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Oscillations of a nominally hemispherical, inviscid drop on a solid plate are considered accounting for the contact line dynamics. Hocking boundary conditions hold on the contact line: the velocity of the contact line is proportional to the deviation of the contact angle from its equilibrium value. Natural oscillations of a drop are studied, and both eigenfrequencies and damping ratios are determined for the axisymmetric modes. The linear oscillations caused by normal vibration of the substrate are considered. Well-pronounced resonant phenomena are revealed. The nonlinear oscillations of a drop are studied.

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International audienceThe paper deals with the numerical investigation of the steady convective flow in a horizontal channel of rectangular cross-section subjected to a uniform longitudinal temperature gradient imposed at the walls. It is shown that at zero Prandtl number the solution of the problem corresponds to a plane-parallel flow along the channel axis. In this case, the fluid moves in the direction of the imposed temperature gradient in the upper part of the channel and in the opposite direction in the lower part. At non-zero values of the Prandtl number, such solution does not exist. At any small values of Pr all three components of the flow velocity differ from zero and in the channel cross-section four vortices develop. The direction of these vortices is such that the fluid moves from the centre to the periphery in the vertical direction and returns to the centre in the horizontal direction. The stability of these convective flows (uniform along the channel axis) with regard to small three-dimensional perturbations periodical in the direction of the channel axis is studied. It is shown that at low values of the Prandtl number the basic state loses its stability due to the steady hydrodynamic mode related to the development of vortices at the boundary of the counter flows. The growth of the Prandtl number results in the strong stabilization of this instability mode and, beyond a certain value of the Prandtl number depending on the cross-section aspect ratio, a new steady hydrodynamic instability mode becomes the most dangerous. This mode is characterized by the localization of the perturbations near the sidewalls of the channel. At still higher values of the Prandtl number, the spiral perturbations (rolls with axis parallel to the temperature gradient) become the most dangerous modes, at first the oscillatory spiral perturbations and then the Rayleigh-type steady spiral perturbations. The influence of the channel width on these different instabilities is also emphasized

The fractal mobile–immobile model (MIM) is intermediate between advection–dispersion (ADE) and fractal Fokker–Planck (FFKPE) equations. It involves two time derivatives, whose orders are 1 and γ (between 0 and 1) on the left-hand side, whereas all mentioned equations have identical right-hand sides. The fractal MIM model accounts for non-Fickian effects that occur when tracers spread in media because of through-flow, and can get trapped by immobile sites. The solid matrix of a porous material may contain such sites, so that non-Fickian spread is actually observed. Within the context of the fractal MIM model, we present a mapping that allows the computation of fluxes on the basis of the density of spreading particles. The mapping behaves as Fickian flux at early times, and tends to a fractional derivative at late times. By means of this mapping, we recast the fractal MIM model into conservative form, which is suitable to deal with sources and bounded domains. Mathematical proofs are illustrated by comparing the discretized fractal p.d.e. with Monte Carlo simulations.