Instabilities in Falling Liquid Films

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

The dynamics of such ¿lms are quite complex and display rich behavior and this attracted many mathematicians, physicists, and engineers to the ¿eld. 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 ¿lm where the ¿ow is driven by various forces such as gravity, capillarity, thermocapillarity, centrifugation, and inter- molecular. The ¿ow 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 ¿lm can exhibit.

Direct analysis of the equations of the model of the interfacial ¿ows is a very complicated mathematical exercise due to the existence of a free, evolving interface that bounds the liquid ¿lm. 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 ¿elds apart from the ¿uid equations; (c) Several nonlinear boundary conditions should be speci¿ed at the unknown interface(s) and (d) The solutions may not exist for all times. In fact in thin ¿lm problems, one may encounter ¿nite-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 di¿erent length scales appearing in thin ¿lm ¿ows to our advantage. Thin ¿lms 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.