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We find a quantitative approximation which explains the appearance and amplification of surface waves in a highly viscous fluid when it is subjected to vertical accelerations (Faraday's instability). Although stationary surface waves with frequency equal to half of the frequency of the excitation are observed in fluids of different kinematical viscosities we show here that the mechanism which produces the instability is very different for a highly viscous fluid as compared with a weakly viscous fluid. This is achieved by deriving an exact equation for the linear evolution of the surface which is non-local in time. We show that for a highly viscous fluid this equation becomes local and of second order and is then a Mathieu equation which is different from the one found for weak viscosity. Analysing the new equation we find an intimate relation with the Rayleigh–Taylor instability.

We derive an exact equation which is nonlocal in time for the linear evolution of the surface of a viscous fluid, and show that this equation becomes local and of second order in an interesting limit. We use our local equation to study Faraday's instability in a strongly dissipative regime and find a new scenario which is the analog of the Rayleigh-Taylor instability. Analytic and numerical calculations are presented for the threshold of the forcing and for the most unstable mode with impressive agreement with experiments and numerical work on the exact Navier-Stokes equations.

Thin fluid films can have surprising behavior depending on the boundary conditions enforced, the energy input and the specific Reynolds number of the fluid motion. Here we study the equations of motion for a thin fluid film with a free boundary and its other interface in contact with a solid wall. Although shear dissipation increases for thinner layers and the motion can generally be described in the limit as viscous, inertial modes can always be excited for a sufficiently high input of energy. We derive the minimal set of equations containing inertial effects in this strongly dissipative regime.

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The effect of additive noise on a static front that connects a stable homogeneous state with an also stable but spatially periodic state is studied. Numerical simulations show that noise induces front propagation. The conversion of random fluctuations into direct motion of the front's core is responsible of the propagation; noise prefers to create or remove a bump, because the necessary perturbations to nucleate or destroy a bump are different. From a prototype model with noise, we deduce an adequate equation for the front's core. An analytical expression for the front velocity is deduced, which is in good agreement with numerical simulations.

2014 Nous établissons les équations aux dérivées partielles non linéaires qui gouvernent la stabilité à grande échelle d'une structure cellulaire unidimensionnelle oscillante, apparaissant dans un système hors équilibre, invariant par translations d'espace et de temps, et par réflexion d'espace. Nous montrons l'existence d'une instabilité oscillatoire, conduisant à un régime quasipériodique possédant deux échelles spatiales distinctes. Abstract. 2014 We present the nonlinear phase equations describing the stability of a time-periodic onedimensional spatial pattern, that arises in a system which is invariant by space and time translations and space reflection symmetry. We show that a large scale oscillatory instability can occur, leading to a quasiperiodic temporal regime with two different spatial scales.