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.

Instabilities in Falling Liquid Films - pocket, Engelska, 2023