Falling Liquid Films

We give the motivations for studying film flows, both isothermal and heated, and expose the main features of the phenomena occurring in their evolution. We present, on the one hand, related experimental results from the relevant literature and, on the other hand, how theoretical modeling for these systems is approached. The structure and contents of the monograph are outlined at the end of the chapter.

We give the governing evolution equations and the boundary conditions for a film flowing down a heated wall. These are the continuity, momentum (Navier–Stokes) and energy (Fourier) equations. We also discuss the boundary conditions applied at the wall and at the free surface. The former consist of the no-slip and no-penetration condition for the velocity field and a choice between two temperature boundary conditions: the wall can be heated with a Dirichlet or specified temperature (ST) or through a Robin/mixed or heat flux (HF). The interfacial conditions consist of the kinematic boundary condition, normal and tangential stress balances (with the Marangoni effect when applicable) and the so-called Newton’s law of cooling. We identify the appropriate scales of the problem and obtain the governing dimensionless groups and parameters.

We analyze the stability of the Nusselt flat film solution with respect to infinitesimal perturbations using the governing equations and boundary conditions presented in Chap. 2. We first present the governing equations for the primary instability (Orr–Sommerfeld eigenvalue problem) for both the specified temperature (ST) and the heat flux (HF) conditions. We then deal with transverse modulations triggered by the Marangoni effect (referred to in Chap. 1 as the S-mode) and, subsequently, with stream-wise waves triggered by both hydrodynamics (referred to in Chap. 1 as the H-mode) and the Marangoni effect. We obtain conditions for the onset of the instability in the presence of both modes. Via an energy method/energy balance for the disturbances we analyze the mechanism that triggers the H-mode in isothermal films.

We derive the boundary-layer equations for falling liquid films. The assumptions used in their derivation are similar in spirit to those in the classical boundary-layer theory in aerodynamics. The key in their derivation is the elimination of the pressure by integrating the y-component of the momentum equation where the inertia terms are neglected while at the same time maintaining the inertia terms in the x- and z-components of the momentum equation. We introduce the “Shkadov scaling,” which makes apparent the balance between all forces necessary to sustain strongly nonlinear waves, and we show that the speed of single-hump solitary waves for an isothermal film shows a steep increase as a function of the Shkadov parameter δ, precisely at δ≃1 which then demarcates two distinct flow regimes: the “drag-gravity regime” where δ is small and the “drag-inertia” regime where δ=O(1). Finally, we summarize the different levels of approximations utilized in the description of the falling film problem and the different scalings.

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.

Nearly all low-dimensional models for isothermal films at moderate Reynolds numbers found in the literature rely on a fundamental closure assumption for the stream-wise velocity field: a simple self-similar velocity profile with the variables (x,t) and y/h separated. This is the basis for the classical Kapitza–Shkadov model introduced first in Chap. 1. Here we discuss a systematic methodology to relax the self-similar assumption while maintaining separation of variables: it is based on a combination of an expansion for the velocity field in terms of polynomial test functions, the gradient expansion and an elaborate averaging technique that utilizes the method of weighted residuals. The result is two “optimal” models in the sense that the models are always the same, independently of the particular averaging methodology employed. The two models are: a two-equation system consistent at O(ϵ), referred to as the “first-order model,” and a four-equation system consistent at O(ϵ ²), referred to as the “full second-order model.” An ad hoc compromise between the two in both complexity and accuracy is provided by the “simplified second-order model,” whereas a regularization procedure enables us to reduce the dimension of the four-equation system and to obtain a two-equation model consistent at O(ϵ ²), referred to as the “regularized model.” These models are capable of describing the drag-inertia regime, i.e., they do not suffer from the unphysical loss of the solitary wave branch of solutions observed with the long-wave theory/BE at δ≳1, and hence they cure the deficiencies of the long-wave theory/BE in the drag-inertia regime. Traveling-wave solutions of the averaged models are compared favorably with DNS, demonstrating that indeed low-dimensional modeling of films flows in the drag-inertia regime can be achieved in terms of a small number of coupled evolution equations.

