The axisymmetric contraction–expansion: the role of extensional rheology on vortex growth dynamics and the enhanced pressure drop

The flow of a polystyrene Boger fluid through axisymmetric contraction–expansions having various contraction ratios (2≤β≤8) and varying degrees of re-entrant corner curvatures are studied experimentally over a large range of Deborah numbers. The ideal elastic fluid is dilute, monodisperse and well characterized in both shear and transient uniaxial extension. A large enhanced pressure drop above that of a Newtonian fluid is observed independent of contraction ratio and re-entrant corner curvature. Streak images, laser Doppler velocimetry (LDV) and digital particle image velocimetry (DPIV) are used to investigate the flow kinematics upstream of the contraction plane. LDV is used to measure velocity fluctuation in the mean flow field and to characterize a global elastic flow instability which occurs at large Deborah numbers. For a contraction ratio of β=2, a steady elastic lip vortex is observed while for contraction ratios of 4≤β≤8, no lip vortex is observed and a corner vortex is seen. Rounding the re-entrant corner leads to shifts in the onset of the flow transitions at larger Deborah numbers, but does not qualitatively change the overall structure of the flow field. We describe a simple rescaling of the deformation rate which incorporates the effects of lip curvature and allows measurements of vortex size, enhanced pressure drop and critical Deborah number for the onset of elastic instability to be collapsed onto master curves. Transient extensional rheology measurements are utilized to explain the significant differences in vortex growth pathways (i.e. elastic corner vortex versus lip vortex growth) observed between the polystyrene Boger fluids used in this research and polyisobutylene and polyacrylamide Boger fluids used in previous contraction flow experiments. We show that the role of contraction ratio on vortex growth dynamics can be rationalized by considering the dimensionless ratio of the elastic normal stress difference in steady shear flow to those in transient uniaxial extension. It appears that the differences in this normal stress ratio for different fluids at a given Deborah number arise from variations in solvent quality or excluded volume effects.     

Large Deborah number flows around confined microfluidic cylinders

Viscoelastic flow around a confined cylinder at high Deborah numbers is studied using microfluidic channels. By varying fluid properties and flow rates, a systematic study of the roles of elasticity and inertia is accomplished. Two new elastic flow instabilities that occur at high Deborah numbers are identified. A downstream instability of disordered and temporally varying streamlines is observed at a Deborah number above 10. This instability is a precursor to an unsteady vortex that develops upstream of the cylinder at higher Deborah numbers. Both instabilities occur at moderate Reynolds numbers but are fundamentally elastic. The size and steadiness of the upstream vortex are primarily controlled by the Deborah and the elasticity number.     

Three-dimensional flow of Newtonian and Boger fluids in square–square contractions

The flow of a Newtonian fluid and a Boger fluid through sudden square–square contractions was investigated experimentally aiming to characterize the flow and provide quantitative data for benchmarking in a complex three-dimensional flow. Visualizations of the flow patterns were undertaken using streak-line photography, detailed velocity field measurements were conducted using particle image velocimetry (PIV) and pressure drop measurements were performed in various geometries with different contraction ratios. For the Newtonian fluid, the experimental results are compared with numerical simulations performed using a finite volume method, and excellent agreement is found for the range of Reynolds number tested (Re₂ ≤ 23). For the viscoelastic case, recirculations are still present upstream of the contraction but we also observe other complex flow patterns that are dependent on contraction ratio (CR) and Deborah number (De₂) for the range of conditions studied: CR = 2.4, 4, 8, 12 and De₂ ≤ 150. For low contraction ratios strong divergent flow is observed upstream of the contraction, whereas for high contraction ratios there is no upstream divergent flow, except in the vicinity of the re-entrant corner where a localized atypical divergent flow is observed. For all contraction ratios studied, at sufficiently high Deborah numbers, strong elastic vortex enhancement upstream of the contraction is observed, which leads to the onset of a periodic complex flow at higher flow rates. The vortices observed under steady flow are not closed, and fluid elasticity was found to modify the flow direction within the recirculations as compared to that found for Newtonian fluids. The entry pressure drop, quantified using a Couette correction, was found to increase with the Deborah number for the higher contraction ratios.     

