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A U niversal Field Equation for Dispersive Processes. J. H. Cushman, Purdue University Former Students M. Park , University of Alabama N. Kleinfelter-Domelle , Brown University M . Moroni , University of Rome, I B. Stroud , Private Industry M. Schoen , Berlin Technical.
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A Universal Field Equation for Dispersive Processes J. H. Cushman, Purdue University Former Students M. Park, University of Alabama N. Kleinfelter-Domelle, Brown University M. Moroni, University of Rome, I B. Stroud, Private Industry M. Schoen, Berlin Technical
Anomalous Diffusion in Slit Nanopores Computational Statistical Mechanics: Quasi-static experiments
Slit Pore Expansion Each frame of the animation shows a single configuration generated in a Monte Carlo simulation in the grand-canonicalisostrain ensemble (μ, T, h*, α) for a fixed pore width. Played in succession, the frames show configurations for the quasistatic pore expansion. It is important to note that the animation does not portray the time evolution of an expanding pore. Periodic boundary conditions are imposed in the plane of the pore. The configurations shown represent small sub-regions of the fluid in an infinite slit pore. The pore walls are five atoms thick, but only a single layer of atoms is rendered for each wall with the remainder of the walls being indicated in outline. As the pore expands you should observe that the configurations are highly ordered for particular wall separations. This order is the result of freezing of the pore fluid.
Simulation Cell Snapshot of a single configuration in the simulation cell.
Thick Nano-Wire Expansion Maps of two dimensional slices in the plane y=0 through the three dimensional, ensemble averaged particle density for a Lennard-Jones 10-wire • reduced chemical potential, μ = -11.7 • reduced temperature, T= 1.0 • wall registration, alpha = 0.0 and reduced pore serpation, h*, as indicated. Points on the rendered density surfaces are displaced towards the viewer in proportion to the density.
Thin Nano-Wire Expansion Maps of two dimensional slices in the plane y=0 through the three dimensional, ensemble averaged particle density for a Lennard-Jones 5-wire • reduced chemical potential, mu = -11.7 • reduced temperature, T= 1.0 • wall registration, alpha = 0.0 and reduced pore serpation, h*, as indicated. Points on the rendered density surfaces are displaced towards the viewer in proportion to the density.
Nano-Wire and Slit Pore Expansion Maps of two dimensional slices through the three dimensional, ensemble averaged particle density in the plane y=0 for reduced chemical potential, μ = -11.7, reduced temperature, T= 1.0,wall registration, α = 0.0 and reduced pore serpation, h*, for (from left to right) a Lennard Jones 5-wire, 10-wire, and an infinite-wire (slit pore). Points on the rendered density surfaces are displaced towards the viewer in proportion to the density.
Anomalous Diffusion When the mean-free-path of a molecule is on the same order as a characteristic dimension of the pore, the classical Fickian expression relating the diffusive flux to the activity gradient breaks down. That is, both diffusivity and the related viscosity become wave vector and frequency dependent.
Panel b is especially interesting as it shows sub-diffusion. Here the mean-square displacement scales with d =0.64 as opposed to the Brownian limit of d =1.
General Dispersion theory without scale constraints G is the conditional probability of finding a particle at x at time t given itwas initially at X(0). If all particles are identical and are released from the origin, then G represents the concentration. The Fourier transform of G is Ĝ
Letting G´ in the last slide be g above, then upon differentiating and assuming dP/dt=0 where P is underlying probability and G'(k,0)=0 we obtain the following equation in Fourier space General Dispersion theory without scale constraints For suitably well behaved g(t) we have where Here we’ve introduced the Exponential Differential Displacement:
General Dispersion theory without scale constraints With a little work can be rewritten
General Dispersion theory without scale constraints Now, invert into real space and obtain where Di' is the inverse transform of (Di'Δ)^. This is an equation for dispersion without the need for scale constraints.
Small wave vector limit-retrieving the classical ADE Assume constant mean velocity so that depends only on k and τ making a convolution. Using a Taylor series expansion about |k|=0, on G^ it can be shown that If the system is stationary, then the mean square displacement can be replaced by twice velocity covariance.
Anomalous Dispersion in Porous Media 3-D Porous Media PTV Experiments
Porous media experimental set-up Particle Tracking Velocimetry (2D-PTV)
Baricenters from the coupled cameras (Ex. 1) Baricenters recognized on the image acquired by camera 1 (21741 baricenters) Baricenters recognized on the image acquired by camera 2 (16740 baricenters)
3D-PTV3-D Trajectory reconstruction from two 2-D projected trajectories
General Dispersion theory without scale constraints This equation for dispersion is just as applicable in porous media as in the nano-film.
Generalized dispersion coefficient Generalized dispersion coefficient in the transverse directions, k=2p/d, compared with the velocity covariance .
Bacterial Swimming Videos courtesy of Howard Berg, Molecular and Cellular Biology, Harvard University
Bacterial mortor and drive train • courtesy of David DeRosier, Brandeis university
Microbe vs. Levy Motion E coli swimming Berg, Phys Today, Jan 2000. / Levy motion with α=1.2
25 20 15 10 5 0 -5 -10 -15 -20 0 10 20 30 40 LévyversusBrownianmotion a = 2 fractal dimension of the trace of Lévy motion (including Brownian motion) = a DISTANCE FROM ORIGIN a = 1.7 DISTANCE FROM ORIGIN
Levy trajectory dispersion A particle which follows an α-stable Levy motion has a transition density given by the following equation (for a divergence free velocity field) where the general (asymmetric) fractional derivative is defined in Fourier space by the following with M the mixing measure, S the d-dimensional unit sphere
Rewrite the fractional derivative as: Now take the Inverse Fourier transform : Inverse FT Levy trajectory dispersion
Levy trajectory dispersion The equation can be written in the divergence form with convolution-Fickian flux
The Finite-Size Lyapunov Exponent:Determining the exponent for Levy motion Definethe Finite-Size Lyapunov Exponent (FSLE) The FSLE describes the exponential divergence of two trajectories that start a specified distance, r, apart and depends on that initial starting scale as well as the threshold ratio, a. or la: is the expected exponential rate two particles separated by a distance r initially, separate to ar at time t. Td is the expected doubling time, i.e. the time it takes two particles to separate from r to ar.
A Statistical Mechanical Formulation of the Finite-Size Lyapunov Exponent Assume for simplicity that the system is completely ‘expansive’. Let <<>> indicate integration over the product space with respect to a joint probability density ,f, defined on (τ) x (τ+t). Here is the joint probability particles i and j have separation in at time τ and separation in at time
A Statistical Mechanical Formulation of the Finite-Size Lyapunov Exponent where G(x,y,t) is the probability of a particle going from separation x to separation y in time t. Therefore the a-time is:
FSLE for α-stable Lévy Motion Varying α Slope-1.2 Slope -1.5 Slope -2
For the case when (1+α)/2 is an integer (i.e., α=1), the summation P(2,1)=2/16
Turbulent Flows • Fluid Flow: Eulerian approach Lagrangian approach (1) Laminar Flow: Re = small (2) Turbulent Flow: Re = very large
Turbulent mixing layer growth and internal waves formation: laboratory simulations
The flux through the interface between the mixing layer and the stable layer plays a fundamental role in characterizing and forecasting the quality of water in stratified lakes and in the upper oceans, and the quality of air in the atmosphere.
