The motion of long bubbles through viscoelastic fluids in capillary tubes

The penetration of long gas bubble through a viscoelastic fluid in a capillary tube has been studied in order to investigate the influence of viscoelastic material properties on the hydrodynamic coating thickness and local flow kinematics. Experiments are conducted for three tailored ideal elastic (Boger) fluids, designed to exhibit similar steady shear properties but substantially different elastic material functions. This allows for the isolation of elastic and extensional material effects on the bubble penetration process. The shear and extensional rheology of the fluid is characterized using rotational and filament stretching rheometers (FSR). The fluids are designed such that the steady-state extensional viscosity measured by the FSR at a Deborah number (De) greater than 1 differs over three orders of magnitude (Trouton ratio = 10³–10⁶). The experiment set up to measure the hydrodynamic coating thickness is designed to provide accurate data over a wide range of capillary numbers (0.01 < Ca < 100). The results indicate that the coating thickness in this process increases with an increase in the extensionally thickening nature of the fluid. Experiments are also conducted using several different capillary tube diameters (0.1 < D < 1 cm), in order to compare responses at similar Ca but different flow De. Suitable scaling methods and nonlinear viscoelastic constitutive equations are explored to characterize the displacement process for polymeric fluids. Bubble tip shapes at different De are recorded using a CCD camera, and measured using an edge detection algorithm. The influence of the mixed flow field on the bubble tip shape is examined. Particle tracking velocimetry experiments are conducted to compare the influence of viscoelastic properties on the velocity field in the vicinity of the bubble tip. Local shear and extension rates are calculated in the vicinity of the bubble tip from the velocity data. The results provide quantitative information on the influence of elastic and extensional properties on the bubble penetration process in gas-assisted injection molding. The bubble shape and velocity field information provides a basis for evaluating the performance of constitutive equations in mixed flow. Received: 19 January 1999 Accepted: 30 June 1999     

Flow-induced crystallization of high-density polyethylene: the effects of shear and uniaxial extension

The effects of shear, uniaxial extension and temperature on the flow-induced crystallization of two different types of high-density polyethylene (a metallocene and a ZN-HDPE) are examined using rheometry. Shear and uniaxial extension experiments were performed at temperatures below and well above the peak melting point of the polyethylenes in order to characterize their flow-induced crystallization behavior at rates relevant to processing (elongational rates up to 30 s^− 1 and shear rates 1 to 1,000 s^− 1 depending on the application). Generally, strain and strain rate found to enhance crystallization in both shear and elongation. In particular, extensional flow was found to be a much stronger stimulus for polymer crystallization compared to shear. At temperatures well above the melting peak point (up to 25°C), polymer crystallized under elongational flow, while there was no sign of crystallization under simple shear. A modified Kolmogorov crystallization model (Kolmogorov, Bull Akad Sci USSR, Class Sci, Math Nat 1:355–359, 1937) proposed by Tanner and Qi (Chem Eng Sci 64:4576–4579, 2009) was used to describe the crystallization kinetics under both shear and elongational flow at different temperatures.     

Application of the NS-α Model to a Recirculating Flow

In this paper we investigate a subgrid model based on an anisotropic version of the NS-α model using a lid-driven cavity flow at a Reynolds number of 10,000. Previously the NS-α model has only been used numerically in the isotropic form. The subgrid model is developed from the Eulerian-averaged anisotropic equations (Holm, Physica D 133:215, 1999). It was found that when α ² was based on the mesh numerical oscillations developed which manifested themselves in the appearance of streamwise vortices and a ‘mixing out’ of the velocity profile. This is analogous to the Craik–Leibovich mechanism, with the difference being that the oscillations here are not physical but numerical. The problem could be traced back to the discontinuity in α ² encountered when α ² = 0 on the endwalls. A definition of α ² based on velocity gradients, rather than mesh spacing, is proposed and tested. Using this definition the results with the model show a significant improvement. The splitting of the downstream wall jet, rms and shear stress profiles are correctly captured a coarse mesh. The model is shown to predict both positive and negative energy transfer in the jet impingement region, in qualitative agreement with DNS results.     

Smart Baffle Placement for Chaotic Mixing

It is well known that fluid mixing can often be improved by the introduction of ‘baffles’ into the flow – the problem of baffle placement is examined here for chaotic fluid mixing of a highly viscous fluid. A simple model for a planetary mixer, with one stirring element, is modified by the introduction of one or more stationary baffles. Regular regions of poor mixing in the unbaffled flow are shown to be significantly reduced in size if the location of the baffles is chosen so that the flow necessarily generates ‘topological chaos’. By contrast, the positioning of baffles in superficially similar ways that do not generate such ‘topological chaos’ fails to provide a similar improvement.     

