Scalar Diffusion Research Articles

Nanoparticle Diffusion in Miscible Polymer Nanocomposite Melts

X-ray photon correlation spectroscopy measurements were used to quantify the dynamics of bare and bimodal grafted silica nanoparticles mixed with PEO melts of different molecular weights. In dilute polymer nanocomposite (PNC) samples, we find diffusive NP behavior as described by the Stokes–Einstein relationship so long as the adsorbed PEO polymer layer is taken into account in determining both the effective NP size and its role on composite viscosity. The size of this bound layer was found to be approximately 2Rg, where Rg is the chain radius of gyration. We also expanded our system to investigate how the dynamics of grafted NPs differ from bare NPs with an adsorbed layer. We showed that the dynamics again can be determined by an effective NP radius at a scale smaller than the effective interparticle spacing; however, at larger length scales, the morphology and grafting parameters play a major role in the system dynamics. These results allow us to quantify NP ordering driven by polymer crystallization. It has previously been speculated that behavior is controlled by the relative ratio of time scale of crystal growth and the diffusive time scale of the NPs, a Peclet number. When the former time scale is longer, then the NPs are expected to be segregated into the interlamellar amorphous zones, while the NPs are too slow to be reorganized in the opposite case. We show here that this conjecture is quantitatively correct and the demarcation in behavior occurs for Pe = 1. Thus, we provide a way to estimate a critical spherulite growth rate for any semicrystalline PNC, at which a given NP can be ordered. Together, the results of this study permit us to tunably design PNCs through directed dispersion of NPs in a polymer matrix.

Perivascular network segmentations derived from high-field MRI and their implications for perivascular and parenchymal mass transport in the rat brain

A custom segmentation workflow was applied to ex vivo high-field MR images of rat brains acquired following in vivo intraventricular contrast agent infusion to generate maps of the perivascular spaces (PVS). The resulting perivascular network segmentations enabled analysis of perivascular connections to the ventricles, parenchymal solute clearance, and dispersive solute transport within PVS. Numerous perivascular connections between the brain surface and the ventricles suggest the ventricles integrate into a PVS-mediated clearance system and raise the possibility of cerebrospinal fluid (CSF) return from the subarachnoid space to the ventricles via PVS. Assuming rapid solute exchange between the PVS and CSF spaces primarily by advection, the extensive perivascular network decreased the mean clearance distance from parenchyma to the nearest CSF compartment resulting in an over 21-fold reduction in the estimated diffusive clearance time scale, irrespective of solute diffusivity. This corresponds to an estimated diffusive clearance time scale under 10 min for amyloid-beta which suggests that the widespread distribution of PVS may render diffusion an effective parenchymal clearance mechanism. Additional analysis of oscillatory solute dispersion within PVS indicates that advection rather than dispersion is likely the primary transport mechanism for dissolved compounds greater than 66 kDa in the long (> 2 mm) perivascular segments identified here, although dispersion may be significant for smaller compounds in shorter perivascular segments.

High Order IMEX Stochastic Galerkin Schemes for Linear Transport Equation with Random Inputs and Diffusive Scalings

High Order IMEX Stochastic Galerkin Schemes for Linear Transport Equation with Random Inputs and Diffusive Scalings

Lagrangian model for passive scalar gradients in turbulence

The equation for the fluid velocity gradient along a Lagrangian trajectory immediately follows from the Navier–Stokes equation. However, such an equation involves two terms that cannot be determined from the velocity gradient along the chosen Lagrangian path: the pressure Hessian and the viscous Laplacian. A recent model handles these unclosed terms using a multi-level version of the recent deformation of Gaussian fields (RDGF) closure (Johnson & Meneveau, Phys. Rev. Fluids, vol. 2 (7), 2017, 072601). This model is in remarkable agreement with direct numerical simulations (DNS) data and works for arbitrary Taylor Reynolds numbers $\textit {Re}_\lambda$ . Inspired by this, we develop a Lagrangian model for passive scalar gradients in isotropic turbulence. The equation for passive scalar gradients also involves an unclosed term in the Lagrangian frame, namely the scalar gradient diffusion term, which we model using the RDGF approach. However, comparisons of the statistics obtained from this model with DNS data reveal substantial errors due to erroneously large fluctuations generated by the model. We address this defect by incorporating into the closure approximation information regarding the scalar gradient production along the local trajectory history of the particle. This modified model makes predictions for the scalar gradients, their production rates, and alignments with the strain-rate eigenvectors that are in very good agreement with DNS data. However, while the model yields valid predictions up to $\textit {Re}_\lambda \approx 500$ , beyond this, the model breaks down.

