"From Navier-Stokes to the biharmonic equation in planar irregular domains"
Abstract: The biharmonic equation is of great importance in many fields of applied mathematics. In particular, the classical (Lagrange , 1768) streamfunction formulation of flow problems involves the biharmonic operator as the highest-order term. We give a brief review of the Navier-Stokes theory. We then present a finite difference scheme which approximates the biharmonic equation in general two-dimensional irregular domains. The scheme includes discretizing the values of the function and its derivatives at grid points, and then studying a class of interpolations by sixth-order polynomials. The main interest is that it allows to construct the time-dependent solver, which fits closely to the pure streamfunction formulation of the Navier-Stokes equation.