Drag reduction change of polyethyleneoxide solutions in pipe flow

Change of drag reduction (DR) along a tube (D=2 mm, L=4 m) was experimentally investigated. To attain turbulent flow with Re=8 × 10⁴, a tank operated under high pressure up to 16 MPa. Solutions of different brands of polyethyleneoxide (PEO) with concentrations from 1 ppm to 100 ppm were tested. The results indicate that DR is not a constant value but depends on the time and intensity of interaction between the polymer and the turbulent flow. There are three regions with different behaviors of DR: growth, maximum, and slope down. Maximum DR coincides with the Virk ultimate DR and can be described by the suggested simple formula . A decrease in the DR maximum has not been found even for high shear stresses τ _p < 800 Pa. DR dynamics for four brands of PEO with different molecular weight was studied. Direct experimentally determined DR may be greater than the Virk ultimate value if the change in velocity profile is not taken into account. The corrected DR never exceeds the ultimate DR. Received: 10 April 2000/Accepted: 24 May 2001     

Three views of viscoelasticity for Cox–Merz materials

A slight rearrangement of the classical Cox and Merz rule suggests that the shear stress value of steady shear flow, , and complex modulus value of small amplitude oscillatory shear, G^∗(ω) = (G^′2 + G^″2)^1/2, are equivalent in many respects. Small changes of material structure, which express themselves most sensitively in the steady shear stress, τ, show equally pronounced in linear viscoelastic data when plotting these with G^∗ as one of the variables. An example is given to demonstrate this phenomenon: viscosity data that cover about three decades in frequency get stretched out over about nine decades in G^∗ while maintaining steep gradients in a transition region. This suggests a more effective way of exploiting the Cox–Merz rule when it is valid and exploring reasons for lack of validity when it is not. The τ −G^∗ equivalence could also further the understanding of the steady shear normal stress function as proposed by Laun.     

Synthesis and electrorheological properties of polar molecule-dominated TiO 2 particles with high yield stress

Two types of amorphous TiO₂ particles with different particle sizes were synthesized by a simple sol–gel method and were characterized by X-ray diffraction analysis, field emission scanning electron microscopy, and Fourier transform infrared spectrometry. The electrorheological (ER) results show that the TiO₂/silicone oil suspensions exhibited a remarkable ER effect. The static shear stress can be up to 130 kPa (shear rate 0.2 s^− 1) under the DC electric field of 4 kV/mm at room temperature. The polar molecules present on the particles’ surface play a decisive role for the observed giant ER effect, which arises from the alignment of polar molecules in the gap between neighboring particles.     

Dynamic rheology of the immiscible blends of liquid crystalline polymers and flexible chain polymers

Immiscible blends containing liquid crystalline polymers (LCP) as dispersed phases show different dynamic rheological properties than those composed of flexible polymers. The widely used Palierne’s model was shown by many authors to be insufficient to describe the frequency dependence of dynamic modulus of such blends. A new model was presented to describe the dynamic rheology of the immiscible blend containing LCP as a dispersed phase. The flexible chain polymer matrix was assumed to be a linear viscoelastic material under small amplitude oscillatory shear flow, and the LCP was assumed to be an Ericksen’s transversely isotropic fluid. The Rapini-Papoular equation of anisotropic interfacial energy was used to account for the effect of nematic orientation on the interfacial tension. It was found that the orientation of the director and the anchoring energy greatly influenced the storage modulus at the “shoulder” regime. The overall dynamic modulus of the blend can be well described by the model with suitable choice of the orientation of the director and anchoring energy of LCP.     

Simultaneous determination of shear and extensional viscosities and wall slip by on-line rheometry with parameter fitting: an application to EPDM rubber

