scholarly journals A dynamical systems approach for the contact-line singularity in thin-film flows

2016 ◽  
Vol 144 ◽  
pp. 204-235 ◽  
Author(s):  
Fethi Ben Belgacem ◽  
Manuel V. Gnann ◽  
Christian Kuehn
Keyword(s):  
Thin Film ◽  
Dynamical Systems ◽  
Contact Line ◽  
Systems Approach ◽  
Line Singularity ◽  
Film Flows

Nonlinear instability of a contact line driven by gravity

2000 ◽  
Vol 413 ◽  
pp. 355-378 ◽  
Author(s):  
SERAFIM KALLIADASIS
Keyword(s):  
Thin Film ◽  
Differential Equation ◽  
Partial Differential Equation ◽  
Dynamical Systems ◽  
Contact Line ◽  
Nonlinear Instability ◽  
Inclined Plane ◽  
Translational Invariance ◽  
Weakly Nonlinear ◽  
Fingering Instability

A thin liquid mass of fixed volume spreading under the action of gravity on an inclined plane develops a fingering instability at the front. In this study we consider the motion of a viscous sheet down a pre-wetted plane with a large inclination angle. We demonstrate that the instability is a phase instability associated with the translational invariance of the system in the direction of flow and we analyse the weakly nonlinear regime of the instability by utilizing methods from dynamical systems theory. It is shown that the evolution of the fingers is governed by a Kuramoto–Sivashinsky-type partial differential equation with solution a saw-tooth pattern when the inclined plane is pre-wetted with a thin film, while the presence of a thick film suppresses fingering.

Stability of free-surface thin-film flows over topography

2001 ◽  
Vol 448 ◽  
pp. 387-410 ◽  
Author(s):  
SERAFIM KALLIADASIS ◽  
G. M. HOMSY
Keyword(s):  
Thin Film ◽  
Free Surface ◽  
Contact Line ◽  
Flow Direction ◽  
Strong Stability ◽  
Spanwise Direction ◽  
Planar Substrate ◽  
Wide Range ◽  
The Stability ◽  
Film Flows

We consider the stability of the steady free-surface thin-film flows over topography examined in detail by Kalliadasis et al. (2000). For flow over a step-down, their computations revealed that the free surface develops a ridge just before the entrance to the step. Such capillary ridges have also been observed in the contact line motion over a planar substrate, and are a key element of the instability of the driven contact line. In this paper we analyse the linear stability of the ridge with respect to disturbances in the spanwise direction. It is shown that the operator of the linearized system has a continuous spectrum for disturbances with wavenumber less than a critical value above which the spectrum is discrete. Unlike the driven contact line problem where an instability grows into well-defined rivulets, our analysis demonstrates that the ridge is surprisingly stable for a wide range of the pertinent parameters. An energy analysis indicates that the strong stability of the capillary ridge is governed by rearrangement of fluid in the flow direction flowing to the net pressure gradient induced by the topography at small wavenumbers and by surface tension at high wavenumbers.

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