The inverse problem in creeping film flows

Abstract

We consider the inverse problem for gravity-driven free surface flows at vanishing Reynolds numbers. In contrast to the direct problem, where information about the underlying topographic structure is given and the steady free surface shape and the flow field are unknown, the inverse problem deals with the flow along unknown topographies. The bottom shape and the corresponding flow field are reconstructed from information at the steady free surface only. We discuss two different configurations for the inverse problem. In the first case, we assume a given free surface shape, and by simplifying the field equations, we find an analytical solution for the corresponding bottom topography, velocity field, wall shear stress, and pressure distribution. The analytical results are successfully compared with experimental data from the literature and with numerical data of the Navier-Stokes equations. In the second inverse problem, we prescribe a free surface velocity and then solve numerically for the full flow domain, i.e. the free surface shape, the topography and simultaneously the wall shear stress and the pressure field. The results are validated with the numerical solution of the corresponding direct problem.

Viscous free-surface flows past cylinders

“…In addition to surface tension, inertia may be significant in some interactions between thin films and obstructions [13,14]. Significant progress has also been made in the inverse problem; determining the underlying topography given a known free-surface, which is important for determining the required bottom topography for a desired free-surface profile [15,16].…”

Flow of a yield-stress fluid past a topographical feature

“…In many applications, the free surface of the fluid is known but the underlying topography or the rheology is unknown (e.g. glaciers and lava flows) [47]. [48] showed that the method of characteristics can be used to reconstruct the topography required to produce a particular free-surface profile for a thin Newtonian.…”

Inferring rheology from free-surface observations

“…The method of characteristics furnishes where , and parameterises the characteristic curves in the plane, which are the streamlines of the flow. A similar method was used by Sellier & Panda (2010) for reconstructing bottom topography from the free-surface flow of thin films of Newtonian fluid (see also Heining, Sellier & Aksel 2012). The direction of the streamlines is given by the (known) direction of .…”