Shear and extensional viscosities and wall slip are determined simultaneously under extrusion processing conditions using an on-line rheometer. Because it is not possible to independently control flow rate and temperature, classical methods for interpretation of capillary data cannot be used with on-line rheometry. This limitation is overcome using computational optimization to fit parameters in a flow model. This consists of three parts, representing shear viscosity, extensional viscosity, and wall slip. Three-parameter, power law forms, based on local instantaneous deformation rates and including temperature dependence, are used for each, and analytic solutions applied for entry flow and flow in the capillary. For entry flow, the Cogswell–Binding approach is used, and for developed flow in the capillary a solution incorporating wall slip is derived. The rheometer, with interchangeable capillaries, is mounted in place of the die on a rubber profile extrusion line. Pressure drops and temperatures for extrusion of an EPDM rubber through 2 mm diameter capillaries of length 0, 2, 3, 4, and 5 mm are logged and flow rates determined for a range of extruder speeds (5 to 20 rpm). Pressures ranged from 60 to 75 bar and temperatures from 86 to 116 °C. Mean flow velocity in the capillaries was between 5 × 10⁻³ and 5 × 10⁻¹ m s⁻¹. The nine material parameters are optimized for best fit of the analytic pressure drops to experimental data, using about 100 data points, with the Levenberg–Marquardt method. It is concluded that flow is dominated by extension and wall slip. Shear flow appears to play little part. The slip model indicates that slip velocity increases much more rapidly than the wall shear stress (in the range 0.5–1 MPa) and decreases with temperature for a given stress level. Results for the (uniaxial) extensional viscosity represent an engineering approximation to this complex phenomenon at the high strains (approximately 200) and high extension rates (up to 800 s⁻¹) applying in the extrusion. Results indicate a slight extension hardening and a decrease with temperature. Results are put into the context of the available studies in the literature, which, particularly with regard to wall-slip and extensional flow, consider conditions far removed from those applying in industrial extrusion. The present methods provide a powerful means for flow characterization under processing conditions, providing data suitable for use in computer simulations of extrusion and optimization of die design.     

Steady shear rheology of dilute polystyrene particle gels

This paper describes an experimental study on dispersions of monodisperse polystyrene (PS) spheres with a typical radius of 1 μm, dispersed in an electrolyte at high ionic strength, screening the electrostatic repulsion. These suspensions gelate at rest even at low volume fractions of PS particles. The density of the particles is matched with the solvent by using deuterium oxide for volume fractions φ≤0.117. Steady-state flow curves, viscosity as a function of shear rate, are measured and reported for 0.014<φ<0.322. The measured flow curves are analyzed on the basis of two models: 1. In the giant floc model (van Diemen and Stein 1983, 1984; Schreuder et al. 1986, 1987; Laven et al. 1988), at low shear rates, the shear is not distributed homogeneously but is limited to certain shear planes; the energy dissipation during steady flow is due primarily to overcoming the viscous drag on the suspended particles during motion caused by encounters of particles in the shear planes. Though this model was developed for higher solid volume fractions (0.35–0.425), we found that it also describes the rheology of dilute particle gels for 0.15≤φ≤0.3, using the same values for the parameters in the model as in the high solid volume fraction region. For φ<0.15, the model also describes the data if the fraction of distance by which a moving particle entrains its neighbors, is assumed to increase in this φ region. 2. The model of de Rooij (de Rooij et al. 1993, 1994) considers aggregates in shear flow to be monodisperse impermeable spheres with a fractal structure. The permeability is taken into account by considering a hydrodynamic radius smaller than the gyration radius in the Krieger-Dougherty expression for the hydrodynamic contribution to the viscosity. Through the use of a yield criterion the aggregate radius is modeled as a function of shear rate. We found that the model describes our experimental results, with a combination of parameter values used already by de Rooij, but only for φ<0.15. Received: 7 May 1998 Accepted: 22 December 1998     

Use of a sliding plate rheometer to measure the first normal stress difference at high shear rates

The use of a sliding plate rheometer (SPR) to determine the first normal stress difference of molten polymers and elastomers at high shear rates is demonstrated. The simple shear flow in this instrument is not subject to the flow instabilities that limit the use of rotational rheometers to shear rates often below 1 s⁻¹. However, issues of secondary flow and wall slip must be addressed to obtain reliable data using an SPR. A highly entangled, monodisperse polybutadiene and a commercial polystyrene were the polymers studied. The inclusion of the polystyrene made it possible to compare data with those obtained by Lodge using a stressmeter, which is an instrument based on the measurement of the hole pressure. The data from the two instruments are in good agreement and are also close to the predictions of an empirical equation of Laun based on the storage and loss moduli.     

