The basic idea of simulating fluid dynamics with cellular automata is implementing particle dynamics as cellular automata rules and fluid dynamics would emerge as mean properties of

Differential equation model for incompressible fluids

We are interested in the flow velocity ${\displaystyle u}$ that is defined by two differential equations

Navier-Stokes equation

${\displaystyle {\frac {\partial u}{\partial t}}+(u\nabla )u=-\nabla P+\nu \nabla ^{2}u}$

continuity equation

${\displaystyle \nabla \cdot u=0}$

${\displaystyle P=p/\rho _{0}}$ is the kinematic pressure, ${\displaystyle p}$ is the pressure, ${\displaystyle \rho _{0}}$ constant mass density and ${\displaystyle \nu }$ is the kinematic share viscosity.

References

1. Dieter A. Wolf-Gladrow, Lattice-Gas Cellular Automata and Lattice Boltzmann Models An Introduction, Springer 2000