Recently, Y. Nambu formulated a new hydrodynamics in which
incompressible fluid dynamics is connected to Hamiltonian dynamics
in terms of area preserving diffeomorphism. If the equations of
motion of two-dimensional fluid are expressed using Poisson brackets,
then it is allowed to understand a stream function plays a role of
Hamiltonian. According to this standpoint, three-dimensional
incompressible fluid theory can be related to the dynamics of Nambu brackets.
In this talk, we investigate a hydrodynamics on non-commutative space
based on the formulation of dynamics by Nambu. Replacing Poisson or
Nambu brackets by Moyal brackets with a parameter θ, a new hydrodynamics
on non-commutative space is derived. It has an additional term of O(θ2),
which does not exist in the usual Navier-Stokes equation.
In hydrodynamics, to introduce Moyal brackets corresponds to a kind of
quantization procedure regarding position coordinates x and y, so that
it makes the position coordinates to non-commutative ones. This procedure
may be a step toward finding the hydrodynamics of granular materials
whose minimum size is given by θ. In order to examine the non-commutative
effect, I compare the behaver of flows which have different size of θ by
computer simulation. In all the discussions in this talk, incompressibility
and non-relativistic flow are supposed.