Steady and transient shear flows of polystyrene solutions I: Concentration and molecular weight dependence of non-dimensional viscometric functions

Start up from rest and relaxation from steady shear flow experiments have been performed on monodisperse polystyrene solutions with molecular weight ranging from 1.3 × 10⁵ to 1.6 × 10⁶ and concentration c ranging from 5% to 40%. A method of reduced variables based on the use of a characteristic time τ_w is proposed. τ_w is defined as the product of zero shear viscosity with the steady state elastic compliance.Reduced steady and transient viscometric functions so obtained depend on the ratio M/M_e (where M_e is the entanglement molecular weight). Limiting forms are obtained when M/M_e ? 18. In steady flow, a simple correlation is found between shear and normal stresses.In stress relaxation experiments, independent of shear rate, the long-time behaviour can be characterised by a single relaxation time τ₁, which is identical for shear and normal stresses. τ₁ can be simply related to the zero shear rate viscosity and the limiting elastic compliance.     

Shear viscosity of slightly-elastic concentrated suspensions at low and high shear rates

A quasi-static asymptotic analysis is employed to investigate the elastic effects of fluids on the shear viscosity of highly concentrated suspensions at low and high shear rates. First a brief discussion is presented on the difference between a quasi-static analysis and the periodic-dynamic approach. The critical point is based on the different order-of-contact time between particles. By considering the motions between a particle withN near contact point particles in a two-dimensional “cell” structure and incorporating the concept of shear-dependent maximum packing fraction reveals the structural evolution of the suspension under shear and a newly asymptotic framework is devised. In order to separate the influence of different elastic mechanisms, the second-order Rivlin-Ericksen fluid assumption for describing normal-stress coefficients at low shear rates and Harnoy's constitutive equation for accounting for the stress relaxation mechanism at high shear rates are employed. The derived formulation shows that the relative shear viscosity is characterized by a recoverable shear strain,S _R at low shear rates if the second normal-stress difference can be neglected, and Deborah number,De, at high shear rates. The predicted values of the viscosities increase withS _R , but decrease withDe. The role ofS _R in the matrix is more pronounced than that ofDe. These tendencies are significant when the maximum packing fraction is considered to be shear-dependent. The results are consistent with that of Frankel and Acrivos in the case of a Newtonian suspension, except for when the different divergent threshhold is given as [1 ? (Φ/Φ _m )^1/2] ? 1.     

Shear wave velocity in sand considering the effects of frequency based on particle contact theory

Since the shear waves involved in in-situ and laboratory measurement methods vary significantly in terms of the frequency range, it is necessary to consider the effects of frequency on the shear wave velocity. In this study, sand particles are assumed to be spherical solid particles with an equal radius and identical material properties, and sand skeletons are regarded as granular aggregations generated through the random packing of sand particles. It is also assumed that the sand particles only undergo elastic deformation during shear wave propagation. Based on a spherical particle model, a formula is obtained for calculating the shear wave velocity in sand, with the shear wave frequency as an extra influencing parameter. The quantitative calculations demonstrate that the shear wave velocity decreases with an increase of sand porosity, and accelerates with increases of vertical effective stress and elastic modulus of the sand particles. It is also indicated that both the particle density and Poisson’s ratio of the sand particles have negligible effects on the shear wave propagation. The frequency dispersion characteristics of shear wave propagating in sand are also discussed. Moreover, the critical frequency is defined and its analytical expression is derived. The calculation results obtained using the proposed equations agree well with the in-situ measurement results and bender element test data.     

