Physicists have re-evaluated the principles behind the formation of snowflakes, raindrops, and Saturn's rings.

The research findings have been published in the journal Physical Review Letters. Aggregation processes in gaseous environments are remarkably diverse: they are observed in atmospheric phenomena, industrial production, and even in space. This includes, for example, the formation of rain from mist droplets and snowflakes from microcrystals of ice. They are also responsible for the formation of Saturn's rings and those of other gas giants from small particles that end up in orbit. This phenomenon is relevant to a number of technologies, such as aerosol painting, the transport of powdered substances, controlled explosions, and more. To understand and predict these processes, as well as to manage them, scientists need adequate mathematical models of aggregation in gaseous environments.

In the early 20th century, Polish physicist Marian Smoluchowski formulated equations describing aggregation processes in terms of the number of aggregates of different sizes and aggregation rates—kinetic coefficients that reflect how quickly aggregates combine to form larger particles. However, Smoluchowski's classical equations are valid for systems without any spatial inhomogeneities and flows. Real processes, of course, do not occur in perfectly homogeneous systems.

When describing the behavior of aggregating particles in the real conditions of the Earth's atmosphere, space, or industrial sites, it is necessary to "mechanically" combine Smoluchowski's formulas with the Euler equations or (in a more general case) the Navier-Stokes equations. The former were derived in the mid-18th century, while the latter were developed in the mid-19th century. Both provide a fundamental description of the motion of liquids and gases. Nevertheless, in the form of a "hybrid" with Smoluchowski's equations, both lead to inconsistencies, which in several applications result in unacceptably high errors or even qualitative discrepancies with reality.

A solution to this issue was proposed in a recent article in Physical Review Letters by senior researcher Alexander Osinsky and Professor Nikolai Brilliantov from the Skoltech Center for Artificial Intelligence. Instead of continuing to seek ways to reconcile the two sets of old equations, the Skoltech researchers derived new hydrodynamic equations with new kinetic coefficients based on a mathematical approach and fundamental principles.

"Surprisingly, the resulting coefficients are neither reaction rate coefficients, as in Smoluchowski's equations, nor transport coefficients, as in the Navier-Stokes equations. These kinetic coefficients of a new nature combine the properties of transport and reaction coefficients. Moreover, for aggregating fluids, they hold the same fundamental significance as viscosity or thermal conductivity does for ordinary liquids," said Brilliantov. "Our detailed computer simulation showed that the proposed Smoluchowski-Euler hydrodynamic equations with new coefficients are quite accurate and adequate for modeling technologically significant aggregating fluids."

The new equations will enhance the accuracy of models used in analyzing air pollution from solid phase particles, in aerosol and powder technologies, rapid transport of fine dispersions, and in certain tasks related to the design of airplanes and automobiles.

The research is supported by a grant from the Russian Science Foundation.