The stationary AKPZ equation: Logarithmic superdiffusivity

AbstractWe study the two‐dimensional Anisotropic KPZ equation (AKPZ) formally given by where ξ is a space‐time white noise and λ is a strictly positive constant. While the classical two‐dimensional KPZ equation, whose nonlinearity is , can be linearised via the Cole‐Hopf transformation, this is not the case for AKPZ. We prove that the stationary solution to AKPZ (whose invariant measure is the Gaussian Free Field (GFF)) is superdiffusive: its diffusion coefficient diverges for large times as up to corrections, in a Tauberian sense. Morally, this says that the correlation length grows with time like . Moreover, we show that if the process is rescaled diffusively (), then it evolves non‐trivially already on time‐scales of order approximately . Both claims hold as soon as the coefficient λ of the nonlinearity is non‐zero. These results are in contrast with the belief, common in the mathematics community, that the AKPZ equation is diffusive at large scales and, under simple diffusive scaling, converges to the two‐dimensional Stochastic Heat Equation (2dSHE) with additive noise (i.e., the case ).

Hilbert expansion based fluid models for kinetic equations describing neutral particles in the plasma edge of a fusion device

Neutral particles in the plasma edge of fusion devices based on magnetic confinement are described by a transient kinetic equation incorporating ionization, recombination, and charge-exchange collisions. In charge-exchange dominated regimes, the neutral particle velocity distribution approaches the drifting Maxwellian defined by the mean velocity and temperature of the plasma. This enables model order reduction from the kinetic equation to approximate fluid models. We derive transient fluid models consistent with the kinetic equation by exploring a splitting based approach. We split the kinetic equation in sources and sinks on the one hand, and transport combined with charge-exchange on the other hand. Combining transport with charge-exchange collisions allows for deriving Hilbert expansion based fluid models. The retrieved fluid models depend on the assumed importance (scaling) of the different terms in the split equation describing transport and charge-exchange. We explore two scalings: the hydrodynamic scaling and the diffusive scaling. The diffusive scaling fluid model closely resembles phenomenological fluid models for describing neutral particles in the plasma edge that have been derived in the past. Therefore, the Hilbert expansion based fluid models can serve as a theoretical basis for such phenomenological fluid models and elucidate some of their properties. The performance of the fluid models with respect to a discrete velocity model and a Monte Carlo reference solver is assessed in numerical experiments. The code used to perform the numerical experiments is openly available.

Advection-enhanced heat and mass transport from neutrally suspended droplet in simple shear flow

Advection-enhanced heat and mass transport from a single droplet neutrally suspended in a simple shear flow has been studied using high-fidelity numerical simulation. The capillary number ranges from 0.01 to 0.5, which encompasses the entire range of small deformation, large deformation, and breakup of the droplets. The Reynolds number is from 0.01 to 1, including regions of both weak and strong advection. The temperature and mass concentration are modeled as the concentration of a passive scalar released at the droplet surface. Two Schmidt numbers, 10 and 100, are considered, for which flow advection plays a role in the transport of passive scalar. For unbroken droplets, the interaction between the carrier fluid and the suspended droplet leads to several different flows around the droplet. The fluid motions together with scalar diffusion constitute a coupled transport mechanism for passive scalar. The dependence of scalar release rate on Reynolds and Peclet numbers can be roughly described by the correlation for a rigid sphere. For broken droplets, the basic flow features around the droplet during the process of elongation and breakup are similar to those of an unbroken droplet. The variation of the scalar release rate can be decomposed into several stages, corresponding to the process of droplet elongation and breakup. The variation of the scalar release rate exhibits a high correlation with the capillary, Reynolds, and Peclet numbers. This suggests that it is feasible to develop an empirical model that incorporates the effects of the number and size distributions of child droplets after breakup.

