Dense shallow granular flows

Simplified equations are derived for a granular flow in the `dense' limit where the volume fraction is close to that for dynamical arrest, and the `shallow' limit where the stream-wise length for flow development (L) is large compared with the cross-stream height (h). The mass and diameter of the particles are set equal to 1 in the analysis without loss of generality. In the dense limit, the equations are simplified by taking advantage of the power-law divergence of the pair distribution function chi proportional to (phi(ad) - phi)(-alpha), and a faster divergence of the derivativ rho(d chi/d rho) similar to (d chi/d phi), where rho and phi are the density and volume fraction, and phi(ad) is the volume fraction for arrested dynamics. When the height h is much larger than the conduction length, the energy equation reduces to an algebraic balance between the rates of production and dissipation of energy, and the stress is proportional to the square of the strain rate (Bagnold law). In the shallow limit, the stress reduces to a simplified Bagnold stress, where all components of the stress are proportional to (partial derivative u(x)/partial derivative y)(2), which is the cross-stream (y) derivative of the stream-wise (x) velocity. In the simplified equations for dense shallow flows, the inertial terms are neglected in the y momentum equation in the shallow limit because the are O(h/L) smaller than the divergence of the stress. The resulting model contains two equations, a mass conservation equations which reduces to a solenoidal condition on the velocity in the incompressible limit, and a stream-wise momentum equation which contains just one parameter B which is a combination of the Bagnold coefficients and their derivatives with respect to volume fraction. The leading-order dense shallow flow equations, as well as the first correction due to density variations, are analysed for two representative flows. The first is the development from a plug flow to a fully developed Bagnold profile for the flow down an inclined plane. The analysis shows that the flow development length is ((rho) over barh(3)/B) , where (rho) over bar is the mean density, and this length is numerically estimated from previous simulation results. The second example is the development of the boundary layer at the base of the flow when a plug flow (with a slip condition at the base) encounters a rough base, in the limit where the momentum boundary layer thickness is small compared with the flow height. Analytical solutions can be found only when the stream-wise velocity far from the surface varies as x(F), where x is the stream-wise distance from the start of the rough base and F is an exponent. The boundary layer thickness increases as (l(2)x)(1/3) for all values of F, where the length scale l = root 2B/(rho) over bar. The analysis reveals important differences between granular flows and the flows of Newtonian fluids. The Reynolds number (ratio of inertial and viscous terms) turns out to depend only on the layer height and Bagnold coefficients, and is independent of the flow velocity, because both the inertial terms in the conservation equations and the divergence of the stress depend on the square of the velocity/velocity gradients. The compressibility number (ratio of the variation in volume fraction and mean volume fraction) is independent of the flow velocity and layer height, and depends only on the volume fraction and Bagnold coefficients.

Wall-mode instability in plane shear flow of viscoelastic fluid over a deformable solid

The linear stability analysis of a plane Couette flow of an Oldroyd-B viscoelastic fluid past a flexible solid medium is carried out to investigate the role of polymer addition in the stability behavior. The system consists of a viscoelastic fluid layer of thickness R, density rho, viscosity eta, relaxation time lambda, and retardation time beta lambda flowing past a linear elastic solid medium of thickness HR, density rho, and shear modulus G. The emphasis is on the high-Reynolds-number wall-mode instability, which has recently been shown in experiments to destabilize the laminar flow of Newtonian fluids in soft-walled tubes and channels at a significantly lower Reynolds number than that for flows in rigid conduits. For Newtonian fluids, the linear stability studies have shown that the wall modes become unstable when flow Reynolds number exceeds a certain critical value Re c which scales as Sigma(3/4), where Reynolds number Re = rho VR/eta, V is the top-plate velocity, and dimensionless parameter Sigma = rho GR(2)/eta(2) characterizes the fluid-solid system. For high-Reynolds-number flow, the addition of polymer tends to decrease the critical Reynolds number in comparison to that for the Newtonian fluid, indicating a destabilizing role for fluid viscoelasticity. Numerical calculations show that the critical Reynolds number could be decreased by up to a factor of 10 by the addition of small amount of polymer. The critical Reynolds number follows the same scaling Re-c similar to Sigma(3/4) as the wall modes for a Newtonian fluid for very high Reynolds number. However, for moderate Reynolds number, there exists a narrow region in beta-H parametric space, corresponding to very dilute polymer solution (0.9 less than or similar to beta < 1) and thin solids (H less than or similar to 1.1), in which the addition of polymer tends to increase the critical Reynolds number in comparison to the Newtonian fluid. Thus, Reynolds number and polymer properties can be tailored to either increase or decrease the critical Reynolds number for unstable modes, thus providing an additional degree of control over the laminar-turbulent transition.