Bulk rheology of dilute suspensions in viscoelastic liquids

A theoretical relation is derived for the bulk stress in dilute suspensions of neutrally buoyant, uniform size, spherical drops in a viscoelastic liquid medium. This is achieved by the classic volume-averaging procedure of Landau and Lifschitz which excludes Brownian motion. The disturbance velocity and pressure fields interior and exterior to a second-order fluid drop suspended in a simple shear flow of another second-order fluid were derived by Peery [9] for small Weissenberg number (We), omitting inertia. The results of the averaging procedure include terms up to orderWe ². The shear viscosity of a suspension of Newtonian droplets in a viscoelastic liquid is derived as $$\eta _{susp} = \eta _0 \left[ {1 + \frac{{5k + 2}}{{2(k + 1)}}\varphi - \frac{{\psi _{10}^2 \dot \gamma ^2 }}{{\eta _0^2 }}\varphi f_1 (k, \varepsilon _0 )} \right],$$ whereη ₀, andω ₁₀ are the viscosity and primary normal stress coefficient of the medium,ε ₀ is a ratio typically between ?0.5 and ?0.86,k is the ratio of viscosities of disperse and continuous phases, and \(\dot \gamma \) is the bulk rate of shear strain. This relation includes, in addition to the Taylor result, a shear-thinning factor (f ₁ > 0) which is associated with the elasticity of the medium. This explains observed trends in relative shear viscosity of suspensions with rigid particles reported by Highgate and Whorlow [6] and with drops reported by Han and King [8]. The expressions (atO (We ²)) for normal-stress coefficients do not include any strain rate dependence; the calculated values of primary normal-stress difference match values observed at very low strain rates.     

Dissipative particle dynamics simulation of polymer drops in a periodic shear flow

The steady-state and transient shear flow dynamics of polymer drops in a microchannel are investigated using the dissipative particle dynamics (DPD) method. The polymer drop is made up of 10% DPD solvent particles and 90% finite extensible non-linear elastic (FENE) bead spring chains, with each chain consisting of 16 beads. The channel’s upper and lower walls are made up of three layers of DPD particles, respectively, perpendicular to Z-axis, and moving in opposite directions to generate the shear flow field. Periodic boundary conditions are implemented in the X and Y directions. With FENE chains, shear thinning and normal stress difference effects are observed. The “colour” method is employed to model immiscible fluids according to Rothman–Keller method; the χ-parameters in Flory–Huggins-type models are also analysed accordingly. The interfacial tension is computed using the Irving–Kirkwood equation. For polymer drops in a steady-state shear field, the relationship between the deformation parameter (D_def) and the capillary number (Ca) can be delineated into a linear and nonlinear regime, in qualitative agreement with experimental results of Guido et al. [J. Rheol. 42 (2) (1998) 395]. In the present study, Ca<0.22, in the linear regime. As the shear rate increases further, the drop elongates; a sufficiently deformed drop will break up; and a possible coalescence may occur for two neighbouring drops. Dynamical equilibrium between break-up and coalescence results in a steady-state average droplet-size distribution. In a shear reversal flow, an elongated and oriented polymer drop retracts towards a roughly spherical shape, with a decrease in the first normal stress difference. The polymer drop is found to undergo a tumbling mode at high Schmidt numbers. A stress analysis shows that the stress response is different from that of a suspension of solid spheres. An overshoot in the strain is observed for the polymer drop under extension due to the memory of the FENE chains.     

Viscosity,first normal-stress coefficient,and molecular stretching in dilute polymer solutions

Warner's numerical method for finitely extensible nonlinear elastic (“FENE”) dumbbells in a dilute suspension undergoing steady-state shear flow has been improved by assuming a form of the distribution function that removes the singularity at R = 0 and improves the behavior of the weight function in the Galerkin expression for large b. The comparison of the results of the present method to those of Christiansen and Bird's extrapolation and Warner's numerical solution indicates the success of this method. The material functions, the dumbbell elongation, and the distribution function for steady-state shear flow are given. In addition the newly obtained results are used to assess the accuracy of two approximate methods referred to as the FENE-P model and the FENE-P-B model.     

Orientation and rheology of rodlike particles with strong Brownian diffusion in a second-order fluid under simple shear flow

An analysis of orientation in a dilute suspension of rodlike macromolecules in a second-order fluid is presented and the effect of the elasticity of the fluid on the orientation of the suspended particles is examined. Distributions of particle orientation under a simple shear flow have been obtained for small β where β is the ratio of the intrinsic relaxation time of the fluid to the rotational relaxation time of the particle, the latter being inversely proportional to the Brownian rotation diffusion coefficient D_r of the particle. The parameter β represents also the ratio of the Weissenberg number of the fluid to the non-dimensional shear rate, g/D_r. An expression of the stress tensor of the suspension is derived and used in conjuntion with the orientation distribution to obtain the rheological properties of the mixture subjected to a simple shear.     

