Persisting asymmetry in the probability distribution function for a random advection–diffusion equation in impermeable channels

Camassa, Roberto; Lee, Ding; Kilic, Zeliha; McLaughlin, Richard M.

doi:10.1016/j.physd.2021.132930

Cited by 6 publications

(6 citation statements)

References 41 publications

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“…With our established evolution equation of ensemble average, we studied the passive scalar transport problem with two different types of flows, the periodic flow, and strain flow. For the periodic flow, by using the homogenization method, we show that the N-point correlation function of the random scalar flow is governed by a diffusion equation at long times, which generalized the conclusion in [10,14] from time-independent case to time-dependent case. For the strain flow, we have explicitly calculated the mean of the random scalar field and shown the statistic of this random field is related to the time integral of geometric Brownian motion.…”

Section: Conclusion and Discussionsupporting

confidence: 63%

“…The diffusion tensor would be extremely simplified. In this case, we have Λ 2N = I 2N + Pe 2 v(y, τ ) T v(y, τ ) y,τ , which reproduces the result in [10,14,7]. Then one can compute the PDF of the scalar field by using the procedure described in [14,6].…”

Section: Therefore We Havesupporting

confidence: 64%

“…Then one can compute the PDF of the scalar field by using the procedure described in [14,6]. We remark that [10,14] only consider the case where u(y) is independent of t. Our result presented in this section generalizes it to the time dependent case, u(y, t).…”

Section: Therefore We Havementioning

confidence: 67%

“…For the shear flow ξ(t)v(x, t) = ξ(t)(u(y), 0) and the parallel-plate channel domain Ω = {(x, y)|x ∈ R, y ∈ [0, 1]} with no-flux boundary conditions, Camassa et al [10] proved that the N-point correlation function satisfies an effective heat equation at long times. Interestedly, in this example, the limiting distribution of the scalar field is not Gaussian and exhibit intermittency.…”

Section: Random Periodic Flowmentioning

confidence: 99%

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Correlation function of a random scalar field evolving with a rapidly fluctuating Gaussian process

Bronski¹,

Ding²,

McLaughlin³

2022

Preprint

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We consider a scalar field governed by an advection-diffusion equation (or a more general evolution equation) with rapidly fluctuating, Gaussian distributed random coefficients. In the white noise limit, we derive the closed evolution equation for the ensemble average of the random scalar field by three different strategies, i.e., Feynman-Kac formula, the limit of Ornstein-Uhlenbeck process, and evaluating the cluster expansion of the propagator on an n-simplex. With the evolution equation of ensemble average, we study the passive scalar transport problem with two different types of flows, a random periodic flow, and a random strain flow. For periodic flows, by utilizing the homogenization method, we show that the N-point correlation function of the random scalar field satisfies an effective diffusion equation at long times. For the strain flow, we explicit compute the mean of the random scalar field and show that the statistics of the random scalar field have a connection to the time integral of geometric Brownian motion. Interestingly, all normalized moment (e.g., skewness, kurtosis) of this random scalar field diverges at long times, meaning that the scalar becomes more and more intermittent during its decay.

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Section: Conclusion and Discussionsupporting

confidence: 63%

Section: Therefore We Havesupporting

confidence: 64%

Section: Therefore We Havementioning

confidence: 67%

Section: Random Periodic Flowmentioning

confidence: 99%

See 2 more Smart Citations

Correlation function of a random scalar field evolving with a rapidly fluctuating Gaussian process

Bronski¹,

Ding²,

McLaughlin³

2022

Preprint

Self Cite

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show abstract

“…The function u(y, t) vanishes on the boundary wall and exhibits periodic time-varying behaviour with period L t . While steady pressure-driven flow is common in many applications (Price 1988;Leaist & Hao 1993;Rodrigo et al 2021), we maintain the general form and time dependence of the flow to ensure the theoretical framework's applicability to various scenarios, including blood flow (Marbach & Alim 2019) and scalar intermittency (Majda & Kramer 1999;Camassa et al 2021). We impose the no-flux boundary condition for the concentration fields of the ion species n…”

Section: Advection Nernst-planck Equationmentioning

confidence: 99%

Shear dispersion of multispecies electrolyte solutions in the channel domain

Ding

2023

J. Fluid Mech.

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In multispecies electrolyte solutions, even in the absence of an external electric field, differences in ion diffusivities induce an electric potential and generate additional fluxes for each species. This electro-diffusion process is well-described by the advection Nernst–Planck equation. This study aims to analyse the long-time behaviour of the governing equation under electroneutrality and zero current conditions, and to investigate how the diffusion-induced electric potential and shear flow enhance the effective diffusion coefficients of each species in channel domains. The exact solutions of the effective equation with certain special parameters, as well as the asymptotic analyses for ions with large diffusivity discrepancies, are presented. Furthermore, there are several interesting properties of the effective equation. First, it is a generalization of the Taylor dispersion, with a nonlinear diffusion tensor replacing the scalar diffusion coefficient. Second, the effective equation exhibits a scaling relation, revealing that the system with a weak flow is equivalent to the system with a strong flow under scaled physical parameters. Third, in the case of injecting an electrolyte solution into a channel containing well-mixed buffer solutions or electrolyte solutions with the same ion species, if the concentration of the injected solution is lower than that of the pre-existing solution, then the effective equation simplifies to a multi-dimensional diffusion equation. However, when introducing the electrolyte solution into a channel filled with deionized water, the ion–electric interaction results in several phenomena not present in the advection–diffusion equation, including upstream migration of some species, spontaneous separation of ions, and non-monotonic dependence of the effective diffusivity on Péclet numbers. Finally, the dependence of effective diffusivity on concentration and ion diffusivity suggests a method to infer the concentration ratio of each component and ion diffusivity by measuring the effective diffusivity.

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Correlation Function of a Random Scalar Field Evolving with a Rapidly Fluctuating Gaussian Process

Ding,

McLaughlin

2023

J Stat Phys

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Persisting asymmetry in the probability distribution function for a random advection–diffusion equation in impermeable channels

Cited by 6 publications

References 41 publications

Correlation function of a random scalar field evolving with a rapidly fluctuating Gaussian process

Correlation function of a random scalar field evolving with a rapidly fluctuating Gaussian process

Shear dispersion of multispecies electrolyte solutions in the channel domain

Correlation Function of a Random Scalar Field Evolving with a Rapidly Fluctuating Gaussian Process

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