The Dilute Rheology of Swimming Suspensions: A Simple Kinetic Model

A simple kinetic model is presented for the shear rheology of a dilute suspension of particles swimming at low Reynolds number. If interparticle hydrodynamic interactions are neglected, the configuration of the suspension is characterized by the particle orientation distribution, which satisfies a Fokker-Planck equation including the effects of the external shear flow, rotary diffusion, and particle tumbling. The orientation distribution then determines the leading-order term in the particle extra stress in the suspension, which can be evaluated based on the classic theory of Hinch and Leal (J Fluid Mech 52(4):683–712, 1972), and involves an additional contribution arising from the permanent force dipole exerted by the particles as they propel themselves through the fluid. Numerical solutions of the steady-state Fokker-Planck equation were obtained using a spectral method, and results are reported for the shear viscosity and normal stress difference coefficients in terms of flow strength, rotary diffusivity, and correlation time for tumbling. It is found that the rheology is characterized by much stronger normal stress differences than for passive suspensions, and that tail-actuated swimmers result in a strong decrease in the effective shear viscosity of the fluid.     

Orientation dynamics in commercial thermotropic liquid crystalline polymers in transient shear flows

In situ X-ray scattering measurements of molecular orientation under shear are reported for two commercial thermotropic liquid crystalline polymers (TLCPs), Vectra A950® and Vectra B950®. Transient shear flow protocols (reversals, step changes, and flow cessation) are used to investigate the underlying director dynamics. Synchrotron X-ray scattering in conjunction with a high-speed area detector provides sufficient time resolution to limit the total time spent in the melt during testing, whereas a redesigned X-ray capable shear cell provides a more robust platform for working with TLCP melts at high temperatures. The transient orientation response upon flow inception or flow reversal does not provide definitive signatures of either tumbling or shear alignment. However, the observation of clear transient responses to step increases or step decreases in shear rate contrasts with expectations and experience with shear-aligning nematics and suggests that these polymers are of the tumbling class. Finally, these two polymers show opposite trends in orientation following flow cessation, which appears to correlate with the evolution of dynamic modulus during relaxation. Specifically, Vectra B shows an increase in orientation upon flow cessation, an observation that can only be rationalized by the assumption of tumbling dynamics in shear. Together with prior observations of commercial LCP melts in channel flows, these results suggest that this class of materials, as a rule, exhibits director tumbling.     

Molecular orientation in sheared molten thermotropic random copolyester

The macromolecular alignment and texture orientation in sheared thermotropic copolyester were investigated using in situ wide-angle X-ray scattering (WAXS) and polarizing optical microscopy (POM). The molecular behavior was correlated with viscoelastic properties. The polymer is a random copolyester based on 60 mol% 1,4-hydroxybenzoic acid (B) and 40 mol% ethylene terephthalate (ET) units. X-ray scattering showed that the molecular chains were aligned along the flow direction. The degree of molecular orientation, , is an increasing function of the applied shear rate. However, rheo-optics showed that shear flow could not orient the polydomain texture, i.e., neither defect stretching nor elimination of defects was observed. Instead, shear compressed the microdomains and gave rise to long-range orientation correlations. Rheology showed that the nematic melt is viscoelastic, the loss modulus G″ dominates the elastic modulus G′, and the dynamic viscosity η* is shear thinning. Moreover, the steady shear viscosity, η, also behaved shear thinning, while the first normal stress difference N ₁ remained positive. The empirical Cox–Merz rule did not hold, , within the shear rate range studied. The microscopic and rheological properties suggest that B–ET is a flow-aligning nematic polymer.     

Transient rheological responses in sheared biaxial liquid crystals

We study the rheological response of monodomain ellipsoidal biaxial liquid crystal polymers (BLCP) as well as bent-core or V-shaped liquid crystal polymers (VLCP) subject to steady and time-dependent small amplitude oscillatory shear in selected regions of the model as well as flow parameter space. We adopt the two newly developed hydrodynamical kinetic theories for ellipsoidal BLCPs and VLCPs, respectively (Sircar and Wang, PRE 78:061702, 2008, J Rheol 53:819–858, 2009; Sircar et al., Comm Math Sci (in press), 2010), in which a generalized Straley’s potential is used to represent the pairwise mean-field interaction of the mesoscopic system in biaxial phases. Transient shear stresses and normal stress differences corresponding to steady and small amplitude oscillatory shear are investigated; their variations with respect to the strength of the intermolecular potential, types of biaxial interaction, and changes in the aspect ratios for ellipsoidal BLCPs and the bent angle for VLCPs are explored.     

