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 2 . A two-equation model resulted from a subsequent simplification but at the cost of lowering the degree of the approximation to order only (Ruyer-Quil & Manneville 2000). A Padé approximant technique is applied here to derive a refined two-equation model consistent at order 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 sticks to 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 where a two-dimensional wavy motion is reported in experiments.The stability of two-dimensional travelling waves against three-dimensional perturbations is investigated based on the extension of the models to include spanwise dependence. The secondary instability is found to be not much 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 next computed assuming periodic boundary conditions. Transition from 2D to 3D flows is shown to be strongly dependent on initial conditions. The herringbone patterns, the synchronously deformed fronts and the threedimensional solitary waves observed in experiments (Liu et al. 1995;Park & Nosoko 2003;Alekseenko et al. 1994) are recovered using our regularised 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.

Self-similarity of solitary waves on inertia-dominated falling liquid films

“…This presumably stems from the fact that most theoretical studies to date have focused on film flows with low inertia in which the balance of viscous stresses and surface tension plays a dominant role, a regime which can be adequately described using models based on the boundary-layer approximation, e.g. [19,21,36,52,[55][56][57]. The Weber number We given in Eq.…”

“…Furthermore, the asymmetry of the solitary wave is not correlated with the driving capillary number Ca * , suggesting that the balance of surface tension and viscous stresses, which is a main assumption for low-dimensional models valid typically in the region of low Re, see e.g. [19,21,36,[55][56][57], is not really appropriate for inertia-dominated film flows. Hence, surface tension balances the inertia of the liquid film at the wave front, and it is this balance that governs the asymmetry of the solitary wave.…”

“…This leads to a strongly non-parabolic velocity profile at the front of the solitary wave [6,36,37], including flow reversal underneath the trough preceding the solitary wave under certain conditions [4,5,38,39]. Furthermore, if inertia is sufficiently high, the maximum flow velocity of the film exceeds the phase velocity of the solitary waves, leading to a recirculation zone in the main wave hump with respect to the reference frame moving with the wave [39][40][41][42].…”

Spontaneous channeling of solitary pulses in heated-film flows

“…In II m the quasi-2D solitary pulses riding the rivulets experience a transverse instability, which, however, does not lead to their complete disintegration. An analogy can be made here with isothermal films on a vertical planar substrate where isolated 2D solitary waves experience an instability in the transverse direction leading to 3D wave patterns [18]. As Re then increases from II r ( fig.…”

Heat Transfer Phenomena in Laminar Wavy Falling Films: Thermal Entry Length, Thermal-Capillary Metastable Structures, Thermal-Capillary Breakdown

“…a) Typical IR picture of quasi-regular metastable structures within the residual layer between large waves of laminar-wavy falling films, b) temporal evolution of the regular structure's "head" between two large parabola-shaped waves (zone marked in a), Re = 15, q "= 2.2 x 10 4 W/m² (Lel et al, 2008) The structure's "head" effect could be explained through interference of waves at the point of intersection of two large parabola-shaped waves. The visualisation of falling films in the work of (Alekseenko et al, 1994) and (Scheid et al, 2006) shows, that capillary waves in front of large waves generate an interference pattern. Also from a recent work by (Lel et al, 2005) has become clear, that the point with the minimal film thickness is always found directly in front of large waves.…”