Effect of self-issuing jets along the span on the near-wake of a square cylinder

The study herein focuses on the vortex shedding characteristics and near-wake vorticity patterns of a square cylinder having self-issuing jets through holes along its span. Three different values of spacing between the consecutive holes λ with respect to the cylinder diameter D, i.e., λ/D = 1.5, 3 and 4 are studied experimentally via Digital Particle Image Velocimetry for the Reynolds number range extending from 200 to 1,000. It has been observed that the three-dimensionality of the wake flow depends on the spacing between the holes and Re number. For sufficiently low Reynolds numbers, the jet flows issuing from the holes yield a non-uniform distribution of mean flow characteristics like the shedding frequency and the formation length of vortices along the span of the cylinder when the spacing between jets along centerline is close to wavelength of the naturally existing three-dimensional wake instability. Additionally, for Re number up to 500, the self-issuing jets emanating from the holes show an indirect interaction with shear layers originating from upper and lower separation lines of the cylinder. However, for higher Re numbers of 750 and 1,000, they directly interact with and modify the vortices forming from the cylinder.     

Analysis of Taylor vortex flow by means of laser light scattering

Rheological measurements and light-scattering experiments were performed on dilute solutions of high molecular polystyrene. We are able to describe the orientation behavior of chain molecules under shear flow by means of light-scattering. Beyond that these investigations of light-scattering of flowing polymer solutions are an useful and suitable tool for detection and characterization of Taylor vortex formation. We can estimate the appearance of these hydrodynamic instabilities, which overlay the laminar main flow and we can observe a typical influence of the solvent power on it.Presented in part at the meeting of the Deutsche Rheologische Gesellschaft, Berlin, 13–15 May, 1991.     

A new mixed finite element for calculating viscoelastic flow

A new mixed finite element has allowed us to calculate flows of Maxwell-B and Oldroyd-B fluids at very high values of the Deborah number, De. The element is divided into several bilinear sub-elements for the stresses, while streamline-upwinding is used for discretizing the constitutive equation. The method is applied to the stick-slip problem, the flow through a tapered contraction and the flow through four-to-one abrupt plane and circular contractions. Important corner vortices develop at high values of De in the circular contraction. We have not encountered upper limits for the Deborah number in our calculations with Oldroyd-B fluids.     

Use of natural instabilities for generation of streamwise vortices in a laminar channel flow

An analysis of pressure-gradient-driven flows in channels with walls modified by transverse ribs has been carried out. The ribs have been introduced intentionally in order to generate streamwise vortices through centrifugally driven instabilities. The cost of their introduction, i.e. the additional pressure losses, have been determined. Linear stability theory has been used to determine conditions required for the formation of the vortices. It has been demonstrated that there exists a finite range of rib wave numbers capable of creating vortices. Within this range, there exists an optimal wave number which results in the minimum critical Reynolds number for the specified rib amplitude. The optimal wave numbers marginally depend on the rib positions and amplitudes. As the formation of the vortices can be interfered with by viscosity-driven instabilities, the critical conditions for the onset of such instabilities have also been determined. The rib geometries which result in the vortex formation with the smallest drag penalty and without interference from the viscosity-driven instabilities have been identified.     

Influence of elongational properties on the contraction flow of polyisobutylene in a mixed solvent

Over the last decade several international programmes have been developed around different standard fluids, one of which is the so-called S1 fluid. This is a solution of polyisobutylene in a mixed solvent and the aim of the programme has been to study the rheology of polymer solutions from the dilute solution to the melt. The focus of this paper will be on the flow visualisation of contraction flows of S1 through orifice dies and on the estimation of some of its extensional properties. The contraction ratios range from 24.4:1 and 124.3:1. The measured entry pressure drops will be correlated with contraction ratio and apparent wall shear rate. Experimental evidence will show that, when regarded as a function of wall shear rate, the entry pressure drops are independent of the contraction ratios. The flow fields for different contraction ratios, at any constant apparent wall shear rate, however, differ substantially. The evolution of the flow fields is monitored and it is shown that an initial increase in vortex size is followed by a slower decrease, this happening at a constant Weissenberg number. At the same Weissenberg number, small scale instabilities start occurring near the office. As the shear rate is increased further, these instabilities grow in size until, eventually, the flow structure is destroyed. An extensional viscosity is evaluated using a modified form of the Binding analysis for contraction flows and we show that the results are not only in qualitative agreement with those from other groups, but also that the analysis is able to predict exactly the onset of the aforementioned flow instabilities. Received: 20 March 1997 Accepted: 18 September 1997     