Fundamental solution of diffusion equationfor Kappa gas: Diffusion length for suprathermal electrons in solar wind.

A recent numerical treatment of data obtained by the Parker Solar Probe spacecraft describes the electron concentration in solar wind as a function of the heliocentric distance based on a Kappa distribution with spectral index κ=5. In this work, we derive and, subsequently, solve an entirely different class of nonlinear partial differential equationsdescribing the one-dimensional diffusion of a suprathermal gas. The theory is applied to describe the aforementioned data and we find a spectral index κ≳1.5 providing the widely acknowledged identification of Kappa electrons in solar wind. We also find that suprathermal effects increase the length scale of classical diffusion by one order of magnitude. Such a result does not depend on the microscopic details of the diffusion coefficient since our theory is based on a macroscopic formulation. Forthcoming extensions of our theory by including magnetic fields and relating our formulation to nonextensive statistics are briefly addressed.

Reduced order modeling for elliptic problems with high contrast diffusion coefficients

We consider a parametric elliptic PDE with a scalar piecewise constant diffusion coefficient taking arbitrary positive values on fixed subdomains. This problem is not uniformly elliptic, as the contrast can be arbitrarily high, contrarily to the Uniform Ellipticity Assumption (UEA) that is com- monly made on parametric elliptic PDEs. We construct reduced model spaces that approximate uniformly well all solutions with estimates in relative error that are independent of the contrast level. These estimates are sub-exponential in the reduced model dimension, yet exhibiting the curse of dimensionality as the number of subdomains grows. Similar es- timates are obtained for the Galerkin projection, as well as for the state estimation and parameter estimation inverse problems. A key ingredient in our construction and analysis is the study of the convergence towards limit solutions of stiff problems when diffusion tends to infinity in certain domains.

Fourth-order multiple-relaxation-time lattice Boltzmann model and equivalent finite-difference scheme for one-dimensional convection-diffusion equations.

In this paper, we first develop a fourth-order multiple-relaxation-time lattice Boltzmann (MRT-LB) model for the one-dimensional convection-diffusion equation(CDE) with the constant velocity and diffusion coefficient, where the D1Q3 (three discrete velocities in one-dimensional space) lattice structure is used. We also perform the Chapman-Enskog analysis to recover the CDE from the MRT-LB model. Then an explicit four-level finite-difference (FLFD) scheme is derived from the developed MRT-LB model for the CDE. Through the Taylor expansion, the truncation error of the FLFD scheme is obtained, and at the diffusive scaling, the FLFD scheme can achieve the fourth-order accuracy in space. After that, we present a stability analysis and derive the same stability condition for the MRT-LB model and FLFD scheme. Finally, we perform some numerical experiments to test the MRT-LB model and FLFD scheme, and the numerical results show that they have a fourth-order convergence rate in space, which is consistent with our theoretical analysis.

Assessing passive scalar dynamics in bubble-induced turbulence using direct numerical simulations

By using direct numerical simulations (DNS) of bubbly flows with passive scalars, we show a transition in the scalar spectra from a $k^{-5/3}$ to a $k^{-3}$ scaling with the wavenumber $k$ , in contrast with those of single-phase isotropic turbulence. For cases with a mean scalar gradient in the horizontal direction, the scalar spectrum decays faster than $k^{-3}$ at high wavenumbers. While the $k^{-3}$ scaling is well established in the bubbly flow velocity spectrum, the scalar spectrum behaviour is not fully understood. We find that the transition length scale of the scalar spectra is comparable to or below the bubble diameter and decreases with the molecular diffusivity of the scalar in the liquid. We use DNS to compute the scalar spectrum budget and show that the scalar fluctuations are produced by the mean scalar gradient at length scales above the bubble diameter, contrary to the velocity fluctuations. At length scales below the bubble diameter, the net scalar transfer scales as $k^{-1}$ inducing the $k^{-3}$ scaling of the scalar spectra. This finding is consistent with the hypothesis proposed by Dung et al. (J. Fluid Mech., vol. 958, 2023, p. A5) about the physical mechanism behind the $k^{-3}$ scaling. We also show dependencies of the bubble suspension's convective scalar diffusivity on the gas volume fraction and molecular diffusivity that differ based on the direction of the mean scalar gradient. For a mean scalar gradient in the vertical direction, we find and qualitatively explain a significant effect of the molecular diffusivity in the gas on the convective scalar diffusivity.

