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.

The Benney equation including thermocapillary effects is considered to study a liquid film flowing down a homogeneously heated inclined wall. The link between the finite-time blow-up of the Benney equation and the absence of the one-hump travelling-wave solution of the associated dynamical system is accurately demonstrated in the whole range of linearly unstable wavenumbers. Then the blow-up boundary is tracked in the whole space of parameters accounting for flow rate, surface tension, inclination and thermocapillarity. In particular, the latter two effects can strongly reduce the validity range of the Benney equation. It is also shown that the subcritical bifurcation found for falling films with the Benney equation is related to the blow-up of solutions and is unphysical in all cases, even with the thermocapillary effect though in contrast to horizontally heated films. The accuracy of bounded solutions of the Benney equation is determined by comparison with a reference weighted integral boundary layer model. A distinction is made between closed and open flow conditions, when calculating travelling-wave solutions; the former corresponds to the conservation of mass and the latter to the conservation of flow rate. The open flow condition matches experimental conditions more closely and is explored for the first time through the associated dynamical system. It yields bounded solutions for larger Reynolds numbers than the closed flow condition. Finally, solutions that are conditionally bounded are found to be unstable to disturbances of larger periodicity. In this case, coalescence is the pathway yielding finite-time blow-up.

We consider the dynamics of a thin liquid film falling down a uniformly heated wall. The heating sets up surface tension gradients that induce thermocapillary stresses on the free surface, thus affecting the evolution of the film. We model this thermocapillary flow by using a gradient expansion combined with a Galerkin projection with polynomial test functions for both velocity and temperature fields. We obtain equations for the evolution of the velocity and temperature amplitudes at first- and second-order in the expansion parameter. These equations are fully compatible close to criticality with the Benney long-wave expansion. Models of reduced dimensionality for the evolution of the local film thickness, flow rate and interfacial temperature only, are proposed.

The present theoretical study focuses on the dynamics of a thin liquid film falling down a vertical plate with a nonuniform, sinusoidal temperature distribution. The results are compared to those obtained in the case of the uniform temperature distribution. The governing evolution equation for the film thickness profile based on long-wave theory accounts for two instability mechanisms related to thermocapillarity. The first mechanism is due to an inhomogeneity of the temperature at the liquid–gas interface induced by perturbations of the film thickness, when heat transfer to the gas phase is present, while the second one is due to the nonuniform heating imposed at the plate and leads to steady-state deformations of the liquid–gas interface. For a moderate nonuniform heating the traveling waves obtained in the case of a uniform heating are modulated by an envelope. When the temperature modulation along the plate increases the shape of the liquid–gas interface becomes “frozen” and the oscillatory traveling wave regime is suppressed. The enhancement of the heat transfer due to permanent deformations and traveling waves is also assessed. The latter is found to have no significant effect on the heat transfer coefficient, while the former can increase it significantly. A good agreement between the theoretical model and the experimental data obtained for a step-wise temperature distribution at the plate is found and the reason for discrepancies is explained.

We analyse the regularized reduced model derived in Part 1 (Ruyer-Quil et al. 2005). Our investigation is two-fold: (i) we demonstrate that the linear stability properties of the model are in good agreement with the Orr–Sommerfeld analysis of the linearized Navier–Stokes/energy equations; (ii) we show the existence of nonlinear solutions, namely single-hump solitary pulses, for the widest possible range of parameters. We also scrutinize the influence of Reynolds, Prandtl and Marangoni numbers on the shape, speed, flow patterns and temperature distributions for the solitary waves obtained from the regularized model. The hydrodynamic and Marangoni instabilities are seen to reinforce each other in a non-trivial manner. The transport of heat by the flow has a stabilizing effect for small-amplitude waves but promotes the instability for large-amplitude waves when a recirculating zone is present. Nevertheless, in this last case, by increasing the shear in the bulk and thus the viscous dissipation, increasing the Prandtl number decreases the amplitude and speed of the waves.