Diffusion of the dicumyl peroxide in molten polymer probed by rheology

The mutual coefficient of diffusion of the dicumyl peroxide (DCP), which diffuses into an ethylene–octene copolymer above its T _g was measured from an innovative rheological experiment. The experiments were carried out on a parallel plate geometry rheometer. The method is based on the cross-linking of a two-layer sample; the upper layer contains 2 wt% of DCP, and the lower layer is free of DCP. Actually, this experiment is based on the competition between the reaction of cross-linking and the diffusion of DCP in the lower layer. Comparing this rheological behavior with the rheological kinetic of cross-linking of an homogeneous sample with 1 wt% DCP, we are able, from an inverse fitting procedure, to calculate the mutual coefficient of diffusion. Our hypothesis is that the diffusion of DCP in the copolymer above T _g, can be described by Fick’s classical law. Using Fick’s law, the concentration of the DCP was established for any given point of the thickness of the two-layer sample at any time. Using a one-dimensional grid to solve continuous equations that describe the different rheological contributions of each abscissa, we determined the linear viscoelastic response of the whole sample. Comparing the experimental storage modulus of the two layer sample to the values measured from an homogeneous sample, we found the values of the mutual coefficient of diffusion. Finally, a simple relation, which describes the mutual coefficient of diffusion of DCP into melt ethylene–octene copolymer was established according to an Arrhenius law as:

A Direct Determination of the Transient Exchange Term of Fractured Media Using a Continuous Time Random Walk Method

In two recent papers, Ntinger and Estébenet, 2000; Ntinger et al., submitted, we set-up a method allowing to compute both the transient and steady-state exchange terms between the matrix and fractured regions of a naturally fractured porous medium using continuous time random walk methods (CTRW). The goal of the present paper is to show that a new version of the CTRW algorithm provides a direct determination of the so called transient exchange function f(t) (or its Laplace transform f(s)) widely used in well test interpretation. It is shown that this function is directly linked with the probability density of the first escape time in the fractured region of a Brownian particle launched initially in the matrix region. This new interpretation allows relating directly the exchange coefficient with the mean escape time of brownian particles in the matrix. From a practical point of view, these new results allow to derive a simpler version of the CTRW method. In addition, we obtain a considerable speed up of the CTRW method for up-scaling fractured reservoirs.     

A rheological approach to the stability of humic acid/clay colloidal suspensions

Steady-state, oscillatory, and transient rheological determinations were used to assess the stability of homoionic sodium montmorillonite (NaMt) suspensions at constant ionic strength (10^–2 mol/l NaCl) and different pH values, after adsorption of humic acid (HA) on the particles. The adsorption of the latter was first spectrophotometrically determined, at pH 3 and 9. While at pH 9 adsorption saturation was observed, at pH 3 the adsorption density continued to grow up to the maximum equilibrium HA concentration reached (∼200 mg/l). Considering the similarity between the structure of edge surfaces of NaMt particles and the surfaces of silica and alumina, the adsorption of HA was also investigated on the latter solids. The results suggest that at pH 3 humic acids adsorb preferentially on edge surfaces, mainly through electrostatic attraction with positively charged aluminol groups. This hypothesis is indirectly confirmed by zeta potential, ζ, values: while HA concentration has little effect on ζ for silica, the addition of HA yields the zeta potential of alumina increasingly negative for all pH values. Using shear stress vs shear rate plots, the yield stress of NaMt was determined as a function of particle concentration, C, for pH 3, 5, 7, and 9, with and without addition of 50 mg/l HA. The yield stress, σ_y, was fitted with a power law σ_y∝C ⁿ ; it was found that n values as high as 12 are characteristic of NaMt suspensions at pH 9 in the presence of HA. This indicates a strong stabilizing effect of humic acid. This stabilization was confirmed by oscillometric measurements, as the storage modulus G′ in the viscoelastic linear region also scales with C, displaying large n values at neutral and basic pHs in the presence of HA. The modulus (in the viscoelastic linear region, for a frequency ν=1 Hz) was found to increase with time, but G′ was lower at any time when HA was added, a consequence of the stabilization provided by HA. Similarly, creep-recovery experiments demonstrated that NaMt suspensions containing HA displayed a less elastic behavior, and a permanent deformation. Modeling the results as a Kelvin-Voigt model allowed one to establish a new scaling law of the reciprocal instantaneous deformation with C. As before, high values of n were found for suspensions at pH 9 in the presence of HA.     

