## 3 Results and Discussion
### Instantaneous Velocity Fields in a Horizontal Section of Sessile Drop
As a result of such chaotic trajectory, the length of the entire trajectory from the contact line of the drop to its center will be 2-3 times longer than for the shortest line on the curved surface. As a result, the time \(t_{m}=3t_{c(s)}=6\) s, which corresponds well to the experiment. Figure 13b illustrates the trajectories of the other two particles. The initial position of the particles was in the vicinity of the contact line. The particles moved to region 3, where the toroid has its center. In region 3, the liquid velocity is close to zero, since the flow lines turn around, and the velocity changes direction by \(90^{\circ}\) (the velocity is directed inside the drop). Figure 13c provides experimental data for the particle velocity on the drop surface (curves 1 and 2). The distance l corresponds to the drop surface (\(l/R=0\) refers to the contact line of the drop, \(l/R=1\) for the center of the toroid, where the liquid velocity  
Figure 13: (**a,b**) Trajectory of the particle on the drop surface: 1, 2—particle trajectory; 3—the area of the toroid center; (**c**) particle velocity on the drop surface: 1, 2—experiment; 3—modeling by (4); (**d**) forces acting on the particle. is zero, \(l\) is the projection on a straight line (for example, on line 1 in Figure 13a), i.e., \(l\) is not the length of the particle trajectory). The movement of the salt crystal near the contact line under the action of capillary force is considered in [60]. To date, there is only an empirical expression describing the movement of a micron-sized particle on the surface of the drop [61]. Let us consider the equation of motion of a particle in a simple model approximation, taking into account basic forces applied to a particle (Figure 13d (the Stokes force (\(F_{st}\)) and the capillary force (\(F_{\alpha l}\))). The acceleration of a particle is determined by the equality of forces in accordance with (4):  
\[mdV_{p}/dt=F_{\alpha}-F_{st}=2\sigma S_{1}((R_{2}-R_{1})/R_{2}R_{1})-6\pi r\mu( V_{p}-V_{l}), \tag{4}\]  
where \(m\) is a particle mass; \(V_{p}\) is the velocity of a particle; \(\sigma\) is the water surface tension; \(R_{1}\) and \(R_{2}\) are the radii of curvature of the meniscus; \(S_{1}\) is the area of a particle, corresponding to the meniscus; \(r\) is the radius of the Al\({}_{2}\)O\({}_{3}\) particle (\(r=0.1\) mm); \(\mu\) is dynamic viscosity; and \(V_{l}\) is the fluid velocity at the drop surface. The boundary conditions of Equation (4) are at \(l=0\) (the location on the contact line) \(\Delta R=R_{1}-R_{2}=\) max, \(V_{l}=0\); at \(l/l_{max}=1\) (the center of the toroid), the difference \(\Delta R=R_{1}-R_{2}=0\), \(V_{l}=0\). The liquid velocity from zero value reaches the maximum (\(V_{l}=3.5\) mm/s in accordance with the experimental data) and then tends to zero when approaching the center of the toroid). The curve of velocity change \(V_{l}\) (from zero to maximum value of 3.5 mm/s and from maximum to zero value) was set as a cosine dependence, which closely corresponds to the known theoretical calculations for the velocity on the surface of the drop [21; 22]. Estimates show that the thermocapillary force is several orders of magnitude greater than buoyancy and gravity due to the smallness of the particle size. Thus, to describe the particle motion on the free liquid surface, it is sufficient to consider the movement due to curvature, changes in \(\sigma\) over the surface, and also due to the Marangoni flow (taking into account the change in \(\Delta T_{s}\) over the surface of drop 2). Changes in \(\Delta T_{s}\) were taken from the experimental data of the thermal imager, and taking into account that on the edges of drop 2 the temperature of the free surface of the liquid is approximately equal to a predetermined wall temperature (\(T_{w}\)). In approximate calculations, it is assumed that the value of \(\Delta R\) varies depending on the cosine.