The Cahn–Hilliard model is increasingly often being used in combination with the incompressible Navier–Stokes equation to describe unsteady binary fluids in a variety of applications ranging from turbulent two-phase flows to microfluidics. The thickness of the interface between the two bulk fluids and the mobility are the main parameters of the model. For real fluids they are usually too small to be directly used in numerical simulations. Several authors proposed criteria for the proper choice of interface thickness and mobility in order to reach the so-called ‘sharp-interface limit’. In this paper the problem is approached by a formal asymptotic expansion of the governing equations. It is shown that the mobility is an effective parameter to be chosen proportional to the square of the interface thickness. The theoretical results are confirmed by numerical simulations for two prototypal flows, namely capillary waves riding the interface and droplets coalescence. The numerical analysis of two different physical problems confirms the theoretical findings and establishes an optimal relationship between the effective parameters of the model.

We have studied the motion of liquid drops on an inclined plate subject to vertical vibrations. The liquids comprised distilled water and different aqueous solutions of glycerol, ethanol and isopropanol spanning the range 1-39 mm 2 s −1 in kinematic viscosities and 40-72 mN m −1 in surface tension. At sufficiently low oscillating amplitudes, the drops are always pinned to the surface. Vibrating the plate above a certain amplitude yields sliding of the drop. Further increasing the oscillating amplitude drives the drop upward against gravity. In the case of the most hydrophilic aqueous solutions, this motion is not observed and the drop only slides downward. Images taken with a fast camera show that the drop profile evolves in a different way during sliding and climbing. In particular, the climbing drop experiences a much bigger variation in its profile during an oscillating period. Complementary numerical simulations of 2D drops based on a diffuse interface approach confirm the experimental findings. The overall qualitative behavior is reproduced suggesting that the contact line pinning due to contact angle hysteresis is not necessary to explain the drop climbing.

Vapor bubbles are formed in liquids by two mechanisms: evaporation (temperature above the boiling threshold) and cavitation (pressure below the vapor pressure). The liquid resists in these metastable (overheating and tensile, respectively) states for a long time since bubble nucleation is an activated process that needs to surmount the free energy barrier separating the liquid and the vapor states. The bubble nucleation rate is difficult to assess and, typically, only for extremely small systems treated at atomistic level of detail. In this work a powerful approach, based on a continuum diffuse interface modeling of the two-phase fluid embedded with thermal fluctuations (Fluctuating Hydrodynamics) is exploited to study the nucleation process in homogeneous conditions, evaluating the bubble nucleation rates and following the long term dynamics of the metastable system, up to the bubble coalescence and expansion stages. In comparison with more classical approaches, this methodology allows on the one hand to deal with much larger systems observed for a much longer times than possible with even the most advanced atomistic models. On the other it extends continuum formulations to thermally activated processes, impossible to deal with in a purely determinist setting.

In this Letter a diffuse-interface model featuring phase change, transition to supercritical conditions, thermal conduction, compressibility effects and shock wave propagation is exploited to deal with the dynamics of a cavitation bubble. At variance with previous descriptions, the model is uniformly valid for all phases (liquid, vapor and supercritical) and phase transitions involved, allowing to describe the non-equilibrium processes ongoing during the collapse. As consequence of this unitary description, rather unexpectedly for pure vapor bubbles, the numerical experiments show that the collapse is accompanied by the emission of a strong shock wave in the liquid and by the oscillation of the bubble that periodically disappears and reappears, due to transition to super/sub critical conditions. The mechanism of shock wave formation is strongly related to the transition of the vapor to supercritical state, with a progressive steepening of the compression wave to form the shock which is eventually reflected as an outward propagating wave in the liquid.Vapor bubble collapse is a fascinating classical problem [1] involving vapor-liquid phase transition and extreme pressures and temperatures [2,3]. Typical experiments concern ultra-fast imaging and the analysis of light and sound emitted after the collapse [4][5][6], see also the review [7]. Both free cavitation bubbles and nano bubbles at solid walls are increasingly investigated [8][9][10]. In ordinary conditions, gas bubble nucleation is associated with the metastability of the mixture of liquid and dissolved gas which, once the free energy barrier between the two states is overcome and the critical nucleus is formed, evolves toward a finite size bubble. The same mechanism is at work in forming pure vapor bubbles when the (ultra pure) liquid is kept in metastable conditions [11], i.e. its pressure is below the equilibrium vapor pressure at the given temperature. In these conditions, away from solid walls (see e.g. [12] for the role of asperities on solid surfaces as a catalyst of bubble nucleation), local density fluctuations can generate the critical vapor nucleus from which the eventual bubble is formed.Intermingled phenomenologies, [13,14], such as interface dynamics [15,16], thermodynamics of phase change [17], and dissolved gas diffusion [18], are a challenge to theoretical modeling of cavitation. The available descriptions combine two distinct adjoining regions, liquid and vapor phase, respectively, with vapor pressure taken to be the saturation pressure [19] and phase transition accounted for through suitable kinetic equations and latent heat release [18].Contrary to available models, the diffuse interface approach discussed in the present Letter encompasses all phases (liquid, vapor and supercritical) and phase transitions involved, embedding capillary forces, compressibility effects and shock wave propagation. The approach enables an unprecedented analysis of collapse, where the bubble interface speed may exceed the speed of sound. This leads to the form...

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