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We present an inelastic model of non-Brownian suspension behaviour based on the persistence of straining concept of Thompson and Souza Mendes (TSM model). It is used to compare with experiments and computations in steady shearing, uniaxial and planar elongation and shear reversal flows. Some remarks about small-strain motions are also given. The agreement is good in general, and the important influence of interparticle friction is shown. Friction has been incorporated in the model via a bootstrap concept.

The rheology of noncolloidal suspensions in superposed simple shearing and oscillatory shearing was explored. With a Newtonian matrix fluid, one would expect that G′ would be zero in an oscillatory flow, but this was not found; the action of Coulomb friction between the particles appears to cause an increment of G′ at lower frequencies. To understand this frictional effect, measurements of small and medium strain oscillatory flows, up to 10% strain magnitude, were made. The matrix fluid was 12 Pa s silicone oil, and the polystyrene spheres were on average 40.3 μm in diameter. Hysteresis during tests with varying strain amplitudes was more dominant in the storage modulus than in the loss modulus, and, at a 50% volume fraction, the effect was severe. Because of the observed tendency to hysteresis, the oscillatory flow was then combined with a parallel steady shear flow to try to control or eliminate hysteresis. The hysteresis appears to be a frictional effect, and it was reduced under superposed shearing. The effect of variable oscillatory shear stress and steady shear stress was studied, and a model was proposed for the superposed storage modulus, loss modulus, and shear viscosity responses. Frictional effects are considered in the proposed model, and one observes a generally satisfactory fit to the experimental data. From the model, the average friction coefficient is shown to be less at higher frequencies due to higher relative rubbing speeds and better lubrication between the particles. Clearly, suspension rheology is dominated by friction and is essentially a study in tribology.

This work explores the rheology of suspensions of spheres in a viscoelastic matrix. The volume fraction varies from zero to 0.2. Steady shearing, small strain sinusoidal shearing and uniaxial elongation are considered. We conclude that the matrix response in shear and elongation is fairly well described by a single-mode Oldroyd-B model, but the small-strain storage modulus (G’) response is less well represented by such a model. The use of a two-mode Oldroyd-B model gives significant improvement. For the suspensions, the viscometric functions η, N1 and N2 are given for volume fractions of 5, 10 and 20%, plus the oscillatory responses G’, G” at 1% strain amplitude. The uniaxial elongation data show a very large increase in flow resistance relative to the matrix; for the same applied force, the rate of elongation decreases from about 17.5 s−1 for the matrix to about 3 s−1 for the suspensions. It appears that this large increase in resistance is due to areas of intense extension attached to two adjacent spheres, as has been demonstrated numerically. It is shown that a single-mode Oldroyd-B model cannot describe suspension behaviour. A two-mode Oldroyd-B model can capture the macroscopic behaviour of the suspensions but only if an initial Hencky strain of order 4 is present in the extending suspension filaments. A two-mode model of the matrix fluid also allows one to understand the suspension response from the microscopic view.

This review deals with non-Brownian (noncolloidal) suspension rheology; experimental and computational works are compared where possible. The matrix fluids are non-Newtonian, and the rigid particles have an aspect ratio close to one. Volume fractions of 0.5 and below are considered. Shearing and extensional flows are discussed; the former are fairly well understood but the latter are not prominent in the literature. Unsteady and oscillatory flows are surveyed. A comparison of Newtonian and viscoelastic suspension rheology is made, and some aspects of finding constitutive models for these suspensions are discussed. While progress has been made, it appears that satisfying agreement between computation and experiment is rare. More attention to rheological and frictional modeling is needed, and improved computational methods need to be developed.

The rheology of non-colloidal suspensions of spheres in a viscoelastic matrix, including the viscometric, G′, G′ and uniaxial extensional responses, is explored. Volume fractions of 50% and less were used. By recognizing that filament stretching between beads begins at a distance comparable to the sphere diameter the unexpected large stresses seen in the uniaxial extension experiments can be understood, and a model is presented. In the experiments there is little variation of the dimensionless elongational stresses with the volume fraction ϕ. For a volume fraction of 50% the behaviour was different from the other concentrations. An essential finding is that a single-mode model of the Oldroyd-B type is not able to describe the elongational experiments at all-at least two modes are needed. There is clearly a difference between behaviour in elongation and in shear with regards to the action of interparticle friction. With negligible rotation of the particles in elongation the kind of vigorous interparticle rubbing possible in shear does not happen.

The role of friction in non-colloidal suspensions is examined with a model which splits the viscosity into a frictionless component (τ*) plus a frictional component which depends on the ratio of the particle pressure (P) to the shear stress (τ). The model needs the input by computation of τ* and P and a suitable choice of particle friction coefficient (μ). It can be extended to elongational flows and cases where sphere roughness is important; volume fractions up to 0.5 are considered. It is shown that friction acts in a feedback or “bootstrap” manner to increase the suspension viscosity. The analysis is also useful for deducing the friction coefficient in suspensions from experimental data. It was applied to several sets of experimental data and reasonable correlations of the viscosities were demonstrated. An example of the correlation for spheres in a silicone oil is shown for volume fractions 0.1–0.5.

