Sliding drops in the diffuse interface model coupled to hydrodynamics

Abstract

Using a film thickness evolution equation derived recently combining long-wave approximation and diffuse interface theory [L. M. Pismen and Y. Pomeau, Phys. Rev. E 62, 2480 (2000)] we study one-dimensional surface profiles for a thin film on an inclined plane. We discuss stationary flat film and periodic solutions including their linear stability. Flat sliding drops are identified as universal profiles, whose main properties do not depend on mean film thickness. The flat drops are analyzed in detail, especially how their velocity, advancing and receding dynamic contact angles and plateau thicknesses depend on the inclination of the plane. A study of nonuniversal drops shows the existence of a dynamical wetting transition with hysteresis between droplike solutions and a flat film with small amplitude nonlinear waves.

Droplet motion on inclined heterogeneous substrates

“…Here we neglect any motions that may give rise to instabilities and focus on the dynamics of a contact line which is uniform in the transverse direction. This simplifying assumption is a rather common one and has been invoked in numerous related studies in the past (see, e.g., Hocking 1981;Thiele et al 2001Thiele et al , 2002Savva & Kalliadasis 2009Vellingiri et al 2011). It is also important to emphasize that film-like shapes can also be observed in solutions to (2.2) with (2.3) in the limit when gravity dominates capillarity, which causes the flattening of the droplet in the middle and the development of a pronounced bulge at the advancing edge (see also § 4).…”

“…The coupling of contact line dynamics with long-wave models, such as (2.2) with (2.3), has a long history dating back to the studies of Greenspan (1978) and Hocking (1981), but it is also important to note that the same set of equations governs the motion of thin films flowing down an inclined plane (see Buckingham, Shearer & Bertozzi 2003;Kondic 2003, for more details on the derivation), which is a subject that has also been widely investigated by many authors since the seminal work of Huppert (1982). In this context, the droplet we consider can be viewed as a liquid ridge/rivulet which is known to be prone to instabilities in the transverse direction (see, e.g., Silvi & Dussan 1985;Troian et al 1989;Bertozzi & Brenner 1997;López, Miksis & Bankoff 1997;Hocking 1990;Kalliadasis 2000;Diez & Kondic 2001;Kondic 2003;Diez, González & Kondic 2012). Here we neglect any motions that may give rise to instabilities and focus on the dynamics of a contact line which is uniform in the transverse direction.…”

“…To address the dynamic problem numerically, a broad range of methodologies were utilized, ranging from precursor film models (e.g. Thiele et al 2001Thiele et al , 2002Koh et al 2009), Lattice Boltzmann simulations (e.g. Dupuis & Yeomans 2006), to smoothed particle hydrodynamics (e.g.…”

Upscaled phase-field models for interfacial dynamics in strongly heterogeneous domains

“…Cahn-Hilliard or more generally phase-field energy functionals are for example applied in image processing such as inpainting (Bertozzi et al 2007). Wetting phenomena, of great interest in technological applications, especially motivated by recent developments in micro-fluidics, enjoy a wide-spread use of phase-field modelling (Pomeau 2001;Laurila et al 2008;Queralt-Martin et al 2011). Such phenomena have some intriguing features, including the appearance of hysteresis and non-locality, e.g.…”

Statics and dynamics of a cylindrical droplet under an external body force

“…Examples are falling drops on an inclined plane due to gravity 8,9,10 or the motion of droplets on a rotating disk due to centrifugal forces. In these cases much attention has been devoted to the onset of motion and the changes of the shape of the droplets and their instabilities in response to strong body forces.…”