Transient anomalous diffusion in Poiseuille flow

Latini, Marco; Bernoff, Andrew J.

doi:10.1017/s0022112001004906

Cited by 98 publications

(94 citation statements)

References 24 publications

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“…It is caused by the interaction of the transport term and dissipation: conservative transport transfers information to high frequencies where the dissipation is increasingly dominant. The effect has been studied by many authors in the mathematics community [21,61,5,57,15] and in the fluid mechanics community [48,23,36]. That a similar effect is predicted to happen in plasmas due to the second-order smoothing of Coulomb collisions is classical in the physics community [37,52,43].…”

Section: Msc: 35b35 35b34 35b40 35q83 35q84mentioning

confidence: 88%

Suppression of Plasma Echoes and Landau Damping in Sobolev Spaces by Weak Collisions in a Vlasov-Fokker-Planck Equation

Bedrossian

2017

Ann. PDE

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In this paper, we study Landau damping in the weakly collisional limit of a Vlasov-FokkerPlanck equation with nonlinear collisions in the phase-space (x, v) ∈ T n x × R n v . The goal is four-fold: (A) to understand how collisions suppress plasma echoes and enable Landau damping in agreement with linearized theory in Sobolev spaces, (B) to understand how phase mixing accelerates collisional relaxation, (C) to understand better how the plasma returns to global equilibrium during Landau damping, and (D) to rule out that collision-driven nonlinear instabilities dominate. We give an estimate for the scaling law between Knudsen number and the maximal size of the perturbation necessary for linear theory to be accurate in Sobolev regularity. We conjecture this scaling to be sharp (up to logarithmic corrections) due to potential nonlinear echoes in the collisionless model.

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Section: Msc: 35b35 35b34 35b40 35q83 35q84mentioning

confidence: 88%

Suppression of Plasma Echoes and Landau Damping in Sobolev Spaces by Weak Collisions in a Vlasov-Fokker-Planck Equation

Bedrossian

2017

Ann. PDE

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“…At times earlier than these "asymptotic" times, the behavior is more complicated as the rate of spreading of the plume and mixing are not the same. 14,20 A variety of works exist studying these pre-asymptotic times for the cylindrical-or parallel plate-configurations, [21][22][23] and also when density-driven coupling of flow and transport is present. 24 However, all these prior studies address advecting flows with no significant inertial effects (Stokes flow).…”

Section: -2mentioning

confidence: 99%

The impact of inertial effects on solute dispersion in a channel with periodically varying aperture

et al. 2012

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We investigate solute transport in channels with a periodically varying aperture, when the flow is still laminar but sufficiently fast for inertial effects to be nonnegligible. The flow field is computed for a two-dimensional setup using a finite element analysis, while transport is modeled using a random walk particle tracking method. Recirculation zones are observed when the aspect ratio of the unit cell and the relative aperture fluctuations are sufficiently large; under non-Stokes flow conditions, the flow in non-reversible, which is clearly noticeable by the horizontal asymmetry in the recirculation zones. After characterizing the size and position of the recirculation zones as a function of the geometry and Reynolds number, we investigate the corresponding behavior of the longitudinal effective diffusion coefficient. We characterize its dependence on the molecular diffusion coefficient D m , the Péclet number, the Reynolds number, and the geometry. The proposed relation is a generalization of the well-known Taylor-Aris relationship relating the longitudinal dispersion coefficient to D m and the Péclet number for a channel of constant aperture at sufficiently low Reynolds number. Inertial effects impact the exponent of the Péclet number in this relationship; the exponent is controlled by the relative amplitude of aperture fluctuations. For the range of parameters investigated, the measured dispersion coefficient always exceeds that corresponding to the parallel plate geometry under Stokes conditions; in other words, boundary fluctuations always result in increased dispersion. The transient approach to the asymptotic regime is also studied and characterized quantitatively. We show that the measured characteristic time to attain asymptotic conditions is controlled by two competing effects: (i) the trapping of particles in the near-immobile zone and, (ii) the enhanced mixing in the central zone where most of the flow takes place (mainstream), due to its thinning. C 2012 American Institute of Physics. [http://dx

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“…For τ L = 0, the effect of initial particle position has been considered (Latini & Bernoff 2001;Camassa et al 2010). The shear-induced diffusivity D S (t|y 0 ) is calculated for y 0 = L/2 and y 0 = L/4 with (4.20), and for an initially uniform particle distribution given by (5.3).…”

Section: Effect Of Initial Distributionmentioning

confidence: 99%

The effect of a non-zero Lagrangian time scale on bounded shear dispersion

Spydell

Feddersen

2011

J. Fluid Mech.

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Previous studies of shear dispersion in bounded velocity fields have assumed random velocities with zero Lagrangian time scale (i.e. velocities are$\delta $-function correlated in time). However, many turbulent (geophysical and engineering) flows with mean velocity shear exist where the Lagrangian time scale is non-zero. Here, the longitudinal (along-flow) shear-induced diffusivity in a two-dimensional bounded velocity field is derived for random velocities with non-zero Lagrangian time scale${\tau }_{L} $. A non-zero${\tau }_{L} $results in two-time transverse (across-flow) displacements that are correlated even for large (relative to the diffusive time scale${\tau }_{D} $) times. The longitudinal (along-flow) shear-induced diffusivity${D}_{S} $is derived, accurate for all${\tau }_{L} $, using a Lagrangian method where the velocity field is periodically extended to infinity so that unbounded transverse particle spreading statistics can be used to determine${D}_{S} $. The non-dimensionalized${D}_{S} $depends on time and two parameters: the ratio of Lagrangian to diffusive time scales${\tau }_{L} / {\tau }_{D} $and the release location. Using a parabolic velocity profile, these dependencies are explored numerically and through asymptotic analysis. The large-time${D}_{S} $is enhanced relative to the classic Taylor diffusivity, and this enhancement increases with$ \sqrt{{\tau }_{L} } $. At moderate${\tau }_{L} / {\tau }_{D} = 0. 1$this enhancement is approximately a factor of 3. For classic shear dispersion with${\tau }_{L} = 0$, the diffusive time scale${\tau }_{D} $determines the time dependence and large-time limit of the shear-induced diffusivity. In contrast, for sufficiently large${\tau }_{L} $, a shear time scale${\tau }_{S} = \mathop{ ({\tau }_{L} {\tau }_{D} )}\nolimits ^{1/ 2} $, anticipated by a simple analysis of the particle’s domain-crossing time, determines both the${D}_{S} $time dependence and the large-time limit. In addition, the scalings for turbulent shear dispersion are recovered from the large-time${D}_{S} $using properties of wall-bounded turbulence.

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Transient anomalous diffusion in Poiseuille flow

Cited by 98 publications

References 24 publications

Suppression of Plasma Echoes and Landau Damping in Sobolev Spaces by Weak Collisions in a Vlasov-Fokker-Planck Equation

Suppression of Plasma Echoes and Landau Damping in Sobolev Spaces by Weak Collisions in a Vlasov-Fokker-Planck Equation

The impact of inertial effects on solute dispersion in a channel with periodically varying aperture

The effect of a non-zero Lagrangian time scale on bounded shear dispersion

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