Professor R. Byron Bird used Oldroyd models of rheological behavior over a span of around 50 years. In this paper, it is suggested that a modified Oldroyd-B model can also be used to describe the rheology of non-colloidal suspensions that have a viscoelastic matrix.

It would be convenient to have a model, albeit approximate, of particle-laden materials (suspensions) that would not need large amounts of computing and/or experimentation to implement for design purposes. There are now adequate models of the pure matrix fluid behaviour, but there are no such models for suspensions with large particles (non-colloidal suspensions). One of the obstacles has been the single-minded devotion to shearing motions of suspensions; experience with the matrix modelling has shown that it is not possible to formulate widely usable models if only shear is considered. Here some new results of axially symmetric elongational tests on suspensions are compared with shearing data. Some suggestions for modelling these and other observations based on using strain rate and strain in a modified Reiner-Rivlin constitutive equation are presented. The model generally works quite well, but it does not predict the positive storage modulus seen in small and medium amplitude oscillatory shear flows.

The rheology of a system must be explored not only in viscometric flows, but also in other flow classes, and so, we present some results for the axisymmetric elongational flow of non-colloidal suspensions of spheres. We compare our results with data from shear flows using the same matrices and spheres. We have experimented with non-colloidal suspensions of 40-μm diameter polystyrene spheres with volume fractions (ϕ) varying from 0.3 to 0.5. Two matrix fluids were used—one was a near-Newtonian polydimethyl siloxane of 12 Pa-s viscosity and the other was a variant of the M1 Boger fluid sample of Sridhar which we call M1*. We did not find that the Trouton ratio for either of these fluids was 3; generally, the ratio was larger. We investigated the role of sphere roughness using spheres roughened to 5.3 % of the radius in a 50 % suspension in silicone oil and found an increase of elongational viscosity of about 65 % which is comparable with the 60 % increase in shear viscosity with roughness noted previously. For the silicone oil matrix, we found no rate effect, with very little strain-hardening. By contrast, the M1-type matrix suspensions showed strain-hardening and an increase of elongational viscosity with elongation rate.

We study the rheology of a non-colloidal suspension of rigid spherical particles interacting with a viscoelastic matrix. Three-dimensional numerical simulations under shear flow are performed using the smoothed particle hydrodynamics method and compared with experimental data available in the literature using different constant-viscosity elastic Boger fluids. The rheological properties of the Boger matrices are matched in simulation under viscometric flow conditions. Suspension rheology under dilute to semi-concentrated conditions (i.e. up to solid volume fraction $\unicode[STIX]{x1D719}=0.3$) is explored. It is found that at small Deborah numbers $De$ (based on the macroscopic imposed shear rate), relative suspension viscosities $\unicode[STIX]{x1D702}_{r}$ exhibit a plateau at every concentration investigated. By increasing $De$, shear thickening is observed, which is related to the extensional thickening of the underlying viscoelastic matrix. Under dilute conditions ($\unicode[STIX]{x1D719}=0.05$), numerical results for $\unicode[STIX]{x1D702}_{r}$ agree quantitatively with experimental data in both the $De$ and $\unicode[STIX]{x1D719}$ dependences. Even under dilute conditions, simulations of full many-particle systems with no a priori specification of their spatial distribution need to be considered to recover precisely experimental values. By increasing the solid volume fraction towards $\unicode[STIX]{x1D719}=0.3$, despite the fact that the trend is well captured, the agreement remains qualitative with discrepancies arising in the absolute values of $\unicode[STIX]{x1D702}_{r}$ obtained from simulations and experiments but also with large deviations existing among different experiments. With regard to the specific mechanism of elastic thickening, the microstructural analysis shows that elastic thickening correlates well with the average viscoelastic dissipation function $\unicode[STIX]{x1D703}^{elast}$, requiring a scaling as $\langle \unicode[STIX]{x1D703}^{elast}\rangle \sim De^{\unicode[STIX]{x1D6FC}}$ with $\unicode[STIX]{x1D6FC}\geqslant 2$ to take place. Locally, despite the fact that regions of large polymer stretching (and viscoelastic dissipation) can occur everywhere in the domain, flow regions uniquely responsible for the elastic thickening are well correlated to areas with significant extensional component.

The response of non-colloidal suspensions to sudden strains and sudden changes of strain rate is explored. The matrix fluids were silicone oil and glycerol/water. A Oldroyd-B model of the suspensions augmented by a friction element gives a description of the shear stresses which agrees with experiment. Friction becomes dominant for volume fractions greater than 0.3.

