If the requirement for accurate fluid pressure is relaxed, then the equations of motion for a liquid can be reduced from the full Navier-Stokes to a simple differential formula which takes account of just convection and diffusion. Thus we can completely remove terms from Eq.(1) that include the pressure or viscous effects. In one dimension, this produces a simplified wave equation which can be written in terms of gravity g, the local depth of fluid d, and the rate of change of fluid height h. The result is the one dimensional wave equation which can been used to approximate simple non-rotational wave mechanics. This equation is written as
When approximated as a finite difference expression the wave equation leads to a tridiagonal linear system that can be solved very efficiently in just a single iteration. This type of solution is out of the scope of this report, but a very clear description of the process is given in [21].
Equation (2) captures the basic characteristics of external blood flow such as mass transport and pooling, without incurring a large computational overhead. It may also be used to quickly calculate a pseudo-three dimensional flow by solving it in two orthogonal directions over a height field. Thus, although not as accurate as the full three-dimensional Navier-Stokes equations, the wave equation is ideal for modeling the visual aspects of bleeding necessary for a real-time simulation.