Experiments on sedimentation beneath downward-facing inclined walls

It has been known since Boycott (1920) that sedimentation rates can be several times higher within inclined walls than in vertical vessels. Beneath the downward-facing inclined wall a thin boundary-layer of clear liquid is formed with an upward velocity. Mass continuity brings about the observed effect. Ponder (1925), and Nakamura & Kuroda (1937) proposed a global kinematic model without any detail of the flow field to predict the sedimentation rate. Recently, two asymptotic theories for describing the flow field became available. The investigation by Acrivos & Herbolzheimer (1979) requires both Re²/Gr and Re⁴/Gr to be small, with Re being a sedimentation Reynolds number and Gr a sedimentation Grashof number. The analysis due to Schneider (1982) is valid for small values of Re²/Gr and large values of Re⁴/Gr. This paper presents experimental sedimentation data in a²/Gr and large values of symmetrical, roof-shaped vessel. The experiments with spherical glass beads in a variable mixture of glycerine and water cover the whole parameter range from very small to very large values of Re⁴/Gr. Therefore, both theories could be verified. In the case of small values of Re⁴/Gr, strong waves beneath the downward-facing wall were observed and graphically evaluated in order to find a basis for further theoretical approach. Furthermore, an upward flow of particles in a sublayer of the boundary-layer leads te protruding “borns” and an oscillation of the originally horizontal discontinuity separating the suspension from the clear liquid. The measured particle distribution and deviations from a monodispersed suspension seems to be a possible explanation for this effect, which is investigated theoretically in a separate paper (Schaflinger 1984). A further experimental shortcoming, the shallow depth of the settling vessel, that was necessary for an optoelectronic measuring of the actual volume fraction solids within the bulk, may affect the convective motion when the sedimentation Reynols number is small. However, in this case the experimental data are in conformity with the theoretical predictions which do not consider the influence of wall friction on the sedimentation behaviour.     

Spatially periodic suspensions of convex particles in linear shear flows. II. Rheology

Following the purely kinematical developments of Part 1, a rigorous analysis is presented of the “almost” time-periodic low Reynolds number hydrodynamics of a spatially periodic suspension of identical convex particles in a Newtonian liquid undergoing a macroscopically homogeneous linear shear flow. By considering the case of a single particle within a unit cell of the instantaneous spatially periodic configuration, the quasistatic dynamical analysis of this infinite-particle system is effected in much the same way as for a single particle suspended in an unbounded fluid. This is accomplished via the introduction of a partitioned hydrodynamic Stokes resistance matrix, linearly relating the force, couple and stresslet on the particle in the unit cell to the translational and rotational particle-(mean) suspension slip velocities and the mean rate-of-strain of the suspension. In contrast with the unbounded fluid case for a given geometry of the individual particles, the (purely geometric) elements of the resistance matrix depend upon the instantaneous lattice configuration.These dynamic quasistatic calculations for a given instantaneous lattice conformation, in particular that for the stresslet, are then appropriately averaged over both space and time for the class of almost time-periodic, lattice-reproducing, flows discussed in Part I. (In actually performing the time average, an important distinction is drawn between the ergodic and deterministic shear processes whose kinematical basis was laid in Part I.) In turn, this averaged dynamical information is translated into knowledge of the rheological properties of the macroscopically homogeneous suspension.A rigorous asymptotic, lubrication-theory analysis is performed during the course of an illustrative calculation of the rheological properties of a concentrated suspension of almost-touching spheres in a simple shear flow. Contrary to the findings of a previous heuristic treatment of this same lubrication-theory problem—one that ignores evolutionary variations in the instantaneous geometrical configuration of the spatially periodic suspension as the shear proceeds—the time-average properties of the suspension are found to be nonsingular in the limit.Finally, brief comments are offered on potential extensions of the scheme to include nonlinear phenomena, such as nonNewtonian fluids and inertial effects.     