Design and Validation of On-chip Planar Mixer Based on Advection and Viscoelastic Effects

Mixing at low Reynolds number is usually due to diffusion and requires longer channel lengths for complete mixing. In order to reduce the mixing lengths, advective flow can be induced by varying the channel geometry. Additionally, in non-newtonian fluids, appropriate modifications to channel geometry can be used to aid the mixing process by capitalizing on their viscoelastic nature. Here we have exploited the advection and viscoelastic effects to implement a planar passive micro-mixer. Microfluidic devices incorporating different blend of mixing geometries were conceived. The optimum design was chosen based on the results of the numerical simulations performed in COMSOL. The chosen design had sudden expansion and contraction along with teeth patterns along the channel walls to improve mixing. Mixing of two different dyes was performed to validate the mixing efficiency. Particle dispersion experiments were also carried out. The results indicated effective mixing. In addition, the same design was also found to be compatible with electrical power free pumping mechanism like suction. The proposed design was then used to carry out on-chip chemical cell lysis with human whole blood samples to establish its use with non-newtonian fluids. Complete lysis of the erythrocytes was observed leaving behind the white blood cells at the outlet.

国内非牛顿流体力学进展

非牛顿流体;;本构方程;;流变测量;;流动稳定性;;收缩流动;;聚合物加工;;石油工业;;生物流体力学

Improved models for drop-on-demand ink-jet shortening

A previous 1-D model for the shortening of an unbroken drop-on-demand ink-jet ligament has been extended to the case of an arbitrary attached tail mass, and can also include extensional viscosity (which has ∼ 2% effect) as well as linear elasticity in the fluid. Predictions from the improved model are shown to be very similar to results from 2-D axisymmetric numerical simulations of DoD ink-jet ligaments and also to the results of recent experiments on Newtonian fluids jetted without satellite formation.

Self-similar breakup of near-inviscid liquids

The final stages of pinchoff and breakup of dripping droplets of near-inviscid Newtonian fluids are studied experimentally for pure water and ethanol. High-speed imaging and image analysis are used to determine the angle and the minimum neck size of the cone-shaped extrema of the ligaments attached to dripping droplets in the final microseconds before pinchoff. The angle is shown to steadily approach the value of 18.0 ±0.4, independently of the initial flow conditions or the type of breakup. The filament thins and necks following a τ2 /3 law in terms of the time remaining until pinchoff, regardless of the initial conditions. The observed behavior confirms theoretical predictions. © 2012 American Physical Society.

Self-similar breakup of near-inviscid liquids.

The final stages of pinchoff and breakup of dripping droplets of near-inviscid Newtonian fluids are studied experimentally for pure water and ethanol. High-speed imaging and image analysis are used to determine the angle and the minimum neck size of the cone-shaped extrema of the ligaments attached to dripping droplets in the final microseconds before pinchoff. The angle is shown to steadily approach the value of 18.0 ± 0.4°, independently of the initial flow conditions or the type of breakup. The filament thins and necks following a τ(2/3) law in terms of the time remaining until pinchoff, regardless of the initial conditions. The observed behavior confirms theoretical predictions.

