Stochastic Fluid Dynamics Simulations of the Velocity Distribution in Protoplasmic Streaming

Protoplasmic streaming in plant cells is directly visible in the cases of Chara corallina and Nittella flexilis, and this streaming is understood to play the role of transportation of biological materials. For this reason, related studies have focused on molecular transportation from a fluid mechanical viewpoint. However, the experimentally observed distribution of the velocity along the flow direction $x$, which exhibits two peaks at $V_x\!=\!0$ and at a finite $V_x(\not=\!0)$, remains to be studied. In this paper, we numerically study whether this behavior of the flow field can be simulated by a two-dimensional stochastic Navier-Stokes (NS) equation for Couette flow, in which random Brownian force is assumed. We present the first numerical evidence that these peaks are reproduced by the stochastic NS equation, which implies that the Brownian motion or the stochastic nature of the fluid particles plays an essential role in the emergence of the peaks in the velocity distribution. We also find that the position of the peak at $V_x(\not=\!0)$ moves with the variation in the strength $D$ of the random Brownian force, which also changes depending on physical parameters such as the kinematic viscosity, boundary velocity and diameter of plant cells.