Structure and properties of sheared fiber suspensions with mechanical contacts

The properties of fiber suspensions are highly sensitive to the suspension microstructure. In dilute or semi-dilute suspensions, nL²d≪1, the fibers' orientation distribution is controlled by hydrodynamic interactions among the fibers. However, direct mechanical contacts among the fibers play an important role in semi-concentrated suspensions, nL²d=O(1). Here, n is the number of fibers per unit volume, L is the fiber length and d is the fiber diameter. We have performed dynamic simulations of fiber suspensions including contact forces that prevent any two fibers from passing through one another. Collisions between the fibers cause them to flip more frequently in the shear flow, leading to a spread of the orientation distribution away from the flow direction. Both this increased orientational dispersion and the direct stress transmitted through the contacts enhance the shear viscosity of the suspension significantly. The contacts also give rise to normal stress differences. The results of the simulation are compared with experiments and the relative importance of contacts and hydrodynamic interactions is discussed.     

Mean-field hydrodynamics brownian dynamics simulations of stabilized colloidal liquids under shear

We extend a recently proposed mean-field hydrodynamics (MFH) discrete element simulation technique to consider the effects of a shear velocity profile on a model colloidal liquid containing monodisperse spherical particles in the non-newtonian shear thinning regime. The MFH method adapts Ermak's free draining brownian dynamics algorithm to include a local density approximation for the friction coefficient, semiempirically parametrized to reproduce the experimentally determined short-time diffusion coefficient. We have also generalized further the previous treatment to allow for a friction coefficient that is dependent on local density anisotropy. The behaviour of Ermak's equations of motion and also the “isotropic” and “anisotropic” MFH schemes with shear flow are compared. We show that, at equilibrium, the MFH approaches generate the same static averages as Ermak's method, and give good agreement with the Percus-Yevick prediction for hard-sphere structure factors using an r⁻³⁶ soft-sphere interaction. However, under shear, the three equations of motion give quite different rheological behaviour. The MFH methods produce higher viscosities, although the structures remain similar (e.g. all give a “string” phase) but at different shear rates. Variation in the specific details of the MFH equations of motion can promote or delay the development of long-range order with Péclet number.     

Hydrodynamic diffusivity of spherical particles in polymer solution

In this research experiments were performed to examine the hydrodynamic diffusion of spherical particles in a highly filled suspension. The suspension consisted of nearly monodisperse polymethylmethacrylate spheres in a density matched polymer solution. The polymer solution was prepared by dissolving 0–700 ppm of polyacrylamide in a mixture of ethyleneglycol and glycerine. The polymer solution did not show appreciable shear thinning. The particle loading was varied from 30 to 55%. The hydrodynamic diffusivity was estimated by measuring the time-dependent viscosity when the suspension was subjected to a circular Couette flow with an air bubble trapped under the rotor of the Couette apparatus. The results show that the dimensionless diffusivity (D/γ˙a ²) of particles in polymer solution is not proportional to shear rate (γ˙), as in the case of a Newtonian fluid, but that it decreases with increasing shear rate. The diffusivity also decreases with increasing polymer concentration. It is suggested that the elongational thickening behaviour and the increased lubrication force due to the first normal stress difference may be responsible for the reduction of diffusivity in the polymer solution. Received: 18 January 2000 Accepted: 6 April 2000     

Stochastic functional expansion in elasticity of heterogeneous solids

The Volterra-Wiener functional expansion is employed to the analysis of statistic properties for random heterogeneous solids. For simplicity, the technique is displayed on an elastic suspension of spheres. The basis function in the expansion is chosen as that corresponding to the so-called “perfect disorder” of spheres (PDS), recently introduced by the authors. An infinite hierarchy of equations for the kernels in the expansion is derived whose truncating after the nth equation is shown to yield results for the averaged statistical characteristics which are valid to order cⁿ_f, where c_f is the volume fraction of the spheres. The kernels for the first and the second approximations, n = 1, 2, are found and related to the displacement fields in an infinite elastic body containing, respectively, one and two spherical inhomogeneities. Within the frame of the so-called singular approximation the overall tensor of elastic moduli for a suspension of perfectly disordered spheres is shown to coincide to the order c²_f with a formula, earlier obtained by means of the method of the effective field.     

