A simulation of hydrodynamics on non-commutative space

Abstract

A simulation of the hydrodynamics on the two dimensional non-commutative space is performed, in which the space coordinates $(x, y)$ are non-commutative, satisfying the commutation relation $[x, y]=i \theta$. The Navier-Stokes equation has an extra force term which reflects the non-commutativity of the space, being proportional to $\theta^2$. This parameter $\theta$ is related to the minimum size of fluid particles which is implied by the uncertainty principle, $\Delta x \Delta y \ge \theta/2$. To see the effect of this parameter on the flow, following situation is considered. An obstacle placed in the middle of the stream, separates the flow into small slit and large slit, but the flow is joined afterwards in the down stream. For the Reynolds number 700, the behavior of the flows with and without $\theta$ is observed to differ, and the difference is seen to be correlated to the difference of the activity of vortices in the down stream. The oscillation of the flow rate at the small slit diminishes after the certain time in the usual flow when the "two attached eddies" appear. In the non-commutative flow this two attached eddies appear from the beginning and the behavior of the flows does not fluctuate largely. The irregularity in the flow existing in the beginning disappears after the certain time.