Supply Chain Risk Diffusion in Partially Mapping Double-Layer Hypernetworks.

The impact of COVID-19 is global, and uncertain information will affect product quality and worker efficiency in the complex supply chain network, thus bringing risks. Aiming at individual heterogeneity, a partial mapping double-layer hypernetwork model is constructed to study the supply chain risk diffusion under uncertain information. Here, we explore the risk diffusion dynamics, drawing on epidemiology, and establish an SPIR (Susceptible-Potential-Infected-Recovered) model to simulate the risk diffusion process. The node represents the enterprise, and hyperedge represents the cooperation among enterprises. The microscopic Markov chain approach (MMCA) is used to prove the theory. Network dynamic evolution includes two removal strategies: (i) removing aging nodes; (ii) removing key nodes. Using Matlab to simulate the model, we found that it is more conducive to market stability to eliminate outdated enterprises than to control key enterprises during risk diffusion. The risk diffusion scale is related to interlayer mapping. Increasing the upper layer mapping rate to strengthen the efforts of official media to issue authoritative information will reduce the infected enterprise number. Reducing the lower layer mapping rate will reduce the misled enterprise number, thereby weakening the efficiency of risk infection. The model is helpful for understanding the risk diffusion characteristics and the importance of online information, and it has guiding significance for supply chain management.

Study of turbulent transport in magnetized plasmas with flow using symplectic maps

Turbulent transport in a magnetized plasma of the kind found in tokamaks is modeled by a 2D wave spectrum that allows reduction to a symplectic map. The properties of particle transport when chaos sets in are analyzed in various circumstances including finite Larmor radius (FLR) effects and a background plasma flow. For large wave amplitudes, regular particle orbits become chaotic, which represents a type of Lagrangian turbulence. When chaos becomes global, it leads to the loss of particle confinement. Poloidal flows tend to decrease the chaos in some regions, and they can give rise to the formation of transport barriers. FLR effects not only reduce chaos but also give rise to non-local behavior. Thus, when the particles have a thermal distribution of Larmor radii, a non-Gaussian particle distribution function in space is obtained. However, the transport preserves its diffusive scaling when there is no flow. Previous results about the dependence of the diffusion coefficient with amplitude are re-derived analytically and numerically taking into account FLR effects. In the presence of general poloidal flows, the transport has to be described by a two-step map. They modify the nature of transport in the direction of the flow from diffusive to ballistic to super-ballistic, depending on the type of flow. The transverse transport, in turn, shows suppression of the oscillations with wave amplitude that are present in the absence of flow. When the plasma flow varies linearly with radius, the transport can be studied with a similar single-step map, and the transverse diffusion coefficient is reduced while parallel transport can become super-ballistic. For non-monotonic flows, there are accelerating modes that can produce ballistic-like particles while the bulk of the particles behaves diffusively.

Particle dynamics in a turbulent electric field

Charged particle velocity-space diffusion in a prescribed one-dimensional turbulent electric field is investigated through numerical trajectories in phase-space and compared against quasi-linear theory (QL), including resonance broadening (RB). A Gaussian spectrum electric field of variable amplitude E is studied in conjunction with two plasma dispersion relations, namely, the Langmuir and ion-acoustic dispersion. A first parameter scan shows that RB effects become significant for a Kubo number K of a few percent. A Kubo number scan shows that diffusion increases as a power law of D∝K3∝E3/2 for large Kubo numbers. Moreover, at large Kubo numbers, transport processes include significant diffusion measured at velocities much higher than the resonant region, where QL and RB predict negligible diffusion. For times much larger than the trapped particle flight time τb and the autocorrelation time τ0, the velocity distribution departs from a Gaussian. Nevertheless, measurements show that the variance increases linearly in time, with a Hurst parameter of H∼0.5, where the diffusion scales as K5/2∝E5/4 and K3/2∝E3/4 for small and large Kubo number, respectively.