Turbulence kinetic energy budget in bubbly flows in a vertical duct

Understanding turbulence kinetic energy (TKE) budget in gas–liquid two-phase bubbly flows is indispensable to develop and improve turbulence models for the bubbly flows. In this study, a molecular tagging velocimetry based on photobleaching reaction was applied to turbulent bubbly flows with sub-millimeter bubbles in a vertical square duct to examine the applicability of the k–ε models to the bubbly flows. Effects of bubbles on TKE budget are discussed and a priori tests of the standard and low Reynolds number k–ε models are carried out to examine the applicability of these models to the bubbly flows. The conclusions obtained are as follows: (1) The photobleaching molecular tagging velocimetry is of use for validating turbulence models. (2) The bubbles increase the liquid velocity gradient in the near wall region, and therefore, enhance the production and dissipation rates of TKE. (3) The k–ε models can reasonably evaluate the production rate of TKE in the bubbly flows. (4) The modulations of diffusion due to the bubbles have different characteristics from the diffusion enhancement due to shear-induced turbulence. Hence, the k–ε models fail in evaluating the diffusion rate in the near wall region in the bubbly flows. (5) The k–ε models represent the trends of the production, dissipation, and diffusion rates of ε in the bubbly flow, although more accurate experimental data are required for quantitative validation of the ε equation.     

A general method for obtaining diffusion coefficients by inversion of measured torque from diffusion experiments under small amplitude oscillatory shear

A numerical approach based on Tikhonov regularization is developed to invert torque curves from time-dependent small amplitude oscillatory shear (SAOS) experiments in which diffusion occurs to determine the diffusion coefficient. Diffusion of a solvent into a polymer melt for example causes the measured torque to decrease over time and is thus dependent on diffusion kinetics and the concentration profile. Our numerical approach provides a general method for retrieving local viscosity profiles during diffusion with reasonable accuracy, depending only on the linear viscoelastic constitutive equation and a general power law dependency of the diffusion process on time. This approach also allows us to identify the type of diffusion (Fickian, pseudo-Fickian, anomalous, and glassy) and estimate the diffusion coefficient without the a priori identification of a specific diffusion model. Retrieving local viscosity profiles from torque measurements in the presence of a concentration gradient is an ill-posed problem of the second type and requires Tikhonov regularization. The robustness of our approach is demonstrated using a number of virtual experiments, with data sets from Fickian and non-Fickian theoretical concentration and torque profiles as well as real experimental data.

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A finite element formulation satisfying the discrete geometric conservation law based on averaged Jacobians

In this article, a new methodology for developing discrete geometric conservation law (DGCL) compliant formulations is presented. It is carried out in the context of the finite element method for general advective–diffusive systems on moving domains using an ALE scheme. There is an extensive literature about the impact of DGCL compliance on the stability and precision of time integration methods. In those articles, it has been proved that satisfying the DGCL is a necessary and sufficient condition for any ALE scheme to maintain on moving grids the nonlinear stability properties of its fixed‐grid counterpart. However, only a few works proposed a methodology for obtaining a compliant scheme. In this work, a DGCL compliant scheme based on an averaged ALE Jacobians formulation is obtained. This new formulation is applied to the θ family of time integration methods. In addition, an extension to the three‐point backward difference formula is given. With the aim to validate the averaged ALE Jacobians formulation, a set of numerical tests are performed. These tests include 2D and 3D diffusion problems with different mesh movements and the 2D compressible Navier–Stokes equations. Copyright © 2011 John Wiley & Sons, Ltd.     

