Analysis of a Nonisothermal Viscous Flow Problem with Free Boundaries and a Dynamic Contact Line

Abstract

In the present paper, we consider the plane steady-state nonisothermal flow problem with two free boundaries describing a coating process with a heavy viscous incompressible fluid onto a horizontally moving rigid wall (cf. Figure 1). One of the a priori unknown free surfaces is noncompact and the other one ends at an a priori unknown contact line on the rigid wall. The flow domain is unbounded and thermocapillary convection takes place because the surface tension depends on temperature. This type of convection is important in many technological and scientific applications; interesting examples can be found in the field of materials processing, particularly in coating and solidification processes or in crystal growth processes (see Cuvelier and Driessen, 1986; Shen, Neitzel, Jankowski and Mittelmann, 1990; Ehrhard and Davis, 1991; Puknachov, 1985). We assume that the viscosity of the liquid is constant, although in general case it may depend on temperature. The Boussinesq approximation for similar problems was used by several authors (see the references cited above, and also Cuvelier and Schulkes, 1990; Lagunova, 1986; Rivkind and Il’in, 1988; Socolowsky, 1994a).