Skip to main content Accessibility help

Flow domain identification from free surface velocity in thin inertial films

  • C. Heining (a1), T. Pollak (a1) and M. Sellier (a2)


We consider the flow of a viscous liquid along an unknown topography. A new strategy is presented to reconstruct the topography and the free surface shape from one component of the free surface velocity only. In contrast to the classical approach in inverse problems based on optimization theory we derive an ordinary differential equation which can be solved directly to obtain the inverse solution. This is achieved by averaging the Navier–Stokes equation and coupling the function parameterizing the flow domain with the free surface velocity. Even though we consider nonlinear systems including inertia and surface tension, the inverse problem can be solved analytically with a Fourier series approach. We test our method on a variety of benchmark problems and show that the analytical solution can be applied to reconstruct the flow domain from noisy input data. It is also demonstrated that the asymptotic approach agrees very well with numerical results of the Navier–Stokes equation. The results are finally confirmed with an experimental study where we measure the free surface velocity for the film flow over a trench and compare the reconstructed topography with the measured one.


Corresponding author

Email address for correspondence:


Hide All
Aksel, N. 2000 Influence of the capillarity on a creeping film flow down an inclined plane with an edge. Arch. Appl. Mech. 70, 8190.
Argyriad, K., Vlachogiannis, M. & Bontozoglou, V. 2006 Experimental study of inclined film flow along periodic corrugations: the effect of wall steepness. Phys. Fluids 18, 012102.
Chang, H.-C. 1994 Wave evolution on a falling film. Annu. Rev. Fluid Mech. 26, 103136.
Craster, R. V. & Matar, O. K. 2009 Dynamics and stability of thin liquid films. Rev. Mod. Phys. 81, 11311198.
D’Alessio, S. J. D., Pascal, J. P. & Jasmine, H. A. 2009 Instability in gravity-driven flow over uneven surfaces. Phys. Fluids 21, 062105.
Dávalos-Orozco, L. A. 2007 Nonlinear instability of a thin film flowing down a smoothly deformed surface. Phys. Fluids 19, 074103.
Decré, M & Baret, J. C. 2004 Gravity-driven flows of viscous liquids over two-dimensional topographies. J. Fluid Mech. 487, 147166.
Engl, H. W., Hanke, M. & Neubauer, A. 2000 Regularization of Inverse Problems. Kluwer.
Gessese, A. & Sellier, M. 2012 A direct solution approach to the inverse shallow-water problem. Math. Problems Engng 2012, 417950.
Gessese, A. F., Sellier, M., VanHouten, E. & Smart, G. 2011 Reconstruction of river bed topography from free surface data using direct numerical approach in one-dimensional shallow water flow. Inverse Problems 27, 025001.
Häcker, T. & Uecker, H. 2009 An integral boundary layer equation for film flow over inclined wavy bottoms. Phys. Fluids 21, 092105.
Heining, C. 2011 Velocity field reconstruction in gravity-driven flow over unknown topography. Phys. Fluids 23, 032101.
Heining, C. & Aksel, N. 2009 Bottom reconstruction in thin-film flow over topography: steady solution and linear stability. Phys. Fluids 21, 083605.
Heining, C. & Aksel, N. 2010 Effects of inertia and surface tension on a power-law fluid flowing down a wavy incline. Intl J. Multiphase Flow 36, 847857.
Heining, C., Bontozoglou, V., Aksel, N. & Wierschem, A. 2009 Nonlinear resonance in viscous films on inclined wavy planes. Intl J. Multiphase Flow 35, 7890.
Heining, C., Pollak, T. & Aksel, N. 2012 Pattern formation and mixing in three-dimensional film flow. Phys. Fluids 24, 042102.
Heining, C., Sellier, M. & Aksel, N. 2012 The inverse problem in creeping film flows. Acta Mechanica 223, 841847.
