Introduction and History

Hsueh-Chia Chang    University of Notre Dame, IL, USA

Evgeny A. Demekhin    Kuban State University, Russia

The scientific community was first introduced to wave dynamics on a falling film when the father-son team of the Kapitza family (Kapitza & Kapitza, 1949) took the elegant photographs shown in Figure 4.3. Under house arrest, they were compelled to make do with the simplest experimental apparati in their kitchen. Yet, a rich variety of wave forms were elicited by their humble experiments – sinusoidal waves with rounded crests and troughs, solitary waves of lonely drops, trains of pearly capillary drops that repel, coalesce and otherwise exhibit fascinating motions. They can be boringly rythmatic and also be unpredictably irregular - traits that have been exploited in fountains and table-top sculptures and extolled by poets. In short, they represent the simplest hydrodynamic instability that can produce the wonderfully rich spatio-temporal patterns of more elaborate instabilities. Their simplicity lies in the fact that, although it is an instability driven by inertia, it can occur at extremely low Reynolds numbers under very common conditions. In fact, the most practical application of falling film waves is at a Reynolds number of about 100. Hence, simple tap water flow is sufficient to produce a wavy film- a fact important to the Kapitzas. Car windshields and window panes on a rainy day are other common examples. Their simplicity also lies in the fact that, for a vertical column and plane, they can be parameterized by a single dimensionless parameter with some mild approximations. All the rich wavy dynamics hence depend only on a single dimensionless number.

Aesthetics aside, interfacial waves on a falling film are also of the utmost practical importance. Interfacial heat and mass transfer is known to increase by an order of magnitude when the waves are present (Dukler, 1977, Nakoryakov et al., 1987). Cooling towers, condenser tubes, compact multi-phase heat exchangers, scrubbers etc all benefit from the interfacial waves. As a direct result, while empirical correlations for wave effects have existed for over a century, earnest engineering research on film waves has also been around since mid twentieth century. Engineering researchers like Dukler of Houston, who first introduced one of the authors to the subject, have devoted their careers to the subject. Dukler, Alekseenko in Novosibirsk, Whitaker at UC Davis, Stainthrop and Allen at Manchester, Portalski and Clegg at Surrey, Goren at Berkeley, Brauner and Maron at Tel-Aviv etc. carried out the definitive experimental characterization of wave statistics under natural conditions. Their data are still among the best in the literature.

Kapitzas’ report arrived when the applied mathematics community was actively formulating the basic linear Orr-Sommerfeld theories for the now-classical flow instabilities – Kelvin-Helmholtz, Tollmein-Schlicting, Couette, Poiseuille, Rayleigh etc. Falling-film instability has the added complexity of involving a free surface and hence became even more appealing. In the late fifties and early sixties, some of the top hydrodynamicists and applied mathematicians pursued this problem with zest. In Cambridge, Benjamin first understood the origin of instability with a long-wave expansion. His theory was later improved by Yih at Michigan who explained the all-important role of surface tension. In an attempt to extend the theory to include nonlinear effects, as the Ginzburg- Landau theory has done for short-wave instabilities, Benney at MIT pioneered the lubrication approach, Shkadov of Moscow State began the averaging method (culminating in the Shkadov model) and Sivashinsky, first in Russia and then in Haifa, derived the famed Kuramoto-Sivashinsky equation for the falling film instability.

With such intense scrutiny by both the scientific and engineering communities (as ably documented by Lin in 1983), our understanding of falling-film waves was still woefully lacking by the late seventies. Like all classical instability theories, only the primary instability of the flat-film basic state was fully explored by the linear theories available at that point. Nonlinear theories, began by Benney, Shkadov and Sivashinsky, were still at their infancy and their application still uncertain. This implies that all interesting wave dynamics, those that first appealed to the Kapitzas and to the romantics, remain a mystery. Equivalently, primary instability theory can only describe the wave dynamics of the first 10 cm of a vertical column or plane. Waves continue to evolve for typically another meter with much richer dynamics that escape the linear theory (Figure 1.1). With all that fanfare, only the most boring 10% had been captured by the best theoretical hydrodynamicists. The same assessment applied to almost all other classical instabilities. The sequence of rich evolution depicted in Figure 1.1 is generic to many open-flow instabilities. Yet, theoretical descriptions were, and still are, only available for the first primary instability region.

Drastic improvements began in the eighties, particularly for the falling-film instability. Dramatically improved computational power allows simulation of wave dynamics within a one-meter channel to fully capture the entire sequence of wave evolution. Although only model equations and not the full equations of motion can be simulated, such numerical efforts by one of the authors at Moscow State, Davis and Bankoff at Northwestern, Villadsen at Lyngby, Scriven at Minnesota, Patera and Brown at MIT etc. began to produce wave dynamics and statistics unavailable from empirical measurements. More importantly, the fledgling field of nonlinear dynamics made a fundamental impact. A pioneering numerical paper by Pumir, Manneville and Pomeau (1983) pointed out that the complex interfacial dynamics on a falling film often consist of quasi-stationary travelling waves that propagate for long distances without changing their speed or shape. In particular, such travelling waves can be associated with limit -cycles, homoclinic orbits and heteroclinic orbits of dynamical systems theory. Several efforts in the late eighties, by both authors independently and by Kevrekidis at Princeton, then raced to classify all traveling-wave solutions both theoretically and numerically. The complete zoo was only recently completed by the collaborative efforts of both authors.

However, quasi-stationary travelling waves must still evolve on the falling film to give rise to the rich dynamics in Figure 1.1. Such evolution is the target of our collaboration in the last six years. Our approach is analogous to that of coherent structure theory in shear turbulence – the spatio-temporal dynamics are driven by the relatively slow and long-range interaction of several localized traveling waves. However, although the theory remains controversial for shear turbulence, our effort has established its validity for the falling film. We are helped in our pursuit by the parallel experimental efforts of Gollub at Haverford and Alekseenko at Novosibirsk.

The dominant travelling waves are shown to be solitary waves. Their formation, mutual attraction, repulsion, coalescence and interaction with the substrate essentially contribute to all the spatio-temporal dynamics beyond the first primary instability region in Figure 1.1. To quantify such dynamics, we have invoked discrete and essential spectral theory for solitary waves, a new weighted resonance pole spatial theory, Center Manifold Theory and a host of new mathematical theories. These new weapons complement the existing arsenals of matched asymptotics, two-scale averaging, bifurcation theory and Shilnikov homoclinic theory. In essence, a new library of mathematical tools have been assembled to study the dynamics of these solitary waves that do not conserve energy. They are the counterparts of the inverse scattering theory and transformation methods for intergrable solitons that do.

We summarize most known results on falling film waves, including those from the new approach of our collaboration, in this book. We shall begin with a complete linear Orr-Sommerfeld analysis of the flat-film basic state. This is followed by derivation of simple model equations from the Navier-Stokes equation, numerical solution of which is still impossible in an extended domain. We then carry out numerical simulation and approximate and exact construction of the localized structures by asymptotic and numerical methods. The solution branches of these structures will be classified by dynamic singularity theories (double-zero, steady-Hopf and Shilnikov homoclinity) and normal form theories. The stability of such structures will then be analyzed by a spectral analysis. Their spectra include a finite number of dominant discrete modes and a continuum of essential modes. A weighted spectral theory is used to collpse the latter continuum into a single discrete resonance pole. The existence and degeneracy of the resonance poles and discrete spectra will be shown to be related to mass conservation and translational symmetries of the structures. They also capture several unique hydrodynamic features like receptivity, susceptibility, transient...