The analysis of coating flows near the contact line
The coating of a surface by a fluid that flows from a container onto a surface that is to be covered can be described by a free boundary problem for the Navier-Stokes equations or some other equations of motion. The free boundary which separates fluid from air is a capillary surface. It meets the surface to be coated in the contact line which separates the coated from the uncovered part. This paper is concerned with the analysis of the solution near the contact line. To investigate this behavior is particularly important for this type of free boundary problems, because examples show that context angles \(\gamma\) different from 0 or \(\pi\) together with the fluid adhering to the boundary \((\underline v= \underline v_\Sigma)\) allow only for solutions with infinite energy. The authors show the existence of a regular solution under the hypotheses \(\gamma= 0\) and \(\underline v= \underline v_\Sigma\), which is the physically relevant case. Despite the long and intricate analysis of this paper the presentation is always clear because the authors not only give detailed proofs but also explain their approach. The assumption of a two-dimensional flow with small velocity leads to a free-boundary problem for the homogeneous Stokes equations. This problem is linearized about the velocity \(\underline v= \text{const}\) in a neighborhood of the contact point; the corresponding solutions can be constructed explicitly. Then the perturbation equations are formulated; as a stream function is introduced, this leads to a nonlinear boundary value problem for an elliptic equation of fourth order. Important steps in obtaining a solution are representation formulas for biharmonic functions in the upper half-plane with rather special boundary values as well as the corresponding estimates which finally allow for an contraction mapping argument to be applied.
analysis of the solution near the contact line
representation formulas for biharmonic functions
scientific article; zbMATH DE number 769001