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Publication# Noise-induced transitions past the onset of a steady symmetry-breaking bifurcation: The case of the sudden expansion

Abstract

We consider fluid flows, governed by the Navier-Stokes equations, subject to a steady symmetry-breaking bifurcation and forced by a weak noise acting on a slow timescale. By generalizing the multiple-scale weakly nonlinear expansion technique employed in the literature for the response of the Duffing oscillator, we rigorously derive a stochastically forced Stuart-Landau equation for the dominant symmetry-breaking mode. The probability density function of the solution, and of the escape time from one attractor to the other, are then determined by solving the associated Fokker-Planck equation. The validity of this reduced order model is tested on the flow past a sudden expansion for a given Reynolds number and different noise amplitudes. At a very low numerical cost, the statistics obtained from the amplitude equation accurately reproduce those of long-time direct numerical simulations.

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Stokes flow (named after George Gabriel Stokes), also named creeping flow or creeping motion, is a type of fluid flow where advective inertial forces are small compared with viscous forces. The Reynolds number is low, i.e. . This is a typical situation in flows where the fluid velocities are very slow, the viscosities are very large, or the length-scales of the flow are very small. Creeping flow was first studied to understand lubrication. In nature, this type of flow occurs in the swimming of microorganisms and sperm.

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