## 3 Results and Discussion
### Instantaneous Velocity Fields in a Horizontal Section of Sessile Drop
The characteristic diffusion time on drop 2 surface (diffusion time of relaxation) \(t_{D}=l^{2}/D_{s}=(5.5\cdot 10^{-3}\)m)\({}^{2}/10^{-10}=30\cdot 10^{4}\) s (\(l\)--distance on drop 2 surface). Based on the estimates of the diffusion time, it turns out that the rate of spread of surfactant molecules on drop 2 surface is determined not by diffusion, but by the surface flow of Marangoni, which causes convection inside drop 2. Then, the convective relaxation time \(t_{c(s)}\) when using surfactant will be \(t_{c(s)}=l/U_{aver}=5.5\) mm/1.5 mm/s = 2.8 s (where the average velocity \(U_{aver}\) (Figure 8c) for the time interval \(\Delta t=4\)-5 s corresponds to 1.4-1.5 mm/s. Thus, already in 2 s after the fall of droplet of water with surfactant, a surface layer of surfactant should be formed near the contact line of the sessile drop 2. The contact line should move in 2-3 s, and in reality, the increase in the contact radius occurs approximately 7 s after the fall, i.e., with a noticeable delay (the characteristic time \(t_{m}\) is 2-2.5 times more than \(t_{c(s)}\)). Figure 11: Velocity field in horizontal section of drop 2; \(V_{01}=2.5\)\(\mu\)L; \(V_{02}=40\)\(\mu\)L; \(T_{w}=80\)\({}^{\circ}\)C; \(T_{01}=20\)\({}^{\circ}\)C; water (drop 2), water + surfactant AF 9-12 (NEONOL AF 9-12: oxyethylated monoalkyl phenol) (droplet 1). To explain such a noticeable delay, an additional experiment was conducted. Several metal particles with a diameter \(d\) = 0.1-0.2 mm were placed on a surface of a water drop after its positioning on a hot wall (\(T_{w}\) = 80 \({}^{\circ}\)C). The surface Marangoni flow and the capillary forces led to the displacement of the particles that were monitored using video cameras at multiple magnification of the image. Figure 13a shows the trajectory of the particle in time. In the presence of a toroid inside the drop, the motion of the particle must be realized along the trajectory 1 (red line). The real trajectory of motion according to experimental measurements corresponds to trajectory 2 (blue line). The direction of the velocity vector has a general tendency towards the center of the toroid. As noted earlier, the Marangoni flow is directed from the center of the toroid to the edge of the drop. Thus, the particle motion due to capillary forces is impeded by Marangoni forces and the friction force of the liquid. Therefore, the capillary force exceeds the sum of Marangoni forces and friction. For points of trajectory 2, the velocity values are given in mm/s. When the particle moves from the drop edge to the center and along the quasi-straight line, the acceleration occurs, and the particle reaches the maximum velocity (2.9 mm/s). As a result of the turn, the velocity decreases almost three times. When moving in a direction parallel to the contact line, the velocity is almost constant (1.2 mm/s). When approaching the center of the toroid, the velocity is reduced to a minimum value (less than 0.5 mm/s). The average particle velocity for the entire trajectory is 1.5-2.0 mm/s, which closely corresponds to the velocity value in the drop cross section \(U_{\text{ever}}\) = 1-1.2 mm/s (Figure 2, \(t>5\) s). The velocity on the drop surface according to numerical calculations) is about 2 times higher than the velocity in the horizontal section, which is close to the wall. The results of the  
Figure 12: Velocity field in horizontal section of drop 2; \(V_{01}\) = 2.5 \(\upmu\)L; \(V_{02}\) = 40 \(\upmu\)L; \(T_{w}\) = 80 \({}^{\circ}\)C; \(T_{01}\) = 20 \({}^{\circ}\)C; water (drop 2), water + surfactant SDS (sodium dodecyl sulfate) (droplet 1). trajectory measurement have shown an amazing result. The real trajectory is curved and has a random character. Indeed, there is a general tendency of particle displacement to the center of the drop. However, the particle motion is superimposed with random oscillations, which are caused by three-dimensional vortices inside the drop. These vortices move from one place to another, change the direction of rotation and interact with each other.