On a vertically vibrating fluid interface, a droplet can remain bouncing indefinitely. When approaching the Faraday instability onset, the droplet couples to the wave it generates and starts propagating horizontally. The resulting wave–particle association, called a walker, was shown previously to have remarkable dynamical properties, reminiscent of quantum behaviours. In the present article, the nature of a walker's wave field is investigated experimentally, numerically and theoretically. It is shown to result from the superposition of waves emitted by the droplet collisions with the interface. A single impact is studied experimentally and in a fluid mechanics theoretical approach. It is shown that each shock emits a radial travelling wave, leaving behind a localized mode of slowly decaying Faraday standing waves. As it moves, the walker keeps generating waves and the global structure of the wave field results from the linear superposition of the waves generated along the recent trajectory. For rectilinear trajectories, this results in a Fresnel interference pattern of the global wave field. Since the droplet moves due to its interaction with the distorted interface, this means that it is guided by a pilot wave that contains a path memory. Through this wave-mediated memory, the past as well as the environment determines the walker's present motion.

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10 p.We propose a quantitative criterion for the merging of a pair of equal two-dimensional co-rotating vortices. A cross-validation between experimental and theoretical analyses is performed. Experimental vortices are generated by the roll-up of a vortex sheet originating from the identical and impulsive rotation of two plates. The phenomenon is then followed up in time until a rapid pairing transition occurs for which critical parameters are measured. In the theoretical approach, the nonlinear Euler solution representing a pair of equal vortices is computed for various nonuniform vorticity distributions. The stability analysis of such a configuration then provides critical values for the onset of merging. From this data set, a criterion depending on global impulse quantities is extracted for different shapes of the vorticity distribution. This theoretical statement agrees well with our experimentally based criterion

In the framework of linear stability theory, we analyze how a liquid-gas mixing layer is affected by several parameters: viscosity ratio, density ratio, and several length scales. These scales reflect the presence of a velocity defect induced by the wake behind the splitter plate and the presence of boundary layers which develop ahead of the plate trailing edge. Incorporating such effects, we compute the various temporal and spatial instability modes and identify their driving instability mechanism based on their Reynolds number dependence, spatial structure, and energy budget. It is examined how the velocity defect modifies the temporal and the spatial stability properties. In addition, the transition from convective to absolute instability occurs at lower velocity contrast between gas and liquid free streams when a defect is present. This transition is also promoted by surface tension. Compared to inviscid stability computations, our spatial stability analysis displays a better agreement with measured growth rates obtained in two recent air-water experiments.

The main objective of the study is to examine the spatio-temporal instability properties of the Batchelor q-vortex, as a function of swirl ratio q and external axial flow parameter a. The inviscid dispersion relation between complex axial wave number and frequency is determined by numerical integration of the Howard–Gupta ordinary differential equation. The absolute-convective nature of the instability is then ascertained by application of the Briggs–Bers zero-group-velocity criterion. A moderate amount of swirl is found to promote the onset of absolute instability. In the case of wakes, transition from convective to absolute instability always takes place via the helical mode of azimuthal wave number m=−1. For sufficiently large swirl, co-flowing wakes become absolutely unstable. In the case of jets, transition from absolute to convective instability occurs through various helical modes, the transitional azimuthal wave number m being negative but sensitive to increasing swirl. For sufficiently large swirl, weakly co-flowing jets become absolutely unstable. These results are in good qualitative and quantitative agreement with those obtained by Delbende et al. through a direct numerical simulation of the linear response. Finally, the spatial (complex axial wave number, real frequency) instability characteristics are illustrated for the case of zero-external flow swirling jets.

The convective instability in a plane liquid layer with time-dependent temperature profile is investigated by means of a general method suitable for linear stability analysis of an unsteady basic flow. The method is based on a non-normal approach, and predicts the onset of instability, critical wavenumber and time. The method is applied to transient Rayleigh-Bénard-Marangoni convection due to cooling by evaporation. Numerical results as well as theoretical scalings for the critical parameters as function of the Biot number are presented for the limiting cases of purely buoyancy-driven and purely surface-tensiondriven convection. Critical parameters from calculations are in good agreement with those from experiments on drying polymer solutions, where the surface cooling is induced by solvent evaporation.

Model identifications based on orbit tracking methods are here extended to stochastic differential equations. In the present approach, deterministic and statistical features are introduced via the time evolution of ensemble averages and variances. The aforementioned quantities are shown to follow deterministic equations, which are explicitly written within a linear as well as a weakly nonlinear approximation. Based on such equations and the observed time series, a cost function is defined. Its minimization by simulated annealing or backpropagation algorithms then yields a set of best-fit parameters. This procedure is successfully applied for various sampling time intervals, on a stochastic Lorenz system.

We discuss a numerical scheme to solve the continuum Kardar-Parisi-Zhang equation in generic spatial dimensions. It is based on a momentum-space discretization of the continuum equation and on a pseudo-spectral approximation of the non-linear term. The method is tested in (1 + 1)-and (2 + 1)-dimensions, where it is shown to reproduce the current most reliable estimates of the critical exponents based on Restricted Solid-on-Solid simulations. In particular it allows the computations of various correlation and structure functions with high degree of numerical accuracy. Some deficiencies which are common to all previously used finite-difference schemes are pointed out and the usefulness of the present approach in this respect is discussed.