Comparison of a quasi-Newtonian fluid with a viscoelastic fluid in planar contraction flow

The flow of a 5.0 wt.% solution of polyisobutylene in tetradecane through a planar 4 : 1 contraction exhibiting a shear thinning viscosity is simulated using the flow-type sensitive quasi-Newtonian fluid model. The shear viscosity is fitted by the Giesekus model, which, with the chosen parameters, leads to an extension thickening elongational viscosity. The stress and velocity fields of the numerical simulations are compared with the experimental results of Quinzani et al. [J. Non-Newtonian Fluid Mech. 52 (1994) 1–36] and the numerical results of the viscoelastic simulation using the Giesekus model of Azaiez et al. [J. Non-Newtonian Fluid Mech. 62 (1996) 253–277]. It can be shown that the quasi-Newtonian fluid qualitatively predicts the essential features of the flow in the vicinity of the contraction.     

Numerical simulation of entry flow of the IUPAC-LDPE melt

Numerical simulations have been undertaken for the creeping entry flow of a well-characterized polymer melt (IUPAC-LDPE) in a 4:1 axisymmetric and a 14:1 planar contraction. The fluid has been modeled using an integral constitutive equation of the K-BKZ type with a spectrum of relaxation times (Papanastasiou–Scriven–Macosko or PSM model). Numerical values for the constants appearing in the equation have been obtained from fitting shear viscosity and normal stress data as measured in shear and elongational data from uniaxial elongation experiments. The numerical solutions show that in the axisymmetric contraction the vortex in the reservoir first increases with increasing flow rate (or apparent shear rate), goes through a maximum and then decreases following the behavior of the uniaxial elongational viscosity. For the planar contraction, the vortex diminishes monotonically with increasing flow rate following the planar extensional viscosity. This kinematic behavior is not in agreement with recent experiments. The PSM strain-memory function of the model is then modified to account for strain-hardening in planar extension. Then the vortex pattern shows an increase in both axisymmetric and planar flows. The results for planar flow are compared with recent experiments showing the correct trend.     

A 1-D theory for extensional deformation of a viscoelastic filament under exponential stretching

We consider a viscoelastic filament placed between two coaxial discs, with the bottom plate fixed and the top plate pulled at an exponential rate. Using a slender rod approximation, we derive a one-dimensional (1-D) model which describes the deformation of a viscoelastic filament governed by the Oldroyd-B constitutive model. It is assumed that the flow is axisymmetric and that inertia and gravity are negligible. One solution of the model equations corresponds to ideal uniaxial elongation. A linear stability analysis shows that this solution is unstable for a Newtonian fluid and for viscoelastic filaments with small Deborah number (De ≤ 0.5). For Deborah number greater than 0.5, ideal uniaxial elongation is linearly stable. Numerical solution of the nonlinear equations confirms the result of the linear stability analysis. For initial conditions close to ideal uniaxial flow, our results show that if De > 0.5, the central portion of the filament undergoes considerable strain hardening. As a result, the sample remains almost cylindrical and the deformation approaches pure uniaxial extension as the Hencky strain increases. For De ≤ 0.5, the Trouton ratio based on the effective extension rate at the mid-plane radius gives a much better approximation to the true extensional viscosity than that based on the imposed stretch rate.     

Positive frictional pressure gradient in vertical gas-high viscosity oil slug flow

The present study describes the wall shear stress and the falling liquid film behavior in upward vertical slug flow of air and high viscosity oil. The frictional pressure gradient is directly related to the wall shear stress, and it is usually negative (opposite to the overall flow direction). However, in vertical slug flow, the average total wall shear stress of a slug unit may be negative (in the same direction of the overall flow), resulting in a positive frictional pressure gradient. However, this does not mean, by any way, generation of additional energy or violation of the second law of thermodynamics.The positive frictional pressure gradient phenomenon, reasons and required conditions were explained in this paper. A simplified model was developed and validated against recent experimental data of air-high viscosity oil slug flow in a 50.8 mm ID vertical pipe. The oil viscosity was in the range of 127 mPa s to 580 mPa s. Positive frictional pressure gradient appears when the liquid film wall shear stress supersede the wall shear stress in the slug body. The rate of increase of both wall shear stresses (with respect to the mixture Reynolds number) depend, not only, on the mixture Reynolds number but also, highly, on the liquid viscosity.     

