Steady free-surface thin film flows over topography

Abstract

We consider the slow motion of a thin viscous film flowing over a topographical feature (trench or mound) under the action of an external body force. Using the lubrication approximation, the equations of motion simplify to a single nonlinear partial differential equation for the evolution of the free surface in time and space. It is shown that the problem is governed by three dimensionless parameters corresponding to the feature depth, feature width and feature steepness. Quasi-steady solutions for the free surface are reported for a wide range of these parameters. Our computations reveal that the free surface develops a ridge right before the entrance to the trench or exit from the mound and that this ridge can become large for steep substrate features of significant depth. Such capillary ridges have also been observed in the contact line motion over a planar substrate where the buildup of pressure near the contact line is responsible for the ridge. For flow over topography, the ridge formation is a manifestation of the effect of the capillary pressure gradient induced by the substrate curvature. In addition, the minimum film thickness is always found near the concave corner of the feature. Both the height of the ridge and the minimum film thickness are found to be strongly dependent on both the profile depth and steepness. Finally, it is found that either finite feature width or a significant vertical component of gravity can suppress these effects in a way that is made quantitative and which allows the operative physical mechanism to be explained.

Validity domain of the Benney equation including the Marangoni effect for closed and open flows

“…We believe, this ensures that the Benney equation will remain a helpful model to study thin film flows, especially to identify new phenomena but also in a limited parameter range for quantitative predictions. Actually, many other effects may be added to the Benney equation like evaporation (Joo et al 1991), Van der Waals force (Tan et al 1966), chemical reaction (Trevelyan et al 2002), topolographical effects (Kalliadasis et al 2000), non-uniform heating (Miladinova et al 2002;Kabov et al 2001;Scheid et al 2002;Skotheim et al 2002;Kalliadasis et al 2003b), etc. The validity of the Benney equation should in the future also be addressed including these additional effects.…”

Stabilization of thin liquid films flowing over locally heated surfaces via substrate topography

“…flowing over various features has been investigated extensively for isothermal flows using lubrication theory. 39 It has been shown that the height of the capillary ridge induced by the feature asymptotes to a constant value as ␦ → 0. A detailed investigation of liquid films flowing over locally heated flat surfaces based on Eqs.…”

“…This ridge is unstable to transverse perturbations above a critical value of the Marangoni parameter. 24 Liquid films flowing over surfaces with topographical features also exhibit capillary ridges, 39 as shown in Fig. 1͑b͒.…”

“…One method of modifying the film curvature is through gradients in capillary pressure induced by topographical features added to the substrate. [37][38][39] Noninertial coating flows over isothermal substrates with topographical features exhibit similar capillary ridges to the locally heated films but are stable to perturbations due to the restoring flow generated by the capillary pressure gradient induced by the features. 40,41 It was shown from boundary integral solutions of the unsimplified equations of Stokes flow 42 that the film closely follows the topography for capillary numbers ͑Ca͒ that are O͑1͒ or larger but a pronounced capillary ridge develops near the feature for smaller Ca.…”

“…III. 39 It has been shown, however, through comparisons to Stokes flow 42 and finite element 43 solutions and experiment 43,44 that the evolution equation derived using lubrication theory ͓Eq. ͑2͔͒ is a valid approximation for small Ca ͑approximately smaller than 10 −2 ͒ even for steep features.…”