Thermocapillary drift velocity and effective thermal conductivity of a suspension

The thermocapillary drift of a homogeneous suspension of spherical drops with the constant properties of a carrier fluid and the fluid inside the drops is considered. Several formulas are obtained for the drift velocity and the effective thermal conductivity of the suspension.     

Particle motions in sheared suspensions XXII: Interactions of rigid spheres (experimental)

Summary The behaviour of rigid spheres suspended in viscous liquids undergoing laminar shear flow was compared experimentally with the approximate theory developed earlier. Good agreement was obtained for a sphere near a wall and for equatorial interactions between two spheres of both equal and unequal diameters. In the region close to but not touching the wall, the theory byGoldman et al. accurately described the motion of a sphere. The prediction that a sphere touching the wall should stick to it could not be confirmed. A viewing device, designed to study motions in two planes simultaneously was used to make several interesting observations on permanent and separating doublet. Some similarities between collisions in the equatorial plane and two-body interactions of vertically oriented rigid circular cylinders were noted. The asymmetry and reversibility of triple collisions as well as the ease of formation of permanent doublets in flowing suspensions were established.

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Zusammenfassung Das Verhalten von starren, in viskosen Flüssigkeiten suspendierten Teilchen unter dem Einfluß von laminarem Scherfluß wurde experimentell mit der früher entwickelten, genäherten Theorie verglichen. Gute Übereinstimmung wurde erhalten für eine Kugel in Wandnähe und für äquatoriale Einflüsse zwischen zwei Kugeln von entweder gleichem oder verschiedenem Durchmesser. Die Theorie vonGoldmann et al. gab eine genaue Beschreibung der Bewegung einer Kugel in Wandnähe, aber ohne Wandberührung. Die Voraussage, daß eine Kugel an einer berührenden Wand kleben würde, konnte nicht bestätigt werden. Eine Beobachtungs-Einrichtung, die für das Studium der gleichzeitigen Bewegung in zwei Ebenen konstruiert war, wurde für verschiedene interessante Beobachtungen an permanenten und an sich trennenden Dubletts benutzt. Einige Änlichkeiten zwischen Zusammenstößen in der Äquatorial-Ebene und Zweikörperbeinflussungen von vertikal orientierten, starren kreisförmigen Zylindern wurde festgestellt. Die Asymmetrie und Reversibilität von dreifachen Stößen wie auch die leichte Bildung von permanenten Dubletts wurde im Fluß von Suspensionen festgestellt.

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This work was also supported by the National Heart Institute of the United States Public Health Service (Grant HE-05911).     

Anisotropic thermal conductivity in sheared polypropylene

We discuss the anisotropy of the thermal conductivity tensor in polymer flow in this paper. Isotactic polypropylene (iPP) specimens were deformed by injection moulding at high shear rates and by steady shear at low shear rates, and were then quenched. The thermal conductivities parallel and perpendicular to the shear direction were measured using modulated differential scanning calorimetry (MDSC) in accordance with the ASTM E1952-01. The measured results showed that the thermal conductivity of the sheared polymer was anisotropic with an increase in the shear direction. The thermal conductivity can be regarded as varying either with the strain or the stress, as suggested by Van den Brule (1989). In addition to the Van den Brule mechanism, crystallization during flow also changes the thermal conductivity and this effect may often be dominant. Suggestions for procedures in processing computations, based on both effects, are given.     

An experimental study on turbulent coherent structures near a sheared air-water interface

The turbulence structures near a sheared air-water interface were experimentally investigated with the hydrogen bubble visualization technique. Surface shear was imposed by an airflow over the water flow which was kept free from surface waves. Results show that the wind shear has the main influence on coherent structures under air-water interfaces. Low- and high- speed streaks form in the region close to the interface as a result of the imposed shear stress. When a certain airflow velocity is reached, “turbulent spots” appear randomly at low-speed streaks with some characteristics of hairpin vortices. At even higher shear rates, the flow near the interface is dominated primarily by intermittent bursting events. The coherent structures observed near sheared air-water interfaces show qualitative similarities with those occurring in near-wall turbulence. However, a few distinctive phenomena were also observed, including the fluctuating thickness of the instantaneous boundary layer and vertical vortices in bursting processes, which appear to be associated with the characteristics of air-water interfaces. The project supported by the National Natural Science Foundation of China (Grant No.19672070)     

Particle motions in sheared suspensions XXI: Interactions of rigid spheres (theoretical)

Summary Based on a method developed byFaxén, approximate expressions were developed for the rotational and translational velocities of a single rigid sphere near a plane rigid wall bounding Couette flow, and near another sphere of either different or the same size in Couette flow. The solutions to the latter problem provide equations for the orbits of each sphere about the other, some of which are closed. In many respects the results are similar to the corresponding two-dimensional problems for cylinders.