Influence of the flow history on stress growth and structure changes in the thermotropic liquid crystalline polymer Vectra B950

The structure changes in the start-up flow of the thermotropic liquid crystalline polymer Vectra 8950 are probed by performing transient experiments after various flow histories. The shear and normal stress growth curves of a squeezed sample and of a randomly oriented sample show a pronounced overshoot at low strains, whereas the stress growth curve of a sample pre-sheared until steady state shows a gradual increase. This first peak is associated with the re-orientation of the director into the shearing plane. All stress transients show a second broad maximum at large strains that results from the generation of a steady defect network. The effect of varying the relaxation period after pre-shearing is reflected in the appearance of two peaks in the subsequent stress growth curves. One of these peaks shifts linearly with re laxation period and the other is more or less fixed in position. The orientation of the molecules during steady shear flow is on average in the flow direction. Intermediate orientation levels may exist in the transient depending on the amount of strain. The material is able to maintain the flow-induced orientation distribution for a long time after cessation of flow. This is reflected in a similar fashion in the initial magnitudes of the stresses and the dynamic moduli after various preshear strains. Moreover, the dynamic moduli decrease with time after cessation of steady shear flow, indicating that the orientation increases during relaxation.     

The flow-phase diagram of Doi-Hess theory for sheared nematic polymers II: finite shear rates

The purpose of this paper is to extend the rheological predictions of the Doi-Hess kinetic theory for sheared nematic polymers from the anomalous weak shear regime (Forest et al. 2004a) to arbitrary shear rates, and to associate salient rheological and optical properties with the solution space of kinetic theory. Using numerical bifurcation software (AUTO), we provide the phase diagram of all stable monodomain orientational probability distribution functions (PDFs) and their phase transitions (bifurcations) vs nematic concentration (N) and normalized shear rate (Peclet number, Pe) for Pe1. Shear stresses, normal stress differences, the peak direction of the orientational distribution, and birefringence order parameters are calculated and illustrated for each type of PDF attractor: steady flow-aligning, both in and out of the flow deformation plane and along the vorticity axis; unsteady limit cycles, where the peak orientation direction rotates in-plane or around the vorticity axis or in bi-stable orbits tilted between them; and chaotic attractors first observed in kinetic simulations by Grosso et al. (2001). We pay particular attention to correlations between rheological features and the variety of monodomain phase transitions. Together with the weak flow regime, these results provide a nearly complete picture of the rheological consequences of the Doi-Hess kinetic theory for sheared monodomains of rigid, extreme aspect ratio, nematic rods or platelets.     

Brownian dynamics simulation of multiphase suspension of disc-like particles and polymers

A computational model is proposed for simulating the flow of polymer nanocomposites. This model is based on a multiphase suspension of disc-like particles and polymers. The particles are represented by oblate spheroid particles that interact with each other via the Gay-Berne (GB) potential, and the polymers are modeled by finitely extensible nonlinear elastic (FENE) chains that interact with each other via the repulsive Lennard-Jones potential. The interaction between an oblate spheroid particle and a FENE chain is also considered using a modified GB potential. A Brownian dynamics simulation of the shear flows of this system was conducted to investigate the orientation behavior of disc-like particles and the rheological properties of this system. The orientation of disc-like particles was affected by polymers, and the particles in a suspension were well aligned in flows because of the flow orientation property of polymers. The predicted shear viscosity exhibited shear thinning, and the normal stress differences agree qualitatively with experimental measurements of polymer/clay nanocomposites. The simulation results suggest that the present model has the potential to be used as a computational model for polymer nanocomposites.     

Strain recovery of model immiscible blends without compatibilizer

Strain recovery after the cessation of shear was studied in model immiscible blends composed of polyisobutylene drops (10–30% by weight) in a polydimethylsiloxane matrix. Blends of viscosity ratio (viscosity of the drops relative to the matrix viscosity) ranging from 0.3 to 1.7 were studied. Most of the strain recovery was attributable to interfacial tension, and could be well-described by just two parameters: the ultimate recovery and a single retardation time. Both these parameters were found to increase with the capillary number of the drops prior to cessation of shear. For blends that had reached steady shear conditions, the ultimate recovery decreased with increasing viscosity ratio, whereas the retardation time increased with increasing viscosity ratio. The retardation time was well-predicted, but the ultimate recovery was over-predicted by a linear viscoelastic model developed previously by Vinckier et al. (Rheol Acta 38:65–72, 1999).     

