Inertial thin film flow on planar surfaces featuring topography

Abstract

Published paperAbstract A range of problems is investigated, involving the gravity-driven inertial flow of a thin viscous liquid film over a planar surface containing topographical features, modelled via a depth-averaged form of the governing unsteady Navier-Stokes equations. The discrete analogue of the resulting coupled equation set, employing a staggered mesh arrangement for the dependent variables, is solved accurately using an efficient Full Approximation Storage (FAS) algorithm and Full Multigrid (FMG) technique together with adaptive timestepping and proper treatment of the nonlinear convective terms. A unique, comprehensive set of results is presented for both one-and two-dimensional topographical features, and errors quantified via detailed comparisons drawn with complementary experimental data and predictions from finite element analyses where they exist. It is found in the case of one-dimensional (spanwise) topography that for small Reynolds number and shallow/short features the depth-averaged form produces results that are in close agreement with corresponding finite element solutions of the full free-surface problem. For the case of flow over two-dimensional (localised) topography the free-surface disturbance observed is influenced significantly by the presence of inertia. It leads, as in the case of spanwise topography, to an increase in the magnitude and severity of the capillary ridge/trough patterns which form.

Modelling the Flow of Droplets of Bio-Pesticide on Foliage

“…The discretised equations are solved using a fixed number of Full Approximation Storage V-cycles on intermediate grid levels and up to 10 V-cycles on the first grid level so that residuals are reduced below a specified tolerance. Further details concerning the spatial and temporal discretisation schemes and the multigrid solution method are available in Veremieiev et al (2010).…”

“…The resulting Depth-Averaged Form (DAF) of the governing continuity and Navier-Stokes equations, see Veremieiev et al (2010) for the full details, retains the inertia terms:…”

“…Since the influence of inertia on free-surface flow can be important in terms of surface deformation and stability, lubrication-type models for continuous thin film flows at non-zero Reynolds number have appeared, see for example Veremieiev et al (2010) who developed a depth-averaged approach, akin to the integral boundary layer equations first reported by Shkadov (1967Shkadov ( , 1968, for exploring such problems. Related experiments, other than in the low Reynolds number regime, have lagged behind; however Puthenveettil et al (2013) has recently addressed this issue and reported a comprehensive set of experimental results, data and visualisations, for the motion of water and mercury droplets on an inclined plane surface in, as they term it, the inertial regime.…”

Gravity-driven thin film flow: The influence of topography and surface tension gradient on rivulet formation

“…the Navier-Stokes and continuity equations, for no slip at the substrate together with the usual free-surface stress and kinematic boundary conditions [11], reduce to the following coupled equation set:…”

Free Surface Thin Film Flow of a Sisko’s Fluid over a Surface Topography

“…In addition, they were able to establish the appropriateness of the theory underpinning the earlier linear analysis of Hayes et al (2000); namely, that when inertia is negligibly small, superposition can be used to construct an appropriate free-surface response to complex topography from the knowledge of the responses to regular elementary topographies. The full Stokes equations for a gravity driven film flow over trench topography was studied by Veremieiev et al (2010) and Veremieiev et al (2011).…”