A Representation of Bounded Viscous Flow Based on Hodge Decomposition of Wall Impulse

A Lagrangian representation of bounded incompressible flow is introduced in which viscous boundary conditions are given kinematic expression by the generation of impulse at the wall. The relationship between such a process and the boundary conditions is deduced from two complementary Hodge decompositions. The orientation of the created impulse vector may be chosen to be parallel at the wall (this being equivalent to a thin vortex doublet sheet) or normal at the wall (this being a thin monopole vortex sheet). Although the representation is developed here for two dimensions, it can be generalized in a natural way to three dimensions. The case of tangentially oriented wall impulse is applied to flow over a semi-infinite plate; the case of normally oriented wall impulse is applied to flow past a circular cylinder.