Thin Films of Non-Newtonian Fluids with Jump Discontinuities
V.Chugunov, L.D.Eskin (Kazan State University, Kazan, Russia); K.Hutter (Institut fur Mechanik, Technische Universitat, Darmstadt, Germany)
Many physical processes associated with film flows are described by sharp changes of their main characteristics -- fluid thickness and mass flux in some small regions. In such situations it is natural to consider models which allow solutions in which a field variable or its space or time derivative may experience a jump discontinuity. Solutions with jumps are natural for hyperbolic equations, and their existence is evident. The situation is different when more complex models, described by parabolic equations, are considered. In this case, the existence of a solution with jumps is not evident. Let us restrict attention to models describing the "diffusive" spreading of a film, which starts initially with a jump in its surface elevation. This choice is motivated by the following reasoning. First, one particular model demonstrates that, if the boundary conditions are regular, then the solution is smooth, even for an initially discontinuous surface elevation; second, such models describe possible realistic situations, and third, they possess certain symmetries allowing a qualitative analysis. Therefore, we can formulate the aim of this work: to consider models admitting the symmetry, in order to show existence of solutions with jumps and to discuss their physical interpretation.
Section : 12