Three dimensional simulations of coating flows: Methods for large free surface deformations and moving three-phase lines

Computational fluid mechanics techniques for examining free surface problems for coating flows in two dimensions is now a well-established procedure, owing much to the efforts of Scriven and his researchers.(1,2) Extending these methods to three dimensions requires a reconsideration of some of the same long-standing difficult issues for coating flows as well as special algorithms designed for the added geometric complexity. This paper presents a non-linear elastic pseudo-solid approach for deforming meshes in three dimensions. Special boundary conditions on the pseudo-solid mesh motion constrain ifs motion in the normal direction according to the relevant fluid physics, but allow shear-free motion of the mesh tangential to bounding surfaces. Within the standard Galerkin Finite Element formulation of the problem, these goals are achieved by locally rotating the pseudo-solid momentum vector equations at the boundaries. Moving or static contact lines provide a challenge in 3D because contact angle conditions must be imposed so that the fictitious solid, in which the mesh is embedded, may slide tangentially along three-phase lines unhindered. The computational techniques discussed in this talk are illustrated with two applications: solid-body rotation of fluid, and extrusion of an incompressible Newtonian fluid from a square nozzle.