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Mixing in Incompressible Flows: Advancing our Understanding of Fluid Dynamics
Mixing in fluid flows—whether liquids or gases—is a ubiquitous process that can be observed in a wide range of scenarios. From everyday occurrences, like mixing of cream into coffee, to more intricate systems such as chemical reactions, combustion, and even large-scale natural phenomena like ocean and atmospheric circulation, fluid mixing plays a key role.
At the Department of Mathematics and Computer Science at the University of Basel, Professor Gianluca Crippa’s research group is actively studying the complex phenomenon of fluid mixing. Their work delves into mathematical aspects of fluid dynamics, as highlighted in their recent paper, “Mixing in Incompressible Flows: Transport, Dissipation, and Their Interplay.” This publication focuses on the mathematical aspects of fluid dynamics, including the regularity of solutions and the behavior of turbulent fluids.
Fluid mixing occurs on two scales: the molecular level, where individual molecules move and interact, and the macroscopic level, where the fluid is treated as a whole. These two levels are closely interconnected. For example, stirring a liquid at the macroscopic level creates turbulence, which brings molecules into contact more frequently, speeding up the mixing process.
The interdependence of these levels makes modeling and understanding fluid mixing a complex challenge. In physics, natural sciences, and applied mathematics, mixing is typically explored through experiments and simulations. However, due to the complexity of many systems, these methods often hit their limits. This is where mathematical theory, particularly differential equations, becomes crucial.
Professor Crippa’s group specializes in studying differential equations derived from fluid dynamics models. These equations link changes in physical properties like density and velocity of the fluid over time and space. As a result, partial differential equations (PDEs) can capture the intricate behaviors of fluids, including chaotic and turbulent patterns. Since exact solutions to these equations are rarely achievable, even with advanced computational methods, rigorous mathematical analysis is crucial for understanding fluid behaviors in complex situations. Professor Crippa’s research provides highly precise predictions of such phenomena, even in very complex scenarios.
At the University of Basel, Professor Crippa’s team has made significant contributions to understanding fluid mixing. In their collaborative work, they have established analytical lower bounds on the mixing rate and assessed the effectiveness of these bounds. This provides an analytical, rigorous answer to the following question: how rapidly can a fluid get mixed if there is a limitation to the energy available for the stirring process?
Their research also draws connections to the theory of scalar turbulence, a concept that dates back to the pioneering work of physicists Obukhov and Corrsin in the 1950s.
The review paper “Mixing in Incompressible Flows: Transport, Dissipation, and Their Interplay” was published in the prestigious journal Notices of the American Mathematical Society, a widely-read, generalist journal that reaches mathematicians across all fields. Publication in Notices is a significant achievement, recognizing the authors as leaders in their field.
Original publication:Mixing in Incompressible Flows: Transport, Dissipation and Their Interplay, by M. Coti Zelati, G. Crippa, G. Iyer, and A. L. Mazzucato. Preprint: arXiv:2308.00358. Notices of the American Mathematical Society, Vol. 71, May 2024, No. 5, pp. 593–604.