The steady, creeping and isothermal sedimentation of a rigid sphere in an incompressible viscoelastic fluid which follows the exponential Phan–Thien and Tanner constitutive equation is studied analytically. The solution of the governing equations is expanded as a regular perturbation series for small values of the Deborah number, and the resulting sequence of two-dimensional partial differential equations is solved analytically up to eighth order. Although the domain is unbounded, the solution is able to resolve features of the flow that cannot be revealed by a low-order theoretical analysis, such as very fine flow structures and a stress boundary layer close to the surface of the sphere. The calculation of the drag force exerted on the sphere by the fluid was done by developing two formulas. The first is based on the flow-field close to the sphere and the second is based on the far flow-field. Both formulas produce the same analytical expressions verifying the correctness and consistency of the series solution. At small Deborah numbers, a decrease of the drag force is predicted (i.e., an increase of the sedimentation velocity), followed by a significant drag enhancement at higher Deborah numbers. Investigation of the solution for the existence of a negative wake close to the rear stagnation point did not reveal such a phenomenon when physical constraints on the solutions were posed. On the contrary, a negative wake is predicted only as an artifact due to the loss of positive definiteness of the conformation tensor. It is also demonstrated that the effect of viscoelasticity is maximized in a region around the sphere with a radius which is about ten times the radius of the sphere. Last, it is shown that the fluid disturbances due to the viscoelasticity of the matrix fluid decay very slowly with the distance from the sphere, depending on the magnitude of the Deborah number. Streamlines, extra-stress and pressure contours as well as the extension of the polymer molecules are also presented and discussed.

We explore the effect of deliberately increased particle roughness on the rheology of noncolloidal suspensions of spheres, both in Newtonian (polydimethylsiloxane or silicone oil) and non-Newtonian (Boger fluid) matrices. The object of the experiment is to change only the roughness of the spheres, while leaving the density and the material of the particles unchanged, so as to isolate the effect of roughness on rheology. Two sphere materials, polystyrene (PS) and polymethylmethacrylate (PMMA) were used. The PS spheres were of 40 and 80 μm nominal diameters, and the PMMA spheres were 40 μm in diameter. Roughness ratios (average roughness/sphere radius) of 0.1%–5% were explored. With silicone matrices, there was up to 50% increase in viscosity with a 50% volume fraction suspension and an increase in the normal stress differences of a similar magnitude. Two polybutene-based Boger fluids were also used. The increases of viscosity with the polybutene matrices were somewhat larger than those with the Newtonian matrix; at 40% volume concentration, we saw approximately a 35% increase in viscosity with a roughness ratio of 5.3%. We compared the experimental results with computations for spheres in Newtonian matrices, and we found reasonable agreement with the computations of Mari et al. [J. Rheol. 58, 1693–1724 (2014)] if a friction coefficient of about 0.5 was assumed. We conclude that friction and roughness must be considered in computational work, or no agreement with experiment will be found. We suggest that the shear-thinning seen with Newtonian matrices is due to a lessening of friction with shear rate. We also show that the unexpected success of the Maron–Pierce formula for Newtonian suspensions is due to the fact that it mimics well a frictional suspension with a friction coefficient of ∼0.5.

We explore the rheology of corn syrup and non-colloidal suspensions of spheres in matrices based on corn syrup. We also comment on the effect of sphere roughness on suspension rheology. Two sphere materials, polystyrene (PS) and polymethyl methacrylate (PMMA), were used. The spheres were of 40 μm nominal diameter. Roughness ratios (average roughness/sphere radius) of 0.15 to 5 % were explored. The viscosity increase with the corn syrup matrix was very large and appears to be due to drying of the corn syrup at the free surfaces of the sample. A corn syrup/glycerine/polyacrylamide Boger fluid was also used. The increases of viscosity with the Boger fluid matrix were again large with the roughened spheres. This appears to be due to the deformation of the long polyacrylamide molecules in the narrow channels between the roughened spheres.

This paper presents a current view of non-Brownian (non-colloidal) suspension rheology; experimental and computational works are compared where possible. The matrix fluids are either Newtonian or viscoelastic; the rigid particles are compact and close to spherical. Volume fractions of 0.5 and below are considered. It is important to move beyond steady simple shearing, so unsteady shearing and uniaxial extensional flows are discussed. Steady shearing is fairly well understood but the extensional flows are not prominent in the literature. In steady shearing with Newtonian matrices the role of friction and the estimation of the average friction coefficient, which reduces as the macroscopic shear stress (or shear rate) increases, is discussed. The ratio of the two normal stress differences to the shear stress is modelled well provided a reasonable value for the average friction coefficient is assumed. With viscoelastic matrices agreement between experiment and theory is less satisfactory. Unsteady and oscillatory flows are surveyed briefly; the prominent hysteresis in oscillatory flows appears to be a frictional effect. While progress has occurred it appears that satisfying agreement between computation and experiment is rare. More attention to rheological and frictional modelling is needed, and improved computational methods may need to be developed.

