Abstract
The problem of low Reynolds number turbulence in active nematic fluids is theoretically addressed. Using numerical simulations, I demonstrate that an incompressible turbulent flow, in two-dimensional active nematics, consists of an ensemble of vortices whose areas are exponentially distributed within a range of scales. Building on this evidence, I construct a mean-field theory of active turbulence by which several measurable quantities, including the spectral densities and the correlation functions, can be analytically calculated. Because of the profound connection between the flow geometry and the topological properties of the nematic director, the theory sheds light on the mechanisms leading to the proliferation of topological defects in active nematics and provides a number of testable predictions. A hypothesis, inspired by Onsager’s statistical hydrodynamics, is finally introduced to account for the equilibrium probability distribution of the vortex sizes.
- Received 24 September 2014
DOI:https://doi.org/10.1103/PhysRevX.5.031003
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Published by the American Physical Society
Popular Summary
Turbulence is ubiquitous in the macroscopic world and represents a hallmark of complexity in classical physical systems. In viscous fluids, turbulence sets in when the Reynolds number, which represents the ratio between inertial and viscous forces, is on the order of thousands. In the inertialess world of “active fluids” such as bacterial suspensions or cytoskeletal mixtures, turbulence may spontaneously occur at Reynolds numbers as small as in the absence of external forces. Here, we use a combination of numerical and analytical calculations to show that low Reynolds number turbulence in two-dimensional active nematic liquid crystals consists of an ensemble of vortices. The geometry of these vortices is strongly correlated with the topological structure of the nematic phase.
We demonstrate that, as for inertial turbulence, low Reynolds number turbulence in active nematics is in fact a multiscale phenomenon characterized by the formation of vortices spanning a range of length scales. Within this active range, the areas of the vortices are exponentially distributed; their vorticity is approximately constant. Building on these observations, we then formulate a mean-field theory that makes it possible to analytically calculate several measurable quantities, including the mean kinetic energy and related enstrophy, their corresponding spectral densities, and the velocity and vorticity correlation functions. We elucidate the connection between the topological structure of the nematic phase and the geometry of the flow using a quantitative description of defect statistics.
Our hypothesis, inspired by Onsager’s statistical hydrodynamics, accounts for the probability distribution of vortex sizes.
