A macrotransport equation for the particle distribution in the flow of a concentrated, non-colloidal suspension through a circular tube

AbstractA two-time-scale perturbation expansion is used to derive a cross-section-averaged convection–dispersion equation for the particle distribution in the flow of a concentrated suspension of neutrally buoyant, non-colloidal particles through a straight, circular tube. Since the cross-streamline motion of particles is governed by shear-induced migration, the Taylor-dispersion coefficient ${\mathscr{D}}_{eff} $ scales as ${U}^{\prime } {R}^{3} / {a}^{2} $, ${U}^{\prime } $, $R$ and $a$ being the characteristic velocity scale, the tube radius and the particle radius, respectively. Here ${\mathscr{D}}_{eff} $ is found to decrease monotonically with an increase in the particle concentration. The linear dependence of ${\mathscr{D}}_{eff} $ on ${U}^{\prime } $ implies that changes in the cross-section averaged axial concentration profile are dependent only on the total axial strain experienced by the suspension. This stipulates that the spatial evolution of a fluctuation in the concentration of particles in the flowing suspension, or the width of the mixing zone between two regions of different concentrations in the tube will be independent of the suspension velocity in the tube. A second interesting feature in particulate dispersion is that the effective velocity of the particulate phase is concentration-dependent, which, by itself (i.e. without considering Taylor dispersion), can produce either sharpening or relaxation of concentration gradients. In particular, shocks with positive concentration gradients along the flow direction can asymptotically evolve into time-independent distributions in an appropriately chosen frame of reference, and concentration pulses relax asymmetrically. These trends are contrasted with those expected from the classical problem of Taylor dispersion of a passive tracer in the same geometry. The results in this paper are especially relevant for suspension flows through microfluidic geometries, where the induction lengths for shear-induced migration are short.

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Forced spreading of films and droplets of colloidal suspensions

Journal of Fluid Mechanics ◽

10.1017/jfm.2014.27 ◽

2014 ◽

Vol 742 ◽

pp. 495-519 ◽

Cited By ~ 17

Author(s):

Leonardo Espín ◽

Satish Kumar

Keyword(s):

Particle Concentration ◽

Colloidal Particles ◽

Hard Spheres ◽

Colloidal Suspension ◽

Colloidal Suspensions ◽

Concentration Gradients ◽

Vertical Diffusion ◽

Interface Evolution ◽

Dependent Viscosity ◽

Liquid Dynamics

AbstractWhen a thin film of a colloidal suspension flows over a substrate, uneven distribution of the suspended particles can lead to an uneven coating. Motivated by this phenomenon, we analyse the flow of perfectly wetting films and droplets of colloidal suspensions down an inclined plane. Lubrication theory and the rapid-vertical-diffusion approximation are used to derive a coupled pair of one-dimensional partial differential equations describing the evolution of the interface height and particle concentration. Precursor films are assumed to be present, the colloidal particles are taken to be hard spheres, and particle and liquid dynamics are coupled through a concentration- dependent viscosity and diffusivity. We find that for sufficiently high Péclet numbers, even small initial concentration inhomogeneities produce viscosity gradients that cause the film or droplet front to evolve continuously in time instead of travelling without changing shape as happens in the absence of colloidal particles. At high enough particle concentrations, particle diffusion can lead to the formation of long-lived secondary flow fronts in films. Our results suggest that particle concentration gradients can have a dramatic influence on interface evolution in flowing films and droplets, a finding which may be relevant for understanding the onset of patterns that are observed experimentally.

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The initial development of longitudinal dispersion in straight tubes

Journal of Fluid Mechanics ◽

10.1017/s0022112077001529 ◽

1977 ◽

Vol 80 (1) ◽

pp. 33-48 ◽

Cited By ~ 27

Author(s):

P. C. Chatwin

Keyword(s):

Laminar Flow ◽

Cross Section ◽

Circular Tube ◽

Tube Wall ◽

Random Motion ◽

Fluid Particle ◽

Straight Tube ◽

Initial Development ◽

Molecular Diffusivity ◽

The Cross

When a contaminant molecule is released in a laminar flow in a straight tube its motion differs from that of the fluid particle with which it initially coincided because of its random motion, whose intensity is measured by the molecular diffusivity κ. For T = kt/a2 ≤ 0·25, where t is the time after release and a is a length characteristic of the cross-section, the statistics of its motion can be determined in the way described by Taylor (1953), Aris (1956) and Chatwin (1970). However, in many applications, including blood flow, the values of T which are attained are much smaller, and the purpose of this paper is to present first approximations for T [Gt ] 1 to some of the statistics measuring the deviations between the motion of the molecule and that of the fluid particle with which it initially coincided. To obtain these a technique due to Saffman (1960) is used for molecules released well away from the tube wall, and an extension of the same technique is used for molecules released near the tube wall. It is shown how the results can be used to describe the initial stages of dispersion of a cloud of contaminant molecules distributed arbitrarily over the cross-section in any laminar flow, steady or unsteady. Comparisons with some exact results for steady Poiseuille flow in a circular tube confirm that the approximations are formally correct, but show that in certain cases their practical use is limited to very small values of T, because of the high coefficients of the first two terms neglected in deriving the approximation.