Rheology and flow-birefringence from viscoelastic polymer-clay solutions

 The shear orientation of viscoelastic clay-polymer solutions was investigated by means of rheology and flow birefringence (Δn). The polymer chains are in dynamic adsorption/desorption equilibrium with the clay particles to form a “network”. The elastic behavior of the network was characterized by constant stress, oscillatory shear, and stress relaxation experiments. Constant stress experiments indicated a yield stress upon which shear flow started and no strain recovery could be observed. Oscillatory shear experiments showed a broad elastic region followed by flow when a critical strain was reached. Stress relaxation experiments showed several relaxation times when the same critical strain was reached. Experiments under steady flow characterized the transient behavior of the network. With increasing steady shear rate a pronounced minimum in birefringence was observed at a critical shear rate. The shear rate dependent viscosity showed near power law behavior and no corresponding critical feature. While birefringence detects orientational effects on a microscopic length scale, rheology averages over macroscopic changes in the sample. The same degree of orientation could be achieved under constant shear rate or constant stress conditions. Received: 25 January 2001 Accepted: 22 May 2001     

Anisotropic thermal conductivity in sheared polypropylene

We discuss the anisotropy of the thermal conductivity tensor in polymer flow in this paper. Isotactic polypropylene (iPP) specimens were deformed by injection moulding at high shear rates and by steady shear at low shear rates, and were then quenched. The thermal conductivities parallel and perpendicular to the shear direction were measured using modulated differential scanning calorimetry (MDSC) in accordance with the ASTM E1952-01. The measured results showed that the thermal conductivity of the sheared polymer was anisotropic with an increase in the shear direction. The thermal conductivity can be regarded as varying either with the strain or the stress, as suggested by Van den Brule (1989). In addition to the Van den Brule mechanism, crystallization during flow also changes the thermal conductivity and this effect may often be dominant. Suggestions for procedures in processing computations, based on both effects, are given.     

Molecular orientation, “Region I” shear thinning and the cholesteric phase in aqueous hydroxypropylcellulose under shear

Birefringence and flow visualization are used to study molecular orientation, texture, and the cholesteric nature of a 60 wt% aqueous hydroxypropylcellulose solution at low to moderate shear rates. There is a sharp transition in behavior at a shear rate near 0.5 s^–1. Below this rate, the sample shows “Region I” shear thinning, takes on a frosted appearance, has low flow-induced orientation, and exhibits faint optical diffraction characteristic of cholesteric liquid crystals. Upon flow cessation from low rates, a highly organized cholesteric phase showing bright optical diffraction is re-formed in around 5min. Above the critical shear rate, the sample exhibits much higher orientation and a striped texture which is readily apparent with or without polarized light illumination. Upon flow cessation, molecular orientation decays significantly for up to around 400 s; however, the striped textures established during shear persist. It is suggested that persistence of the cholesteric phase under shear is responsible for the occurrence of Region I shear thinning at low shear rates in this solution.     

Modelling elongational and shear rheology of two LDPE melts

Experimental data of two low-density polyethylene (LDPE) melts at 200°C for both shear flow (transient and steady shear viscosity as well as transient and steady first normal stress coefficient) and elongational flow (transient and steady-state elongational viscosity) as published by Pivokonsky et al. (J Non-Newtonian Fluid Mech 135:58–67, 2006) were analysed using the molecular stress function model for broadly distributed, randomly branched molecular structures. For quantitative modelling of melt rheology in both types of flow and in a very wide range of deformation rates, only three nonlinear viscoelastic material parameters are needed: Whilst the rotational parameter, a ₂, and the structural parameter, β, are found to be equal for the two melts considered, the melts differ in the parameter describing maximum stretch of the polymer chains.     

