Instabilities in Falling Liquid Films

The flows of thin film form the core of a large number of scientific, technological, and engineering applications. The occurrence of such flows can be observed in nature, for example on the windshield of vehicles in rainy weather. Thin film flows are also found in various engineering, geophysical, and biophysical ap- plications. Specific examples are nanofluidics, microfluidics, coating flows, intensive processing, tear-film rupture, lava flows, and dynamics of continental ice sheets. Important industrial applications of thin films include nuclear fusion research - for cooling the chamber walls surrounding the plasma, complex coating flows - where a thin film adheres to a moving substrate, distillation units, condensers, and heat exchangers, microfluidics, geophysical settings, such as gravity currents, mud, granular and debris flows, snow avalanches, ice sheet models, lava flows, biological and biophysical scenarios, such as flexible tubes, tear-film flows and many more.

The dynamics of such films are quite complex and display rich behavior and this attracted many mathematicians, physicists, and engineers to the field. In the past three decades, the work in the area has progressed a lot with considerable stress on revealing the stability and dynamics of the film where the flow is driven by various forces such as gravity, capillarity, thermocapillarity, centrifugation, and inter- molecular. The flow may happen over structured or smooth and impermeable or slippery surfaces. The investigation approaches include modeling and analytical work, numerical simulations, and performing experiments to explain the instabilities that the film can exhibit.

Direct analysis of the equations of the model of the interfacial flows is a very complicated mathematical exercise due to the existence of a free, evolving interface that bounds the liquid film. The mathematical complexity emerges from a number of things: (a) The Navier-Stokes (or Stokes or Euler) equations need to be solved in changing domains; (b) In certain applications one has to solve for the temperature or electrostatic or electromagnetic fields apart from the fluid equations; (c) Several nonlinear boundary conditions should be specified at the unknown interface(s) and (d) The solutions may not exist for all times. In fact in thin film problems, one may encounter finite-time singularities accompanied by topological transitions. The breakup of liquid jets is an example of that. However, in the subsequent chapters, we shall see that it is possible to use the different length scales appearing in thin film flows to our advantage. Thin films are characterized by much smaller length scales in the vertical direction as compared to those in the stream-wise direction. This gives rise to a small aspect ratio which makes the problem amenable for small amplitude perturbation expansions.