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A New Approach to the Demonstration of Oxidase-Localization Using Tannic Acid Or Catechin

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100073933 ◽

1973 ◽

Vol 31 ◽

pp. 698-699

Author(s):

V. Mizuhira ◽

Y. Futaesaku

Keyword(s):

Cross Section ◽

Tannic Acid ◽

Elastic Fiber ◽

The Other ◽

New Approach ◽

Tissue Block ◽

Active Reaction ◽

The Cross

Previously we reported that tannic acid is a very effective fixative for proteins including polypeptides. Especially, in the cross section of microtubules, thirteen submits in A-tubule and eleven in B-tubule could be observed very clearly. An elastic fiber could be demonstrated very clearly, as an electron opaque, homogeneous fiber. However, tannic acid did not penetrate into the deep portion of the tissue-block. So we tried Catechin. This shows almost the same chemical natures as that of proteins, as tannic acid. Moreover, we thought that catechin should have two active-reaction sites, one is phenol,and the other is catechole. Catechole site should react with osmium, to make Os- black. Phenol-site should react with peroxidase existing perhydroxide.

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Energy Distribution in Electron Beams

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100096096 ◽

1972 ◽

Vol 30 ◽

pp. 568-569

Author(s):

Tamotsu Ohno

Keyword(s):

Electron Beam ◽

Energy Distribution ◽

Cross Section ◽

Integral Curve ◽

Electron Beams ◽

Electron Gun ◽

Angular Divergence ◽

Apparent Increase ◽

Integral Width ◽

The Cross

The energy distribution in an electron; beam from an electron gun provided with a biased Wehnelt cylinder was measured by a retarding potential analyser. All the measurements were carried out with a beam of small angular divergence (<3xl0-4 rad) to eliminate the apparent increase of energy width as pointed out by Ichinokawa.The cross section of the beam from a gun with a tungsten hairpin cathode varies as shown in Fig.1a with the bias voltage Vg. The central part of the beam was analysed. An example of the integral curve as well as the energy spectrum is shown in Fig.2. The integral width of the spectrum ΔEi varies with Vg as shown in Fig.1b The width ΔEi is smaller than the Maxwellian width near the cut-off. As |Vg| is decreased, ΔEi increases beyond the Maxwellian width, reaches a maximum and then decreases. Note that the cross section of the beam enlarges with decreasing |Vg|.

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Research in the cross-section of community psychology and global climate change: A call for action

PsycEXTRA Dataset ◽

10.1037/e628592012-083 ◽

2009 ◽

Author(s):

Marci Culley ◽

Holly Angelique ◽

Courte Voorhees ◽

Brian John Bishop ◽

Peta Louise Dzidic ◽

...

Keyword(s):

Climate Change ◽

Cross Section ◽

Global Climate Change ◽

Community Psychology ◽

Global Climate ◽

The Cross

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Experimental Studies Of Multilayer Glass Structures On Compression, Compression With Bending And Simple Bending

Promyshlennoe i Grazhdanskoe Stroitel stvo ◽

10.33622/0869-7019.2019.01.22-30 ◽

2019 ◽

pp. 22-30

Keyword(s):

Cross Section ◽

Cross Sections ◽

Applied Load ◽

Experimental Studies ◽

Strength Properties ◽

Research Topic ◽

Normative Values ◽

Laminated Glass ◽

The Cross ◽

Experimental Values

The work of multilayer glass structures for central and eccentric compression and bending are considered. The substantiation of the chosen research topic is made. The description and features of laminated glass for the structures investigated, their characteristics are presented. The analysis of the results obtained when testing for compression, compression with bending, simple bending of models of columns, beams, samples of laminated glass was made. Overview of the types and nature of destruction of the models are presented, diagrams of material operation are constructed, average values of the resistance of the cross-sections of samples are obtained, the table of destructive loads is generated. The need for development of a set of rules and guidelines for the design of glass structures, including laminated glass, for bearing elements, as well as standards for testing, rules for assessing the strength, stiffness, crack resistance and methods for determining the strength of control samples is emphasized. It is established that the strength properties of glass depend on the type of applied load and vary widely, and significantly lower than the corresponding normative values of the strength of heat-strengthened glass. The effect of the connecting polymeric material and manufacturing technology of laminated glass on the strength of the structure is also shown. The experimental values of the elastic modulus are different in different directions of the cross section and in the direction perpendicular to the glass layers are two times less than along the glass layers.

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