Truncation errors and modified equations for the lattice Boltzmann methodviathe corresponding Finite Difference schemes

Lattice Boltzmann schemes are efficient numerical methods to solve a broad range of problems under the form of conservation laws. However, they suffer from a chronic lack of clear theoretical foundations. In particular, the consistency analysis and the derivation of the modified equations are still open issues. This has prevented, until today, to have an analogous of the Lax equivalence theorem for lattice Boltzmann schemes. We propose a rigorous consistency study and the derivation of the modified equations for any lattice Boltzmann scheme under acoustic and diffusive scalings. This is done by passing from a kinetic (lattice Boltzmann) to a macroscopic (Finite Difference) point of view at a fully discrete level in order to eliminate the non-conserved moments relaxing away from the equilibrium. We rewrite the lattice Boltzmann scheme as a multi-step Finite Difference scheme on the conserved variables, as introduced in our previous contribution. We then perform the usual analyses for Finite Difference by exploiting its precise characterization using matrices of Finite Difference operators. Though we present the derivation of the modified equations until second-order under acoustic scaling, we provide all the elements to extend it to higher orders, since the kinetic-macroscopic connection is conducted at the fully discrete level. Finally, we show that our strategy yields, in a more rigorous setting, the same results as previous works in the literature.

Theoretical Two Dimensional Infrared Spectroscopy of Aqueous Solutions of tert-Butyl Alcohol: Variation of the Dynamics of Spectral Diffusion along the Percolation Transition.

Binary mixtures of water and tert-butyl alcohol (TBA) are known to exhibit the so-called percolation transition where small clusters of TBA molecules span into large aggregates beyond a threshold concentration of the alcohol. In the present study, we have investigated the linear and two-dimensional infrared spectral features of aqueous solutions of TBA for varying concentration of the alcohol along the percolation transition. The percolation transition is characterized through calculations of intermolecular radial distribution functions and average size of the largest cluster of TBA molecules. It is found that, with variation of alcohol concentration, the radial distribution functions of the central carbon atoms of TBA molecules show a nonmonotonic change in the height of the first peak and also the size of the largest cluster of TBA molecules show a jump in the increase of its size for TBA mole fraction between 0.04 and 0.06 corresponding to a transition from smaller clusters to larger spanning aggregates. However, it is found that the linear infrared spectrum of water does not exhibit any noticeable changes on variation of TBA concentration along the percolation transition. Subsequently, two-dimensional infrared (2DIR) spectra and vibrational frequency time correlation function of water are calculated for all the TBA-water solutions considered in this study. The spectral diffusion of water calculated from 2DIR is found to slow down with increase of the TBA concentration. The time scales of spectral diffusion of water, as characterized by the relaxation of frequency time correlation function, 2DIR metric of central line slope, and also the hydrogen bond time correlation functions, are found to exhibit a noticeable jump along the percolation transition. The hydrophilic group of TBA is found to retard the water dynamics more effectively than the hydrophobic groups. Also, the jump in the dynamical slowdown along the percolation transition is found to be more significant for water molecules at the hydrophilic sites.

Aspect ratios of far-infrared and H I filaments in the diffuse interstellar medium at high Galactic latitudes

Context. Dusty magnetized structures observable in the far-infrared (FIR) at high Galactic latitudes are ubiquitous and found to be closely related to H I filaments with coherent velocity structures. Aims. Considering dimensionless morphological characteristics based on Minkowski functionals, we determine the distribution of filamentarities ℱ and aspect ratios 𝒜 for these structures. Methods. Our data are based on Planck FIR and HI4PI H I observations. Filaments have previously been extracted by applying the Hessian operator. We trace individual filamentary structures along the plane of the sky and determine 𝒜 and ℱ. Results. Filaments in the diffuse interstellar medium (ISM) are seldom isolated structures, but are rather part of a network of filaments with a well-defined, continuous distribution in 𝒜 and ℱ. This distribution is self-replicating, and the merger or disruption of individual filamentary structures leads only to a repositioning of the filament in 𝒜 and ℱ without changing the course of the distribution. Conclusions. FIR and H I filaments identified at high Galactic latitudes are a close match to model expectations for narrow filaments with approximately constant widths. This distribution is continuous without clear upper limits on the observed aspect ratios. Filaments are associated with enhanced column densities of CO-dark H2. Radial velocities along the filaments are coherent and mostly linear with typical dispersions of ∆υLSR = 5.24 km s−1. The magnetic field strength in the diffuse turbulent ISM scales with hydrogen volume density as B ∝ nH0.58. At high Galactic latitudes, we determine an average turbulent magnetic field strength of 〈δB〉 = 5.3 µG and an average mean strength of the magnetic field in the plane of the sky of 〈BPOS〉 = 4.4 µG.