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Zusammenfassung Angenäherte Gleichungen wurden aufgrund einer vonFaxén entwickelten Methode für die Rotations- und Translations-Geschwindigkeiten einer einzelnen, starren Kugel in der Nähe einer ebenen, starren Wand, die einenCouette- Fluß begrenzte, aufgestellt, und außerdem für eine zweite Kugel von gleicher oder verschiedener Größe imCouette-Fluß. Die Lösungen für dieses zweite Problem geben Gleichungen für die Bahnen jeder Kugel um die andere und einige davon stellen geschlossene Ausdrücke dar. In vieler Hinsicht sind die Resultate änlich wie für die entsprechenden zweidimensionalen Zylinder-Probleme.

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This work was supported in part by the National Heart Institute of the United States Public Health Service (Grant HE-05911).     

On the rheology of a dilute suspension of rigid spheres in a weakly viscoelastic matrix fluid

The complete solution for the pressure and the velocity field up to O(De) of a dilute suspension of neutrally buoyant, non-Brownian rigid spheres suspended in an unbounded, weakly viscoelastic matrix fluid, where is the solid volume fraction and De is the Deborah number of the matrix fluid, is presented. The spheres are subjected to an arbitrary linear velocity profile at infinity. The analytical solution is used for the prediction of the bulk stress, and specifically for the calculation of the first and the second normal stress differences in simple shear and uniaxial elongational flows. A comparison of the results with available values reported in the literature is also offered. The final expressions for the bulk normal stress differences in shear and uniaxial elongational flow fully agree with those reported earlier by Greco et al., J. Non-Newton. Fluid Mech., 147 (2007) 1–10.     

An experimental study of multimodal glass suspension rheology to test and validate a polydisperse suspension viscosity model

Experimental measurements of non-colloidal multimodal suspension viscosities are performed over a wide range of mixing ratios and used to test the robustness and predictive capability of a recent viscosity model (Mwasame et al. in Phys Fluids 28:061701, 2016b), subsequently referred to as the MWB model. Three unimodally distributed particle suspensions with narrow size distributions are blended to make the bimodal and trimodal suspensions used in the rheological experiments. We demonstrate how predictions for mixture viscosities can be made using the MWB model only requiring the volume-weighted average particle sizes and viscosity correlations of the individual unimodal suspensions comprising the blend. The resultant model predictions are found to be in good agreement with measured bimodal and trimodal viscosity data to within expected experimental uncertainty. The datasets provided here can be used to validate future modeling efforts, and the MWB model can be used to optimize the viscosity of multimodal suspension mixtures for specific performance criteria.     

On the viscosity of a concentrated suspension of solid spheres

Explanations of the very high viscosities of concentrated suspensions of spheres based on the dissipation in squeezing flow between particles pairs are shown to be in error. The dissipation in pair interactions is always of the order of that generated by shearing motions, and this dissipation is of too low an order in concentration for single pair interactions to explain the observed viscosities.     

An experimental study of the regimes of motion of spheres falling or ascending freely in a Newtonian fluid

This paper presents the results of an experimental investigation aimed at verifying some of the interesting conclusions of the numerical study by Jenny et al. concerning the instability and the transition of the motion of solid spheres falling or ascending freely in a Newtonian fluid. The phenomenon is governed by two dimensionsless parameters: the Galileo number G, and the ratio of the density of the spheres to that of the surrounding fluid ρ_s/ρ. Jenny et al. showed that the (G, ρ_s/ρ) parameter space may be divided into regions with distinct features of the trajectories followed eventually by the spheres after their release from rest. The characteristics of these ‘regimes of motion’ as described by Jenny et al., agree well with what was observed in our experiments. However, flow visualizations of the wakes of the spheres using a Schlieren optics technique raise doubts about another conclusion of Jenny et al., namely the absence of a bifid wake structure.     

An experimental study on the thermal performance of ground heat exchanger

A knowledge of ground thermal properties is most important for the proper design of large GHE (ground heat exchanger) systems. Thermal response tests have so far been used primarily not only for in situ determination of design data for GHE systems, but also for the evaluation of grout material, heat exchanger types and groundwater effects. The main purpose has been to determine in situ values of effective ground thermal conductivity, including the effect of groundwater flow and natural convection in boreholes.     

