Improved modeling of flows down inclined planes
Laboratoire d'Hydrodynamique, CNRS UMR 7646, École Polytechnique, 91128 Palaiseau, France
Revised: 6 January 2000
Published online: 15 May 2000
New models of film flows down inclined planes have been derived by combining a gradient expansion at first or second order to weighted residual techniques with polynomials as test functions. The two-dimensional formulation has been extended to account for three-dimensional flows as well. The full second-order two-dimensional model can be expressed as a set of four coupled evolution equations for four slowly varying fields, the thickness h, the flow rate q and two other quantities measuring the departure from the flat-film semi-parabolic velocity profile. A simplified model has been obtained in terms of h and q only. Including viscous dispersion effects properly, it closely sticks to the asymptotic expansion in the appropriate limit. Our new models improve over previous ones in that they remain valid deep into the strongly nonlinear regime, as shown by the comparison of our results relative to travelling-wave and solitary-wave solutions with those of both direct numerical simulations and experiments.
PACS: 47.20.Ma – Interfacial instability / 47.20.Ky – Nonlinearity (including bifurcation theory)
© EDP Sciences, Società Italiana di Fisica, Springer-Verlag, 2000