Pre-asymptotic dispersion of active particles through a vertical pipe: the origin of hydrodynamic focusing

When motile algal cells are exposed to gyrotactic torques, their swimming directions are guided to form radial accumulation, well known as hydrodynamic focusing. The origin of hydrodynamic focusing from the effects of active swimming, ambient flow and particle anisotropy is elucidated in the present study on the pre-asymptotic dispersion of active particles through a vertical pipe. With an extension of the Galerkin method to pipe flows, time-dependent solutions directly from the Smoluchowski equation in the position and orientation space are derived by series expansions of spherical harmonics and Bessel functions. Ballistic and diffusive scaling laws are examined with the predominance of self-propelled swimming, and computation is validated against an explicit benchmark solution and Lagrangian particle simulation. In the limit of extreme shear, the competitive roles of shear dispersion and Brownian rotation are reflected concretely in the pre-asymptotic phase of hydrodynamic focusing. For flows with various shear strengths, a concentration peak in near-wall regions with a smooth transition to hydrodynamic focusing is illustrated with richer phenomena in upwelling and downwelling flows. A newly observed regime through a vertical pipe, named transient effective trapping, is revealed as a transitional mode towards hydrodynamic focusing. The pre-asymptotic approach to hydrodynamic focusing is elaborated intensively through extensive solutions of concentration moments and macroscopic transport coefficients characterised by swimming and flow Péclet numbers. The unique findings for the origin of hydrodynamic focusing could provide insight into related micro-algae reactor technology and contribute to flow control and biomass transfer in confined environments.

Hydrodynamics for the ABC model with slow/fast boundary

In this article, we consider the ABC model in contact with slow/fast reservoirs. In this model, there is one particle per site, which can be of type α∈{A,B,C} and particles exchange positions in the discrete set of points {1,…,N−1} with a rate which is weakly asymmetric and depends on the type of particles involved in the exchange mechanism. At the boundary points x=1,N−1 particles can be injected or removed, and the rate at which this happens depends on the type of particles involved. We prove the hydrodynamic limit, in the diffusive time scale, given by a system of non-linear coupled equations with several boundary conditions, that depend on the strength of the reservoir’s action.

From adherence to slip in nanofluidics: a mathematical justification based on a drop of viscosity—the scalar case

The question of the origin of the unusually large slip length that may appear in some nanofluidic experimental settings is the object of much debate in the physics community. An idea suggested by Myers (Microfluid Nanofluid 10(5):1141–1145, 2011) is that in carbon nanotubes, this phenomenon may originate from a viscosity drop of the fluid near the wall of the tube, in a very thin region called ”depletion layer”. In order to investigate this claim mathematically, we introduce as a first step a scalar diffusion model whose coefficient of diffusion tends to zero in the depletion layer, as the thickness of this region goes to zero. Using the natural energy bound satisfied by weak solutions $$u_\varepsilon $$ of the system, we develop an adaptation of the unfolding method (Arbogast et al in SIAM J Math Anal 21:823–836, 1990; Casado-Díaz in Proc R Soc Edinb 130(A):249–276, 2000; Cioranescu et al in C R Acad Sci Paris Sér I 335:99–104, 2002), (which is related to the two-scale convergence method (Allaire in SIAM J Math Anal 23:1482–1518, 1992; Nguetseng in SIAM J Math Anal 20:608–623, 1989)), based on the introduction of a fast variable and on a compactness argument applied to the corresponding sequence of rescaled solutions in the depletion region. This fine analysis of $$u_\varepsilon $$ allows us to construct an adapted sequence of test functions and to pass to the limit in the variational formulation of the initial problem, in order to derive an effective model. We conclude that a viscosity drop may result in a change of boundary condition akin to the apparition of a slip length.