very precisely, especially the melt flow velocity at the hole's edge. Thus, a 2-D axisymmetric transient model is used and accordingly the momentum and continuity equations used.

Ganesh's model for melt ejection is comprehensive and can be used for different stages of the hole drilling process. However, the calculation is very time consuming and Solana, et al. (2001),^[14] presented a simplified time dependent model that assumes that the melt expulsion velocity is only along the hole wall, and can give results with a minimum computational effort.

The liquid will move upwards with velocity u as a consequence of the pressure gradient along the vertical walls, which is given in turn by the difference between the ablation pressure and the surface tension divided by the penetration depth x.

Assuming that the drilling front is moving at a constant velocity, the following linear equation of liquid motion on the vertical wall is a good approximation to model the melt expulsion after the initial stage of drilling.

${\displaystyle \rho {\frac {\partial u(r,t)}{\partial t}}=P(t)+\mu {\frac {\partial ^{2}u(r,t)}{\partial r^{2}}}}$ (2)

where p is the melt density, μ is the viscosity of the liquid, P(t)=(ΔP(t)/x(t)) is the pressure gradient along the liquid layer, ΔP(t) is the difference between the vapor pressure P_v and the surface tension ${\displaystyle 2\sigma \over {\bar {\delta }}}$.