Large Eddy Simulation of Compressible Subsonic Turbulent Jet Starting From a Smooth Contraction Nozzle

Compressible subsonic turbulent starting jet with a relatively large Reynolds number of significant practical importance is investigated using large eddy simulation (LES), starting from a smooth contraction nozzle. The computational domain of truncated conical shape is determined through the comparison of the time-averaged numerical solution with the particle imaging velocimetry measurements for the steady jet. It is shown that the starting jet consists of a leading vortex ring followed by a quasi-steady jet, and the instantaneous velocity field exhibits contraction and expansion zones, corresponding to the high pressure (HP) and low pressure (LP) regions formed by the convecting vortex rings, and are related to the Kelvin-Helmholtz instability. The thin boundary layer inside the smooth contraction nozzle evolves into a shear layer at the nozzle exit and develops with the downstream penetration of the jet. Using λ ₂ criterion, the formation and evolution of the vortical structures are temporally visualized, illustrating distortion of vortex rings into lobed shapes prior to break-down. Rib-shape streamwise vortex filaments exist in the braid region between a pair of consecutive vortex rings due to secondary instabilities. Finally, formation and dynamics of hairpin vortices in the shear layer is identified.     

Numerical breakdown at high Weissenberg number in non-Newtonian contraction flows

 Planar contraction flows of non-Newtonian fluids with integral constitutive models are studied to investigate the problem of numerical breakdown at high Weissenberg or Debrorah numbers. Spurious shear stress extrema are found on the wall downstream of the re-entrant corner for both sharp and rounded corners. Moreover, a non-monotonic relation between shear stress and strain rate is found when the Deborah number limit is approached, which correlates with these shear extrema. This strongly suggests that non-monotonicity between shear stress and strain rate may be responsible for the Deborah number limit problem in contraction flow simulations. This non-monotonicity is caused by the inaccuracy of the quadrature, using constitutive equations that do not have shear stress maxima when exactly evaluated. This conclusion agrees with recent analytical findings by others that inaccuracy of the integration along the streamlines – either by numerical integration or asymptotic approximation – makes the problem ill-conditioned, with spurious growth occurring on the wall downstream of the re-entrant corner. Received: 5 March 1999/Accepted: 1 September 1999     

Influence of the formation of associations of macromolecules on the rheo-optical properties of their dilute solutions. I. Theory

Assuming the formation of doublets in the flow according to a mass action law, the shear rate and the concentration dependence of the extinction angle, of the birefringence, and of the average coil expansion are calculated for dilute solutions of flexible macromolecules. It is shown that this reversible association process has a strong influence on the measurable parameters in a flow birefringence experiment. c concentration (g/cm³) - h ² mean square end-to-end distance at shear rate - h ₀ ² mean-square end-to-end distance at zero-shear rate - n refractive index of the solution (not very different from the solvent for a very dilute solution) - E mean coil expansion - K ₀,K constant of the mass action law - M molecular weight - R _G gas constant - T absolute temperature - ₁ – ₂ optical anisotropy of the segment - ₀ Deborah number: - Deborah number: - shear rate - ₀, reduced concentration - _s viscosity of the solvent - [] ₀ intrinsic viscosity at zero-shear rate - [] intrinsic viscosity at shear rate - extinction angle - N _a Avodagro's number - _n magnitude of the birefringence     

New laboratory tests to measure rheological properties of paper coatings in transient and steady-state flows

The paper describes pertinent laboratory tests to characterize the rheological properties of paper coatings with regard to blade coating over a very wide range of shear rates in both transient and steady-state shear flows. Shear rates as high as 10⁶ s^–1 can be reached by means of a gas-driven capillary rheometer. Examples for the evaluation of end effects, wall effects, and coating thixotropy are given. A stiff and fast Couette rheometer is used to determine flow curves and the shear stress overshoot in step shear rate tests. The primary normal stress difference can be measured up to 10⁴ s^–1 by means of a high shear cone-plate rheometer with piezo transducer. A correct evaluation of the measurements has to take into account inertia contributions to the normal force. First results using a sinusoidal modulation of the shear rate are presented.Paper presented at: International Symposium on Pigment Coating Structure and Rheology, Helsinki, Febr. 8–9, 1989     