The weak shear kinetic phase diagram for nematic polymers

We study the shear problem for nematic polymers as modeled by the molecular kinetic theory of Doi (1981), focusing on the anomalous slow flow regime. We provide the kinetic phase diagram of monodomain (MD) attractors and phase transitions vs normalized nematic concentration (N) and weak normalized shear rate (Peclet number, Pe). We then overlay all rheological features typically reported in experiments: alignment properties, normal stress differences and shear stress. These features play a critical role in the synthesis between theory and experiment for nematic polymers (Larson 1999; Doi and Edwards 1986). MD type is routinely used for rheological shear characterization: cf., flow-aligning 5CB (Mather et al. 1996a), tumbling PBT (Srinivasarao and Berry 1991), and 8CB (Mather et al. 1996b), evidence for a wagging regime (Mewis et al. 1997), out-of-plane kayaking modes (Larson and Ottinger 1991), and evidence for chaotic major director dynamics (Bandyopadhyay et al. 2000). MD transitions correlate with sign changes in normal stresses (Larson and Ottinger 1991; Magda et al. 1991; Kiss and Porter 1978, 1980). Furthermore, structure formation in shear devices appears to be correlated with monodomain precursor dynamics (Tan and Berry 2003; Forest et al. 2002a). In this paper we combine seminal kinetic theory results (Kuzuu and Doi 1983, 1984; Larson 1990; Larson and Ottinger 1991; Faraoni et al. 1999; Grosso et al. 2001), symmetry observations (Forest et al. 2002b), and mesoscopic results on the fate of orientational degeneracy in weak shear (Forest and Wang 2003; Forest et al. 2003a), together with our resolved numerical simulations, to provide the kinetic flow-phase diagram of Doi theory in the weak shear regime, 0<Pe<1, for infinitely thin rods. We report the "birth" of key rheological features at the onset of flow: sign changes and local maxima and minima in normal stress differences (N₁ and N₂) associated with MD transitions. These results serve as the basis for continuation of the kinetic phase diagram to Pe>1 ; as the definitive benchmark for any mesoscopic or continuum model; and experimental data can be compared in order to determine accuracy and limitations of the Doi theory in weak shear.     

Viscoelastic plastic rheological model for particle filled polymer melts

A viscoelastic plastic model for suspension of small particles in polymer melts has been developed. In this model, the total stress is assumed to be the sum of stress in the polymer matrix and the filler network. A nonlinear viscoelastic model along with a yield criterion were used to represent the stresses in the polymer matrix and the filler network, respectively. The yield function is defined in terms of differential equations with an internal parameter. The internal parameter models the evolution of structure changes during floc rupture and restoration. The theoretical results were obtained for steady and oscillatory shear flow and compared with experimental data for particle filled thermoplastic melt. The experimental data included the steady state shear strress over a wide range of shear rates, the transient stress in a start up shear flow, stress relaxation after cessation of a steady state shear flow, the step shear and the oscillatory shear flow at various amplitudes.     

Unsteady simple shear flow in a viscoplastic fluid: comparison between analytical and numerical solutions

In this paper, an unsteady flow of a viscoplastic fluid for simple shear flow geometry is solved numerically using two regularizing functions to overcome the discontinuity for zero shear rate of the Bingham constitutive law. The adopted models are the well-known Papanastasiou relation and one based on the error function. The numerical results are compared with the analytical solution of the same problem obtained by Sekimoto (J Non-Newton Fluid Mech 39:107–113, 1991). The analysis of the results emphasizes that the errors are much smaller in the yielded than in the unyielded region. The models approximate closer the ideal Bingham model as the regularization parameters increase. The differences between the models tend to vanish as the regularization parameters are at least greater than 10⁵.     

Prediction of multiple overshoots in shear stress during fast flows of bidisperse polymer melts

We present a differential constitutive model of stress relaxation in polydisperse linear polymer melts and solutions that contains contributions from reptation, contour-length fluctuations, and chain stretching. The predictions of the model during fast start-up and steady shear flows of polymer melts are in accord with experimental observations. Moreover, in accordance with reported experimental literature (Osaki et al. in J Polym Sci B Polym Phys 38:2043–2050, 2000), the model predicts, for a range of shear rates, two overshoots in shear stress during start-up of steady shear flows of bidisperse polymer melts having components with widely separated molar masses. Two overshoots result only when the stretch or Rouse relaxation time of the higher molar mass component is longer than the terminal relaxation time of the lower molar mass component. The “first overshoot” is the first to appear with increasing shear rate and occurs as a result of the stretching of longer chains. Transient stretching of the short chains is responsible for the early time second overshoot. The model predictions in steady and transitional extensional flows are also remarkable for both monodisperse and bidisperse polymer solutions. The computationally efficient differential model can be used to predict rheology of commercial polydisperse polymer melts and solutions.     