The rheology of high volume fraction (30–40%) suspensions with hollow glass microspheres (radius r = 15 µm) in a matrix of silicone oil has been studied experimentally. The surface of the hollow glass microspheres was roughened by wet chemical processing, using hydrochloric (HCl) acid and also sodium hydroxide etching. We obtained particles of various roughnesses in the range of Ra = 62–655 nm. The effect of particle roughness on the overall rheology is investigated, demonstrating its clear impact. The effect depends on the volume fraction, the magnitude of roughness and shear rate. At a 30% volume fraction, increasing the roughness ratio (Ra/r) from 0.42% to 4.44% does not result in a measurable change in viscosity. At a higher volume fraction of 40%, even a modest increase of the roughness ratio to 1.5% causes a significant increase in the viscosity of 117%, 97%, 50% and 39%, respectively at shear rates of 0.47, 1, 10 and 100 s−1. Increasing the roughness to 4.44% results in an increase of 272%, 105%, 41%, 23% in viscosity over the same range of shear rates. In all cases, we show that the effect of roughness diminished by increasing the shear rate. Some estimates of the average interparticle coefficient of friction are given- showing roughness increases friction. The friction coefficient is shown to increase with the volume fraction. Friction coefficient shows a power-law dependence on shear rate and decreases with the shear rate. The rate of decrease with shear rate is smaller for higher volume fraction of 40%.

We present the results of measuring the viscometric flow of concentrated rigid-sphere suspensions with constant-viscosity matrices, both Newtonian (silicone oil) and non-Newtonian (Boger fluid: corn syrup (79.4%)–glycerin (19.8%)–4%PAA water (0.78%)). Various problems with the measurements are discussed, including settling due to density differences. The effect of sphere roughness on shear rheology is a main concern of the work. The viscosities of silicone oil and Boger fluid suspensions were increased by increasing sphere roughness, and the normal stress difference N1 − N2 in the Boger fluid suspension also increased with roughness.

This paper considers the relation between theory, computation and new experiments in noncolloidal suspensions with Newtonian and viscoelastic matrices in shearing flow. One expects from creeping flow theory that a suspension of rigid spheres in a Newtonian fluid matrix at negligible Reynolds number and large Peclet number would exhibit a constant shear viscosity. However, one usually sees shear-thinning behaviour in experiments for larger volume fractions (φ > 0.3). Experiments with a Boger fluid matrix of constant shear viscosity on the other hand, show mild shear thickening at higher shear rates. Explanations for this behaviour are unsatisfying. There are also puzzles for the normal stresses in sheared suspensions. For a suspension with a Newtonian matrix in a simple shear flow various workers have found experimentally that the bulk stress field is no longer a shear plus an isotropic pressure, and that both first and second normal stress differences (N1 and N2) occur and are negative. Our own experiments using an open semi-circular trough to find N2 agree reasonably well with previous experimental results and the Brady–Morris theory of 1997. However, this theory predicts that N1 is zero, whereas it is measured to be negative but smaller in magnitude than N2. The viscometric functions (η, N1 and N2) for non-colloidal suspensions of spheres in a Boger fluid matrix were also measured. Volume fractions (ϕ) from 5% up to 40% were investigated. The relative viscosity (ηr = η/η0) and the (positive) first normal stress difference N1 showed increases with ϕ which were larger than the dilute suspension theory predictions of 1 + 2.5ϕ, indicating semi-dilute suspension behaviour, even down to 5% concentration. A major concern however centres on the second normal stress difference N2. The Boger matrix fluid showed a zero second normal stress difference, and the measurements showed that N2 was always negative for the suspensions. This agrees with the dilute suspension theoretical prediction found using the Landau–Lifshitz averaging procedure, but not with the result from the ensemble averaging method, which predicts a positive N2. The reason for this predictive failure is suggested to lie in the treatment of the stresses induced by the far distant spheres. It is suggested that using a Brinkman approach to model the effect of the far spheres would be useful. For larger concentrations (up to 0.4) the second normal stress difference was always negative. However, we saw a switch from a positive N1 to a negative value when the volume fraction exceeded 0.3, similar to the results of Aral and Kalyon (1997). This can be understood as a contest between the component of (positive) N1 due to the matrix properties and the influence of the component of (negative) N1 due to the proximity of the spheres. A suggestion for a constitutive model based on this idea is given.