Rubber-like behaviour of poly(ethylene oxide) solution in elongational flow

The behaviour of an aqueous poly(ethylene oxide) sucrose solution and of a suspension of glass beads in a similar solution has been examined in elongational flow using a spinline rheometer. Over the accessible strain-rate range of ca. 1 to 10 s^?1 these fluids behaved essentially as elastic materials whereas, at similar strain rates in shear, they show shear-thinning behaviour.     

Computation of rheological properties of suspensions of rigid rods: stress growth after inception of steady shear flow

The orientation distribution and stress growth for a suspension of rigid rods (or dumbbells) in a Newtonian solvent are calculated for inception of steady shear flow. Galerkin's method, with spherical harmonics as trial functions, is used in the spatial coordinates to obtain a system of ordinary differential equations in time which is solved by the spectral method. The method is applicable over a wide range of dimensionless shear rates (Peclet numbers) and has been coded with standard system-solvent and eigensystem packages. For sufficiently large Peclet numbers, the results give the well known rigid-dumbbell prediction of an overshoot in the shear viscosity and normal stress differences. This overshoot is then followed by an undershoot. An explicit analytical approximation for the fluid stresses is presented which is reasonably accurate for Peclet numbers less than unity.     

A critical state sand model with elastic–plastic coupling

Soil elastic moduli are highly pressure-dependent. Experimental findings have indicated that the elastic shear modulus of sands depends on p^χ, where p is mean principal effective stress and χ is a non-dimensional parameter. χ practically remains unchanged for shear strains less than 10⁻⁵ where the mechanical behavior is purely elastic. However, experiments have revealed that the emergence of plasticity for shear strains larger than 10⁻⁵ provokes a gradual increase in χ. Technically, this observation is an elastic–plastic coupling effect in which plasticity causes to change the elastic characteristics. Here, this issue is considered in hyper-elasticity framework in conjunction with a critical state compatible bounding surface plasticity platform for granular soils. To this aim, constitutive equations linking χ to a proper kinematic hardening parameter are presented. Then, using the proposed approach, a hyper-elastic theory is modified to consider the mentioned elastic–plastic coupling effect in the whole domain of the elastoplastic behavior. Adopting the improved hyper-elasticity necessitates the modification of a number of basic plasticity platform elements. In this regard, dilatancy and plastic hardening modulus of the bounding surface platform are modified. Successful performance of the modified constitutive model is presented against experimental data of loading/unloading triaxial tests.     

Droplet deformation of a strongly shear thinning dense suspension of polymeric micelles

We investigated the deformation of a strong shear thinning droplet undergoing simple shear flow in a Newtonian liquid. The droplet was an aqueous solution of poly(ethylene oxide) end capped with an alkyl group that forms spherical micelles in aqueous solution. At high concentrations and below a critical temperature, the jammed micelles showed strong shear thinning behaviour, and neither a yield stress nor a Newtonian viscosity was observed. At small shear rates, the droplet rotated and aligned in the flow, but did not deform or only very weakly. At high shear rates, the droplet deformation increased with increasing shear rate. The deformed droplet did not relax after the shear was stopped except for a modest rounding of the edges. For each shear rate, an apparent viscosity, η _ad, of the equivalent Newtonian droplet was calculated assuming affine deformation. η _ad showed a power law dependence on the capillary number Ca with an exponent of − 1.8 and was larger than the shear viscosity of the micelle suspension at the same shear rates. The results were explained by the existence of a strong gradient of the viscosity inside the droplet leading to a very low viscosity fluid layer near the droplet/matrix interface.     

