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.
References
Hutter K., Svendsen B., Rickenmann D.: Debris flow modelling: a review. Cont. Mech. Thermodyn. 8, 1 (1994)
Luca I., Hutter K., Thai Y.C., Kuo C.Y.: A hierarchy of avalanche models on arbitrary topography. Acta Mech. 205, 121 (2009)
Weinstein S.J., Ruschak K.J.: Coating flows. Ann. Rev. Fluid Mech. 36, 29 (2004)
Webb R.L.: Principles of enhanced heat transfer. Wiley, New York (1994)
Sellier M.: Substrate design or reconstruction from free surface data for thin film flows. Phys. Fluids 20, 062106 (2008)
Sellier M., Panda S.: Beating capillarity in thin film flows. Int. J. Numer. Meth. Fluids 63, 431–448 (2010)
Heining C., Aksel N.: Bottom reconstruction in thin-film flow over topography: steady solution and linear stability. Phys. Fluids 21, 083605 (2009)
Gessese A.F., Sellier M., Van Houten E., Smart G.: Reconstruction of river bed topography from free surface data using direct numerical approach in one dimensional shallow water flow. Inverse Problems 27, 025001 (2010)
Heining C.: Velocity field reconstruction in gravity-driven flow over unknown topography. Phys. Fluids 23, 032101 (2011)
Spurk J.H., Aksel N.: Fluid Mechanics. 2nd edn. Springer, Berlin (2008)
Aksel N.: Influence of the capillarity on a creeping film flow down an inclined plane with an edge. Arch. Appl. Mech. 70, 81–90 (2000)
Wierschem A., Scholle M., Aksel N.: Comparison of different theoretical approaches to experiments on film flow down an inclined wavy channel. Exp. Fluids 33, 429–442 (2002)
Engl H.W., Hanke M., Neubauer A.: Regularization of inverse problems. Kluwer, Dordrecht (2000)
Pak M.I., Hu G.H.: Numerical investigations on vortical structures of viscous film flows along periodic rectangular corrugations. Int. J. Multiph. Flow 37, 369–379 (2011)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Heining, C., Sellier, M. & Aksel, N. The inverse problem in creeping film flows. Acta Mech 223, 841–847 (2012). https://doi.org/10.1007/s00707-011-0599-3
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00707-011-0599-3