Experiments and computations were carried out to explore the origins of shear-thinning in non-colloidal suspensions. Two grades of polydimethylsiloxane (silicone oil) and a glycerine/water mixture were used as matrices for the suspensions. The particles were 40 μm diameter polystyrene (PS) and polymethyl methacrylate (PMMA) spheres. We concentrated on 40% volume fraction suspensions where shear-thinning was clear. The silicone oil matrices were nearly Newtonian: at 24 °C the viscosity of the 1.15 Pa s sample showed a 2% drop in viscosity a shear rate of about 3000 s−1, the 13.2 Pa s sample showed a drop of 2% at a shear rate of approximately 100 s−1, and the glycerine/water sample appeared to be Newtonian at least up to 104 s−1. Mild shear-thinning was seen with all suspensions, beginning at shear rates of order 0.1–1 s−1, followed by a rapid reduction of torque in the parallel-plate system at shear rates of 14, 150 and 1000 s−1 respectively with the three matrices. These rapid reductions are ascribed to edge effects. Matching smoothed particle hydrodynamics (SPH) simulations were made. The silicone matrix viscosities were modelled by a Carreau–Yasuda (CY) fit up to shear rates of order 107 s−1. The agreement between computations and experiments is generally good for 40% volume fraction suspensions up to the shear rate where edge effects intervene in the experiments- there are no edge effects in the simulations. This confirms the suggestion by Vazquez-Quesada et al. [1] that ‘hidden’ high shear rates between particles, where the non-Newtonian matrix viscosity comes into play, can result in shear-thinning at the macroscopic level. For the glycerine/water matrix at low shear rates this mechanism does not apply and a separate mechanism based on variable interparticle friction is suggested; the two mechanisms can co-exist.

We present a model for the shear viscosity of non-colloidal suspensions with Newtonian matrix fluids. The model is based on the original idea first presented by Brinkman (Applied Sci Research A1:27-34. 1947) for the viscous force exerted by a flowing fluid on a dense swarm of spherical particles. In particular, we consider an inertialess suspension in which the mean flow is driven by a pressure difference, and simultaneously, the suspension is subject to simple shear. Assuming steady state, incompressibility and taking into account a resistance force which is generated due to the presence of the particles in the flow, the three-dimensional governing equations for the mean flow around a single spherical particle are solved analytically. Self-consistency of the model provides a relationship between the resistance parameter and the volume fraction of the solid phase. A volume, or an ensemble, averaging of the total stress gives the bulk properties and an expression for the relative (bulk) viscosity of the suspension. The viscosity expression reduces to the Einstein limit for dilute suspensions and agrees well with empirical formulas from the literature in the semi-dilute and concentrated regimes. Since the model is based on a single particle and its average interaction with the other particles is isotropic, no normal stress differences can be predicted. A possible method of addressing this problem is provided in the paper.

We have measured the viscometric functions (η, N1, and N2) for a suspension of noncolloidal spherical particles in a Boger fluid matrix. The polymethyl methacrylate particles were nominally 40 μm in size, and the matrix fluid was similar to that used by Zarraga et al. [J. Rheol. 45, 1065–1084 (2001)]. The volume concentrations (ϕ) ranged from 5% to 40%, and a combination of the open trough and parallel-plate methods was used. The viscosity did not shear-thin for higher concentrations, unlike the Newtonian matrix case; for the larger concentrations we saw a mild shear-thickening. This is in agreement with the work of Zarraga et al. [J. Rheol. 45, 1065–1084 (2001)] and Scirocco et al. [J. Rheol. 49, 551–567 (2005)]. N2 was always negative, and its magnitude increased with concentration. N1 was initially positive as in the matrix fluid, but for ϕ>0.3, it became negative, similar to the result of Aral and Kalyon [J. Rheol. 41, 599–620 (1997)]; also, we found, at a constant shear stress, that |N2/N1| was great...

The phenomenon of edge fracture interferes with measurement in cone-plate and parallel plate rheometers. The instability is due to the effect of the second normal stress difference (N2) at the free edge. With non-colloidal suspensions, even in suspensions with Newtonian matrix fluids, N2 is dominant and the flows are more stable than the viscoelastic flows. There are no guidelines for the onset of fracture in these materials. In this Note we discuss the stability of non-colloidal suspensions and suggest that a useful guide to the onset of instability can be made if the ‘flaw’ size at the edge is assumed to be about 10% of the gap size, as found previously.

Following our previous work [K. D. Housiadas and R. I. Tanner, “Viscoelastic shear flow past an infinitely long and freely rotating cylinder,” Phys. Fluids 30, 073101 (2018)], we study analytically the effect of steady planar elongational flow past an infinitely long circular cylinder using asymptotic methods. The ambient fluid is assumed viscoelastic and modelled with the Upper Convected Maxwell, Oldroyd-B, exponential Phan-Thien and Tanner, Giesekus, and Finite Extensibility Nonlinear Elastic model with the Peterlin approximation constitutive equations under isothermal and creeping flow conditions. The solution for all the dependent variables is expanded as an asymptotic power series with the small parameter being the Weissenberg number, Wi, which is defined as the product of the single relaxation time of the fluid times the constant rate of elongation. The resulting sequence of equations is solved analytically up to fourth order in the Weissenberg number. The solution derived here is the first analytical result in the literature for the planar elongational flow of viscoelastic fluids past a cylinder. It reveals the effect of viscoelasticity and all the relevant rheological parameters on the flow variables and the extensional viscosity of the fluid.Following our previous work [K. D. Housiadas and R. I. Tanner, “Viscoelastic shear flow past an infinitely long and freely rotating cylinder,” Phys. Fluids 30, 073101 (2018)], we study analytically the effect of steady planar elongational flow past an infinitely long circular cylinder using asymptotic methods. The ambient fluid is assumed viscoelastic and modelled with the Upper Convected Maxwell, Oldroyd-B, exponential Phan-Thien and Tanner, Giesekus, and Finite Extensibility Nonlinear Elastic model with the Peterlin approximation constitutive equations under isothermal and creeping flow conditions. The solution for all the dependent variables is expanded as an asymptotic power series with the small parameter being the Weissenberg number, Wi, which is defined as the product of the single relaxation time of the fluid times the constant rate of elongation. The resulting sequence of equations is solved analytically up to fourth order in the Weissenberg number. The solution derived here is the first analytic...

