127 research outputs found
Rigorous coherent-structure theory for falling liquid films: Viscous dispersion effects on bound-state formation and self-organization
We examine the interaction of two-dimensional solitary pulses on falling
liquid films. We make use of the second-order model derived by Ruyer-Quil and
Manneville [Eur. Phys. J. B 6, 277 (1998); Eur. Phys. J. B 15, 357 (2000);
Phys. Fluids 14, 170 (2002)] by combining the long-wave approximation with a
weighted residuals technique. The model includes (second-order) viscous
dispersion effects which originate from the streamwise momentum equation and
tangential stress balance. These effects play a dispersive role that primarily
influences the shape of the capillary ripples in front of the solitary pulses.
We show that different physical parameters, such as surface tension and
viscosity, play a crucial role in the interaction between solitary pulses
giving rise eventually to the formation of bound states consisting of two or
more pulses separated by well-defined distances and travelling at the same
velocity. By developing a rigorous coherent-structure theory, we are able to
theoretically predict the pulse-separation distances for which bound states are
formed. Viscous dispersion affects the distances at which bound states are
observed. We show that the theory is in very good agreement with computations
of the second-order model. We also demonstrate that the presence of bound
states allows the film free surface to reach a self-organized state that can be
statistically described in terms of a gas of solitary waves separated by a
typical mean distance and characterized by a typical density
A new framework for extracting coarse-grained models from time series with multiscale structure
In many applications it is desirable to infer coarse-grained models from
observational data. The observed process often corresponds only to a few
selected degrees of freedom of a high-dimensional dynamical system with
multiple time scales. In this work we consider the inference problem of
identifying an appropriate coarse-grained model from a single time series of a
multiscale system. It is known that estimators such as the maximum likelihood
estimator or the quadratic variation of the path estimator can be strongly
biased in this setting. Here we present a novel parametric inference
methodology for problems with linear parameter dependency that does not suffer
from this drawback. Furthermore, we demonstrate through a wide spectrum of
examples that our methodology can be used to derive appropriate coarse-grained
models from time series of partial observations of a multiscale system in an
effective and systematic fashion
Rate of Convergence of Phase Field Equations in Strongly Heterogeneous Media towards their Homogenized Limit
We study phase field equations based on the diffuse-interface approximation
of general homogeneous free energy densities showing different local minima of
possible equilibrium configurations in perforated/porous domains. The study of
such free energies in homogeneous environments found a broad interest over the
last decades and hence is now widely accepted and applied in both science and
engineering. Here, we focus on strongly heterogeneous materials with
perforations such as porous media. To the best of our knowledge, we present a
general formal derivation of upscaled phase field equations for arbitrary free
energy densities and give a rigorous justification by error estimates for a
broad class of polynomial free energies. The error between the effective
macroscopic solution of the new upscaled formulation and the solution of the
microscopic phase field problem is of order for a material given
characteristic heterogeneity . Our new, effective, and reliable
macroscopic porous media formulation of general phase field equations opens new
modelling directions and computational perspectives for interfacial transport
in strongly heterogeneous environments
Two-dimensional droplet spreading over random topographical substrates
We examine theoretically the effects of random topographical substrates on
the motion of two-dimensional droplets via appropriate statistical approaches.
Different random substrate families are represented as stationary random
functions. The variance of the droplet shift at both early times and in the
long-time limit is deduced and the droplet footprint is found to be a normal
random variable at all times. It is shown that substrate roughness decreases
droplet wetting, illustrating also the tendency of the droplet to slide without
spreading as equilibrium is approached. Our theoretical predictions are
verified by numerical experiments.Comment: 12 pages, 5 figure
A comparison of slip, disjoining pressure, and interface formation models for contact line motion through asymptotic analysis of thin two-dimensional droplet spreading
The motion of a contact line is examined, and comparisons drawn, for a
variety of models proposed in the literature. Pressure and stress behaviours at
the contact line are examined in the prototype system of quasistatic spreading
of a thin two-dimensional droplet on a planar substrate. The models analysed
include three disjoining pressure models based on van der Waals interactions, a
model introduced for polar fluids, and a liquid-gas diffuse-interface model;
Navier-slip and two non-linear slip models are investigated, with three
microscopic contact angle boundary conditions imposed (two of these contact
angle conditions having a contact line velocity dependence); and the interface
formation model is also considered. In certain parameter regimes it is shown
that all of the models predict the same quasistatic droplet spreading
behaviour.Comment: 29 pages, 3 figures, J. Eng. Math. 201
On the moving contact line singularity: Asymptotics of a diffuse-interface model
The behaviour of a solid-liquid-gas system near the three-phase contact line
is considered using a diffuse-interface model with no-slip at the solid and
where the fluid phase is specified by a continuous density field. Relaxation of
the classical approach of a sharp liquid-gas interface and careful examination
of the asymptotic behaviour as the contact line is approached is shown to
resolve the stress and pressure singularities associated with the moving
contact line problem. Various features of the model are scrutinised, alongside
extensions to incorporate slip, finite-time relaxation of the chemical
potential, or a precursor film at the wall.Comment: 14 pages, 3 figure
The contact line behaviour of solid-liquid-gas diffuse-interface models
A solid-liquid-gas moving contact line is considered through a
diffuse-interface model with the classical boundary condition of no-slip at the
solid surface. Examination of the asymptotic behaviour as the contact line is
approached shows that the relaxation of the classical model of a sharp
liquid-gas interface, whilst retaining the no-slip condition, resolves the
stress and pressure singularities associated with the moving contact line
problem while the fluid velocity is well defined (not multi-valued). The moving
contact line behaviour is analysed for a general problem relevant for any
density dependent dynamic viscosity and volume viscosity, and for general
microscopic contact angle and double well free-energy forms. Away from the
contact line, analysis of the diffuse-interface model shows that the
Navier--Stokes equations and classical interfacial boundary conditions are
obtained at leading order in the sharp-interface limit, justifying the creeping
flow problem imposed in an intermediate region in the seminal work of Seppecher
[Int. J. Eng. Sci. 34, 977--992 (1996)]. Corrections to Seppecher's work are
given, as an incorrect solution form was originally used.Comment: 33 pages, 3 figure
- …