The non-Newtonian flow of dilute aqueous polyethylene oxide (PEO) solutions through micro-fabricated planar abrupt contraction-expansions is investigated. The small lengthscales and high deformation rates in the contraction throat lead to significant extensional flow effects even with dilute polymer solutions having time constants on the order of milliseconds. By considering the definition of the elasticity number, El = Wi/Re, we show that the lengthscale of the geometry is key to the generation of strong viscoelastic effects, such that the same flow behaviour cannot be reproduced using the equivalent macro-scale geometry using the same fluid. We observe significant vortex growth upstream of the contraction plane, which is accompanied by an increase of more than 200% in the dimensionless extra pressure drop across the contraction. Streak photography and video-microscopy using epifluorescent particles shows that the flow ultimately becomes unstable and three-dimensional. The moderate Reynolds numbers (0.44 ≤ Re ≤ 64) associated with these high Weissenberg number (0 ≤ Wi ≤ 548) micro-fluidic flows results in the exploration of new regions of the Re-Wi parameter space in which the effects of both elasticity and inertia can be observed. Understanding such interactions will be increasingly important in micro-fluidic applications involving complex fluids and can best be interpreted in terms of the elasticity number, El = Wi/Re, which is independent of the flow kinematics and depends only on the fluid rheology and the characteristic size of the device.

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The inertio-elastic planar entry flow of low-viscosity elastic fluids in micro-fabricated geometries

We present a new dimensionless criterion that can be used to characterize and unify the critical conditions required for onset of purely elastic instabilities in a wide range of different flow geometries. This scaling incorporates both the presence of non-zero elastic normal stresses in the fluid plus the magnitude of the streamline curvature in the flow, and it can be thought of as the viscoelastic complement of the Gortler number. We present detailed experimental and theoretical evidence that justifies and generalizes the form of the dimensionless criterion. We show how this criterion naturally arises from the linearized stability equations governing the viscoelastic flow and apply it to analytical and experimental results in a number of standard benchmark problems. In geometrically simple flows (e.g. torsional flows such as those in a circular Couette cell or a cone-and-plate rheometer) a characteristic radius of curvature of the streamlines may be readily identified and an analytical solution for the undisturbed base flow can be found. However, in the more complex flows characteristic of those found in commercial polymer processing operations, the base flow must typically be determined numerically and the streamline curvature varies in a complex manner throughout the flow. In the former case, we show how our scaling reduces to well-established results in the literature and for the latter case we present a particularly simple approach for understanding and quantifying the sensitivity of the critical conditions for onset of elastic instability to dimensionless geometric design parameters such as the aspect ratio of the test cell. The generality of the scaling is confirmed by applying it to new experimental measurements in a lid-driven cavity and numerical linear stability calculations for flow past a cylinder in a channel. We also show how the scaling may be generalized to incorporate, at least qualitatively, the variation in the critical conditions with other rheological parameters such as changes in the solvent viscosity, shear-thinning in the viscometric functions, a spectrum of relaxation times and a non-zero second normal stress coefficient. In a number of cases, these modifications and the predicted scaling of the critical onset conditions for purely elastic instabilities in other complex geometries, such as planar contractions or eccentric rotating cylinders, remain to be confirmed by future experiments or calculations.

