Abstract
With the goal of providing the first example of application of a recently proposed method, thus demonstrating its ability to give results in principle, global stability of a version of the rotating Couette flow is examined. The flow depends on the Reynolds number and a parameter characterizing the magnitude of the Coriolis force. By converting the original Navier–Stokes equations to a finite-dimensional uncertain dynamical system using a partial Galerkin expansion, high-degree polynomial Lyapunov functionals were found by sum-of-squares of polynomials optimization. It is demonstrated that the proposed method allows obtaining the exact global stability limit for this flow in a range of values of the parameter characterizing the Coriolis force. Outside this range a lower bound for the global stability limit was obtained, which is still better than the energy stability limit. In the course of the study, several results meaningful in the context of the method used were also obtained. Overall, the results obtained demonstrate the applicability of the recently proposed approach to global stability of the fluid flows. To the best of our knowledge, it is the first case in which global stability of a fluid flow has been proved by a generic method for the value of a Reynolds number greater than that which could be achieved with the energy stability approach.
Highlights
Hydrodynamic stability is the field that investigates the transient effects of an initial perturbation of a known steady flow
Using Lyapunov stability theory, a steady flow can be proved to be stable with respect to perturbations of arbitrary amplitude by constructing a Lyapunov functional V[u], which is a positive-definite functional of the velocity perturbation u that decays monotonically on any non-zero solution u(t, x) of the Navier–Stokes equations [3]
The Navier–Stokes equations are first reduced to a finite-dimensional uncertain dynamical system, that is a system of ordinary differential equations (ODEs) with right-hand side containing terms for which only bounds, but not exact expressions, are available
Summary
Hydrodynamic stability is the field that investigates the transient effects of an initial perturbation of a known steady flow. A method was proposed by Goulart & Chernyshenko [6] for exploiting the sumof-squares (SOS) decomposition [7,8] to construct polynomial Lyapunov functionals differing from E, extending the range of Re in which the flow can be proved to be globally stable. In this approach, the Navier–Stokes equations are first reduced to a finite-dimensional uncertain dynamical system, that is a system of ordinary differential equations (ODEs) with right-hand side containing terms for which only bounds, but not exact expressions, are available. To the best of our knowledge, this study is the first case in which global stability of a fluid flow has been proved by a generic method for the value of a Reynolds number greater than that which could be achieved with the energy stability approach
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callMore From: Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
