Experimental evidence of non-unique solutions to a steady non-linear coating flow

The industrial practice of curtain coating, used in a vast number of applications, involves a liquid film falling vertically to coat a solid surface passed underneath at high speed. This solid surface may either be dry prior to reaching the liquid film making contact or be pre-wetted by a thin liquid film, depending upon the nature of the application. Above some critical speed, dependent upon the physical and chemical properties of the solid and liquid and other system parameters, air is entrained at the contact line which is formed at the intersection of the falling film with the moving surface. This particular flow geometry has been exploited previously to provide experimental evidence of non-local hydrodynamic influence on the dynamics of the contact line (Blake et al., 1994, 1999). They showed that the flow field (and especially the flow rate) at the orifice, from which the falling film emerges, can be manipulated to promote higher stable coating speeds; this effect was termed hydrodynamic assist and has been a challenging aspect to incorporate into mathematical models (Blake, 2006). The complexity of this flow was further demonstrated by Marston et al. (2006a) who showed that, under certain conditions, the shape of the instability boundaries in parameter space is more complicated than had previously been expected, meaning that the global maximum speed of stable coating could only be achieved by taking a non-linear route through parameter space. There are numerous competing models of coating processes, based around the Navier–Stokes equations, with different suggested boundary conditions near the contact line. The modelling uncertainty arises in how to correctly describe the physics in the vicinity of the intersection of the falling film and the moving surface. When the moving substrate is dry, there is debate as to how to generalize the no-slip