Drop speeds from drop-on-demand ink-jet print heads

Measured drop speeds from a range of industrial drop-on-demand (DoD) ink-jet print head designs scale with the predictions of very simple physical models and results of numerical simulations. The main drop/jet speeds at a specified stand-off depend on fluid properties, nozzle exit diameter, and print head drive amplitude for fixed waveform timescales. Drop speeds from the Xaar, Spectra Dimatix, and MicroFab DoD print heads tested with (i) Newtonian, (ii) weakly elastic, and (iii) highly shear-thinning fluids all show a characteristic linear rise with drive voltage (setting) above an apparent threshold drive voltage. Jetting, simple modeling approaches, and numerical simulations of Newtonian fluids over the typical DoD printing range of surface tensions and viscosities were studied to determine how this threshold drive value and the slope of the characteristic linear rise depend on these fluid properties and nozzle exit area. The final speed is inversely proportional to the nozzle exit area, as expected from volume conservation. These results should assist specialist users in the development and optimization of DoD applications and print head design. For a given density, the drive threshold is determined primarily by viscosity, and the constant of proportionality k linking speed with drive above a drive threshold becomes independent of viscosity and surface tension for more viscous DoD fluid jetting. © 2013 Society for Imaging Science and Technology.

Inkjet printing of weakly elastic polymer solutions

Fluid assessment methods, requiring small volumes and avoiding the need for jetting, are particularly useful in the design of functional fluids for inkjet printing applications. With the increasing use of complex (rather than Newtonian) fluids for manufacturing, single frequency fluid characterisation cannot reliably predict good jetting behaviour, owing to the range of shearing and extensional flow rates involved. However, the scope of inkjet fluid assessments (beyond achievement of a nominal viscosity within the print head design specification) is usually focused on the final application rather than the jetting processes. The experimental demonstration of the clear insufficiency of such approaches shows that fluid jetting can readily discriminate between fluids assessed as having similar LVE characterisation (within a factor of 2) for typical commercial rheometer measurements at shearing rates reaching 104rads-1.Jetting behaviour of weakly elastic dilute linear polystyrene solutions, for molecular weights of 110-488. kDa, recorded using high speed video was compared with recent results from numerical modelling and capillary thinning studies of the same solutions.The jetting images show behaviour ranging from near-Newtonian to "beads-on-a-string". The inkjet printing behaviour does not correlate simply with the measured extensional relaxation times or Zimm times, but may be consistent with non-linear extensibility L and the production of fully extended polymer molecules in the thinning jet ligament.Fluid test methods allowing a more complete characterisation of NLVE parameters are needed to assess inkjet printing feasibility prior to directly jetting complex fluids. At the present time, directly jetting such fluids may prove to be the only alternative. © 2014 The Authors.

Investigation into the rheological properties of PES/NMP/nonsolvent membrane-forming systems

From the perspective of a polymer solution, the rheological properties of the popular polyethersulfone (PES)/N-methyl-2-pyrrolidone (NMP)/nonsolvent (NS) membrane-forming system were investigated thoroughly with a controlled stress rheometer (HAAKE RS75, Germany). The scope of the study included measurements of the controlled-stress flow curve, creep recovery, and dynamic oscillation. H2O, 1-butanol, ethylene glycol, and diethylene glycol were used as NS additives. The effects of the polymer concentration and the quality of the solvent mixture, as characterized by the approaching ratio, on the rheological behavior of the dopes were studied. Up to 38 wt % PES and extremely adjacent to the phase separation (i.e., the approaching ratio of the dope was 0.95), the viscous property dominated all the dopes, which behaved as Newtonian fluids. Moreover, all the membrane-forming dopes investigated were in the crossover regime in the semidilute region, in which the chains were overlapping but unentangled. (C) 2001 John Wiley & Sons, Inc.

Density-viscosity product of small-volume ionic liquid samples using quartz crystal impedance analysis

