Физики обновили понимание формирования снежинок, капель дождя и колец Сатурна.

The findings of the study have been published in the journal Physical Review Letters. Aggregation processes in gaseous environments are remarkably diverse: they can be observed in atmospheric phenomena, industrial production, and even in outer space. Examples include the formation of rain from mist droplets and snowflakes from microcrystals of ice. These processes are also responsible for the creation of the rings of Saturn and other gas giant planets from small particles that have entered orbit. This phenomenon is also relevant to a range of technologies, including aerosol painting, the transportation of powdered substances, and controlled explosions, among others. To understand, predict, and manage these processes, scientists require adequate mathematical models of aggregation in gaseous environments.

In the early 20th century, Polish physicist Marian Smoluchowski formulated equations that describe aggregation processes in terms of the number of aggregates of different sizes and the rates of aggregation—kinetic coefficients reflecting how quickly aggregates combine to form larger particles. However, Smoluchowski's classical equations are valid for systems devoid of any spatial inhomogeneities and flows. Real processes, of course, occur in systems that are not perfectly homogeneous.

When describing the behavior of aggregating particles under real conditions in the Earth's atmosphere, outer space, or industrial settings, it becomes necessary to "mechanically" combine Smoluchowski's formulas with the equations of Euler or, in a more general case, Navier-Stokes. The former were derived in the mid-18th century, while the latter emerged in the mid-19th century. Both provide a foundational description of the movement of liquids and gases. Nevertheless, as a "hybrid" alongside 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 search for ways to reconcile two sets of old equations, the Skoltech researchers derive new hydrodynamic equations with new kinetic coefficients based on a mathematical approach and fundamental principles.

"Surprisingly, the obtained coefficients are neither reaction rate coefficients, as in Smoluchowski's equations, nor transport coefficients, as in Navier-Stokes equations. These new kinetic coefficients possess properties that combine both transport and reaction coefficients. For aggregating fluids, they hold fundamental significance analogous to viscosity or thermal conductivity for ordinary liquids," explained Brilliantov. "Our detailed computer simulation demonstrated that the proposed Smoluchowski-Euler hydrodynamic equations with new coefficients are quite accurate and appropriate 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, in the rapid transportation of fine dispersions, and in certain tasks related to the design of aircraft and automobiles.

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