On the squeeze flow of a power-law fluid between rigid spheres

The lubrication solution for the squeeze flow of a power-law fluid between two rigid spherical particles has been investigated. It is shown that the radial pressure distribution converges to zero within the gap between the particles for any value of the flow index, n, provided that the gap separation distance is sufficiently small. However, in the case of the viscous force, it is useful to consider that there are two contributions. The first is developed in the inner region of the gap and corresponds to the lubrication limit. The second is due to an integration of the pressure in the adjacent outer region of the gap. The relative contribution to the force in this outer region increases as n decreases and the separation distance increases. In particular, for flow indices in the range n>1/3, the contribution in the outer region is negligible if the separation distance is sufficiently small. For n1/3, this is the dominant term and an accurate prediction of the viscous force is possible only for discrete liquid bridges.Based on “zero” pressure and lubrication criteria for the upper limits of integration, two closed-form solutions have been derived for the viscous force. Both are accurate for n>0.5 and are in close agreement with a previously published asymptotic solution in the range n>0.6. For smaller values of n, the asymptotic solution over-estimates the viscous force and predicts a singularity when n approaches 1/3. The two closed-form solutions show continuous and monotonic behaviour for all values of n. Moreover, the solution satisfying the lubrication limit is valid in the range n<1/3 provided that it is restricted to liquid bridges.     

Model for predicting effective thermal conductivity of composites with aligned continuous fibers of graded conductivity

Governing differential equations in both transverse and longitudinal directions for predicting the effective thermal conductivities of composites with aligned, graded continuous fibers are derived. It is shown that the effective conductivities of composites with graded fibers are predicted by solving the equations. The results by the present approach are applicable to both dilute and non-dilute cases without additional procedures unlike other approaches. The results are compared with those in the literature, and the applicability of the present approach is justified. A solution by the present approach is obtained analytically or numerically as long as thermal conductivity profile of fibers is given.     

Corner flow of a suspension of rigid rods

Stress singularities in the neighbourhood of sharp corners can be a source of severe problems in the numerical simulation of non-Newtonian flows leading to loss of convergence with grid refinement (G.G. Lipscombe, R. Kennings and M.M. Denn, J. Non-Newtonian Fluid Mech., 24 (1987) 85 [1]). For Newtonian flows the nature of this singularity is given by the analysis of Dean and Montagnon (W.R. Dean and P.E. Montagnon, Phil. Trans. R. Soc. London, Ser. A., 308 (1949) 199 [2]) in terms of similarity solutions. In this paper we extend this similarity analysis to a suspension of rigid rods. In the limit of nearly full extension the FENE constitutive model has the same behaviour as such a suspension. Our analysis predicts the possibility of lip vortices but their behaviour is somewhat inconsistent with those observed experimentally.     

Dissipation in a dilute suspension of spheres in a second-order fluid

We examine the effective medium properties of a dilute suspension of spheres in a second-order fluid under linear shear. Since the second-order fluid is the first step toward the general viscoelastic fluid, the results obtained may provide a qualitative feel for the problem in which the suspending fluid obeys a more complicated (and realistic) constitutive relation.The dissipation in the medium is calculated by determining the rate of working by surface forces; this is compared to the dissipation in a homogeneous fluid to give the effective properties. The results show that the term linear in volume fraction increases the corresponding rheological coefficient, just as in the Newtonian case. It is to be noted that the second-order dissipation is zero for simple shear and other weak flows, whereas for strong flows the small correction may increase or decrease the overall dissipation.     

The effect of Brownian motion for a suspension of spheres

Summary A system consisting of an incompressible Newtonian fluid in which solid particles of spherical shape are suspended is investiged. Employing irreducibleCartesian tensors an explicit solution to the Stokes problem for a sphere in an arbitrary flow field is obtained, valid even under conditions that the sphere is so small that the customary no-slip assumption need no longer be valid (Richardson). Since in that case Brownian motion cannot be neglected the effect of thermal agitation is studied. While no influence is found for a Couette flow the results for Poiseuille flow are markedly effected: In a vertical tube flow a nonuniform radial distribution of spheres results while for horizontal Poiseuille flow the effective viscosity becomes dependent upon the sphere radius. It increases with increasing sphere radius and is always larger than the corresponding result for truly neutrally buoyant spheres. These results are obtained from the equations governing the mechanical behavior of such a suspension, equations which are derived by means of ensemble averages. The advantage of these averages over the volume averages is demonstrated.