The viscometric functions ( η , N 1 and N 2 ) for non-colloidal suspensions of spheres in a Boger fluid matrix were measured. Volume fractions ( ϕ ) of 5%, 10% and 20% were investigated. The relative viscosity ( η r = η / η 0 ) and the (positive) first normal stress difference N 1 showed increases with ϕ which were larger than the dilute suspension theory predictions of 1 + 2.5 ϕ , indicating semi-dilute suspension behaviour. The main interest centres on the second normal stress difference N 2 . The matrix fluid showed a zero second normal stress difference, and the measurements showed that N 2 was always negative for the suspensions. This agrees with the dilute suspension prediction found using the Landau-Lifschitz averaging procedure, but not with the ensemble averaging method, which predicts a positive N 2 . Possible causes for this result are discussed.

Due to the potentially anisotropic microstructure of bread dough, the pre-test deformation produced during the making of the specimens significantly affects the rheological properties of dough, and the resistance to deformation is increased. Some new experimental results for JANZ dough are reported in this paper. Lubricated compression tests have been made up to medium-size (order 1) Hencky strains, and we find that the stress response given by the compression test is consistent with the one given by the spherical bubble test if pre-strain is taken into account. Therefore a damage function model can be successfully used to describe the rheological behaviour of the dough for elongation, compression and biaxial stretching. The Mooney–Rivlin term in the model is very important for the biaxial stretching, but not for elongation and shear. The damage function used in compression is the same as that used in elongation and shear, when computed using the maximum positive Hencky strain in all cases. Thus the deformation mode does not make the damage function change; it depends only on the dough variety. Good agreement between the experimental data and the predictions given by the model is observed.

We present the results of measuring the three viscometric functions [the relative viscosity ηr, and the first (N1) and second (N2) normal stress differences] for nominally monosize sphere suspensions in a silicone fluid, which is nominally Newtonian. The measurements of ηr and N1−N2 were made with a parallel-plate rheometer, while we used the open semicircular trough method to give N2 directly. With the trough method measurements of N2 could be made down to a 10% concentration (φ=0.1); measurements were continued up to 45% concentration. The trough surface shows visually that N2 is directly proportional to the shear stress τ, and the measurements of N2 agree quite well with the results of Zarraga et al. [J. Rheol. 44, 185–220 (2000)] in the range where concentrations overlap (0.3–0.45) and with those of later investigators. The results for N1 show greater scatter. In the range 0.1≤φ≤0.45, our best estimate of N2/τ is −4.4φ3 and that of N1/τ is −0.8φ3. Hence, the magnitude of N2 is much greater than that o...

We have collected six sets of results for the ratio of the second normal stress difference to the shear stress $(N_{2}/\tau )$ in non-colloidal suspensions of spheres in Newtonian matrices. They all show a near-cubic dependence on the volume fraction $\varphi $ in the range $0.1 < \varphi < 0.5$ , in contrast to the square law predictions of Brady and Morris (J Fluid Mech 348:103–139, 1997) for dilute suspensions. We suggest that the difference can be resolved by using a dependence on the square of the effective volume fraction $\varphi _{\textrm e}$ , and good agreement is then found.

A red blood cell (RBC) in a microfluidic channel is highly interesting for scientists in various fields of research on biological systems. This system has been studied extensively by empirical, analytical, and numerical methods. Nonetheless, research of predicting the behavior of an RBC in a microchannel is still an interesting area. The complications arise from deformation of an RBC and interactions among the surrounding fluid, wall, and RBCs. In this study, a pressure-driven RBC in a microchannel was simulated with a three-dimensional lattice Boltzmann method of an immersed boundary. First, the effect of boundary thickness on the interaction between the wall and cell was analyzed by measuring the time of passage through the narrow channel. Second, the effect of volume conservation stiffness was studied. Finally, the effect of global area stiffness was analyzed.

We study the steady, creeping viscoelastic flow around a freely rotating rigid sphere in a stream with uniform velocity which is subject to simple shear flow far from the sphere. The sphere moves along the vorticity axis of the shear flow. The viscoelasticity of the stream is modelled using the Oldroyd-B and Upper Convected Maxwell constitutive equations. A spherical coordinate system with origin at the centre of the sphere is used to describe the flow field. The solution of the governing equations is expanded as a series for small values of the Deborah number and the resulting sequence of three-dimensional partial differential equations is solved analytically up to second order in the Deborah number. The solution is used to deduce the resulting drag force on the sphere in terms of the dimensionless parameters of the flow. The expression for the drag shows that there is a significant increase of the drag due to the shearing, except in the Newtonian case.

