Nonlinear Monotone Energy Stability of Plane Shear Flows: Joseph or Orr Critical Thresholds?

SIAM Journal on Applied Mathematics, Volume 84, Issue 1, Page 60-74, February 2024.
Abstract. Critical Reynolds numbers for the monotone exponential energy stability of Couette and Poiseuille plane flows were obtained by Orr in 1907 [Proc. Roy. Irish Acad. A, 27 (1907), pp. 9–68, 69–138] in a famous paper, and by Joseph in 1966 [J. Fluid Mech., 33 (1966), pp. 617–621], Joseph and Carmi in 1969 [Quart. Appl. Math., 26 (1969), pp. 575–579], and Busse in 1972 [Arch. Ration. Mech. Anal., 47 (1972), pp. 28–35]. All these authors obtained their results applying variational methods to compute the maximum of a functional ratio derived from the Reynolds–Orr energy identity. Orr and Joseph obtained different results; for instance, in the Couette case Orr computed the critical Reynolds value of 44.3 (on spanwise perturbations) and Joseph 20.65 (on streamwise perturbations). Recently in [P. Falsaperla, G. Mulone, and C. Perrone, Eur. J. Mech. B Fluids, 93 (2022), pp. 93–100], the authors conjectured that the search of the maximum should be restricted to a subspace of the space of kinematically admissible perturbations. With this conjecture, the critical nonlinear energy Reynolds number was found among spanwise perturbations (a Squire theorem for nonlinear systems). With a direct proof and an appropriate and original decomposition of the dissipation terms in the Reynolds–Orr identity we show the validity of this conjecture in the space of three-dimensional perturbations.