Characterization of the flow properties of sodium carboxymethylcellulose via mechanical and optical techniques

Sodium carboxymethylcellulose (NaCMC) in solution represents a complex rheological system, since it forms aggregates and associations and hence higher-level structures and, depending on the synthesis, is only found in a molecularly dispersed form in exceptional cases. Rheo-mechanical investigations of the viscoelasticity showed that the Cox-Merz rule is not fulfilled. The aim was therefore to examine whether rheo-optics could be employed to provide more detailed conclusions about the parameters that influence the flow behavior of NaCMC than has hitherto been available with mechanical methods. The flow birefringence, Δn ^′, rises as the degree of polymerization increases, and exhibits the same dependence on molar mass as does the viscosity: Δn ^′∝M _w ^3.4. As the degree of polymerization increases while the shear rate remains constant, the polymer segments become more distinctly aligned in the direction of shear. Hence increasing the degree of polymerization also affects the solution structure, i.e. the interaction of the molecules with one another. The stress-optical rule only applies to a limited extent for this system. The stress-optical coefficient, C, is almost independent of the shear rate, but is strongly influenced by the concentration and attains a limiting value of 3 × 10⁻⁸ Pa⁻¹. C was determined for a polymer in dilute solution and the curve obtained also enabled transitions in the solution structure to be recognized. Received: 1 May 1998 Accepted: 5 October 1998     

The orientational behavior of multiwall carbon nanotubes in polycarbonate in simple shear flow

This paper describes the changes in the orientation of multiwall carbon nanotubes (MWCNT) in polycarbonate as determined by transient and oscillatory shear rheology. It is well known from rheological studies on composites with macroscopic fibers that the overshoot in transient shear viscosity is caused by the change in orientation distribution of these fibers. This study shows that although an overshoot in transient shear viscosity of MWCNT/polycarbonate is measured at shear rates as low as 0.1 s^− 1, the MWCNT network is disturbed only at considerably higher shear rates. Scanning electron microscopy micrographs and oscillatory shear show that MWCNT in thermoplastic composites will only be oriented at high shear rates. Simultaneous measurements of the electrical conductivity during rheological start-up shear and oscillatory measurements show large differences between electrical and mechanical relaxation behaviors. The viscosity of the composite seems to depend strongly on the MWCNT network density, whereas the proximity of the tubes at the network points seems to determine the electrical properties of the MWCNT composite.     

The rheological characterisation of bread dough using capillary rheometry

An Australian hard wheat flour–water dough has been characterised using parallel plate and capillary rheometers over an extensive range of apparent shear rates (10^− 3–10³ s^− 1) relevant to process conditions. Torsional measurements showed that the shear viscosity of the dough increased with strain to a maximum value and then decreased, suggesting a breakdown of the dough structure. Both torsional and capillary experiments revealed the shear-thinning behaviour of the dough. The wall slip phenomenon in capillary rheometry was investigated and found to be diameter dependent and occurred at a critical shear stress of approximately 5–10 kPa. A two-regime power law behaviour was observed, with the power law index approximately 0.3 in the low shear rate range increasing to 0.67 in the high shear rate range. Pressure fluctuation was observed in the capillary data and increased with shear rate, in particular, at shear rates approaching 10⁴ s^− 1. The results demonstrate that capillary rheometry is a viable means of rheologically testing dough at high shear rates provided pressure fluctuation is carefully monitored and capillary rheometry corrections, including wall slip, are accounted for.     

Shallow wakes generated on smooth and rough surfaces

This study deals with the behavior of shallow turbulent wakes generated on smooth and rough surfaces. The wake generator used is a flat plate placed normal to the flow. Experiments were conducted at flow depths of 40 and 80 mm. The boundary layer thickness in the approaching flow occupies 60–75% of the flow depth. The Reynolds number based on the plate width and approaching freestream velocity varies from 13.0 × 10³ to 14.5 × 10³. Velocity measurements were carried out in the near-wake region (1–10 plate widths) using a laser-Doppler anemometer. The mean velocity distributions at various axial stations collapse onto a single curve by a proper choice of the length and velocity scales. It is important to note that a sense of self-similarity is attained even in the near-wake region. Attempts were made to clarify the relative effects of the transverse shear and bed friction in shallow open channel wakes. Received: 11 February 1999/Accepted: 30 August 2000     

Rheostructural studies of a discotic mesophase pitch at processing flow conditions

The flow-induced microstructure of a mesophase pitch was studied within custom-made dies for changing wall shear rates from 20 to 1,100 s^− 1, a flow scenario that is typically encountered during fiber spinning. The apparent viscosity values, measured at the nominal wall shear rates ranging from 500 to 2,500 s^− 1 using these dies, remain fairly constant. The microstructure was studied in two orthogonal sections: r–θ (cross section) and r–z (longitudinal mid plane). In these dies, the size of the microstructure gradually decreases toward the wall (to as low as a few micrometers), where shear rate is highest. Furthermore, as observed in the r–θ plane of the capillary, for a significant fraction of the cross section, discotic mesophase has a radial orientation. Thus, the directors of disc-like molecules were aligned in the vorticity (θ) direction. As confirmed from the microstructure in the r–z plane, most of the discotic molecules remain nominally in the flow plane. Orientation of the pitch molecules in the shear flow conditions is consistent with that observed in controlled low-shear rheometric experiments reported earlier. Microstructral investigation suggests that the radial orientation of carbon fibers obtained from a mesophase pitch originates during flow of pitch through the die.     

