Abstract
Thin liquid films are ubiquitous in natural phenomena and technological applications. They have been extensively studied via deterministic hydrodynamic equations, but thermal fluctuations often play a crucial role that needs to be understood. An example of this is dewetting, which involves the rupture of a thin liquid film and the formation of droplets. Such a process is thermally activated and requires fluctuations to be taken into account self-consistently. In this work we present an analytical and numerical study of a stochastic thin-film equation derived from first principles. Following a brief review of the derivation, we scrutinise the behaviour of the equation in the limit of perfectly correlated noise along the wall-normal direction, as opposed to the perfectly uncorrelated limit studied by Grün et al. (J Stat Phys 122(6):1261–1291, 2006). We also present a numerical scheme based on a spectral collocation method, which is then utilised to simulate the stochastic thin-film equation. This scheme seems to be very convenient for numerical studies of the stochastic thin-film equation, since it makes it easier to select the frequency modes of the noise (following the spirit of the long-wave approximation). With our numerical scheme we explore the fluctuating dynamics of the thin film and the behaviour of its free energy in the vicinity of rupture. Finally, we study the effect of the noise intensity on the rupture time, using a large number of sample paths as compared to previous studies.
Highlights
A thin liquid film can be understood as a layer of liquid with thickness ranging from fractions of a nanometer to several micrometers, typically resting or flowing on a substrate
It should be noted that the phenomenological origin of fluctuating hydrodynamics (FH) should not be seen as a weakness, as a great deal of theoretical work has been successfully undertaken to derive it from first principles since the days of Landau and Lifshitz
The starting point of the derivation is the stochastic Navier–Stokes equations for the velocity field of an incompresible fluid on a planar horizontal solid substrate. The relation of these equations with an underlying Hamiltonian dynamics of the constituent particles of the fluid is sketched in Fig. 1, where we highlight our contribution to the state-of-the-art and summarise all possible model equations obtained from the original Hamiltonian system
Summary
The fluctuating dynamics of a two-dimensional thin film of Newtonian fluid flowing on a horizontal substrate can be described by the incompressible Navier–Stokes equations [37] (see Fig. 1):. The operator Dt = (∂t + u · ∇) = (∂t + u∂x + v∂y) is the convective derivative, and S is the fluctuating stress tensor, which represents the effect of random thermal fluctuations on the film dynamics. Whereas at the fluid-air interface y = h(x; t), with h being the film height at the position x and time t, we apply the stress-balance boundary condition:. Where σ is the viscous stress tensor, κ is the mean curvature of the surface, γ is the surface tension coefficient, nis the normal vector to the interface, and. We apply the following kinematic boundary condition at the interface:. Which states that a fluid particle on the interface will remain there for all times, preventing matter from leaving the interface via e.g. evaporation or any other mechanism
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