## Appendix A Generalized Piston Model  
The KH geometry used for describing the KH shape observed at high welding speeds, when the KH front wall is clearly inclined, is shown in Figure 3. A half-cone shape on the KH front has been chosen because it is considered that this is the characteristic shape observed for rather low aspect ratio KHs of this regime. For higher aspect ratios, because of the beam propagation thanks to the multi-reflections, a semi-cylindrical shape should be used. Accordingly, the radius of the half-cone r(z) and the melt thickness \(\delta\)(z) vary linearly along the KH depth e (=AB) with the laws: r(z) = 0.5d z/e and \(\delta\)(z) = \(\delta_{0}\) z/e, where the melt thickness \(\delta_{0}\) is assumed to be given by [38]:  
\[\delta_{0}=\frac{\kappa_{\rm m}}{V_{\rm d}}Y(T_{\rm s}), \tag{1}\]  
where Y(T\({}_{\rm s}\)) is a function of the surface temperature T\({}_{\rm s}\) of the KH, which also depends of the thermo-physical parameters of the material through the relation Y(T\({}_{\rm s}\)) = Ln(1 + (\(\rho_{\rm m}\)C\({}_{\rm pm}\)(T\({}_{\rm s}\) - T\({}_{\rm m}\)))/(\(\rho_{\rm s}\)C\({}_{\rm ps}\)(T\({}_{\rm m}\) - T\({}_{\rm 0}\)) + \(\rho_{\rm s}\)L\({}_{\rm m}\)). (\(\kappa_{\rm m}\): liquid heat diffusivity; \(\rho_{\rm m}\), \(\rho_{\rm s}\) and C\({}_{\rm pm}\), C\({}_{\rm ps}\): respectively, liquid/solid densities and heat capacities; and Lm: latent heat of fusion). It is assumed that the melt thickness of the liquid ejected at V\({}_{\rm m}\) on each side of the KH is also given by Equation (1). We have already seen that the condition of stationarity of the KH front, in the beam frame leads to Equation (8): V\({}_{\rm d}\) = V\({}_{\rm w}\) cos(\(\alpha\)), where \(\alpha\) is the inclination of this KH front (angle EAB on Figure 3) and V\({}_{\rm d}\) the drilling velocity normal to this front. In order to solve the entire problem, one must write the conservative equations for mass, momentum and energy:  
\(\bullet\) Mass conservation:  
The incoming mass flow rate (in kg/s) of the solid material enters the transverse section of the weld seam (defined by the triangle AD'\({}_{1}\)C'\({}_{1}\)) at the welding speed V\({}_{\rm w}\). A part of this mass flow rate is ejected as liquid with the velocity V\({}_{\rm m}\), through the two lateral transverse sections (AD\({}_{1}\)D'\({}_{1}\) and AC\({}_{1}\)C'\({}_{1}\)) and another part is evaporated at the velocity V\({}_{\rm v}\) through the inner inclined surface S\({}_{\rm v}\) defined inside the points C\({}_{1}\), C, E, D, D\({}_{1}\) and A (the entire surface S\({}_{\rm v}\) is assumed to be irradiated by the incoming laser beam, and it can be shown that S\({}_{\rm v}\)\(\approx\) (1.5\(\pi\)/4)(d\({}^{2}\)/sin \(\alpha\)). Taking into account this geometry, this mass conservation equation becomes:  
\[0.5\rho_{\rm s}V_{\rm w}(d+2\delta_{0})\rm e=\rho_{\rm m}V_{\rm m}\ \delta_{0}\rm e+\rho_{\rm m}V_{\rm v}\ S_{\rm v}, \tag{2}\]  
\(\bullet\) Energy conservation:  
This conservation law can be written as:  
\[\rm P_{abs}=P_{\rm m}+P_{\rm vap}+P_{\rm cond}+P_{\rm kin}, \tag{3}\]  
where P\({}_{\rm abs}\) is the absorbed power (in W) of the incoming laser beam inside the KH. P\({}_{\rm m}\) is the power used for melting, P\({}_{\rm vap}\) the power involved in the vaporization process, P\({}_{\rm cond}\) the power lost by conduction inside the solid and P\({}_{\rm kin}\) the power transferred to kinetic energy of the melt flow. The absorbed power P\({}_{\rm abs}\) is given by:  
\[\rm P_{abs}=A_{F}(\alpha)P \tag{4}\]  
where A\({}_{F}(\alpha)\) is the Fresnel absorptivity and P is incident laser power. It is assumed that the all the incident beam impinges the KH surface with the same angle of incidence \(\pi\)/2 - \(\alpha\), and the optical constants n and k used are 3.6 and 5 respectively, corresponding to \(\lambda\) =1.06 \(\mu\)m, on steel [39].