A new method is presented for flow visualization based on an advection reaction model. The key point of the method is to use a flow dependent reaction source which couples advection and reaction dynamics. When embedded in semi-Lagrangian advection this reaction source provides systematic pattern generation and integration along pathlines. Based on this idea and a very simple equation with a stable and efficient numerical solution, a visualization model is developed which allows visualization of both Lagrangian and Eulerian structures in a flexible environment and with user control. This system also allows the emulation of conventional dense texture and reaction-diffusion type visualization techniques easily and provides solutions to the problems in these methods.

In 2D, incompressible flows, the Stokes equations that represent the dynamics of very viscous flows and vorticity formulation of hydrodynamic equations both reduce to a scalar stream-function representation in terms of elliptic equations. By making use of this simplification and the properties of Fourier space representation of elliptic equations, we use a common spectral method to solve both of these equations. Based on this system of equations, we propose a level set based input description which provides a flexible environment for the user to model a wide range of flows and artistic effects in 2D. This input type allows the modeling of vortex sheet patterns and other complex flows with a very practical approach and chaotic, dynamic flows, even with viscous Stokes equations. A user interface is developed for the level set input which allows the user to draw the strokes or edit the level set data by applying transformation functions or perturbations. To sum up, this model can be used for the simulation of very viscous flows, vorticity dynamics, vortex sheet patterns, turbulent and chaotic flows as well as other artistic effects such as the traditional marbling patterns, with a simple, fast and stable system at high resolutions.

This paper presents a multiscale fluid model based on mesoscale dynamics and viscous fluid equations as a generic tool for digital marbling purposes. The model uses an averaging technique on the adaptation of a stochastic mesoscale model to obtain the effect of fluctuations at different levels. It allows various user controls to simulate complex flow behaviors as in traditional marbling techniques, as well as laminar and turbulent flows. Material transport is based on an improved advection solution to be able to match the highly detailed, sharp fluid interfaces in marbling patterns. In the transport model, two reaction models are introduced to create different effects and to simulate density fluctuations.