## 3 Results
### Dimensionless Analysis  
Based on the comparison shown in Figure 4, naturally the question arises: how do the crystallization parameters depend on the shape of the rear part of the weld pool? To answer this question, and for convenience of the further analysis, the following dimensionless parameters are defined:  
\[\xi=\frac{x-x_{c}}{L_{\mathrm{r}}},\ \eta=\frac{y}{b},\ \Lambda=\frac{L_{r}}{b},\ \rho=\frac{R}{v_{w}},\ \omega=\frac{W}{W_{0}}. \tag{11}\]  
Figure 3: Bead-on-plate welded 316L steel sheet. (**left**) metallographic cross-section; (**right**) metallographic horizontal section on the top surface. Figure 5a shows the contours of the upper rear part of the weld pool boundary when its length, \(L\), is three times its half-width, b, i.e., \(\Lambda=3\), in the practical range of shape parameters, \(1.1\leq m,n\leq 2.5\). As seen in Figure 5a, all curves merge seamlessly into the lower and upper part of the weld pool. It can be seen that for lower values of the shape parameters, \(n\leq 1.5\), the rear part of the weld pool boundary is sharpened (the curvature of the boundary is very large near the weld centerline), which is typically the case when welding alloys with a large amount of latent heat of solidification or when welding at high speed [9; 24]. An elliptical shape of the weld pool, i.e., \(m=n=2\), however, corresponds to a smoother curvature with a relatively large radius. The shape parameter, \(m\), affects the shape of the weld pool close to the weld centerline. When \(m\) is high, the radius of the curvature increases, and, consequently, the time during which the metal in the heat affected zone has a temperature close to the melting temperature, increases. The shape of the rear weld pool boundary has a strong influence on the geometry of the crystal axes, especially near the weld centerline, see Figure 5b. It can be seen in the figure, that at the beginning of the crystallization process, all crystals, independent of the shape parameters \(m\) and \(n\), grow perpendicular to the fusion line. However, a closer look at the curves shows that the higher the value of \(m\) and \(n\), the earlier the curve bends and thus reaches its asymptote, i.e., the weld centerline, at higher \(\xi\) values. When the shape of the rear part of the weld pool is close to linear, e.g., \(n=1.1\), the trajectory of the crystal axis is close to a straight line and the angle between the crystals growing towards each other near the weld centerline is large. The growth rate of the crystal varies from zero at the fusion boundary to the welding speed near the weld centerline, see Figure 5c. Based on the obtained results, it can be seen that the kinetics of the crystal growth is also greatly influenced by the shape of the rear weld pool boundary. For a nearly linear weld pool boundary, i.e., \(m=n=1.1\), the crystal growth rate is constant, except for small marginal areas, i.e., at the beginning and end of the crystallization process. For smoother and closer to ellipse-shaped weld pool boundaries, the growth rate increases linearly as it approaches the weld centerline. During crystals' growth, the width of the crystal decreases from 1 to 0, as shown in Figure 5d. Thereby, the rate of decrease depends again on the shape of the crystallization front. Nonetheless, the three main crystallization parameters also depend on the length-to-width ratio of the rear weld pool boundary, i.e., the dimensionless parameter \(\Lambda\). Figure 6a shows several crystal axes for ellipse-shaped weld pools, ranging from \(\Lambda=0.2\), i.e., short and wide weld pool, e.g., typical for transverse laser oscillation welding, to \(\Lambda=5\), i.e., long and narrow weld pool, which is typical for high-speed welding. In Figure 6b, the corresponding crystals' growth rate is shown. As seen, the growth rate increases from 0 at the fusion line, i.e., \(\eta=1\), to the welding speed at the weld centerline, i.e., \(\eta=0\).