Flow of dilute to semi-dilute polystyrene solutions through a benchmark 8:1 planar abrupt micro-contraction

We have studied the flow of thermodynamically ideal solutions of a high molecular weight (M_w = 6.9 MDa) atactic polystyrene in the θ solvent dioctyl phthalate (aPS in DOP) through a micro-fabricated 8:1 planar abrupt contraction geometry. The channel is much deeper than most micro-scale geometries, providing an aspect ratio of 16:1 and a good approximation to 2D flow in the narrow channel. The solutions span a range of concentration 0.03 wt.% < c < 0.6 wt.%, encompassing the dilute to semi-dilute regimes and providing a range of fluid viscosities and relaxation times such that we achieve a range of Weissenberg numbers (8.7 < Wi < 1562) and Reynolds numbers (0.07 < Re < 11.2), giving elasticity numbers between 31 < El < 295. We study the flow using a combination of micro-particle image velocimetry (μ-PIV) to characterize the flow field, pressure measurements to evaluate the non-Newtonian viscosity, and birefringence measurements to assess macromolecular strain. Flow field observations reveal three broad flow regimes characterized by Newtonian-like flow, unstable flow and vortex growth in the upstream salient corners. Transitions between the flow regimes scale with Wi, independent of El, indicating the dominance of elastic over inertial effects in all the fluids. Transitions in the flow field are also reflected by transitions in the relative viscosity (determined from the pressure drop) and the macromolecular strain (determined from birefringence measurements). The strain through the 8:1 contraction saturates at a value of ～4–5 at high Wi. The results of these experiments broaden the limited set of literature on flow through micro-fluidic planar contractions and should be of significant value for optimizing lab-on-a-chip design and for comparison with modeling studies with elasticity dominated fluids.     

3D particle image velocimetry of the flow field around a sphere sedimenting near a wall: Part 1. Effects of Weissenberg number

The flow fields surrounding a sphere sedimenting through a liquid near a vertical wall are characterized using 3D stereoscopic particle-image velocimetry (PIV) experiments. Three different fluids, a Newtonian reference fluid, a constant (shear) viscosity Boger fluid, and a shear-thinning elastic fluid, are used to determine the effects of both elasticity and shear-thinning on the flow field. All three fluids have similar zero shear viscosities. The Weissenberg number is manipulated by varying the diameter and the composition of the ball. Significant differences are found for the different types of fluid, demonstrating both the influence of elasticity and shear-thinning on the velocity fields. In addition, the impact of the wall on the flow field is qualitatively different for each fluid. We find that the flow behind the sphere is strongly dependent on the fluid properties as well as the elasticity. Also, the presence of a negative wake is found for the shear-thinning fluid at high Weissenberg number (Wi > 1).     

An unsymmetric constitutive equation for anisotropic viscoelastic fluid

A continuum constitutive theory of corotational derivative type is developed for the anisotropic viscoelastic fluid–liquid crystalline (LC) polymers. A concept of anisotropic viscoelastic simple fluid is introduced. The stress tensor instead of the velocity gradient tensor D in the classic Leslie–Ericksen theory is described by the first Rivlin–Ericksen tensor A and a spin tensor W measured with respect to a co-rotational coordinate system. A model LCP-H on this theory is proposed and the characteristic unsymmetric behaviour of the shear stress is predicted for LC polymer liquids. Two shear stresses thereby in shear flow of LC polymer liquids lead to internal vortex flow and rotational flow. The conclusion could be of theoretical meaning for the modern liquid crystalline display technology. By using the equation, extrusion–extensional flows of the fluid are studied for fiber spinning of LC polymer melts, the elongational viscosity vs. extension rate with variation of shear rate is given in figures. A considerable increase of elongational viscosity and bifurcation behaviour are observed when the orientational motion of the director vector is considered. The contraction of extrudate of LC polymer melts is caused by the high elongational viscosity. For anisotropic viscoelastic fluids, an important advance has been made in the investigation on the constitutive equation on the basis of which a series of new anisotropic non-Newtonian fluid problems can be addressed. The project supported by the National Natural Science Foundation of China (10372100, 19832050) (Key project). The English text was polished by Yunming Chen.     

