Abstract
In this set of notes I wish to summarize a few ideas about a class of free-boundary problems that arise in fluid dynamics. In particular, I shall consider the time-dependent motion of thin films in cases where viscous effects are significant. This topic may be considered naturally under the theme “reactive flows” since (i) changes in surface tension produce fluid motions (so-called Marangoni motions) - this is a response that occurs at fluid-fluid interfaces - and (ii) recent research has demonstrated the ability to carefully prepare patterned surfaces on the scale of (sub)microns and so control the movement of small droplets along a surface (e.g. [1], [2]) - these are examples of reactive wetting at solid surfaces (e.g. [3]). Since many common configurations, such as coating operations and spreading of fluid films and droplets, have a liquid layer adjacent to a rigid boundary we shall be content here to outline the basic fluid dynamics equations for thin film flows. Several different examples of nonlinear partial differential equations naturally arise in these problems. As this is a subject with a very large literature, we refer the reader to the references at the end (some of which have extensive reference lists) for more details.
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Stone, H.A. (2002). Partial Differential Equations in Thin Film Flows in Fluid Dynamics and Rivulets. In: Berestycki, H., Pomeau, Y. (eds) Nonlinear PDE’s in Condensed Matter and Reactive Flows. NATO Science Series, vol 569. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-0307-0_12
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