幂律型浆液扩散压力环向分布模型

为研究幂律型浆液注浆时注浆压力的变化情况,考虑盾尾断面新注入浆液与已注入浆液间阻碍作用,假设壁后注浆时盾尾形成三维环形空隙,提出了幂律型浆液扩散压力环向分布模型,并利用流体力学理论推导了幂律型浆液扩散压力环向分布式,分析了公式适用范围以及各参数的实际意义。与实际工程数据对比,验证了模型和计算式的正确性。结果表明,计算式可以反映注浆时环向分布各个位置压力值的大小;当公式中稠度系数n=1时,该式即为牛顿流体计算式,环向压力扩散模型同样适用,且幂律型流体环向扩散压力小于牛顿流体;受浆液自重影响,注浆孔注浆时向上表现为减压,向下表现为加压;压力环向分布断面呈现出上窄下宽的不规则环形;同一注浆孔幂律型浆液水灰比越大,浆液扩散压力越小。     

Influence of the surface roughness on the third-order moments of velocity fluctuations

Roughness wall effects in a zero pressure gradient turbulent boundary layers were investigated using hot-wire anemometry. The skewness and diffusion factors of u and v, the longitudinal and normal velocity fluctuations, were measured and represented using wall variables. The results indicate that the wall roughness removes the crossover point between sweep and ejection events to the outer region of the layer for a single Reynolds number Re _θ  > 3,000. This behaviour exhibits that the roughness surface favours the maintaining of sweep events obtained by a quadrant analysis. These results show that communication between the wall region and outer region of a turbulent boundary layer exists and the wall similarity hypothesis for a rough wall is questionable. The effect of the wall roughness on the position of the point crossover from sweep to ejection motions with respect to the wall seems to be the same as that obtained when the Reynolds number is higher. Received: 8 March 2000/Accepted: 15 May 2000     

Thermal de-cohesion model for time of craze failure in cohesive zone

The cohesive parameter corresponding to craze failure time is predicted for thermoplastics material. A craze failure separation criterion is proposed for a cohesive zone subjected to a melt layer formed and thickened by adiabatic deformation heat from a craze drawing. The numerical simulation of cohesive zone separation is based on non-linear thermal conduction and convection in the craze region and bulk region around the active layer, associated with a mechanical craze fibrils drawing in an uniaxial direction. The craze failure time is predicted with the assumption of the constant craze thickening rate and cohesive stress for a pipe-grade polyethylene. The numerically computed model reveals the inverse power law decay of the craze failure time, t_f, with increasing in craze thickening rate, v_c, (almost, t_f∝V_c⁻¹) for the thermoplastics. The full notch impact test experimental results are consistent with the analysis prediction. It is concluded that the craze failure time can be theoretically predicted using the numerical modeling.     

Long-Time Behavior of Scalar Viscous Shock Fronts in Two Dimensions

We prove nonlinear stability in L ¹ of planar shock front solutions to a viscous conservation law in two spatial dimensions and obtain an expression for the asymptotic form of small perturbations. The leading-order behavior is shown rigorously to be governed by an effective diffusion coefficient depending on forces transverse to the shock front. The proof is based on a spectral analysis of the linearized problem.     