Numerical simulation of the fountain flow instability in injection molding

For the first time, the viscoelastic flow front instability is studied in the full non-linear regime by numerical simulation. A two-component viscoelastic numerical model is developed which can predict fountain flow behavior in a two-dimensional cavity. The eXtended Pom-Pom (XPP) viscoelastic model is used. The levelset method is used for modeling the two-component flow of polymer and gas. The difficulties arising from the three-phase contact point modeling are addressed, and solved by treating the wall as an interface and the gas as a compressible fluid with a low viscosity. The resulting set of equations is solved in a decoupled way using a finite element formulation. Since the model for the polymer does not contain a solvent viscosity, the time discretized evolution equation for the conformation tensor is substituted into the momentum balance in order to obtain a Stokes like equation for computing the velocity and pressure at the new time level. Weissenberg numbers range from 0.1 to 10. The simulations reveal a symmetric fountain flow for Wi = 0.1–5. For Wi = 10 however, an oscillating motion of the fountain flow is found with a spatial period of three times the channel height, which corresponds to experimental observations.     

Extensional rheology of a shear-thickening cornstarch and water suspension

A filament-stretching rheometer is used to measure the extensional viscosity of a shear-thickening suspension of cornstarch in water. The experiments are performed at a concentration of 55 wt.%. The shear rheology of these suspensions demonstrates a strong shear-thickening behavior. The extensional rheology of the suspensions demonstrates a Newtonian response at low extension rates. At moderate strain rates, the fluid strain hardens. The speed of the strain hardening and the extensional viscosity achieved increase quickly with increasing extension rate. Above a critical extension rate, the extensional viscosity goes through a maximum and the fluid filaments fail through a brittle fracture at a constant tensile stress. The glassy response of the suspension is likely the result of jamming of particles or clusters of particles at these high extension rates. This same mechanism is responsible for the shear thickening of these suspensions. In capillary breakup extensional rheometry, measurement of these suspensions demonstrates a divergence in the extensional viscosity as the fluid stops draining after a modest strain is accumulated.     

Smoothed particle hydrodynamics simulation of non-Newtonian moulding flow

Smoothed particle hydrodynamics (SPH) has been widely applied in simulating fluid flow because of its attractive properties, for example, it is fully Lagrangian and mesh free. However, this method usually uses the explicit method to solve the conservation equations and in this form it is only suitable to momentum dominated flows with low viscosity. In polymer processing, the fluid is non-Newtonian with high viscosity, O(10³) to O(10⁴) Pa-s say, and the pressure is high as O(10⁶) to O(10¹⁰) Pa. The algorithm of the standard SPH is infeasible in this case, because only very small time steps can be used for a stable simulation. We have developed an implicit SPH for non-Newtonian flow, which is completely matrix free, to solve the equation system iteratively and robustly. The artificial pressure is introduced between particles to stabilize the SPH system avoiding the tensile instability. The fluid is compressible under high pressure. Realistic state equations for polymers, such as the Tait and SSY [16] equations are adopted to describe the density/pressure relations. The method is finally applied to the simulation of moulding flow of a modified power law fluid with the zero shear rate viscosity of 1.22 × 10⁴ Pa-s, Reynolds number of 3 × 10^?4 to 6 × 10^?5 and the highest pressure of O(10⁸) to O(10¹⁰) Pa.     

Non-Newtonian flow in microporous structures under the electroviscous effect

Single phase non-Newtonian microporous flow combined with the electroviscous effect is investigated in the pore-scale under conditions of various rheological properties and electrokinetic parameters. The lattice Boltzmann method is employed to solve both the electric potential field and flow velocity field. The simulation of commonly used power-law non-Newtonian flow shows that the electroviscous effect on the flow depends on both the fluid rheological behavior and pore surface area ratio significantly. For the shear thinning fluid with power-law exponent n < 1, the fluid viscosity near the wall is smaller and the electrovicous effect plays a more important role compared to the Newtonian fluid and shear thickening fluid. The high pore surface area ratio in the porous structure increases the electroviscous force and the induced flow resistance becomes important even to the flow of Newtonian and shear thickening fluids.     

