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International audience; We consider the flow past a sphere held at a fixed position in a uniform incoming flow but free to rotate around a transverse axis. A steady pitchfork bifurcation is reported to take place at a threshold \(Re^\mathrm{OS}=206\) leading to a state with zero torque but nonzero lift. Numerical simulations allow to characterize this state up to \(Re\approx 270\) and confirm that it substantially differs from the steady-state solution which exists in the wake of a fixed, non-rotating sphere beyond the threshold \(Re^\mathrm{SS}=212\). A weakly nonlinear analysis is carried out and is shown to successfully reproduce the results and to give substantial improvement over a previous analysis (Fabre et al. in J Fluid Mech 707:24–36, 2012). The connection between the present problem and that of a sphere in free fall following an oblique, steady (OS) path is also discussed.

Among hyperbolic Initial Boundary Value Problems (IBVP), those coming from a variational principle ‘generically’ admit linear surface waves, as was shown by Serre (J Funct Anal 236(2):409–446, 2006). At the weakly nonlinear level, the behavior of surface waves is expected to be governed by an amplitude equation that can be derived by means of a formal asymptotic expansion. Amplitude equations for weakly nonlinear surface waves were introduced by Lardner (Int J Eng Sci 21(11):1331–1342, 1983), Parker and co-workers (J Elast 15(4):389–426, 1985) in the framework of elasticity, and by Hunter (Nonlinear surface waves. In: Current progress in hyberbolic systems: Riemann problems and computations (Brunswick, 1988). Contemporary mathematics, vol 100. American Mathematical Society, pp 185–202, 1989) for abstract hyperbolic problems. They consist of nonlocal evolution equations involving a complicated, bilinear Fourier multiplier in the direction of propagation along the boundary. It was shown by the authors in an earlier work (Benzoni-Gavage and Coulombel Arch Ration Mech Anal 205(3):871–925, 2012) that this multiplier, or kernel, inherits some algebraic properties from the original IBVP. These properties are crucial for the (local) well-posedness of the amplitude equation, as shown together with Tzvetkov (Adv Math, 2011). Properties of amplitude equations are revisited here in a somehow simpler way, for surface waves in a variational setting. Applications include various physical models, from elasticity of course to the director-field system for liquid crystals introduced by Saxton (Dynamic instability of the liquid crystal director. In: Current progress in hyperbolic systems: Riemann problems and computations (Brunswick, 1988). Contemporary mathematics, vol 100. American Mathematical Society, Providence, pp 325–330, 1989) and studied by Austria and Hunter (Commun Inf Syst 13(1):3–43, 2013). Similar properties are eventually shown for the amplitude equation associated with surface waves at reversible phase boundaries in compressible fluids, thus completing a work initiated by Benzoni-Gavage and Rosini (Comput Math Appl 57(3–4):1463–1484, 2009).

International audience; Gravity- or buoyancy-driven bodies moving in a slightly viscous fluid frequently follow fluttering or helical paths. Current models of such systems are largely empirical and fail to predict several of the key features of their evolution, especially close to the onset of path instability. Here, using a weakly nonlinear expansion of the full set of governing equations, we present a new generic reduced-order model based on a pair of amplitude equations with exact coefficients that drive the evolution of the first pair of unstable modes. We show that the predictions of this model for the style (e.g., fluttering or spiraling) and characteristics (e.g., frequency and maximum inclination angle) of path oscillations compare well with various recent data for both solid disks and air bubbles.