Wave patterns in film flows: modelling and three-dimensional waves

Abstract

In a previous work, two-dimensional film flows were modelled using a weighted-residual approach that led to a four-equation model consistent at order $\epsilon^2$. A two-equation model resulted from a subsequent simplification but at the cost of lowering the degree of the approximation to order $\epsilon$ only. A Padé approximant technique is applied here to derive a refined two-equation model consistent at order $\epsilon^2$. This model, formulated in terms of coupled evolution equations for the film thickness $h$ and the flow rate $q$, accounts for inertia effects due to the deviations of the velocity profile from the parabolic shape, and closely follows the asymptotic long-wave expansion in the appropriate limit. Comparisons of two-dimensional wave properties with experiments and direct numerical simulations show good agreement for the range of parameters in which a two-dimensional wavy motion is reported in experiments.

The stability of two-dimensional travelling waves to three-dimensional pertur-bations is investigated based on the extension of the models to include spanwise dependence. The secondary instability is found to be not very selective, which explains the widespread presence of the synchronous instability observed in the experiments by Liu et al.(1995) whereas Floquet analysis predicts a subharmonic scenario in most cases. Three-dimensional wave patterns are computed next assuming periodic boundary conditions. Transition from two- to three-dimensional flows is shown to be strongly dependent on initial conditions. The herringbone patterns, the synchronously deformed fronts and the three-dimensional solitary waves observed in experiments are recovered using our regularized model, which is found to be an excellent compromise between the complete model, which has seven equations, and the simplified model, which does not include the second-order inertia corrections. Those corrections are found to play a role in the selection of the type of secondary instability as well as of the spanwise wavelength of the emerging pattern.