Electroviscous effect on non-Newtonian fluid flow in microchannels

Understanding non-Newtonian flow in microchannels is of both fundamental and practical significance for various microfluidic devices. A numerical study of non-Newtonian flow in microchannels combined with electroviscous effect has been conducted. The electric potential in the electroviscous force term is calculated by solving a lattice Boltzmann equation. And another lattice Boltzmann equation without derivations of the velocity when calculating the shear is employed to obtain flow field. The simulation of commonly used power-law non-Newtonian flow shows that the electroviscous effect on the flow depends significantly on the fluid rheological behavior. For the shear thinning fluid of the power-law exponent n < 1, the fluid viscosity near the wall is smaller and the electroviscous effect plays a more important role. And its effect on the flow increases as the ratio of the Debye length to the channel height increases and the exponent n decreases. While the shear thickening fluid of n > 1 is less affected by the electroviscous force, it can be neglected in practical applications.     

Rheology of liquid metals and alloys

The flow behavior and viscosity of liquid Zn, Sn, Cd, Bi-42 wt%Sn, Zn-7 wt%Al, and Sn-3 wt%Ag-0.5 wt%Cu were characterized and quantified with rotational rheometry experiments. Evidences from this study shows these liquid systems uniquely exhibit a shear thinning and time-independent (non-thixotropic) flow behavior in all the evaluated shear rate regimes. We have attempted to offer a physical explanation from prior-art for the observed unique flow behavior of the liquid metal systems. The strong short range atomic order in these metals significantly contribute to their flow behavior and at any shear rate the viscosity obeys the standard Arrhenius energy equation for temperature dependence.     

Numerical simulations of Boger fluids through different contraction configurations for the development of a measuring system for extensional viscosity

This paper reports the flow behaviour of Newtonian and Boger fluids through various axisymmetric contraction configurations by means of numerical predictions. A principal aim has been to evaluate the geometrical design choice of the hyperbolic contraction flow. The FENE-CR model has been used to reflect the behaviour of Boger fluids, with constant shear viscosity, finite (yet large) extensional viscosity and less than quadratic first normal stress difference. Numerical calculations have been performed on six different contraction configurations to evaluate an optimized geometry for measuring extensional viscosity in uniaxial extensional flow. The influence of a sharp or rounded recess-corner on the nozzle has also been investigated. Few commercial measuring systems are currently available for measurement of the extensional rheology of medium-viscosity fluids, such as foods and other biological systems. In this context, a technique based on the hyperbolic contraction flow would be a suitable alternative. The pressure drop, the velocity field, the first normal stress difference and the strain rate across the geometry have each been evaluated for Newtonian and Boger fluids. This numerical study has shown that the hyperbolic configuration is superior to the other geometry choices in achieving a constant extension rate. In this hyperbolic configuration, no vortices are formed, the measuring range is broader and the strain rate is constant throughout the geometric domain, unlike in the alternative configurations tested. The difference between sharp and rounded recess-corner configurations proved to be negligible and a rise in excess pressure drop (epd) for increasing deformation rates has been observed.     

Influence of sudden contractions on in situ volume fractions for oil–water flows in horizontal pipes

Oil–water two-phase flow experiments were conducted in horizontal ducts made of Plexiglas® to determine the in situ oil fraction (holdup) by means of the closing valves technique, using mineral oil (viscosity: 0.838 Pa s at 20 °C; density: 890 kg m⁻³) and tap water. The ducts present sudden contractions from 50 mm to 40 mm i.d. and from 50 mm to 30 mm i.d., with contraction ratios of 0.64 and 0.36, respectively. About 200–320 tests were performed by varying the flow rates of the phases. Flow patterns were investigated for both the up- and downstream pipe in order to assess whether relevant variations of the flow patterns across the sudden contraction take place. Data were then compared with predictions of a specific correlation for oil–water flow and some correlations for gas–water flow. A drift-flux model was also applied to determine the distribution parameter.     