Hutter, K., Svendsen, B. & Rickenmann, D. 1994 Debris flow modelling: a review. Cont. Mech. Thermodyn. 8, 1.
Kalliadasis, S., Bielarz, C. & Homsy, G. M. 2000 Steady free-surface thin film flows over topography. Phys. Fluids 12, 18891898.
Kanaris, A. G. & Mouza, A. A. 2006 Flow and heat transfer prediction in a corrugated plate heat exchanger using a CFD code. Chem. Engng Technol. 29, 923930.
Kistler, S. F. & Schweizer, P. M. 1997 Liquid Film Coating. Chapman & Hall.
Lonyangapuo, J. K., Elliott, L., Ingham, D. B. & Wen, X. 1999 Retrieval of the shape of the bottom surface of a channel when the free surface profile is given. Engng Anal. Bound. Elem. 23, 457470.
Lonyangapuo, J. K., Elliott, L., Ingham, D. B. & Wen, X. 2001 Solving free surface fluid flow problems by the minimal kinetic energy functional. Intl J. Numer. Meth. Fluids 37, 577600.
Luca, I., Hutter, K., Thai, Y. C. & Kuo, C. Y. 2009 A hierarchy of avalanche models on arbitrary topography. Acta Mech. 205, 121.
Maxwell, D., Truffer, M., Avdonin, S. & Stuefer, M. 2008 An iterative scheme for determining glacier velocities and stresses. J. Glaciol. 54, 888898.
Oron, A. & Heining, C. 2008 Weighted-residual integral boundary-layer model for the nonlinear dynamics of thin liquid films falling on an undulating vertical wall. Phys. Fluids 20, 082102.
Pak, M. I. & Hu, G. H. 2011 Numerical investigations on vortical structures of viscous film flows along periodic rectangular corrugations. Intl J. Multiphase Flow 37, 369379.
Rogers, S. S., Waigh, T. A., Zhao, X. & Lu, J. R. 2007 Precise particle tracking against a complicated background: polynomial fitting with Gaussian weight. Phys. Biol. 4, 220.
Sellier, M. 2008 Substrate design or reconstruction from free surface data for thin film flows. Phys. Fluids 20, 062106.
Sellier, M. & Panda, S. 2010 Beating capillarity in thin film flows. Int. J. Numer. Meth. Fluids 63, 431448.
Spurk, J. H. & Aksel, N. 2008 Fluid Mechanics, 2nd edn. Springer.
Trifonov, Y. Y. 1998 Viscous liquid film flows over a periodic surface. Intl J. Multiphase Flow 24, 11391161.
Trifonov, Y. Y. 2007 Stability and nonlinear wavy regimes in downward film flows on a corrugated surface. J. Appl. Mech. Tech. Phys. 48, 91100.
Tuffer, M. 2004 The basal speed of valley glaciers: an inverse approach. J. Glaciol. 50, 236242.
Vlachogiannis, M. & Bontozoglou, V. 2002 Experiments on laminar film flow along a periodic wall. J. Fluid Mech. 457, 133156.
Veremieiev, S., Thompson, H. M., Lee, Y.-C. & Gaskell, P. H. 2010 Inertial thin film flow on planar surfaces featuring topography. Comp. & Fluids 39, 431450.
Webb, R. L. 1994 Principles of Enhanced Heat Transfer. John Wiley & Sons.
Weinstein, S. J. & Ruschak, K. J. 2004 Coating flows. Annu. Rev. Fluid Mech. 36, 29.
Wierschem, A. & Aksel, N. 2004 Hydraulic jumps and standing waves in gravity-driven flows of viscous liquids in wavy open channels. Phys. Fluids 16, 38683877.
Wierschem, A., Bontozoglou, V., Heining, C., Uecker, H. & Aksel, N. 2008 Linear resonance in viscous films on inclined wavy planes. Intl J. Multiphase Flow 34, 580590.
Wierschem, A., Lepski, C. & Aksel, N. 2005 Effect of long undulated bottoms on thin gravity-driven films. Acta Mech. 179, 4166.
Wierschem, A., Pollak, T., Heining, C. & Aksel, N. 2010 Suppression of eddies in films over topography. Phys. Fluids 22, 113603.
Wierschem, A., Scholle, M. & Aksel, N. 2002 Comparison of different theoretical approaches to experiments on film flow down an inclined wavy channel. Exp. Fluids 33, 429442.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification

Flow domain identification from free surface velocity in thin inertial films

  • C. Heining (a1), T. Pollak (a1) and M. Sellier (a2)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed