Oseen Flow in Paint Marbling

Abstract

Paint marbling refers to techniques for creating intricate designs in colored paints floating on a liquid surface. If the marbling motions are executed slowly, then this layer of paints can be modeled as a two-dimensional incompressible Newtonian fluid. In this highly constrained model many marbling techniques can be exactly represented by closed form homeomorphisms. Homeomorphisms can be composed and compute the composite mapping at any resolution. Computing homeomorphisms directly is orders of magnitude faster than finite-element methods in solving paint marbling flows. Most marbling patterns involve drawing rakes from one side of the tank to the other; and these can be modeled by exact closed form homeomorphisms. But pictorial designs for flowers and animals use short strokes of a single stylus; presented is an exact velocity field for Oseen fluid flow and its application to creating short stroke marbling homeomorphisms.