Inertial, dimensional and stochastic effects on the instability of a thin liquid film

J. A. DIEZ; A. G. GONZÁLEZ

Florianopolis

Congreso; 10th Ibero-American Workshop on Complex Fluids 2015; 2015

Sociedade Brasileira de Fisica

We consider the effects of inertia as well as those of thermal fluctuations on the instability of a flat liquid film upon a solid substrate. The flow is mainly driven by capillary and intermolecular forces (van der Waals interaction).Firstly, we perform the linear stability analysis within the long wave approximation, which shows that the inclusion of inertia does not produce new regions of instability other than the one previously known from the usual lubrication case. The wavelength, lm, corresponding to the maximum growth, wm, and the critical (marginal) wavelength do not change at all. The most affected feature of the instability under an increase of the Laplace number is the noticeable decrease of the growth rates of the unstable modes. In order to put in evidence the effects of the bidimensional aspects of the flow (neglected in the long wave approximation), we also calculate the dispersion relation of the instability from the linearized version of the complete Navier-Stokes (N-S) equation. Unlike the long wave approximation, the bidimensional model shows that lm can vary significantly with inertia when the aspect ratio of the film is not sufficiently small. We also perform numerical simulations of the nonlinear N-S equations and analyze to which extent the linear predictions can be applied depending on both the amount of inertia involved and the aspect ratio of the film.Secondly, we study the effects of stochastic thermal fluctuations on the instability of the free surface. These fluctuations are represented as a standard Brownian motion that can be added to the deterministic equation for the film thickness within the lubrication approximation. We consider that while the noise term is white in time, it is coloured in space. This allows for the introduction of a finite correlation length in the description of the randomized intermolecular interaction. Together with the expected spatial periodicity of the flow, we find a dimensionless parameter, b, that accounts for the relative importance of the spatial correlation. We perform the linear stability analysis (LSA) of the film under the influence of both terms, and find the corresponding power spectra for the amplitudes of the normal modes of the instability. We compare this theoretical result with the numerical simulations of the complete non-linear problem, and find a good agreement for early times. For late times, we find that the stochastic LSA predictions on the dominant wavelength remains basically valid. We also use the theoretical spectra to fit experimental data from a nanometric melted copper film, and find the corresponding times of the evolution as well as the values of the parameter b.