An analytical and computational study of the incompressible Toner-Tu Equations
arXiv:2206.00414 · doi:10.1016/j.physd.2022.133594
Abstract
The incompressible Toner-Tu (ITT) partial differential equations (PDEs) are an important example of a set of active-fluid PDEs. While they share certain properties with the Navier-Stokes equations (NSEs), such as the same scaling invariance, there are also important differences. The NSEs are usually considered in either the decaying or the additively forced cases, whereas the ITT equations have no additive forcing. Instead, they include a linear, activity term $α\bu$ ($\bu$ is the velocity field) which pumps energy into the system, but also a negative $\bu|\bu|^{2}$-term which provides a platform for either frozen or statistically steady states. Taken together, these differences make the ITT equations an intriguing candidate for study using a combination of PDE analysis and pseudo-spectral direct numerical simulations (DNSs). In the case, we have established global regularity of solutions, but we have also shown the existence of bounded hierarchies of weighted, time-averaged norms of both higher derivatives and higher moments of the velocity field. Similar bounded hierarchies for Leray-type weak solutions have also been established in the case. We present results for these norms from our DNSs in both and , and contrast them with their Navier-Stokes counterparts.
References in corpus (23)
- Novel type of phase transition in a system of self-driven particles
- Meso-scale turbulence in living fluids
- Fluid dynamics of bacterial turbulence
- Active Turbulence
- Spontaneously ordered motion of self-propelled particles
- Continuous data assimilation for the three-dimensional Brinkman-Forchheimer-extended Darcy model
- Universal scaling of active nematic turbulence
- Surprising mappings of 2D polar active fluids to 2D soap and 1D sandblasting
- Phase transition to large scale coherent structures in 2d active matter turbulence
- Spontaneous mirror-symmetry breaking induces inverse energy cascade in 3D active fluids
- Anomalous diffusion and Lévy walks distinguish active from inertial turbulence
- Vorticity moments in four numerical simulations of the 3D Navier-Stokes equations
- Condensate formation and multiscale dynamics in two-dimensional active suspensions
- Incompressible polar active fluids in the moving phase
- Regimes of nonlinear depletion and regularity in the 3D Navier-Stokes equations
- Coarsening in the two-dimensional incompressible Toner-Tu equation: Signatures of turbulence
- A correspondence between the multifractal model of turbulence and the Navier-Stokes equations
- Weak and strong solutions of the Navier-Stokes equations and their relation to a chessboard of convergent inverse length scales
- Intermittency, cascades and thin sets in three-dimensional Navier-Stokes turbulence
- Depletion of Nonlinearity in Magnetohydrodynamic Turbulence: Insights from Analysis and Simulations
- The role of BKM-type theorems in Euler, Navier-Stokes and Cahn-Hilliard-Navier-Stokes analysis
- Phase ordering, topological defects, and turbulence in the 3D incompressible Toner-Tu equation
- Irreversiblity in Bacterial Turbulence: Insights from the Mean-Bacterial-Velocity Model