Sum-of-squares of polynomials approach to nonlinear stability of fluid flows: an example of application

D. Huang, S. Chernyshenko, P. J. Goulart, D. Lasagna, O. Tutty and F. Fuentes

@article{HCGetal:2015,
  author = {D. Huang and S. Chernyshenko and P. J. Goulart and D. Lasagna and O. Tutty and F. Fuentes},
  title = {Sum-of-squares of polynomials approach to nonlinear stability of fluid flows: an example of application},
  journal = {Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences},
  publisher = {The Royal Society},
  year = {2015},
  volume = {471},
  number = {2183},
  url = {http://rspa.royalsocietypublishing.org/content/471/2183/20150622},
  doi = {10.1098/rspa.2015.0622}
}

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{textendash}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.