This animation depicts a liquid spray emerging from a nozzle. It is a two-phase flow: liquid droplets, shown as uniformly sized particles, move through the surrounding gas, represented by faint gray streaks.
Near the nozzle, the droplets are densely packed, forming a thick spray. Farther downstream, they spread into a cone, slowed by drag and deflected by the turbulent gas, which they in turn set in motion. As the droplet concentration decreases, the flow transitions first to a thin spray and then to a very thin spray.
Animation generated with Opus 5.5.
This animation illustrates how particle size—in two dimensions, disk diameter—affects collision frequency.
The three panels contain the same $N=200$ hard disks. They start from identical positions with identical velocities, all moving at unit speed in random directions. The only difference is the disk diameter, chosen so that the disks occupy 0.2%, 2%, and 20% of the total area (area fraction $\Phi=N\pi\varepsilon^2/4$). The domain is periodic ($\mathbb{T}^2$), and the hard-disk dynamics are computed exactly, one collision at a time.
One particle is highlighted in orange together with its recent trajectory, and each dot marks one of its collisions. For the smallest disks, collisions are rare and the mean free path is approximately half the side length of the box: the tagged particle travels along long straight segments, close to free transport. As the disks grow, collisions become more frequent—about 14 times as frequent at 20% as at 0.2% in this simulation—and the trajectory becomes a short-step random walk.
Animation generated with Opus 5.5.
In the Boltzmann–Grad limit, collisions involving more than two particles at the same instant are negligible, so only binary collisions are retained. When particle diameters are no longer negligible, however, simultaneous multiple collisions must be taken into account.
The animation depicts the five distinct ways in which a four-particle hard-sphere collision can occur when spheres 1 and 2 collide through spheres 3 and 4 (cases I–V). All four spheres have the same non-negligible diameter and are in motion. Spheres 1 and 2 are shown in pink, while spheres 3 and 4 are shown in light blue. In each case, every contact in the configuration—indicated by a dashed line joining the sphere centers—occurs at the same instant, with no preceding contact.
The number following each case gives its number of labelings. Exchanging spheres 3 and 4 produces a distinct configuration in cases I–III, but leaves the configuration unchanged in cases IV and V, giving eight configurations in total.
Animation generated with Opus 5.5.