Viscosity of some clay-based coating colors at high shear rates

The shear viscosity of clay-based coating colors containing latex and carboxymethyl cellulose (CMC) has been measured over a relatively large shearrate region. In the shear-rate range of 50–1500 s^–1 the measurements were performed using a rotational viscometer and, at higher shear rates extending into the region 10⁵ – 10⁶ s^–1, a high pressure capillary viscometer was employed. The viscosity of the clay colors increased with increasing CMC-concentration, but the influence of the CMC-content was less pronounced at higher shear rates. The apparent shear-thinning behavior of the investigated colors could, in part, be attributed to the shear-thinning of the corresponding polymer (CMC) solution constituting the liquid phase of the color, but the influence of another factor was also indicated. At low shear rates, the interaction between the color components can produce relatively high viscosity levels, but in the high shear rate region these interactions appear to be less important for the viscosity level. It is also of interest to note that the viscosity dependence on the solids content in the high shear-rate region could be described with reasonable accuracy using an empirical equation neglecting interactions between the color components.     

Viscosity measurements with a magnetoviscometer in the zero-shear and transition region of polypropylenes

Streaming of a non-Newtonian fluid around a sphere is of special importance not only for measuring viscosities with falling spheres, but also for many problems connected with polymer processing. Using the mentioned measuring principle, attention has to be paid to the following points: The sphere is moving in a fluid (melt) of finite extension which requires the application of wall and perhaps end corrections. These are possibly not the same for Newtonian and non-Newtonian fluids. To calculate the viscosity with the help of Stokes law the steady-state velocity is necessary, and it is essential, how long it takes the sphere to reach it. To compare our results with other data, an average shear rate has to be calculated, since there is no uniform shear rate around the sphere. Velocities being very low in our experiments result in very small Reynolds numbers (Re < 10^–3), which allows the application of Stokes law practically without corrections.The experiments were performed at zero shear and in the transition region above. It turned out, that it is usually not possible to extrapolate from shear-dependent viscosity data to zero-shear viscosity.Dedicated to Prof. A. Neckel on the occasion of his 60th birthday     

A linear analysis of instabilities in Couette flow of nematic liquid crystals

This paper presents a new method for determining the critical threshold values at which both homogeneous and roll-type instabilities may appear when thin layers of nematic liquid crystal are subjected to Couette flow. The anisotropic axis of the material is initially uniformly aligned parallel to the axial direction and the effect of a stabilising, axially oriented, uniform magnetic field is allowed for in calculations. Carrying out a linear stability analysis of the continuum equations, we employ a Fourier series method to derive an expression for determining the critical shear rate at which instability occurs. Thresholds for both homogeneous and roll instabilities are obtained and their variation with rotation rate and applied magnetic field strength is examined for the material MBBA.     

Numerical investigation of steady suction control of flow around a circular cylinder

This paper numerically investigates the effectiveness of the control of steady suction on a stationary circular cylinder with several isolated suction holes on the surface at a subcritical Reynolds number. The control effectiveness as a function of the azimuthal position, spanwise spacing and suction flow rate of the suction holes on the control of the aerodynamic forces on the cylinder and the suppression of alternate vortex shedding are taken into account. The study of the azimuthal location of the suction holes indicates that azimuthal angles of θ=90° and 270°, which are close to the separation point, provide the most substantial decreases in the aerodynamic forces. When restricted to the most effective azimuthal angle, a remarkable control effectiveness can be achieved when the axial spacing between two neighboring suction holes is less than a minimum value even under a small suction momentum coefficient. However, if the axial spacing exceeds the minimum spacing, the control effectiveness will not be saturated even under a very large suction momentum coefficient. Thus, the cause of the effective aerodynamic force control is suggested to be a result of obvious three-dimensional phenomenon in the near wake, which is characterized by the generation of a convergent flow between two neighboring suction hole sections and a stronger, larger three-dimensional vortex pair adjacent to the convergent flow. It has been suggested that this strongly three-dimensional flow pattern is induced by the strong interaction between two neighboring but counter-rotating three-dimensional vortices separately produced by two neighboring suction holes. Moreover, the effects of such three-dimensional flow patterns are investigated in detail based on variations in the flow field and sectional aerodynamic forces in different cross sections. Finally, the upper limit of the axial spacing between two neighboring suction holes to form such a three-dimensional flow pattern is suggested to be between 0.75 D and 1.5 D when the suction flow rate exceeds a certain value.     