We refer to the elongational testing of bread dough using a constant strain rate and specimens of varying initial diameter. Previously we found that 5 mm diameter specimens showed a much larger tensile stress for a given strain and strain rate than larger (20 mm diameter) specimens [Tanner et al., in Proceedings of the 5th International Symposium on Food Rheology and Structure, edited by P. Fischer, M. Pollard, and E. J. Windhab (ETH, Zurich, 2009), pp. 348–351]. It is important to understand this result because if it is true it is not possible to write a simple constitutive equation for the dough. In this paper, we examine the roles of ambient humidity and method of specimen preparation. The latter factor was seen to be crucial to obtaining consistent results in elongational testing of dough with different size specimens; it seems essential to limit pretest deformation to a Hencky strain of about 0.5 (extension ratio of about 1.6). Ambient humidity variations also affected the results, but the apparent shift with diameter was much smaller.

We study the steady, three-dimensional creeping and viscoelastic flow around a rigid sphere subject to steady uniaxial extensional flow imposed at infinity. The viscoelastic response of the ambient fluid to the flow deformation is modeled using the second-order-fluid model, the Upper Convected Maxwell, the exponential affine Phan-Thien and Tanner and the Giesekus constitutive equations. A spherical coordinate system with origin at the center of the sphere is used to describe the flow field and the solution of the governing equations is expanded as a series in the Deborah number. The resulting sequence of differential equations is solved analytically up to second order and numerically up to fourth order in Deborah number by employing fully spectral representations for all the primary variables. In particular, Chebyshev polynomials are utilized in the radial coordinate and the Double Fourier Series in the longitudinal and latitudinal coordinates. The numerical results up to second-order agree within machine accuracy with the available analytical solutions clearly indicating the correctness and accuracy of the numerical method used here.

Multimodal suspensions, consisting of non-colloidal spherical particles and a Newtonian matrix, are considered. A new differential (or multi-scale mean field approach) model for the relative viscosity of multimodal suspensions has been developed. The problem of the random close packing fraction of the solid phase is also investigated. We suppose that the multimodal suspension has a dominant large particle composition and that the smaller particles are embedded in the empty space between the larger particles. The relative viscosity model can therefore be based on the theory of monomodal suspensions. Experimental data of Eveson are compared to the predictions given by using three different models of monomodal suspensions respectively. The Maron–Pierce approach appears to give the best agreement with Eveson’s experiments. However, due to experimental uncertainties, we recommend that the Mendoza and Santamaria-Holek (MSH) formula be adopted.

A new differential (or multi-scale, mean field approach) model for the relative viscosity of bimodal suspensions is discussed in this paper. Solid spherical particles with a bimodal size distribution in a Newtonian solvent are considered. The problem of random close packing for a bidisperse system is studied. The bounds on volume fractions are given by

A sphere falling along the vorticity axis of a shear flow in a viscoelastic fluid is known to experience an increase in drag from the value in a quiescent fluid (van den Brule and Gheissary, 1993). Two recent papers have analysed the problem and this paper seeks to provide a simpler explanation of the phenomenon by considering deformation of the streamline pattern and the consequent generation of lift on the sphere by normal stresses.

We describe an improved damage function model for bread dough rheology. The model has relatively few parameters, all of which can easily be found from simple experiments as discussed in this paper. Small deformations in the linear region are described by a gel-like power-law memory function. Then, we consider a set of large non-reversing deformations—stress relaxation after a step of shear, steady shearing and elongation beginning from rest and biaxial stretching. With the introduction of a revised strain measure which includes a Mooney–Rivlin term, all of these motions can be well described by the damage function described previously. For reversing step strains, larger amplitude oscillatory shearing and recoil we present a discussion which shows how the damage function model can be applied in these cases.

We apply the differential method of Roscoe (1952) to the problem of finding the viscosity of a suspension of non-colloidal spheres in a power-law matrix. The results are compared with other theories and published experiments, and reasonable agreement is found up to moderate concentrations ( ϕ ∼ 0.5) when viscoelasticity and other effects are not important. The Roscoe paper depends on using a “crowding” function in the analysis; here two modified crowding functions are discussed, with a view to explaining the success of the Maron–Pierce formula, which does not reduce to the Einstein form for low concentrations.

This review deals with non-Brownian (non-colloidal) suspension rheology; experimental and computational studies are compared where possible. The matrix fluids are Newtonian, and the rigid particles have an aspect ratio close to one. Volume fractions up to and including 0.5 are considered. Shearing and extensional flows are discussed; the former are fairly well understood, but the latter are not prominent in the literature. Unsteady and oscillatory flows are surveyed; more work is needed in this area. Finally some attempts to find constitutive models are discussed, and an empirically based suggestion based on a modified Reiner-Rivlin model is described.