Rheological and geometric scaling of purely elastic flow instabilities

We explore the interplay of fluid inertia and fluid elasticity in planar entry flows by studying the flow of weakly elastic solutions through micro-fabricated planar contraction geometries. The small characteristic lengthscales make it possible to achieve a wide range of Weissenberg numbers (0.4 < Wi < 42) and Reynolds numbers (0.03 < Re < 12), allowing access to a large region of Wi-Re space that is typically unattainable in conventional macroscale entry flow experiments. Experiments are carried out using a series of dilute solutions (0.78 < clc* < 1.09) of a high molecular weight polyethylene oxide, in which the solvent viscosity is varied in order to achieve a range of elasticity numbers, 2.8 < El = WilRe < 68. Fluorescent streak imaging and micro-particle image velocimetry ([L-PIV) are used to characterize the kinematics, which are classified into a number of flow regimes including Newtonian-like flow at low Wi, steady viscoelastic flow, unsteady diverging flow and vortex growth regimes. Progressive changes in the centreline velocity profilt are used to identify each of the flow regimes and to map the resulting stability boundaries in Wi-Re space. The same flow transitions can also be detected through measurements of the enhanced pressure drop across the contraction/expansion which arise from fluid viscoelasticity. The results of this work have significant design implications for lab-on-a-chip devices, which commonly contain complex geometric features and transport complex fluids, such as those containing DNA or proteins. The results also illustrate the potential for using micro-fabricated devices as rheometric tools for measuring the extensional properties of weakly elastic fluids. (C) 2007 Elsevier B.V. All rights reserved.

Role of the elasticity number in the entry flow of dilute polymer solutions in micro-fabricated contraction geometries

Chapter 7 – On the Motion of a Rigid Body in a Viscous Liquid: A Mathematical Analysis with Applications

Laser Doppler velocimetry (LDV) and video flow visualization are used to investigate the creeping motion of a highly elastic, constant-viscosity fluid flowing past a cylinder mounted centrally in a rectangular channel. A sequence of viscoelastic flow transitions are documented as the volumetric flow rate past the cylinder is increased and elastic effects in the fluid become increasingly important. Velocity profiles clearly show that elasticity has almost no effect on the kinematics upstream of the cylinder, but that the streamlines in the wake of the cylinder are gradually shifted further downstream . Finite element calculations with a nonlinear constitutive model closely reproduce the evolution of the steady two-dimensional velocity field. However, at a well defined set of flow conditions the steady planar stagnation ow in the downstream wake is experimentally observed to become unstable to a steady, three-dimensional cellular structure. The Reynolds number at the onset of the flow instability is less than 0.05 and inertia plays little role in the flow transition, LDV measurements in the wake close to the cylinder reveal large spatially periodic fluctuations of the streamwise velocity that extend along the length of the cylinder and more than five cylinder radii downstream of the cylinder. Fourier analysis shows that the characteristic spatial wavelength of these flow perturbations scales closely with the cylinder radius R . Flow visualization combined with LDV measurements also indicates that the perturbations in the velocity field are confined to the narrow region of strongly extensional flow near the downstream stagnation point. A second flow transition is observed at higher flow rates that leads to steady translation of the cellular structure along the length of the cylinder and time-dependent velocity oscillations in the wake. Measurements of the flow instability are presented for a range of cylinder sizes, and a stability diagram is constructed which shows that the onset point of the wake instability depends on both the extensional deformation of the fluid in the stagnation flow and the shearing flow between the cylinder and the channel.

The wake instability in viscoelastic flow past confined circular cylinders

The motion of a slender body falling in quiescent polymer solutions is investigated experimentally. It represents the simplest model of motion of single fibers in the flow of fiber suspensions. The fall behavior in quiescent polymer solutions is compared with that in water. It is demonstrated that a slender body falling in Newtonian liquids rotates to adopt a horizontal orientation, whereas in non-Newtonian liquids it rotates towards a vertical orientation but for less concentrated solutions is not able to reach the vertical orientation and moves sideways with a constant orientation angle. The effects of shear thinning and elasticity on the motion of the body are discussed.

Motion of a slender body in quiescent polymer solutions

The structure and origin of the so-called bundle-like streams which are generated near the salient corners in unstable planar entry flows of rather dilute polyacrylamide solutions have been studied. The flow visualization experiments using the dye-solution injection technique clearly show that a bundle consists of two counterrotating vortex tubes, i.e. Goertler vortices whose cross-section is almost rectangular.