On the elastic properties of model suspensions as investigated by creep recovery measurements in shear

 The elastic properties of model suspensions with spherical monodisperse hydrophilic glass spheres that were dispersed in a Newtonian liquid were determined in creep and creep recovery measurements in shear with a magnetic bearing torsional creep rheometer. The creep and creep recovery measurements were performed depending on the applied level of shear stresses ranging from 0.19 Pa to 200 Pa. Since the recoverable creep compliances of the chosen suspending medium (i.e. a low molecular weight polyisobutylene) were far below the lower limit of the resolution of the creep rheometer it can be considered to behave as purely viscous. By applying a large shear stress in the creep tests the investigated suspensions with a volume fraction of Φ _t =0.35 behave as Newtonian liquids, too. For these suspensions no significant recoverable creep compliances could be detected, as well. In contrast to the Newtonian state of suspensions at high shear stresses, where a shear induced ordering of the particles can be expected, a non-Newtonian behaviour arises by applying a very low shear stress in the creep test. In this state large recoverable creep compliances were detected for the suspensions. The magnitude of the recoverable creep compliances of the suspensions exceeded the largest creep compliances of polymer melts that are reported in the literature by more than two decades. From the results obtained by creep recovery measurements with a magnetic bearing torsional creep rheometer it can clearly be concluded that the particle structure present in the chosen model suspension gives rise to a pronounced elasticity. Received: 21 November 2000 Accepted: 12 July 2001     

Load enhancement effects by polymer-thickened oils in strip squeeze film flow

A modification to existing equipment is described which permits continuous squeeze-film flow to be obtained between parallel-sided strips of material rather than between disc-shaped surface. Squeeze film flow is simulated by having liquid move through one of the surfaces via an array of equispaced holes. The Squeeze-film behavior of a Newtonian base oil is first tested at temperatures of 24°C and 55°C. It is shwon that loads are in reasonable agreement with theretical predictions and that end effects (corrected by means of a guard ring) and fluid maldistribution effects are of small proporitons. At the very highest liquid flowrates, the rapid liquid flow through the holes may influence the measured load. The Polymer-thickened oils, representinhg 10 W/30 and 10 W/50 motor oils, are tested at temperatures of 24°C and 55°C. Both oils five marked load enhancement, compared with Newtonian oil under similar flow conditions, at the higher flowrates used. The 10 W/50 oil gives load enhancement of 76 per cent at 55°C, increasing rapidly with the simulated approach velocity. Fluid inertia effects in the squeeze film flow aslo increase the load significantly. The results confirm earlier data using disc-shaped surfaces; load enhancement is greater in the present work on strip squeeze films because the fluid deformation rates are greater (2000 s^t-1 in planar extention and 2 x 10⁵ s^t-1 in shear). It is suggested that the fuel consumption of cars could be improved by the development of elastic, shear stable oils of lower viscosity than those currently in use.     

Turbulence production in flow over a wavy wall

Measurements of the spatial and time variation of two components of the velocity have been made over a sinusoidal solid wavy boundary with a height to length ratio of 2a/λ = 0.10 and with a dimensionless wave number of α⁺ = (2π/λ)(v/u ^?) = 0.02. For these conditions, both intermittent and time-mean flow reversals are observed near the troughs of the waves. Statistical quantities that are determined are the mean streamwise and normal velocities, the root-meansquare of the fluctuations of the streamwise and normal velocities, and the Reynolds shear stresses. Turbulence production is calculated from these measurements. The flow is characterized by an outer flow and by an inner flow extending to a distance of about α^?1 from the mean level of the surface. Turbulence production in the inner region is fundamentally different from flow over a flat surface in that it is mainly associated with a shear layer that separates from the back of the wave. Flow close to the surface is best described by an interaction between the shear layer and the wall, which produces a retarded zone and a boundary-layer with large wall shear stresses. Measurements of the outer flow compare favorably with measurements over a flat wall if velocities are made dimensionless by a friction velocity defined with a shear stress obtained by extrapolating measurements of the Reynolds stress to the mean levels of the surface (rather than from the drag on the wall).