We analyze the linear and nonlinear stage of the instability of a falling liquid film by using the average models developed in Chap. 6. Their linear stability characteristics, e.g. their description of spatially growing disturbances in relation to the convective nature of the instability, are shown to be in good agreement with the Orr–Sommerfeld eigenvalue problem (Chap. 3). By using the average models, the mechanism of the primary instability, already discussed in Chap. 3, is then re-investigated within the framework of the wave hierarchy analysis proposed by Whitham. We emphasize the similarities between roll waves in open channels and solitary waves in film flows at large Reynolds numbers. In particular, two-equation models of film flows have a structure similar to the Saint-Venant equations for shallow-water flows. In both cases, the mechanism of the primary instability can be understood in terms of a wave hierarchy as the competition between kinematic and dynamic waves. We scrutinize the influence of dispersive effects associated with the stream-wise second-order viscous terms, a phenomenon we refer to as “viscous dispersion,” onto the kinematic waves: viscous damping of high-frequency waves reduces the kinematic wave speed which in turn reduces the gap in speed between kinematic and dynamic waves. As far as the nonlinear stage of the dynamics of a falling liquid film is concerned, it is dominated by a competition between the primary instability of the Nusselt flat film flow and the secondary instabilities of the traveling waves with saturated amplitudes. This competition is characterized by a variety of nonlinear processes (e.g., spatial and temporal modulations, phase locking) which are still not fully understood. Applying a periodic forcing at the inlet may regularize the flow, leading further downstream to regular periodic wave-trains whose properties can be obtained using elements from dynamical systems theory. We construct bifurcation diagrams of permanent-form traveling waves including solitary waves. Particular attention is given to the role of stream-wise viscous effects on the properties, such as shape, speed and solution branches of the traveling waves. Taking into account these effects is crucial for a proper description of the dynamics of wavy film flows.

The three-dimensional wave regime on film flows down an inclined plane has a very rich phenomenology. In particular, isolated synchronous depressions, rugged-modulated waves as well as horseshoe-like three-dimensional solitary waves and oblique solitary waves are observable. This phenomenology is far from being fully understood. However, recent well-controlled experiments in the inclined plane geometry by Gollub and coworkers have enabled us to understand the transition from two-dimensional to three-dimensional flows in terms of two different secondary instability mechanisms, leading to in-phase span-wise/synchronous modulations or to herringbone patterns, and ultimately, to modulated or horseshoe-like solitary waves. To scrutinize the transition from two-dimensional to three-dimensional flows, the two-dimensional averaged models derived in Chap. 6 have been extended to include the span-wise dependence of the flow. The stability of two-dimensional periodic waves based on a Floquet analysis shows that the secondary instability is not selective, which makes the resulting three-dimensional instability strongly dependent on the initial conditions. Provided that initial conditions are appropriately tuned, the experimental results reported by Gollub and coworkers are recovered by numerical simulations. The widespread observation of the synchronous instability in the above experiments is likely to be related to the span-wise nonuniformities at the inlet, favoring in-phase modulations of the wave fronts. In some cases, the three-dimensional patterns emerge from a two-dimensional oscillatory mode rather than from saturated traveling waves, as also observed in DNS. The competition between the growing two-dimensional modulation and the secondary three-dimensional instability makes the evolution of the film even more sensitive to initial conditions. The agreement of the simulations of the three-dimensional low-dimensional models obtained from the weighted residuals methods, to the available experimental data is encouraging. The regularized and the full second-order model are able to recover the synchronous scenario of transition from two-dimensional to three-dimensional wave patterns observed in the experiments by Gollub and coworkers, whereas simulations based on the simplified model systematically show a subharmonic transition scenario (herringbone pattern).

We give the methodology for the development of low-dimensional models of a heated falling film at moderate Reynolds numbers, i.e. in the drag-inertia regime, and accounting for second-order viscous and heat effects. As for isothermal conditions, the full second-order model being of large dimensionality/degree of complexity, necessitates the development of a regularized model that contains the same physical ingredients and also retains the basic dynamic characteristics of the full system, but with only three coupled equations. The regularized model is used to analyze the influence of the different parameters in the linear regime. The linear stability properties of the model are in good agreement with the Orr–Sommerfeld eigenvalue problem (Chap. 3) for a wide range of Reynolds and Marangoni numbers. In the nonlinear regime, the single-hump solitary wave solution branches of the regularized model do not exhibit any unphysical turning points such as those encountered with BE and instead predict the continuous existence of solitary waves for all Reynolds numbers. But the regularized model does have some limitations for large Péclet numbers, the main one being the prediction of unphysical negative temperatures. Nevertheless, the model allows us to identify the different mechanisms involved in the modification of solitary waves due to the thermocapillary effect, in both the drag-gravity and the drag-inertia regimes, and in both two-dimensional and three-dimensional flows. In the latter case, of particular interest is the occurrence of two-dimensional solitary-like waves triggered by the growth of thermocapillary rivulets and riding the rivulets.

In this final chapter, we present a number of open questions that still demand an answer. Our hope is to motivate further research devoted to the fascinating dynamics of falling film flows. We also believe that the methodology developed in Chap. 6, and applied to isothermal film flows (Chaps. 7 and 8) and to heated film flows (Chap. 9), can be extended to a wide variety of problems which have not been described in previous chapters. Some of them are discussed here.