## 4 Analysis of KH Stability Near Its Threshold Around R \(\approx\) 1
### Experimental Observation of These Unstable Behaviors  
#### 4.3.1 Welding Experiments  
In order to test the prediction of our model of the occurrence of some unstable behavior around the KH threshold P\({}_{\rm t}\), a set of very simple experiments has been undertaken. In order to cover the conduction to the KH modes, weld seams have been realized for different incident laser powers, at several welding speeds. A continuous-wave laser light with a wavelength of 1.03 \(\upmu\)m was used, with a uniform beam intensity distribution inside a spot size of 600 \(\upmu\)m diameter. The melt pool was also observed with a PHOTRON Fastcam high speed video camera at 10 kHz frame rate. One expects that this unstable regime is characterized by some fluctuations of the melt pool that could lead to some melt ejection, and therefore, to the occurrence of blow-holes inside the weld seam. Therefore, the number of blow-holes that appeared along the weld seam track should be an indicator of the stability of the welding process. For a welding speed V\({}_{\rm w}\) = 1 m/min, Figure 10 shows the number of blow-holes observed along a 45 mm bead length on mild steel, as a function of the incident laser power. It is remarkable to notice that these blowholes appear only within a range of laser powers extending from 400 to 1000 W. Additionally, the corresponding depths of these blow-holes have been reported. Finally, the corresponding threshold laser power P\({}_{\rm t}\) can be estimated by using Equations (5) and (6), and the evaluation of the constants a and b. This kind of determination is fairly approximate, because there is a rather large uncertainty on the different involved thermophysical parameters. However, for mild steel, by using the following thermophysical data: \(\rho\approx 7800\) kg/m\({}^{3}\), C\({}_{\rm p}\approx 800\) J/kgK, K \(\approx 30\) W/mK, T\({}_{\rm v}\approx 3100\) K, T\({}_{\rm 0}\approx 300\) K, one finds that a \(\approx 2.5\times 10^{5}\) W/m and b \(\approx 2\times 10^{10}\) J/m\({}^{3}\). The constants m and n used are those involved in the thermal model; typically, m \(\approx\) 2.4 and n \(\approx\) 3, and A(1) is about 0.6. As a result, with these parameters one finds that P\({}_{\rm t}\approx 450\) W for a welding speed V\({}_{\rm w}\) = 1 m/min, and if one uses the previous range of 0.8 \(<\) P/P\({}_{\rm t}\) \(<\) 1.7 for the critical zone, it would correspond for our conditions to 360 W \(<\) P \(<\) 765 W, which is in a surprising agreement with the experimental power range where blow-holes are observed. High speed videos of a sequence leading to the occurrence of this blow-hole show that the melt pool surface is first lifted and then explodes, leaving a blow-hole behind the melt pool. One interpretation  
Figure 10: Number of blow-holes on a 45 mm bead length and depth of the blow-holes, as a function of the incident laser power (on mild steel material, 0.6 mm spot diameter; V\({}_{\rm w}\) = 1 m/min). Typical aspects of surface weld seams for these welding regimes: the conduction, unstable and KH regime. would be that the interaction of the metallic vapor ejected perpendicularly from the KH front with the melt pool plays an important role in the complex dynamics of the melt pool [28; 35]. For these operating conditions, as the KH front inclination should be around 45\({}^{\circ}\) and with the rear KH wall being rather close, this vapor jet leaving the KH front could penetrate inside the liquid KH rear wall and cause the observed lifting of the melt pool. But if this effect should occur here, it should be also accompanied by the emission of spatters usually observed to be ejected from the KH rim, which was not seen at all. Therefore, the effect of this vapor jet, if it cannot be completely excluded, may not be entirely satisfying to explain the observed dynamics of the melt pool. The multi-physics simulations of [PERSON] et al.