Stochastic fluid dynamics simulations of the velocity distribution in protoplasmic streaming

Abstract

Protoplasmic streaming in plant cells is directly visible in the cases of Chara corallina and Nitella flexilis, and this streaming is understood to play a role in the transport of biological materials. For this reason, related studies have focused on molecular transportation from a fluid mechanics viewpoint. However, the experimentally observed distribution of the velocity along the flow direction x, which exhibits two peaks at Vx = 0 and at a finite Vx(≠0), remains to be studied. In this paper, we numerically study whether this behavior of the flow field can be simulated by a 2D stochastic Navier-Stokes (NS) equation for Couette flow in which a 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 of the fluid particles plays an essential role in the emergence of these peaks in the velocity distribution. We also find that the position of the peak at Vx(≠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 the plant cells.

Langevin simulations of protoplasmic streaming in non-Euclidean geometry

“…This statement (A) is the same as that in Ref. [20] except the fact that S includes ρ and the extra condition γ = 1 in this paper. The second statement is as follows:…”

“…On the right-hand side, the components of g are given by Gaussian random numbers of mean 0 and variance 1, and the relation between g and η is given by η∆t = √ 2D∆t g (see Ref. [20] for further details). Solving this Poisson's equation and obtaining p(t), we get the condition ∇ • V (t+∆t) = 0 for any t in Eq.…”

“…As discussed in Ref. [20], the solution should be independent of ∆t, ∆x and simulation units, which are always different from the real units for length (m), time (s), and weight (kg), where the final one (kg) is necessary for the LNS equation in Eq. ( 3), because it includes ρ(kgm −3 ), which is not included in LNS equation in Ref.…”

“…The peak at V x = 0 corresponds to the zero-velocity Brownian motion of fluid particles, while the peak at V x = 0 is considered to correspond to the speed of the molecular motor. Recently, these two different peaks have been numerically studied by Langevin Navier-Stokes (LNS) simulations with vortex (ω) and stream function (ψ) for two-dimensional (2D) Couette flow between parallel plates [20].…”

“…However, the technique in Ref. [20] is limited to only 2D flows because the NS equation is for ω and ψ and is not applicable to actual 3D flows. Therefore, studying 2D Couette flow by LNS simulations for velocity ( V ) and pressure (p) is worthwhile.…”