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An analytical and computational study of the incompressible Toner–Tu Equations

Gibbon, JD and Kiran, KV and Padhan, NB and Pandit, R (2023) An analytical and computational study of the incompressible Toner–Tu Equations. In: Physica D: Nonlinear Phenomena, 444 .

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Official URL: https://doi.org/10.1016/j.physd.2022.133594


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 αu(u is the velocity field) which pumps energy into the system, but also a negative u|u|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 d=2 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 d=3 case. We present results for these norms from our DNSs in both d=2 and d=3, and contrast them with their Navier–Stokes counterparts. © 2022 Elsevier B.V.

Item Type: Journal Article
Publication: Physica D: Nonlinear Phenomena
Publisher: Elsevier B.V.
Additional Information: The copyright for this article belongs to Elsevier
Keywords: Additives, Active matter; Analytical studies; Computational studies; Direct-numerical-simulation; Navier Stokes; Navier-Stokes equation; Property; Scaling invariance; Toner�tu; Velocity field, Navier Stokes equations
Department/Centre: Division of Physical & Mathematical Sciences > Physics
Date Deposited: 16 Jan 2023 09:04
Last Modified: 16 Jan 2023 09:04
URI: https://eprints.iisc.ac.in/id/eprint/79179

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