Flow pattern and molecular visualization of DNA solutions through a 4:1 planar micro-contraction

The present work explores unusual flow behavior of entangled fluids in an abrupt contraction flow device. Fluorescent imaging was carried out on four different entangled DNA solutions with concentrations ranging from 0.1 to 1.0% (with a wide range of entanglements per chain Z = 7–55). For weakly entangled solutions (Z < 30), vortex flow was dominant at high flow rates. However, for well-entangled DNA solutions (Z ≥ 30), unusual time dependant shear banding was observed at the contraction entrance. Upon reducing the slip length by adding sucrose to the well-entangled DNA solution, vortex flow became dominant again. In vortex flow, most DNA chains remained coiled at the corner in regular recirculation. However, when jerky-shear-banding flow developed, significant stable stretching of DNA chains occurred at the center-line, with quasi-periodic switching between stretching and recoil at the corner.     

Extracting extensional properties through excess pressure drop estimation in axisymmetric contraction and expansion flows for constant shear viscosity,extension strain-hardening fluids

In this study, hyperbolic contraction–expansion flow (HCF) devices have been investigated with the specific aim of devising new experimental measuring systems for extensional rheological properties. To this end, a hyperbolic contraction–expansion configuration has been designed to minimize the influence of shear in the flow. Experiments have been conducted using well-characterized model fluids, alongside simulations using a viscoelastic White–Metzner/FENE-CR model and finite element/finite volume analysis. Here, the application of appropriate rheological models to reproduce quantitative pressure drop predictions for constant shear viscosity fluids has been investigated, in order to extract the relevant extensional properties for the various test fluids in question. Accordingly, experimental evaluation of the hyperbolic contraction–expansion configuration has shown rising corrected pressure drops with increasing elastic behaviour (De=0～16), evidence which has been corroborated through numerical prediction. Moreover, theoretical to predicted solution correspondence has been established between extensional viscosity and first normal stress difference. This leads to a practical means to measure extensional viscosity for elastic fluids, obtained through the derived pressure drop data in these HCF devices.     

Numerical investigation of transient contraction flows for worm-like micellar systems using Bautista–Manero models

This study is concerned with the numerical modelling of the Modified Bautista–Manero (MBM) model, for both steady-state and transient solutions in planar 4:1 contraction flow. This model was proposed to represent the structured composition and behaviour of worm-like micellar systems which have importance in industrial oil-reservoir recovery applications. A parameter sensitivity analysis for the rheology of this model is presented in both transient and steady response, covering pure shear and uniaxial extension. In addition, some features in evolutionary flow-structure are demonstrated in contraction flows due to the influence and imposition of start-up transient boundary conditions. The different effects of various model parameter choices are described through transient field response, stress and viscosity fields in the contraction flow setting. Distinction may be drawn between fluid response in the strong/moderate extension hardening regimes by matching both steady-state and transient shear and extensional viscosity peaks, contrasting between micellar (MBM) models against network-based counterparts Phan-Thien/Tanner (PTT). Simulations are performed with a hybrid finite volume/element algorithm. The momentum and continuity equations are solved by a Taylor–Galerkin/pressure-correction finite element method, whilst the constitutive equation is dealt with by a cell-vertex finite volume algorithm.     

Analysis of an organized turbulent structure using a pattern recognition technique in a drag-reducing surfactant solution flow

This work presents the investigation for an organized turbulent structure in a drag-reducing flow of dilute surfactant solution by utilizing a particle image velocimetry system to perform the pattern recognition technique on a trajectory in four quadrants of streamwise and wall-normal velocity fluctuations. The pattern recognition is added to a new algorithm in order to directly capture the spatial rotation motion. The Reynolds number based on the channel height and bulk mean velocity was set to 1.5 × 10⁴. Surfactant solution with a weight concentration of 150 ppm was employed and the drag reduction rate was 65%. In the drag-reducing flow, we observe increased frequencies of occurrence of the flow events that correspond to a meandering motion in the wall-normal direction of the high-and low-speed regions. Three findings from investigation of the ensemble-averaged Reynolds shear stress and vortex structure are as follows: (i) the Reynolds shear stress in the large fluctuation range occurs in the narrow region; (ii) Size, strength, arrangement and inclination in the spatial vortex structure in the drag-reducing flow differ from those of the water; and (iii) all trajectory contributions for the wall friction coefficient decrease. Finally, we interpreted that the viscoelasticity characterizing the viscoelastic stress and relaxation time in rheological properties of the flow changes specific elementary vortex for the drag-reducing flow, and the trajectories of each flow pattern change drastically.