Nonlinear pattern formation in thin liquid films under external vibrations

Abstract

We study a thin liquid film with a free surface on a planar horizontal substrate. The substrate is subjected to oscillatory accelerations in the normal and/or in the horizontal direction(s). The description is based on the longwave approximation including inertia effects, which are important due to the large velocities imparted by external vibrations. The linearized system is examined using the Floquet analysis. Pattern formation in the nonlinear regime is computed numerically from the longwave equations for the thickness and the flow rate of the fluid in two and in three spatial dimensions. For certain amplitude and frequency ranges, combined lateral and normal oscillations can give rise to one or more traveling drops, similar to recent experimental findings by Brunet et al. [Phys. Rev. Lett. 99, 144501 (2007)].

Harmonic solutions for polygonal hydraulic jumps in thin fluid films

“…In two dimensions, Roberts (1996) rationally derives equations for the surface interface and the fluid flow down an inclined plane using centre manifold techniques; in this approach the expansion of the velocities and pressure is performed in the bed-slope angle and spatial derivatives ∂/∂x, and thus the final equations for the flow and the fluid depth also differ from (2.6) in Cartesian coordinates. These approaches (Roberts 1996;Ruyer-Quil & Manneville 2000;Bestehorn et al 2013) match different terms with (2.6), consistent with the Shkadov (1967) model and the Benney (1966) equation, and they are correct in the lubrication approximation including fluid inertia, converging to the Reynolds equation in the strongly viscous limit.…”

“…Here Fr = Ω 2 L 2 /gh is the Froude number and Bo = h 2 /l 2 c is the Bond number related to the capillary length l c = (γ /ρg) 1/2 . Equations (2.6) have been generalized (Bestehorn 2013;Bestehorn, Han & Oron 2013) for horizontal and normal forcings, using expansion coefficients (Galerkin method; see Guo, Labrosse & Narayanan (2012)) in the lubrication approximation ( = 0). Ruyer-Quil & Manneville (2000) also utilize the Galerkin method to obtain similar equations describing free surface flows down inclined planes for the horizontal flow components and the fluid height, which can be compared with (2.6) by writing the second-order results in vectorial form.…”

Faraday instability of a two-layer liquid film with a free upper surface

“…Sufficiently strong vibration destabilizes the flat interface, leading to the formation of standing surface waves. The phenomenon has received much attention in the past and has now been extensively studied for various geometries of the system, including two semi-infinite liquid phases [2], two liquid films of finite or infinite thickness confined between two solid plates [3][4][5][6][7][8], one finite thickness liquid layer, supported by a solid plate and exposed to a gas phase [9][10][11][12][13][14][15][16][17][18], and a single unsupported soaplike viscoelastic liquid film [19].…”

Shaping liquid drops by vibration

“…To close this gap, we present in this letter a minimal model that accounts for the worm-like dynamical regimes of a vertically vibrated liquid drop. At finite Reynolds numbers, we employ the reduced model that was derived in the long-wave approximation from the Navier-Stokes equation to describe nonlinear Faraday waves in one-layer liquid films [10]. The model consists of two coupled equations for the film thickness h(x, y, t) and the flow q(x, y, t) = h 0 u(z, x, y, t) dz across the film, where u is the horizontal fluid velocity.…”