Model for avoiding accidents

Christina Lienstromberg is a professor at the University of Stuttgart in Germany. Thanks to a grant from Knut and Alice Wallenberg Foundation, she will be a visiting professor at the Centre for Mathematical Sciences, Lund University.

The planned project involves developing mathematical models for liquid films coating the inside of a rotating cylinder. Flows such as these are widely used in various industrial applications, such as spin coating, spin casting, or rotational moulding. The dynamic behavior of the flow strongly depends on the interplay of external forces that arise from rotation, friction, surface tension and gravity and might cause interesting effects – for example, they can lead to a rupture of the film coating the inner wall of the cylinder.

The point of departure is the Navier-Stokes equations, which are fundamental to describing the behaviour of fluid flows. Using these, it is possible to specifically derive a single model equation for the height of the thin film when the aspect ratio of the fluid film tends to zero. The resulting equation is a fourth-order quasilinear partial differential equation. When the fluid’s surface touches the inner cylinder wall, the film ruptures. Mathematically, this means that the equation degenerates.

One aim of the project is to derive higher dimensional thin-film rimming-flow model equations from the Navier-Stokes system. Another aim is to use rigorous mathematical analysis to develop methods for studying the long-time behavior of solutions to thin-film rimming-flow problems and their (in-)stability with respect to small perturbations.