From rain to Saturn’s rings: New equations push the boundaries of fluid dynamics

Have you ever wondered about the magic behind the formation of raindrops or picturesque snowflakes? Or maybe about the creation of the otherworldly rings surrounding Saturn? These processes are all tied to the intricate dance of aggregating particles in fluid dynamics.

Now, researchers from Skoltech are adding a new chapter to the understanding of this cosmic dance. The team has proposed innovative mathematical equations that promise to push the frontiers of both our natural and technological worlds.

This advancement addresses natural and engineered phenomena, ranging from rain and snow formation to the creation of planetary rings and industrial fluid flows.

Simplified approach to fluid dynamics

Fluid aggregation is a critical process observed in both natural and technological systems.

In nature, water droplets come together to form rain, ice crystals merge into snowflakes, and particles orbiting planets coalesce into structures like Saturn’s rings. These processes demonstrate how particles in fluids aggregate to create larger entities.

In technology, fluid aggregation plays a significant role in activities like aerosol painting (where particles form a smooth layer), powder transport in industries, and controlled explosions that rely on particle behaviors.

Understanding and predicting these phenomena are essential for improving various applications.

Models of aggregating particles

Physicist Marian Smoluchowski’s equations, developed in the early 20th century, were a major step in understanding how particles combine in fluids.

For example, these equations could describe how small droplets in a mist merge to form larger drops, or how tiny ice crystals in clouds come together to create snowflakes. The equations calculated the rate at which particles of different sizes combined, providing insights into the overall aggregation process.

However, these equations assumed that the environment was uniform – meaning that there were no changes in temperature, pressure, or fluid movement.

For instance, the equations worked well in theoretical situations like a calm, evenly distributed mist but failed to account for real-world complexities, such as turbulent winds in a storm cloud or uneven temperature distributions in the atmosphere.

To address these shortcomings, the research team tried combining Smoluchowski’s equations with models of fluid dynamics like the Euler equations (for ideal fluids) and the Navier-Stokes equations (which include viscosity and other forces).

For example, in atmospheric science, they might use Smoluchowski’s equations to predict how raindrops grow while using the Navier-Stokes equations to simulate airflows in a storm cloud.

In industrial processes, they could use these models together to simulate how fine powder particles mix and aggregate in a turbulent airflow inside a factory.

The challenge, however, was that these two sets of equations were developed for entirely different purposes and mathematical frameworks.

Limitations of past models

Smoluchowski’s equations focus on particle merging rates, while the Navier-Stokes equations focus on fluid movement. Merging them without a unified foundation sometimes caused errors.

For example, in air pollution studies, combining the equations might incorrectly predict how soot particles cluster in turbulent air.

In planetary science, using these hybrid models could lead to inaccurate predictions about how particles form rings around planets like Saturn.

The root problem was that the mathematical assumptions of these two frameworks were not compatible, leading to unreliable results in complex scenarios.

For example, in a turbulent environment where particle sizes and motion vary significantly, the models could miscalculate aggregation rates or fail to predict certain behaviors, limiting their usefulness in both research and practical applications.

The Skoltech solution: Hydrodynamic equations

Skoltech researchers Alexander Osinsky and Professor Nikolay Brilliantov addressed these challenges by developing new hydrodynamic equations.

Instead of merging incompatible systems, they derived a unified framework from fundamental principles. Their novel equations incorporate coefficients that account for both reaction rates and fluid transport – essentially blending the best aspects of traditional approaches.

“Surprisingly, these are neither the reaction-rate, nor the transport coefficients familiar from the Navier-Stokes equations, but a combination of both in the form of kinetic coefficients of a new nature,” explained Professor Brilliantov.

“They are as fundamental for aggregating fluids as viscosity and thermal conductivity are for ordinary fluids.”

“Using extensive computer simulations, we have shown the accuracy and relevance of our novel Smoluchowski-Euler hydrodynamic equations with the new coefficients for some of the technologically important aggregating fluids.”

Scientific and industrial applications

This breakthrough has implications for both natural systems and human technologies. By enhancing the precision of fluid aggregation models, the new equations will improve predictions in the areas listed below.

• Air pollution analysis: To understand how solid particles aggregate and disperse in the atmosphere.
• Granular flow dynamics: To optimize processes involving powders and other granular materials.
• Powder technology: To develop more efficient methods for handling and transporting powders.
• Aircraft and automotive design: To refine fluid dynamics for better performance and safety.

The researchers validated their equations through extensive computer simulations, demonstrating their applicability to real-world systems. This unified approach eliminates the errors introduced by previous hybrid models, paving the way for more reliable analyses.

A step forward in fluid dynamics

The new equations represent a fundamental shift in how scientists model aggregation in fluids. By addressing long-standing limitations, Skoltech researchers have provided a tool that bridges gaps between theory and application.

As Brilliantov noted, these kinetic coefficients hold the same significance for aggregating fluids as traditional properties like viscosity do for ordinary fluids.

This advance not only deepens our understanding of natural processes but also enhances engineering capabilities. Whether analyzing atmospheric phenomena, improving industrial processes, or designing vehicles, the implications are vast and far-reaching.

The unified framework developed by Skoltech promises to revolutionize fields where fluid aggregation plays a critical role.

The study is published in the journal Physical Review Letters.

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