Rheological modeling of the diffusion process and the interphase of symmetrical bilayers based on PVDF and PMMA with varying molecular weights

The diffusion process in the molten state at a polymer/polymer interface of symmetrical and model bilayers has been investigated using a small-amplitude oscillatory shear measurement. The polymers employed in this study were poly (vinylidene fluoride) (PVDF) and poly (methyl methacrylate) (PMMA) of varying molecular weights and polydispersities. The measurements were conducted in the linear viscoelastic regime (small deformations) so as to decouple the effect of flow from the diffusion. The focus of this paper has been to investigate the effects of healing time, angular frequency (ω), temperature, and molecular weight on the inter-diffusion and the triggered interphase between the neighboring layers. The kinetics of diffusion, based on the evolution of the apparent diffusion coefficient (D _a) versus the healing time, was experimentally obtained. The transition from the non-Fickian to the normal Fickian region for the inter-diffusion at the interface was clearly observed, qualitatively consistent with the reptation model, but it occurred at a critical time greater than the reptation time (τ _rep). In non-Fickian region, effects of frequency and temperature were studied with regard to the ratio of the apparent diffusion coefficient to the self-diffusion coefficient (D _a/D _s). The D _s determined in the Fickian region was found to be consistent with Graessley’s model as well as with the literatures. And the dependence of the D_s on the frequency agreed well with the Doi–Edwards theory, in particular, scaling as $D_{\rm s} \sim \omega^{1/2}$ at ω?>?1/τ _e and $D_{\rm s} \sim \omega^{0}$ at ω?<?1/τ _rep. Our experimental results also confirmed that the dependence of the D _s on the temperature for PMMA and PVDF can be well described by the Arrhenius law. Moreover, blends of PMMAs have been proposed in order to be able to change the $\overline M_\emph{w} $ . The rheological investigations of these corresponding bilayers rendered it possible to monitor the effect of $\overline M_\emph{w} $ on the diffusion process. The obtained results gave $D_{\rm s} \sim \overline M_\emph{w}^{-1}$ , thus corroborating some earlier studies and some experimental results recently reported by Time-Resolved Neutron Reflectivity Measurements. Lastly, the thickness of the interphase and its corresponding viscoelastic properties could be theoretically determined as a function of the healing?time.     

A Physically Based Model of Dissolution of Nonaqueous Phase Liquids in the Saturated Zone

The design of remediation strategies for nonaqueous phase liquid (NAPL) contaminants involves predicting the rate of NAPL dissolution. A physically based model of an idealized pore geometry was developed to predict nonaqueous phase liquid dissolution rate coefficients. A bundle of parallel pores in series model is used to represent NAPL dissolution as a function of three processes: pore diffusion, corner diffusion, and mixing and multiple contact. The dissolution rate coefficient is expressed in terms of the modified Sherwood number (Sh) and is a function of Peclet (Pe) number. The model captures the complex behavior of Sh versus Pe data for both water-wet (Powers, 1992) and NAPL-wet (Parker et al., 1991) media. For water-wet media, the observed behavior can be broken down into four distinct regions. Each region represents a different physical process controlling NAPL dissolution: the low-Pe region is controlled by pore diffusion; the low- to moderate-Pe region is a transition zone; the moderate-Pe region is controlled by mixing and multiple contact; and the high-Pe region is controlled by corner diffusion. For the high-Pe conditions typical of most column experiments, the model involves only one fitting parameter. For NAPL-wet media, NAPL dissolution is governed exclusively by corner diffusion, and the model again involves only one fitting parameter.     

