Langevin and Navier-Stokes Simulation of Three-Dimensional Protoplasmic Streaming

In this paper, we report the numerical results obtained using the Langevin Navier-Stokes (LNS) simulation of the velocity distribution of three-dimensional (3D) protoplasmic streaming in plant cells, such as those of {\it Nitella flexilis}. The LNS simulations are performed on 3D cylinders discretized by regular cubes in which fluid velocities are activated by boundary velocities parallel and nonparallel to the longitudinal direction and a random Brownian force with strength $D$. We find that, for a finite $D$, the velocity distribution $h(V), V\!=\!|\vec{V}|$, has two different peaks at a small non-zero $V$ and a finite $V$, and the distribution $h(V_z)$ for $|V_z|$ along the longitudinal direction also has a peak at finite $V_z$. These results are in good agreement with the reported velocity distributions observed using laser Doppler velocimetry. Moreover, we study the effects of the Brownian force on biological material mixing and find that mixing along the $\vec{V}$ direction enhanced by the nonparallel circular motion is further improved by the Brownian force in the experimentally relevant region of $D$. In addition, the experimentally relevant $D$ is found to be consistent with the expectation from the fluctuation dissipation relation between the random stress and viscosity in the LNS equation of Landau and Lifschitz for incompressible fluids.