Free surface Stokes flows obstructed by multiple obstacles

Abstract

SUMMARYGravity-driven Stokes flow down an inclined plane over and around multiple obstacles is considered. The flow problem is formulated in terms of a boundary integral equation and solved using the boundary element method. A Hermitian radial basis function (RBF) is used for the interpolation of the free surface, generation of the unit normal and curvature, and to prescribe the far-field conditions. For flow over an obstacle, hemispheres are taken. For flow around an obstacle, circular cylinders are modelled and the contact angle condition on the obstacle/free surface intersection specified using the RBF formulation. Explicit profiles are produced for flow over and around two obstacles placed in various locations relative to one another. Interaction due to two obstacles is given by comparisons made with the profiles for flow over and around individual obstacles. In general, when the obstacles are separated by a sufficiently large distance the flow profiles are identical to a single obstacle analysis. For flow over and around two obstacles in-line with the incident flow, effects of the governing parameters are examined, with variations in plane inclination angle, Bond number, obstacle size, and in the case of obstacles intersecting the free surface, static contact angle is considered. Finally flows over and around three obstacles are modelled.

Modelling the wetting of a solid occlusion by a liquid film

“…This study was then followed by a series of papers based on the same formulation exploring the numerical solution technique, Lee et al (2008), the addition of more complex physics, Lee et al (2011), and the effect of inertia, Veremeiev et al (2011). At the same time, Baxter and co-workers investigated the same problem but the approach they adopted is based on solving the Stokes equations using a Boundary Element technique, Baxter et al (2009Baxter et al ( , 2010. Naturally, this latter method has less restrictive assumptions than the lubrication approximation.…”

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

“…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.…”

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.…”

“…Having established the credentials of the current model and the accuracy of the associated method of solution, predictions obtained with the same are compared with those obtained with the LUBI model; both of which are contrasted with corresponding finite element solutions of the full Navier-Stokes (N-S) equations and attendant boundary conditions, expressions (5), (6), (9) to (14) and (17), obtained in a manner similar to Scholle et al (2008a). The problem chosen for this purpose is that of electrified film flow, with ε = 0.2, Ca = ε 2 = 0.04, θ = 30…”

“…(i) Re-scaling equations (5) - (6) and boundary conditions (9) - (14) in terms of L 0 , which is equivalent to the following change of non-dimensional variables: (x, y, l t , w t , l p , w p , p) → (x, y, l t , w t , l p , w p , p) /ε, w → εw with all terms of O(ε 2 ) or higher neglected; the scalings used for the other variables remain unchanged.…”