# Investigation of mathematical models for thin-film flows

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# The mathematics of thin liquid lines

Research into the dynamics and stability of thin liquid films on a solid surface has advanced through a combination of modelling and mathematical analysis. From the better understanding gained of fluid motion under the influence of surface tension, a wide variety of industrial applications may benefit.

Industrial Technologies

The motion of a liquid such as water under the influence of surface tension is a phenomenon we experience every day when we take a shower or turn on the windshield wipers. Many industrial processes involve the application of a thin layer of viscous liquid to a solid surface under carefully controlled conditions. Typical examples are the process of coating the inside of fluorescent light bulbs or the laying down of different types of coatings on television screens. With the support of EU funding, the 'Investigation of mathematical models for thin-film flows' (TFE) project touched on many aspects of such thin liquid films and their interactions with a solid surface. Researchers modelled the dynamics of thin liquid films using the lubrication approximation, which is employed to simplify the Navier–Stokes equations. In theory, the Navier–Stokes equations reveal the velocity and pressure of a fluid flowing by any point near an object's surface. The TFE researchers started with the study of a droplet of surfactant, a compound that lowers surface tension thus allowing easier spreading on a thin liquid film. This is a classic test bed for modelling to further understand the intricate fluid dynamics observed in laboratory experiments. It also formed the basis for a thorough mathematical analysis of multi-layered polymer films and wall-confined binary alloys, formed by a mixture of three types of atoms. As we know from waxing our cars and cooking with non-stick cookware, the dynamics of the fluid flowing on a solid surface depends heavily on the surface chemistry. This is why the TFE researchers searched and found series of solutions to describe the dynamically evolving contact line under the influence of attractive van der Waals and repulsive Born forces between the molecules. Furthermore, they developed much-needed alternative ways to solve the lubrication equations when the tangential component of force exerted by the fluid on the solid surface diverges. The classical theory of fluids requires that the fluid velocity is set to zero on the solid boundary. While this makes sense for the Navier–Stokes equations in wall-bounded flows, its relevance to a moving contact line was questioned. The systematic study of such model equations may require new numerical methods. TFE project results offer, however, the possibility of better understanding the processes leading to the observed patterns. It is hoped that they will spark more interest in these mathematical problems with significant benefits in biological, medical and industrial applications.

## Keywords

Thin liquid films, solid surface, surface tension, thin-film flows, lubrication approximation, surfactant, fluid dynamics, polymer films, binary alloys, surface chemistry, wall-bounded flows, numerical methods