Nonlinear instability of a thin film flowing down a smoothly deformed surface

Abstract

Here, the nonlinear flow instability of a thin film flowing down an inclined wall with smooth deformations is investigated. A nonlinear evolution equation is obtained under the small wavenumber approximation. This equation describes the film free surface deformation including the effects of inertia, viscosity, surface tension, and deformation of the wall. The equation has a forcing term that corresponds to periodic time-dependent perturbations hitting on the free surface. These time-dependent perturbations are tested against the presence of the free surface waves due to wall deformations. In particular, a wall sinusoidal profile is investigated. It is found that when the wall wavelength is small enough, the valleys of the surface response to the wall profile become very deep, showing a kind of resonance, which has an important stabilizing influence in the evolution of the superposed time-dependent perturbations.

Electrified thin film flow at finite Reynolds number on planar substrates featuring topography

“…Recent experiments, however, in which the flow is taken to be essentially two-dimensional in a streamwise cross-sectional plane, have demonstrated that there is a strong coupling between inertia and topography for the case of gravity-driven flow over surfaces containing spanwise periodic rectangular ; Argyriadi, Vlachogiannis and Bontozoglou, 2006) or wavy (Wierschem, Lepski and Aksel, 2005) features. The experimentally-observed rise in critical Reynolds number with increasing topography steepness that occurs has also been predicted theoretically by several authors, see for example , Trifonov (2007) and Dávalos-Orozco (2007; conversely, it has been reported recently that undulating surfaces may have a destabilising influence on the flow if the surface tension is sufficiently high (Heining and The present work has two main strands in relation to gravity-driven film flow at finite Reynolds number in the presence of an electric field normal to it: two-dimensional flow over dis-crete steep and smooth periodically varying spanwise topography revisited; three-dimensional flow over localised steep topography, the exploration of which is considered for the first time. In both cases, the hydrodynamics is modelled using the approach of Veremieiev et al (2010) and the governing equation set solved numerically.…”

Instabilities of Thin Films Flowing Down Flat and Smoothly Deformed Walls

“…The influence of the time dependent perturbations has been investigated in (Dávalos-Orozco et al 1997) where the impact of the frequency of the perturbation and the magnitude of the Reynolds number on the free surface pattern were made clear. Smooth spatial wall deformations were investigated in (Dávalos-Orozco 2007). In that paper, a review is given of papers which present results related with this problem but under different approximations.…”

“…Functions that satisfy this condition are the sine and cosine. In (Dávalos-Orozco 2007) it was shown that in the case of a sinusoidal wall it is possible to stabilize the time dependent perturbations propagating on the free surface response to the wall. However, in that case, the wall had sinusoidal deformations with infinite extension.…”

“…The solutions of the velocities up to first order are introduced into the kinematic boundary condition. As a result, this gives an evolution equation that has the following form (see Dávalos-Orozco 2007):…”

“…Here, h is the free surface height, β is the wall inclination angle, G is the Reynolds number, S is the surface tension number and P p = A |sin (ωt/2)| exp −a x 2 + y 2 is the surface time dependent pressure due to a turbulent air jet (see Dávalos-Orozco 2007). Q(x,y) is the wall profile.…”