The structure of anomalous entry flow patterns through a planar contraction

The hydrodynamic force on an inclined circular cylinder of finite length in a uniform flow has been calculated numerically from the steady velocity field around the cylinder predicted in Part I, and the influence of shear thinning and elasticity on the hydrodynamic force and the attitude variation of a slender body has been discussed. The shear thinning behavior greatly reduces the fluid viscosity adjacent to the cylinder, in turn reduces drag force. In addition, an increase in viscosity compensates to some extent for a decrease in velocity gradient on the cylinder surface. Therefore, the moment acting to rotate the cylinder into a perpendicular orientation to the incoming flow decreases with an increase in shear thinning. While elasticity has little effect on shear stress, it strongly affects normal stress and the drag force due to the normal stress becomes larger as elasticity increases. This elastic effect is especially remarkable near both ends of the cylinder. When only shear thinning is taken into account, the moment acts in such a way as to rotate the cylinder into a perpendicular orientation to the flow. In contrast to this result, the moment acting to rotate the cylinder into a parallel orientation to the flow can be predicted when not only shear thinning but also elasticity are taken into account. Thus, the present numerical analysis for the hydrodynamic force on an inclined cylinder is valid to explain the mechanism of the attitude variation of a slender body falling in polymer solutions.

Numerical solution for the flow of viscoelastic fluids around an inclined circular cylinder.

The objective of this study is to well understand the migration and orientation of thin micro-particles in a suspension flow by means of both experiments and numerical simulations. In this framework, the evolution of the orientation of thin micro-particles such as talc and mica, which were modeled by a disk-like particle, in a flow through a slit channel was analyzed to obtain the knowledge of the processing operations of thin micro-particle reinforced composites.The thin disk-like particles were subjected by a planar extensional flow in a reservoir, then by a simple shear flow through a slit channel in the experimental apparatus used. The evolution of the orientation of thin disk-like particles was, therefore, studied in both a planar extensional and simple shear flows by numerical calculation of the Jeffery equation: thin disk-like particles aligned in a parallel orientation to upper- and lower-walls of the slit channel in a planar extensional flow through the reservoir, then entered into the inlet of the slit channel. On the other hand, in a simple shear flow through the slit channel, the disk-like particles kept this parallel orientation except the occurrence of a flip-over phenomenon. The period of the flip-over became longer with a decrease in the aspect ratio of the disk-like particles.Furthermore, the measurements of the orientation of the talc particles in a suspension flow through the slit channel clearly showed that almost the same period of the flip-over was found although the particle size was different. These experimental results arise from complex geometries and no accurate data of the thickness of the talc particles.

Orientation of Disk-like Particles in a Microcomposite Processing

The orientation of fibers has been investigated experimentally in fiber suspension flows of both Newtonian and viscoelastic fluids through a parallel plate channel and a planar channel with an abrupt contraction, and the development of the fiber orientation has been examined while the fibers flow along the channel. The effects of shear thinning and elasticity on the fiber orientation have been discussed.The fibers lie tangent to the streamlines with an increase in viscoelasticity in the flow through a parallel plate channel and a channel with an abrupt contraction. This phenomenon is essentially equivalent to the phenomenon that a slender body, namely a straight circular cylinder with a large length to diameter ratio, rotates towards a vertical orientation when falling through a quiescent polymer solution. High fiber alignment to the streamlines, thus, is attributed to the shear thinning and elsticity of the suspending fluids; however, elastic effect is most dominant.The state of fiber orientation is strongly influenced by the drastic change of the velocity field due to viscoelasticity and flow rate: the fibers align in a manner of wine-glass shape as the contraction is approached and then the distribution of fibers abruptly expands in the immediate downstream region of the contraction in a highly elastic fluid flow.

Kunji Chiba

Papers

Motion of a slender body in quiescent polymer solutions

The structure of anomalous entry flow patterns through a planar contraction

Numerical solution for the flow of viscoelastic fluids around an inclined circular cylinder.

Orientation of Disk-like Particles in a Microcomposite Processing

Fiber orientation in Fiber Suspension Flow through a Parallel Plate Channel and an Abrupt Contraction Channel