### Abstract

We present a study of the effect of finite detector integration/exposure time E, in relation to interrogation time interval Δt, on analysis of Brownian motion of small particles using numerical simulation of the Langevin equation for both free diffusion and hindered diffusion near a solid wall. The simulation result for free diffusion recovers the known scaling law for the dependence of estimated diffusion coefficient on E/Δt, i.e. for 0 ≤ E/Δt ≤ 1 the estimated diffusion coefficient scales linearly as 1-(E/Δt/=3) Extending the analysis to the parameter range E/Δt ≥1, we find a new nonlinear scaling behaviour given by.E/Δt/^{-1}[1-E/Δt/^{-1}/)3], for which we also provide an exact analytical solution. The simulation of near-wall diffusion shows that hindered diffusion of particles parallel to a solid wall, when normalized appropriately, follows with a high degree of accuracy the same form of scaling laws given above for free diffusion. Specifically, the scaling laws in this case are well represented by 1 ((1 + ε)(E/Δt))/3, for E/Δt ≤ 1, and (E/Δt)^{-1}[1 - ((1 + ε)(E/Δt)^{-1}/3], for E/Δt ≥ 1, where the small parameter ε depends on the size of the near-wall domain used in the estimation of the diffusion coefficient and value of E. For the range of parameters reported in the literature, we estimate ε >0:03. The near-wall simulations also show a bias in the estimated diffusion coefficient parallel to the wall even in the limit E D 0, indicating an overestimation which increases with increasing time delay δt. This diffusion-induced overestimation is caused by the same underlying mechanism responsible for the previously reported overestimation of mean velocity in near-wall velocimetry.

Language | English (US) |
---|---|

Pages | 65-84 |

Number of pages | 20 |

Journal | Journal of Fluid Mechanics |

Volume | 767 |

DOIs | |

State | Published - 2015 |

### Profile

### Keywords

- low-Reynolds-number flows
- micro-/nano-fluid dynamics
- Stokesian dynamics

### ASJC Scopus subject areas

- Mechanical Engineering
- Mechanics of Materials
- Condensed Matter Physics

### Cite this

**Effect of finite sampling time on estimation of Brownian fluctuation.** / Pouya, Shahram; Liu, Di; Koochesfahani, Manoochehr M.

Research output: Research - peer-review › Article

*Journal of Fluid Mechanics*, vol 767, pp. 65-84. DOI: 10.1017/jfm.2015.41

}

TY - JOUR

T1 - Effect of finite sampling time on estimation of Brownian fluctuation

AU - Pouya,Shahram

AU - Liu,Di

AU - Koochesfahani,Manoochehr M.

PY - 2015

Y1 - 2015

N2 - We present a study of the effect of finite detector integration/exposure time E, in relation to interrogation time interval Δt, on analysis of Brownian motion of small particles using numerical simulation of the Langevin equation for both free diffusion and hindered diffusion near a solid wall. The simulation result for free diffusion recovers the known scaling law for the dependence of estimated diffusion coefficient on E/Δt, i.e. for 0 ≤ E/Δt ≤ 1 the estimated diffusion coefficient scales linearly as 1-(E/Δt/=3) Extending the analysis to the parameter range E/Δt ≥1, we find a new nonlinear scaling behaviour given by.E/Δt/-1[1-E/Δt/-1/)3], for which we also provide an exact analytical solution. The simulation of near-wall diffusion shows that hindered diffusion of particles parallel to a solid wall, when normalized appropriately, follows with a high degree of accuracy the same form of scaling laws given above for free diffusion. Specifically, the scaling laws in this case are well represented by 1 ((1 + ε)(E/Δt))/3, for E/Δt ≤ 1, and (E/Δt)-1[1 - ((1 + ε)(E/Δt)-1/3], for E/Δt ≥ 1, where the small parameter ε depends on the size of the near-wall domain used in the estimation of the diffusion coefficient and value of E. For the range of parameters reported in the literature, we estimate ε >0:03. The near-wall simulations also show a bias in the estimated diffusion coefficient parallel to the wall even in the limit E D 0, indicating an overestimation which increases with increasing time delay δt. This diffusion-induced overestimation is caused by the same underlying mechanism responsible for the previously reported overestimation of mean velocity in near-wall velocimetry.

AB - We present a study of the effect of finite detector integration/exposure time E, in relation to interrogation time interval Δt, on analysis of Brownian motion of small particles using numerical simulation of the Langevin equation for both free diffusion and hindered diffusion near a solid wall. The simulation result for free diffusion recovers the known scaling law for the dependence of estimated diffusion coefficient on E/Δt, i.e. for 0 ≤ E/Δt ≤ 1 the estimated diffusion coefficient scales linearly as 1-(E/Δt/=3) Extending the analysis to the parameter range E/Δt ≥1, we find a new nonlinear scaling behaviour given by.E/Δt/-1[1-E/Δt/-1/)3], for which we also provide an exact analytical solution. The simulation of near-wall diffusion shows that hindered diffusion of particles parallel to a solid wall, when normalized appropriately, follows with a high degree of accuracy the same form of scaling laws given above for free diffusion. Specifically, the scaling laws in this case are well represented by 1 ((1 + ε)(E/Δt))/3, for E/Δt ≤ 1, and (E/Δt)-1[1 - ((1 + ε)(E/Δt)-1/3], for E/Δt ≥ 1, where the small parameter ε depends on the size of the near-wall domain used in the estimation of the diffusion coefficient and value of E. For the range of parameters reported in the literature, we estimate ε >0:03. The near-wall simulations also show a bias in the estimated diffusion coefficient parallel to the wall even in the limit E D 0, indicating an overestimation which increases with increasing time delay δt. This diffusion-induced overestimation is caused by the same underlying mechanism responsible for the previously reported overestimation of mean velocity in near-wall velocimetry.

KW - low-Reynolds-number flows

KW - micro-/nano-fluid dynamics

KW - Stokesian dynamics

UR - http://www.scopus.com/inward/record.url?scp=84923078947&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84923078947&partnerID=8YFLogxK

U2 - 10.1017/jfm.2015.41

DO - 10.1017/jfm.2015.41

M3 - Article

VL - 767

SP - 65

EP - 84

JO - Journal of Fluid Mechanics

T2 - Journal of Fluid Mechanics

JF - Journal of Fluid Mechanics

SN - 0022-1120

ER -