title: Study of the instability of the Poiseuille flow using a thermodynamic formalism.
abstract: The stability of the plane Poiseuille flow is analyzed using a thermodynamic formalism by considering the deterministic Navier-Stokes equation with Gaussian random initial data. A unique critical Reynolds number, Rec ≈ 2,332, at which the probability of observing puffs in the solution changes from 0 to 1, is numerically demonstrated to exist in the thermodynamic limit and is found to be independent of the noise amplitude. Using the puff density as the macrostate variable, the free energy of such a system is computed and analyzed. The puff density approaches zero as the critical Reynolds number is approached from above, signaling a continuous transition despite the fact that the bifurcation is subcritical for a finite-sized system. An action function is found for the probability of observing puffs in a small subregion of the flow, and this action function depends only on the Reynolds number. The strategy used here should be applicable to a wide range of other problems exhibiting subcritical instabilities.
authors.affiliation: The Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544;
authors.forename: Jianchun
authors.initials: J
authors.lastname: Wang
authors.affiliation: The Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544;
authors.forename: Qianxiao
authors.initials: Q
authors.lastname: Li
authors.affiliation: The Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544; School of Mathematical Sciences and Beijing International Center for Mathematical Research, Peking University, Beijing 100871, China; Department of Mathematics, Princeton University, Princeton, NJ 08544 weinan@math.princeton.edu.
authors.forename: Weinan
authors.initials: W
authors.lastname: E
mesh_terms: D017276:Friction; D007700:Kinetics; D060328:Magnetic Phenomena; D008962:Models, Theoretical; D012212:Rheology; D012879:Skin Physiological Phenomena; D012996:Solutions; D013816:Thermodynamics
chemical_list: D012996:Solutions
keywords: Poiseuille flow; free energy; phase transition; statistical mechanics; subcritical transition