Methodologies for Low-Reynolds Number Flows

  • S. KalliadasisEmail author
  • C. Ruyer-Quil
  • B. Scheid
  • M. G. Velarde
Part of the Applied Mathematical Sciences book series (AMS, volume 176)


We develop the long-wave theory for a film falling down a heated wall. The theory is based on a gradient expansion of the governing equations and wall and free-surface boundary conditions and leads to the Benney equation (BE—Chap. 1) for the evolution in time and space of the film thickness. A weakly nonlinear expansion of the equation leads to either the Kuramoto–Sivashinsky or the Kawahara equation depending on the distance from criticality, the orders-of-magnitude assignments of the different parameters and whether the film is inclined or vertical. BE fully resolves the behavior of the film close to the instability threshold, but it blows up in finite time at δ≃1, i.e., precisely where the transition between the drag-gravity and drag-inertia regimes takes place, as shown in Chap. 4. This in turn suggests that δ is the natural parameter for validation purposes/assessment of the validity of a model that aims to describe the dynamics of the film. The blow up behavior is strongly connected with the nonexistence of solitary waves of BE for δ≳1. A regularization procedure based on an extension of the Padé approximants technique to differential operators provides an evolution equation free of singularities which nevertheless seriously underestimates the amplitude and phase speed of the solitary waves at moderate Reynolds numbers. This is a direct consequence of slaving the dynamics of the film to its kinematics. It then seems that in the drag-inertia regime it is not possible to describe the dynamics of the flow with a single evolution equation for the film thickness.


Solitary Wave Travel Wave Solution Homoclinic Orbit Solitary Wave Solution Homoclinic Solution 
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Copyright information

© Springer-Verlag London Limited 2012

Authors and Affiliations

  • S. Kalliadasis
    • 1
    Email author
  • C. Ruyer-Quil
    • 2
  • B. Scheid
    • 3
  • M. G. Velarde
    • 4
  1. 1.Department of Chemical EngineeringImperial College LondonLondonUK
  2. 2.Laboratoire FASTUniversité Pierre et Marie Curie (UPMC)ParisFrance
  3. 3.TIPs—Fluid Physics UnitUniversité Libre de BruxellesBrusselsBelgium
  4. 4.Instituto PluridisciplinarUniversidad ComplutenseMadridSpain

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