Effect of finite sampling time on estimation of Brownian fluctuation

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    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.

    Original languageEnglish (US)
    Pages (from-to)65-84
    Number of pages20
    JournalJournal of Fluid Mechanics
    Volume767
    DOIs
    StatePublished - 2015

    Profile

    diffusion coefficient
    simulation
    scaling laws
    Scaling laws
    interrogation
    domain wall
    time lag
    sampling
    intervals
    scaling
    detectors
    estimates
    Brownian movement
    Domain walls
    Velocity measurement
    Time delay
    Sampling
    Detectors
    Computer simulation

    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

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    title = "Effect of finite sampling time on estimation of Brownian fluctuation",
    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.",
    keywords = "low-Reynolds-number flows, micro-/nano-fluid dynamics, Stokesian dynamics",
    author = "Shahram Pouya and Di Liu and Koochesfahani, {Manoochehr M.}",
    year = "2015",
    doi = "10.1017/jfm.2015.41",
    volume = "767",
    pages = "65--84",
    journal = "Journal of Fluid Mechanics",
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    publisher = "Cambridge University Press",

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    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

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    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 -