Spreading Problems in Fluid Mechanics

Northwestern University, Evanston IL

Investigators

Abstract

0104935 Miksis In this proposal, it is planned to investigate both the problem of the spreading of a liquid along a solid substrate and the spreading of a liquid along a liquid interface. The Navier-Stokes equations will govern the motion of the fluids. The investigation of the spreading of a liquid along a solid substrate is interesting and difficult because of the singularities that can occur at the contact line. Spreading along a liquid interface is a challenging problem where multiple free boundaries can exist. Here, in addition to the liquid/liquid and liquid/gas boundaries, a triple junction can exist at the liquid/liquid/gas line of intersection. Predicting the dynamics of these three phase lines is one of the aims of this proposal. We will be interested in situations where the interfaces are clean and situations where surfactants are present. Mathematically we are faced with problems having coupled moving interfaces along which singularities can exits. Both asymptotic and numerical methods will be used to study these complex free boundary problems. For example, we will consider the thin film limit where the complete three-dimensional system of equations can be reduced to a coupled system of nonlinear evolution equations. We will also apply the level-set numerical method to solve the complete nonlinear system of equations. When a liquid drop is resting on a solid surface, the free boundary of the drop is characterized by the gas/liquid interface, and the line of contact between the liquid/gas/solid phases. This three-phase line is usually referred to as the contact line. It occurs whenever three phases come into contact. It is a familiar phenomena of everyday life, and can be observed when one pours oil into a frying pan or in a partially filled wine glass. It also occurs in many areas of practical interest. Examples are industrial coating processes, the transport of gas/liquid mixtures in pipes, the spreading of droplets of medication in the lungs and the spreading of liquid wastes (e.g. oil or chemicals) on the sea. The latter problem is an example of a situation where the contact line (refereed to as the triple junction in this case) occurs at the intersection of a liquid/liquid/gas interface. Even though the contact line occurs in these many areas, our understanding of it is very limited. Models are still being developed and because of the mathematical singularities associated with the behavior of the solutions in the neighborhood of the contact line, solution techniques, both analytical and numerical, need to be identified.