Three-dimensional thin film flow over and around an obstacle on an inclined plane

Abstract

Steady Stokes flow driven by gravity down an inclined plane over and around an attached obstacle is considered. The effects of the obstacle are examined for various flow configurations and results produced for flow over hemispherical obstacles. Comparison is made with previously published papers that assume that the obstacle is small and/or the free surface deflection and disturbance velocity are small. Values for the unit normal and curvature of the free surface are found using both finite difference approximations and Hermitian radial basis function interpolations, with the resulting solutions compared. Free surface profiles for thin film flows over hemispherical obstacles that approach the film surface are produced and the effects of near point singularities considered. All free surface profiles indicate an upstream peak, followed by a trough downstream of the obstacle with the peak decaying in a “horseshoe” shaped surface deformation. Flow profiles are governed by the plane inclination, the Bond number, and the obstacle geometry. An extension of this approach provides a new class of solutions where a thin film flows around a cylindrical obstacle. Notably, the static contact line angle between the free surface and the obstacle is introduced as an extra flow parameter and its effect investigated for a given set of flow parameters and fixed boundary conditions. Solutions are obtained where steady flow profiles can be found both over and around a cylindrical obstacle raising the awareness of possible multiple solutions.

Free-surface film flow over topography: Full three-dimensional finite element solutions

“…For the case of film flow over hemispheriods, the predictions obtained using the N-S solver are shown to produce results in excellent agreement with those of [25] for Stokes flow at different capillary numbers, in terms of both the free-surface disturbance and underpinning internal flow.…”

“…In the absence of any experimental data with which to compare, other than that of [10] for flow over trench like topography, comparisons are drawn with results for related Stokes flow problems available in the literature -namely flow over a hemispheroid as investigated by [25]. In all cases it is ensured that the flow does not violate, the arguably conservative, classical stability criterion for gravity-driven film flow down an inclined plane.…”

“…As might be construed, there are very few such film flow solutions in the literature; the exceptions being boundary element solutions obtained for Stokes flow, as reported by [23] and [24] for flow over a small particle and a three-dimensional obstacle, respectively, and by [25,26] for flow past hemispheriod-shaped obstacles with large free-surface disturbances. Latterly, [14] obtained solutions with inertia present for film flow over a bi-periodically repeating substrate using a Volume of Fluid algorithm to investigate pattern formation and mixing; see also the work of [27] which addresses the capillary flow problem of dynamic wetting as an interface forming process.…”

“…Initially, solutions were obtained for Stokes flow over a hemispheric obstacle with diameter l t = 1.8 (w t = 1.8 and s 0 = 0.9) as considered by [25], enabling direct comparison with the predictions they achieved using a boundary element (BE) discretisation of the governing equations -see their Figures 15 and 16. The boundary element method has similarly been employed by [23] and [24] to investigate Stokes flow over three-dimensional obstacles but for cases of small free-surface deflections.…”

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

“…The complexity of the general flow problem as specified above can be reduced to one which is more tractable, while retaining the nonlinear inertial terms, by depth-averaging equations (5) and (6). Since the approach is described in detail elsewhere, Veremieiev et al (2010), an outline only is provided.…”