Rheological behaviour of spray-dried egg yolk/xanthan gum aqueous dispersions

This paper provides information on the microstructure of, and reports particle size distributions and rheological results for, aqueous dispersions of spray-dried egg yolk and xanthan gum prepared on a laboratory scale using two types of homogenisers. Laser light scattering results demonstrated that higher energy input during homogenisation yielded a dispersion with a lower average particle size and a wider polydispersity, slightly influencing the linear dynamic viscoelastic functions due to the low concentration of egg yolk particles. These dispersions exhibited weak gel properties at the composition studied. The mechanical spectrum and the corresponding relaxation spectrum were dominated by the xanthan/gum-water matrix which controls the structure of the continuous phase. This fact explained the lack of any wall depletion effects. Several controlled-rate and controlled-stress rotational rheometers and a capillary rheometer were used to obtain information on flow properties. The shear rate dependence of steady state viscosity was determined through twelve decades, and was fitted using the Carreau equation. The kinetics of structural recovery after steady-state shear was studied by start-up at the inception of shear and flow interrupted experiments under controlled shear history. The results were analysed in terms of the ratio of a time-dependent amount of overshoot to the amount of overshoot of the original sample, using the addition of two first order equations. Additionally, combined steady state flow properties at fixed shear stress/oscillatory shear experiments were also used. The increase of the storage modulus with time, checking a linear viscoelastic response, tracked the structural recovery after steady shear. Laser light scattering of sheared samples helped gain a better understanding of the role of egg yolk particles on the rheology of these dispersions. Received: 6 February 2000 Accepted: 5 September 2000     

Strain-thickening transition in ferric-oxide suspensions under oscillatory shear

Shear-strain-thickening transition under oscillatory flow was observed in flocculated ferric-oxide suspensions in mineral oil. The value of the dynamic modulus of the suspensions that was measured at small strain amplitude after cessation of shear also became higher when the strain amplitude of the applied shear had been within or above the transition region.The ferric-oxide powders used were an acicular submicron maghemite (magnetic) and the hematite (non-magnetic) that was converted from the maghemite by heat treatment. The powders were treated with a dispersing agent and the suspensions were prepared in 33% by particle weight. The strain-thickening transition was observed in both the magnetic and the non-magnetic suspensions. However, the onset of the strain-thickening in the magnetic suspension was found at about one decade larger strain amplitude than that in the non-magnetic analog suspension, indicating particle interactions affect to the appearance of the phenomenon.A qualitative interpretation was made in view of site percolation for the enhancement of modulus at rest after the application of the large-amplitude oscillatory shear, where the process of the strain-thickening transition under shear and the development of the modulus after stopping the shear was described with a floc model in which the flocculation phase dilates as a result of the reduction of the particle linkages under higher shear.     

Flow structure and skin friction in the vicinity of a streamwise-angled injection hole fed by a short pipe

The velocity field and skin friction distribution around a row of five jets issuing into a crossflow from short (L/D ≃ 1) pipes inclined by 35° with respect to the streamwise direction, (i.e., “short holes”) are presented for two different jet supply flow directions. Velocity was measured using PIV, while the skin friction was measured with oil-film interferometry. The flow features are compared with previously published data for jets issuing through holes oriented normal to the crossflow and with numerical simulations of similar geometries. The distinguishing features of the flow field include a reduced recirculation region in comparison to the 90° case and markedly different in-hole flow physics. The jetting process caused by in-hole separations force the bulk of the jet fluid to issue from the leading half of the streamwise-angled injection hole, as previously reported by Brundage et al. (Tech Rep ASME 99-GT-35, 1999) and predicted by Walters and Leylek (ASME J Turbomach 122:101–112, 2000). The flow structure impacts the skin friction distribution around the holes, resulting in higher near-hole shear stress for a counter-flow supply plenum (jet fluid supplied by a high speed plenum flowing opposite to the free stream direction). In contrast, the counter-flow supply plenum was previously found to have the lowest near-hole wall shear stress for normal injection holes (Peterson and Plesniak in Exp Fluids 37:497–503, 2004b). Streamwise-angled injection generally reduces the near-hole skin friction due to the reduced jet trajectory resulting from the lower wall-normal jet momentum. Far downstream, the skin friction distributions are similar for the two injection angle cases.