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Zusammenfassung Ein System, bestehend aus einer inkompressiblen newtonschen Flüssigkeit mit darin suspendierten starren kugelförmigen Teilchen wird untersucht. Mit Hilfe irreduzibler kartesischer Tensoren erhält man eine explizite Lösung des Stokesschen Problems einer Kugel in einem beliebigen Strömungsfeld. Diese Lösung behält ihre Gültigkeit sogar dann, wenn die Kugeln so klein sind, daß die Haftbedingung der Flüssigkeit nicht länger gültig zu sein braucht (Richardson). Da dann die Brownsche Bewegung nicht länger vernach-lässigt werden kann, wird der Einfluß der thermischen Bewegung untersucht. Während kein Einfluß in einer Couette-Strömung feststellbar ist, ändern sich die Resultate für eine Poiseuille-Strömung drastisch: In einer vertikalen Kapillarströmung stellt sich eine ungleichförmige radiale Verteilung von Teilchen ein, und in einer horizontalen Poiseuille-Strömung wird die scheinbare Viskosität eine Funktion des Kugelradius. Sie nimmt nicht nur mit größer werdendem Radius zu, sondern ist auch immer größer als im Fall auftriebloser Kugeln.Diese Resultate werden über die Bewegungsgleichungen einer Kugelsuspension gewonnen, wobei die Gleichungen mit Hilfe von Ensemble-Mittelwerten hergeleitet werden. Der Vorteil solcher Mittelwerte über Volumenmittelwerte wird aufgezeigt.

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With 2 figures     

Effect of polydispersity on the viscosity of a suspension of hard spheres

A simple procedure is suggested for calculating the concentration dependence of the viscosity of a suspension of two types of hard spheres, which have considerably different diameters. It is shown that for the calculations it is sufficient to know the concentration dependence of the suspension viscosity of monodisperse hard spheres.     

Effect of temperature on the effective thermal conductivity of n-tetradecane-based nanofluids containing copper nanoparticles

Nanofluids were prepared by dispersing Cu nanoparticles (∼20 nm) in n-tetradecane by a two-step method. The effective thermal conductivity was measured for various nanoparticle volume fractions (0.0001–0.02) and temperatures (306.22–452.66 K). The experimental data compares well with the Jang and Choi model. The thermal conductivity enhancement was lower above 391.06 K than for that between 306.22 and 360.77 K. The interfacial thermal resistance increased with increasing temperature. The effective thermal conductivity enhancement was greater than that obtained with a more viscous fluid as the base media at 452.66 K because of nanoconvection induced by nanoparticle Brownian motion at high temperature.     

Effect of temperature on the effective thermal conductivity of n-tetradecane-based nanofluids containing copper nanoparticles

Nanofluids were prepared by dispersing Cu nanoparticles(~20nm) in n-tetradecane by a two-step method.The effective thermal conductivity was measured for various nanoparticle volume fractions(0.0001-0.02) and temperatures(306.22-452.66 K).The experimental data compares well with the Jang and Choi model.The thermal conductivity enhancement was lower above 391.06 K than for that between306.22 and 360.77 K.The interfacial thermal resistance increased with increasing temperature.The effective thermal conductivity enhancement was greater than that obtained with a more viscous fluid as the base media at 452.66 K because of nanoconvection induced by nanoparticle Brownian motion at high temperature.     

Analytical design of effective thermal conductivity for fluid-saturated prismatic cellular metal honeycombs

A comparative optimal design of fluid-saturated prismatic cellular metal honeycombs(PCMHs) having different cell shapes is presented for thermal management applications. Based on the periodic topology of each PCMH, a unit cell(UC) for thermal transport analysis was selected to calculate its effective thermal conductivity. Without introducing any empirical coefficient, we modified and extended the analytical model of parallel–series thermal–electric network to a wider porosity range(0.7 ~ 0.98) by considering the effects of two-dimensional local heat conduction in solid ligaments inside each UC. Good agreement was achieved between analytical predictions and numerical simulations based on the method of finite volume. The concept of ligament heat conduction efficiency(LTCE) was proposed to physically explain the mechanisms underlying the effects of ligament configuration on effective thermal conductivity(ETC).Based upon the proposed theory, a construct strategy was developed for designing the ETC by altering the equivalent interaction angle with the direction of heat flow: relatively small average interaction angle for thermal conduction and relatively large one for thermal insulation.