Stability of planar stagnation flow of a highly viscoelastic fluid

Linear stability analysis is used to predict the onset of instabilities in inertialess viscoelastic planar stagnation flow. Beyond a critical value of the dimensionless flow rate, or Deborah number, the creeping base flow of similarity type, which is valid in the limit of vanishingly small Reynolds numbers, becomes unstable to localized three-dimensional disturbances. Stability calculations of the local similarity type viscoelastic flow in a small region near the stagnation plane are reported for the quasi-linear Oldroyd-B constitutive equation. The stability results for a range of Deborah numbers and viscosity ratio are presented to explore systematically the effects of elasticity and other rheological properties. The onset of instability and the temporal and spatial characteristics of the secondary flow predicted here resemble other purely elastic instabilities measured and predicted for viscoelastic flows in other simple and complex geometries with curved streamlines.     

Direct measurement of static yield properties of cohesive suspensions

A technique of yield stress investigation based upon the combined use of two devices (an applied stress rheometer and an instrument for measuring the propagation velocity of small amplitude, torsional shear waves) is described. Investigations into the low shear rate rheological properties of illitic suspensions are reported for shear rates, typically, in the range 10^–4— 10^–1 s^–1 under applied stresses in the range 0.01 — 10 Nm^–2 and involving shear strains between 10^–1 and 10^–4. Results are presented which demonstrate that the technique does not invoke the excessive structural disruption of material associated with applied shear rate based methods (direct and otherwise) and the widely encountered problem of wall slip at the surface of rotational measuring devices is avoided using miniature vane geometries. Results are compared with those obtained using smooth-walled cyclindrical measuring devices in both applied stress and applied shear rate instruments.Yield measurements are considered in relation to the structural properties of the undisturbed material state and shear moduli obtained by studying the propagation of small amplitude (10^–5 rad), high frequency (~ 300 Hz) torsional shear waves through the test materials are reported. Experimental techniques and instrument modifications to permit these measurements are described.     

Measurement of the first normal-stress difference at high shear rates for a polyisobutylene/decalin solution “D2”

Reasonable agreement is found between values of the first normal-stress differenceN ₁ for samples of D2, a polyisobutylene/decalin solution, measured in steady shear flow using three different instruments: a Weissenberg Rheogoniometer (a cone-plate rotational rheometer), a Torsional Balance (plate-plate rotational) Rheometer, and a Stressmeter (a transverse-slot slit-die rheometer). Viscosity values are also in reasonable agreement. Ranges of variables common to at least two rheometers include values of shear stress up to 3,700 Pa and shear rate up to 20,700 s^–1 near 25 °C. The agreement supports the approximate validity of the semi-empirical HPBL equation used to calculateN ₁ from Stressmeter data over a range of shear rates up to 20,700 s^–1 near 25 °C. Time-temperature superposition behavior exhibited by Stressmeter data at temperatures in the range 21 °C to 111 °C suggest that the range of validity of the Stressmeter method for determiningN ₁ approximately may extend up to shear rates of 290,000 s^–1 at 111 °C.     

Rheology of concentrated microgel solutions

Viscosity, modulus, and yield stress for 0–6 wt% aqueous solutions of Carbopol 941 were investigated using constant shear rate, constant shear stress, and dynamic oscillatory experiments. The microgel character of the polymer was evident from the solid-like behavior of the solutions above 1 wt%. Yield stress increased with concentration, but yield occurred at a critical shear strain of 40%, independent of concentration. The static stress-strain relationship became non-linear at ~ 25% strain, in fair agreement with the onset of non-linear response in the storage modulus at ~ 10% strain. Small strain moduli from static and low frequency measurements agreed rather well; modulus values obtained from the recoverable strain after yielding were 30–40% smaller. Solutions flowed at near-constant stress in the low shear rate regime; at higher rates the stress increases with shear rate more rapidly. The viscosity did not obey the Cox-Merz rule. Steady-state viscosity scaled with polymer concentration to the 3/4 power. Results were interpreted using a cellular, deformable sphere model for the polymer, in analogy to emulsions and foams.