Water Vapor Diffusion Through Soil as Affected by Temperature and Aggregate Size

Water vapor diffusion through the soil is an important part in the total water flux in the unsaturated zone of arid or semiarid regions and has several significant agricultural and engineering applications because soil moisture contents near the surface are relatively low. Water vapor diffusing through dry soil is absorbed for both long and short terms. Long-term absorption allows more water to enter than exit the soil, as reflected in the concentration gradient. Short-term absorption leads to an apparent reduction in the diffusion rate, as reflected in the diffusion coefficient. This investigation studied the effects of soil temperature and porosity on the isothermal diffusion of water vapor through soil. The diffusion model consisted of 25.4 cm × 8.9 cm × 20.3 cm Plexiglas box divided into two compartments by a partition holding a soil reservoir. Water vapor moved from a container suspended by a spring in one compartment, through the porous medium in the center of the model, to calcium chloride in a container suspended by a spring in the other compartment. The porous materials consisted of aggregates of varying size (2–2.8, 1–2, and 0.5–1 mm) of a Fayatte silty clay loam (a fine-silty, mixed mesic Typic Hapludalf). The flow rates of water vapor were measured at temperatures of 10, 20, 30, and 40°C. Warmer temperatures increased the rate of diffusion through dry soil while reduced the amount of water absorbed by that soil. Reducing porosity slowed the rate of diffusion and increased the amount of water absorbed. The dry soil in this study absorbed from 1/8 to 2/3 of the diffusing water. Maximum absorption rates occurred with the most compact soil samples at the highest temperature, though the maximum absorption as a percentage of the diffusing water was in the compact samples at the lowest temperature. The diffusivity equation D/D ₀ = [(S – 0.1)/0.9]² fit the D/D ₀ values obtained from these data if a coefficient of 1/3 or 1/3.5 is added to correct for the time delays caused by temporary sorption of the diffusing water vapor. The data, influenced by the interaction of water vapor and soil materials, represent a diffusion rate lower than the diffusion rate that would have resulted without this interaction. Mention of trade names, proprietary products, or specific equipment is intended for reader information only and does not constitute a guarantee or warranty by the USDA-ARS nor does it imply approval of the product named to the exclusion of other products. An erratum to this article can be found at     

Experimental evidence of diffusion-induced bias in near-wall velocimetry using quantum dot measurements

Near-surface velocity measurements are carried out with quantum dot (QD) nanoparticles using evanescent wave illumination. Relying on the small size of QDs, their correspondingly small hydrodynamic radius and high Brownian diffusion coefficient, we consider the situation where the tracer diffusion length over the inter-frame time Δt is large compared to the size of the interrogation region next to the wall. While keeping all other experimental parameters fixed, we systematically increase Δt by as much as a factor of 25, resulting in an increase of the QD diffusion length by a factor of 5. Data indicate a significant overestimation of the “apparent” mean velocity measured experimentally. These results provide a direct confirmation of the phenomenon of diffusion-induced bias described by the simulations of Sadr et al. (2007).     

Boundary discretization for high‐order discontinuous Galerkin computations of tidal flows around shallow water islands

In this paper some preliminary results concerning the application of the high‐order discontinuous Galerkin (DG) method for the resolution of realistic problems of tidal flows around shallow water islands are presented. In particular, tidal flows are computed around the Rattray island located in the Great Barrier Reef. This island is a standard benchmark problem well documented in the literature providing useful in situ measurements for validation of the model. Realistic elements of the simulation are a tidal flow forcing, a variable bathymetry and a non‐trivial coastline. The computation of tidal flows in shallow water around an island is very similar to the simulation of the Euler equations around bluff bodies in quasi‐steady flows. The main difference lies in the high irregularity of islands' shapes and in the fact that, in the framework of large‐scale ocean models, the number of elements to represent an island is drastically limited compared with classical engineering computations. We observe that the high‐order DG method applied to shallow water flows around bluff bodies with poor linear boundary representations produces oscillations and spurious eddies. Surprisingly those eddies may have the right size and intensity but may be generated by numerical diffusion and are not always mathematically relevant. Although not interested in solving accurately the boundary layers of an island, we show that a high‐order boundary representation is mandatory to avoid non‐physical eddies and spurious oscillations. It is then possible to parametrize accurately the subgrid‐scale processes to introduce the correct amount of diffusion in the model. The DG results around the Rattray island are eventually compared with current measurements and reveal good agreement. Copyright © 2008 John Wiley & Sons, Ltd.     

