## Appendix A Generalized Piston Model
The power P\({}_{\rm m}\) used for melting:  
\[\rm P_{m}=\rho_{m}\ V_{m}\ \delta_{0}\ e\ \Delta H_{m}, \tag{5}\]where \(\Delta\rm{H_{m}}=[C_{ps}(T_{m}-T_{0})+L_{m}+C_{pm}(T_{s}-T^{*})]\) is the fusion enthalpy for bringing the solid from \(T_{0}\) to a mean exit temperature assumed to be: \(T^{*}=0.5(T_{m}+T_{s})\) (\(L_{m}\): latent heat of fusion) [23]. The power \(P_{\rm vap}\) involved in vaporization:  
\[P_{\rm vap}=\rho_{m}\ V_{\rm v}\ S_{\rm v}\ \Delta\rm{H_{v}}, \tag{10}\]  
where \(\Delta\rm{H_{v}}=[C_{ps}(T_{m}-T_{0})+L_{m}+C_{pm}(T_{s}-T_{m})+L_{v}]\) is the vaporization enthalpy for bringing the solid from \(T_{0}\) to the surface temperature \(T_{s}\), with \(T_{s}>T_{v}\) (\(L_{v}\): latent heat of vaporization). The evaporated mass flux \(\rho_{m}V_{v}\) is discussed below. The determination of the power \(P_{\rm cond}\) lost by conduction inside the solid is more delicate, and in order to have an analytically tractable solution, some assumptions are necessary that restrict the range of its applicability. By using FEM simulations of heat conduction for this half-cone geometry, and because of the high Peclet numbers involved for these operating conditions, the heat flux around the KH can be considered as 2D for aspect ratios \(R>\approx\)0.5. Accordingly, the power \(dP_{\rm cond}\) lost by a slice of thickness dz, from the solid-liquid interface (at \(T_{m}\)) of the half-cone shape of Figure 3, can follow a relation similar to Equation (1) given by:  
\[\frac{dP_{\rm cond}}{dz}=g(Pe(z))K_{s}(T_{m}-T_{0}), \tag{11}\]  
In Equation (11), following the same procedure as in Section 2.1, the linear approximation \(g(Pe(z))\approx m_{0}Pe(z)+n_{0}\) is then assumed, where the constants \(m_{0}\) and \(n_{0}\) are representative of the heat conduction mechanism from a semi-cylindrical shape. FEM simulations of heat conduction from a front half-cylinder also show that \(m_{0}\approx n_{0}\approx 2.3\). (In fact, these constants \(m_{0}\) and \(n_{0}\) are not so different from those for a complete cylinder, because the heat flux gradients that contribute to heat losses are mainly located on the front side of the moving shape). Equation (11) can be now integrated, knowing that the Peclet number is given by \(Pe(z)=V_{w}r^{*}(z)/\kappa_{s}\), and the radius \(r^{*}(z)\) of the solid-liquid interface varies with z as: \(r^{*}(z)=r(z)\) + \(\delta(z)=0.5d(1+2\delta_{0}/d)(z/e)\). It follows that:  
\[\frac{P_{\rm cond}}{K_{s}(T_{m}-T_{0})}=\int_{0}^{e}(m_{0}Pe(z)+n_{0})dz=[0.5m _{0}\Big{(}\frac{V_{w}d}{2\kappa_{s}}\Big{)}\Big{(}1+\frac{2\delta_{0}}{d} \Big{)}+n_{0}]e, \tag{12}\]  
Finally, the power \(P_{\rm kin}\) transferred to the kinetic energy of the melt flow ejected at \(V_{m}\), is given by:  
\[P_{\rm kin}=\rho_{m}V_{m}\delta_{0}e\Big{(}\frac{V_{m}^{2}}{2}\Big{)}, \tag{13}\]  
In fact, even if \(P_{\rm kin}\) is varying as \(V_{m}\)[3], it appears that \(P_{\rm kin}\) is quite negligible compared to \(P_{m}\), \(P_{\rm cond}\) or \(P_{\rm vap}\). Typically, \(P_{\rm kin}\) represents less than 0.1% of the absorbed power \(P_{\rm abs}\) for these operating conditions. \(\bullet\) Momentum conservation:  
In fact, this corresponds to the use of the Bernouilli's law which expresses the speed of the ejected melt at \(V_{m}\) resulting from the pressure difference between the KH front which is undergoing the recoil pressure \(P_{r}\) due to the vaporization process, and the KH sides at ambient pressure \(P_{\rm amb}\). One can then write:  
\[\delta_{0}=\frac{\kappa_{\rm m}}{V_{\rm d}}Y(T_{\rm s}), \tag{1}\]0  
In Equation (14), \(P_{r}(T_{s})=0.5(1+\beta)P_{\rm CC}(T_{s})\), with the condensation factor \(\beta\) (\(\beta=0.2\) will be used here, because of high evaporation rate) and \(P_{\rm CC}(T_{s})\) is the well-known Clausius-Clapeyron pressure that depends of the surface temperature \(T_{s}\)[40, 41]:  
\[\delta_{0}=\frac{\kappa_{\rm m}}{V_{\rm d}}Y(T_{\rm s}), \tag{1}\]1where \(\rm P_{0}=10^{5}\) Pa and \(\rm T_{v0}\) is the vaporization temperature defined at vapor pressure \(\rm P_{0}\).