Quartz crystal impedance analysis has been developed as a technique to assess whether room-temperature ionic liquids are Newtonian fluids and as a small-volume method for determining the values of their viscosity-density product, rho eta. Changes in the impedance spectrum of a 5-MHz fundamental frequency quartz crystal induced by a water-miscible room-temperature ionic liquid, 1-butyl-3-methylimidazolium. trifluoromethylsulfonate ([C(4)mim][OTf]), were measured. From coupled frequency shift and bandwidth changes as the concentration was varied from 0 to 100% ionic liquid, it was determined that this liquid provided a Newtonian response. A second water-immiscible ionic liquid, 1-butyl-3-methylimidazolium bis(trifluoromethanesulfonyl)imide [C(4)mim][NTf2], with concentration varied using methanol, was tested and also found to provide a Newtonian response. In both cases, the values of the square root of the viscosity-density product deduced from the small-volume quartz crystal technique were consistent with those measured using a viscometer and density meter. The third harmonic of the crystal was found to provide the closest agreement between the two measurement methods; the pure ionic liquids had the largest difference of similar to 10%. In addition, 18 pure ionic liquids were tested, and for 11 of these, good-quality frequency shift and bandwidth data were obtained; these 12 all had a Newtonian response. The frequency shift of the third harmonic was found to vary linearly with square root of viscosity-density product of the pure ionic liquids up to a value of root(rho eta) approximate to 18 kg m(-2) s(-1/2), but with a slope 10% smaller than that predicted by the Kanazawa and Gordon equation. It is envisaged that the quartz crystal technique could be used in a high-throughput microfluidic system for characterizing ionic liquids.

Determination of the Physical Properties of Room Temperature Ionic Liquids Using a Love Wave Device

In this work, we have shown that a 100 MHz Love wave device can be used to determine whether room temperature ionic liquids (RTILs) are Newtonian fluids and have developed a technique that allows the determination of the density-viscosity product, rho eta of a Newtonian RTIL. In addition, a test for a Newtonian response was established by relating the phase change to insertion loss change. Five concentrations of a water-miscible RTIL and seven pure RTILs were measured. The changes in phase and insertion loss were found to vary linearly with the square root of the density-viscosity product for values up to (rho eta)(1/2) similar to 10 kg m(-2) s(-1/2). The square root of the density-viscosity product was deduced from the changes in either phase or insertion loss using glycerol as a calibration liquid. In both cases, the deduced values of rho eta agree well with those measured using viscosity and density meters. Miniaturization of the device, beyond that achievable with the lower-frequency quartz crystal microbalance approach, to measure smaller volumes is possible. The ability to fabricate Love wave and other surface acoustic wave sensors using planar metallization technologies gives potential for future integration into lab-on-a-chip analytical systems for characterizing ionic liquids.

Etude des instabilités d'interface du procédé de coextrusion

Ecole polytechnique de Montréal, département de mathématiques, André Fortin, et Pierre Carreau du département de génie chimique

Numerical study of 2D heat transfer in a scraped surface heat exchanger

A numerical study of fluid mechanics and heat transfer in a scraped surface heat exchanger with non-Newtonian power law fluids is undertaken. Numerical results are generated for 2D steady-state conditions using finite element methods. The effect of blade design and material properties, and especially the independent effects of shear thinning and heat thinning on the flow and heat transfer, are studied. The results show that the gaps at the root of the blades, where the blades are connected to the inner cylinder, remove the stagnation points, reduce the net force on the blades and shift the location of the central stagnation point. The shear thinning property of the fluid reduces the local viscous dissipation close to the singularity corners, i.e. near the tip of the blades, and as a result the local fluid temperature is regulated. The heat thinning effect is greatest for Newtonian fluids where the viscous dissipation and the local temperature are highest at the tip of the blades. Where comparison is possible, very good agreement is found between the numerical results and the available data. Aspects of scraped surface heat exchanger design are assessed in the light of the results. (C) 2003 Elsevier Ltd. All rights reserved.

FORCED CONVECTION TO LAMINAR FLOW OF LIQUID EGG YOLK IN CIRCULAR AND ANNULAR DUCTS

The steady-state heat transfer in laminar flow of liquid egg yolk - an important pseudoplastic fluid food - in circular and concentric annular ducts was experimentally investigated. The average convection heat transfer coefficients, determined by measuring temperatures before and after heating sections with constant temperatures at the tube wall, were used to obtain simple new empirical expressions to estimate the Nusselt numbers for fully established flows at the thermal entrance of the considered geometries. The comparisons with existing correlations for Newtonian and non-Newtonian fluids resulted in excellent agreement. The main contribution of this work is to supply practical and easily applicable correlations, which are, especially for the case of annulus, rather scarce and extensively required in the design of heat transfer operations dealing with similar shear-thinning products. In addition, the experimental results may support existing theoretical analyses.