A Discussion of the Effect of Tortuosity on the Capillary Imbibition in Porous Media

In the past decades, there was considerable controversy over the Lucas–Washburn (LW) equation widely applied in capillary imbibition kinetics. Many experimental results showed that the time exponent of the LW equation is less than 0.5. Based on the tortuous capillary model and fractal geometry, the effect of tortuosity on the capillary imbibition in wetting porous media is discussed in this article. The average height growth of wetting liquid in porous media driven by capillary force following the [`(L)] _s(t) ~ t^1/2D_T{\overline L _{\rm {s}}(t)\sim t^{1/{2D_{\rm {T}}}}} law is obtained (here D _T is the fractal dimension for tortuosity, which represents the heterogeneity of flow in porous media). The LW law turns out to be the special case when the straight capillary tube (D _T = 1) is assumed. The predictions by the present model for the time exponent for capillary imbibition in porous media are compared with available experimental data, and the present model can reproduce approximately the global trend of variation of the time exponent with porosity changing.     

ASYMPTOTIC ANALYSIS OF MODE Ⅱ STATIONARY GROWTHCRACK ON ELASTIC-ELASTIC POWER LAWCREEPING BIMATERIAL INTERFACE

A mechanical model was established for mode Ⅱ interfacial crack static growingalong an elastic-elastic power law creeping bimaterial interface. For two kinds of boundaryconditions on crack faces, traction free and frictional contact, asymptotic solutions of thestress and strain near tip-crack were given. Results derived indicate that the stress andstrain have the same singularity, there is not the oscillatory singularity in the field; thecreep power-hardening index n and the ratio of Young‘s module notably influence the crack-tip field in region of elastic power law creeping material and n only influences distribution ofstresses and strains in region of elastic material. When n is bigger, the creepingdeformation is dominant and stress fields become steady, which does not change with n.Poisson‘s ratio does not affect the distributing of the crack-tip field.     

Asymptotic solutions of the problem of heat and mass transfer of particles in a fluid

Some questions related to asymptotic analysis (as P , where P is the Péclet number) of problems involving heat and mass transfer of particles in a fluid are considered. The first part of the paper investigates the stationary convective diffusion of a solute to a particle near its front critical point (incidence point). An explicit expression is obtained for the concentration in the region of the front critical point of a solid or liquid particle around which a Stokes flow occurs. In the second part of the paper, a unified formula is obtained for the concentration distribution behind the particle. Appropriate limits in this formula determine the concentration in the mixing region and the inner and convective boundary-layer regions of the diffusion wake. In the final part of the paper, a study is made of the diffusion to a chain of absorbing solid spheres of equal radius a at distances1, 1 1/a P^1/3, from each other on the axis of an oncoming Stokes flow; an integral equation is obtained for the local diffusion flux when a chemical reaction with arbitrary kinetics takes place on the surfaces of the spheres. A certain heterogeneity of the material in the paper is due to the investigation in it of various questions that arise in the solution of more general problems (see, for example, [1–14]) which have not been considered hitherto.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza., No. 1, pp. 89–96, January–February, 1979.     

Constructal Design of Vascular Porous Materials and Electrokinetic Mass Transfer

According to constructal theory, the “generation of flow configuration” is a universal phenomenon in physics. This phenomenon is covered by the constructal law: “For a finite-size flow system to persist in time (to live) it must evolve such that it provides greater and greater access to the currents that flow through it.” This paper shows how the constructal law can be used to (1) predict and explain features of “design” in nature, and (2) design effective strategies and configurations for engineering. Many natural flow designs rely on two flow mechanisms: channels with relatively low resistivity, interwoven with diffusion across the interstices. The “design” is the balance between the two mechanisms. The flow from line to line (or plane to plane) through a sufficiently fine porous medium encounters less resistance than the flow through parallel channels when it is configured as trees that alternate with upside down trees: from this follows the prediction that natural porous media (e.g., hill slope) should be multiscale (bidisperse) and non-uniformly distributed. A porous medium contaminated with ionic species is decontaminated the fastest when the ionic flow is configured as two flow mechanisms in balance: “channeling” driven by potential differences between optimally positioned electrodes, and diffusion driven by concentration differences across the interstices between the channels.