Energy dissipation of Oldroyd-B fluids in plane Couette flow
File(s)
Author(s)
Chen, Grange
Chernyshenko, Sergei
Wynn, Andrew
Type
Journal Article
Abstract
This paper establishes a rigorous upper bound on the infinite-time-averaged energy dissipation rate of Oldroyd-B fluids in plane Couette flow. The bound depends only on
system parameters—the Reynolds number, Weissenberg number, and viscosity ratio—and applies to all steady and unsteady solutions within a certain region in parameter space. The bound is proven by extending the ‘background flow method’ to the case where the system energy is no longer a quadratic functional of the underlying flow fields, and is obtained by using a non-polynomial auxiliary functional related to the free polymeric energy. Within the range of the flow parameters in which the steady solution is known to be globally stable, the dissipation rate of the steady flow is recovered, and in the Newtonian limit the result reduces to the best-known bound for Newtonian Couette flow. Our analysis also identifies a range of parameters for which the total energy of the viscoelastic flow must be bounded, thus ruling out the possibility of energy blow-ups in these situations.
system parameters—the Reynolds number, Weissenberg number, and viscosity ratio—and applies to all steady and unsteady solutions within a certain region in parameter space. The bound is proven by extending the ‘background flow method’ to the case where the system energy is no longer a quadratic functional of the underlying flow fields, and is obtained by using a non-polynomial auxiliary functional related to the free polymeric energy. Within the range of the flow parameters in which the steady solution is known to be globally stable, the dissipation rate of the steady flow is recovered, and in the Newtonian limit the result reduces to the best-known bound for Newtonian Couette flow. Our analysis also identifies a range of parameters for which the total energy of the viscoelastic flow must be bounded, thus ruling out the possibility of energy blow-ups in these situations.
Date Issued
2026-07-10
Date Acceptance
2026-05-25
Citation
Journal of Fluid Mechanics, 2026, 1038
ISSN
0022-1120
Publisher
Cambridge University Press
Journal / Book Title
Journal of Fluid Mechanics
Volume
1038
Copyright Statement
© The Author(s), 2026. Published by Cambridge University Press. This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.
License URL
Identifier
10.1017/jfm.2026.11731
Publication Status
Published
Article Number
ARTN A29
Date Publish Online
2026-07-07