The effect of mixing particles of different size on the electrorheological response under steady shear flow

 The effect of mixing particles of different sizes on the electrorheological response of suspensions under steady shear flow was investigated. Two sizes, 15 μm and 50 μm, of monodisperse spherical sulfonated poly(styrene-co-divinylbenzene) particles were used. Several electrorheological fluids were made containing different proportions of small and large particles dispersed in silicone oil, but with constant overall particulate concentration. It was found that the mixed size system produced the highest electrorheological response under the shear rates used (10 s⁻¹ to 500 s⁻¹), which is the opposite trend to previous studies of bimodal systems with larger size ratios. Received: 21 December 2000 Accepted: 29 March 2001     

Plane jets impinging on porous walls

The flow of a two-dimensional plane turbulent jet impinging on a porous screen has been studied experimentally. It is shown how the overall flow structure depends on the porosity of the surface. For low screen porosity (β < 0.41, say), transverse wall jets can be formed on both sides of the screen and in extreme cases the axial momentum flux some way downstream of the screen falls to zero, so that the screen has the same drag as would a solid wall. For high screen porosity (β > 0.57, say) the axial volume flux is largely preserved through the screen, but the dominant eddy structures present in the upstream jet are largely destroyed, so that entrainment rates downstream of the screen can be very low. The relatively small, intermediate range of porosities (0.41 < β < 0.57, where β is the screen open area ratio) is associated with dramatic changes in flow pattern and recirculating regions can exist on the upstream side of the screen. These flows, although all geometrically very simple, provide a serious challenge for computational modelling. Received: 25 May 2000 / Accepted: 22 February 2001     

Structural rearrangements in hairy-rod polymer solutions undergoing shear

 We report on a rheooptical investigation of hairy-rod poly(p-phenylene) solutions at different concentrations and temperatures. These polymers have a reasonably high persistence length (about 28 nm) and behave as worm-like chains in dilute solutions, whereas they form nearly spherical fractal aggregates with internal anisotropy at higher concentrations. By exposing these systems to time-dependent simple shear and following the evolution of birefringence in start-up and its subsequent relaxation upon the cessation of shear, we find a substantial broadening of the cluster size distribution, resulting from flow-induced cluster deformation and break-up. In contrast to the very dilute solutions, where polymers align in the flow direction, the deformed clusters main axes are aligned in the vorticity direction, presumably due to their strong steric local pretransitional type of ordering, with the constituent polymers following the velocity vector. At the highest concentration, which corresponds to a weak gel, shear is shown to break-up the gel and the steady-state response of a broad-size aggregate suspension is eventually recovered. Received: 18 February 1999/Accepted: 6 July 1999     

Convective mass transfer across fluid interfaces in straight angular pores

Steady convective mass transfer to or from fluid interfaces in pores of angular cross-section is theoretically investigated. This situation is relevant to a variety of mass transport process in porous media, including the fate of residual non-aqueous phase liquid ganglia and gas bubbles. The model incorporates the essential physics of capillarity and solute mass transfer by convection and diffusion in corner fluid filaments. The geometry of the corner filaments, characterized by the fluid–fluid contact angle, the corner half-angle and the interface meniscus curvature, is accounted for. Boundary conditions of zero surface shear (‘perfect-slip’) and infinite surface shear (‘no-slip’) at the fluid–fluid interface are considered. The governing equations for laminar flow within the corner filament and convective diffusion to or from the fluid–fluid interface are solved using finite-element methods. Flow computations are verified by comparing the dimensionless resistance factor and hydraulic conductance of corner filaments against recent numerical solutions by Patzek and Kristensen (J. Colloid Interface Sci 236, 305–317 2001). Novel results are obtained for the average effluent concentration as a function of flow geometry and pore-scale Peclet number. These results are correlated to a characteristic corner length and local pore-scale Peclet number using empirical equations appropriate for implementation in pore network models. Finally, a previously published “2D-slit” approximation to the problem at hand is checked and found to be in considerable error.