An analytical and computational study of the incompressible Toner-Tu Equations
File(s)
Accepted version
Author(s)
Gibbon, John D
Kiran, Kolluru Venkata
Padhan, Nadia Bihari
Pandit, Rahul
Type
Journal Article
Abstract
We consider the incompressible Toner-Tu (ITT) partial differential equations (PDEs), which are
an important example of a set of active-fluid PDEs. They share many properties with the NavierStokes equations (NSEs) but there are also important differences. The NSEs are usually considered either in the decaying or the additively forced cases, whereas the ITT equations have
no additive forcing, but instead have a linear, activity term αu (with u the velocity field), which
pumps energy into the system; they also have a negative u|u|
2
-term that stabilizes growth and provides a platform for either frozen or statistically steady states. These differences make the ITT
equations an intriguing candidate for study using a combination of PDE analysis and pseudospectral direct numerical simulations (DNSs). In the d = 2 case, we have established global
regularity of solutions. 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. For
d = 3 there are equivalent bounded hierarchies for Leray-type weak solutions. We present results for these norms from our DNSs in both d = 2 and d = 3, and contrast them with their
counterparts for the d = 3 NSEs.
an important example of a set of active-fluid PDEs. They share many properties with the NavierStokes equations (NSEs) but there are also important differences. The NSEs are usually considered either in the decaying or the additively forced cases, whereas the ITT equations have
no additive forcing, but instead have a linear, activity term αu (with u the velocity field), which
pumps energy into the system; they also have a negative u|u|
2
-term that stabilizes growth and provides a platform for either frozen or statistically steady states. These differences make the ITT
equations an intriguing candidate for study using a combination of PDE analysis and pseudospectral direct numerical simulations (DNSs). In the d = 2 case, we have established global
regularity of solutions. 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. For
d = 3 there are equivalent bounded hierarchies for Leray-type weak solutions. We present results for these norms from our DNSs in both d = 2 and d = 3, and contrast them with their
counterparts for the d = 3 NSEs.
Date Issued
2023-02-01
Date Acceptance
2022-11-13
Citation
Physica D : Non-linear phenomena, 2023, 444
ISSN
0167-2789
Publisher
Elsevier
Journal / Book Title
Physica D : Non-linear phenomena
Volume
444
Copyright Statement
Copyright © Elsevier Ltd. All rights reserved. This manuscript version is made available under the CC-BY-NC-ND 4.0 license https://creativecommons.org/licenses/by-nc-nd/4.0/
Subjects
Active matter
ATTRACTORS
CONTINUOUS DEPENDENCE
EXISTENCE
FORCHHEIMER
Mathematics
Mathematics, Applied
Navier-Stokes
NAVIER-STOKES EQUATIONS
PDE
Physical Sciences
Physics
Physics, Fluids & Plasmas
Physics, Mathematical
Physics, Multidisciplinary
Science & Technology
Toner-Tu
TURBULENCE
WEAK
Publication Status
Published
Article Number
133594
Date Publish Online
2022-11-23