We study analytically the effect of steady shear flow on an infinitely long and freely rotating circular cylinder using asymptotic methods. The ambient fluid is assumed viscoelastic and modeled with the Oldroyd-B constitutive equation under isothermal and creeping flow conditions. The solution for all the dependent variables is expanded as an asymptotic power series with the small parameter being the Weissenberg number, Wi, which is defined as the product of the single relaxation time of the fluid times the externally imposed shear rate. The resulting sequence of equations is solved analytically up to the eighth order in the Weissenberg number, leading to a series solution, the accuracy and validity of which is shown to be linked to the loss of positive definiteness of the conformation tensor. The solution derived here is the first analytical result in the literature for viscoelastic fluids past a freely rotating cylinder. It reveals the slowdown of the rotation of the cylinder with respect to the Newtonian case, as previously found for the same type of flow past a rigid sphere. It is also seen that the first correction to the velocity profile is of second order in Wi, while that for the pressure is of first order. This contrasts with the solution for the viscoelastic flow past a sphere for which the correction for both the velocity and pressure profiles are of first order in Wi. The analytical solution also shows that the region of closed streamlines around the cylinder is enlarged compared to the Newtonian case. Last, it is seen that the up-and-down and fore-and-aft symmetries of the streamlines and the vorticity contours, observed for Newtonian fluids, break in the viscoelastic case.

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

A model for the rheological properties of a concentrated suspension in weakly viscoelastic fluid matrices is proposed. The model is derived according to the Roscoe differential procedure described in 1952. The analytical results produced recently by Greco et al. (J Non-Newton Fluid Mech 147:1–10, 2007) and Housiadas and Tanner (J Non-Newton Fluid Mech 162:88–92, 2009) for dilute suspensions of neutrally buoyant, non-Brownian rigid spheres in weakly viscoelastic matrix fluids are the key results which are used as a base to predict the properties of concentrated suspensions. The results are compared with the few available experimental data from the literature, showing promising trends for the viscometric properties of the suspensions. In particular, one sees the rapidly increasing value of −N2/N1 as concentration increases.

Viscosity behavior is used to illustrate the relative sensitivity of the rheological properties to the morphological structure imposed by the presence of organic copper phthalocyanine and inorganic ultramarine blue pigments in isotactic polypropylene. While the inorganic pigment showed a small effect, the organic pigment was responsible for greatly increasing the nucleation rate and rapidly increasing the viscosity. This was also accompanied by a lowering of the degree of supercooling and the formation of a fibrous morphology. The results indicate that copper phthalocyanine is an excellent nucleating agent, drastically reducing the onset time of crystal formation and increasing the crystallization kinetics. Dynamic rheological properties also suggest that the organic pigment acts as a softening agent in the supercooled melt. Percent concentration and shear rate were not found to be major factors accelerating the rate of crystallization. This study details contributing factors including the particle size differences of the pigments under scanning electron microscopy, which provides further insight into the non-spherulitic morphology observed under a Linkam device. A possible correlation between the pigment morphology, the crystallization kinetics and rheology is discussed.

A simulation algorithm was developed to predict the rheological properties of oblate spheroidal suspensions. The motion of each particle is described by Jeffery’s solution, which is then modified by the interactions between the particles. The interactions are considered to be short range and are described by results from lubrication theory and by approximating locally the spheroid surface by an equivalent spherical surface. The simulation is first tested on a sphere suspension, results are compared with known experimental and numerical data, and good agreement is found. Results are then presented for suspensions of oblate spheroids of two mean aspect ratios of 0.3 and 0.2. Results for the relative viscosity η r, normal stress differences N 1 and N 2 are reported and compared with the few available results on oblate particle suspensions in a hydrodynamic regime. Evolution of the orientation of the particles is also observed, and a clear alignment with the flow is found to occur after a transient period. A change of sign of N 1 from negative to positive as the particle concentration is increased is observed. This phenomenon is more significant as the particle aspect ratio increases. It is believed to arise from a change in the suspension microstructure as the particle alignment increases.

The complete solution for the pressure and the velocity field up to O ( De ) of a dilute suspension ofneutrally buoyant, non-Brownian rigid spheres suspended in an unbounded, weakly viscoelastic matrixfluid, where is the solid volume fraction and De is the Deborah number of the matrix fluid, is presented.The spheres are subjected to an arbitrary linear velocity profile at infinity. The analytical solution isused for the prediction of the bulk stress, and specifically for the calculation of the first and the secondnormal stress differences in simple shear and uniaxial elongational flows. A comparison of the resultswith available values reported in the literature is also offered. The final expressions for the bulk normalstress differences in shear and uniaxial elongational flow fully agree with those reported earlier by Grecoet al., J. Non-Newton. Fluid Mech., 147 (2007) 1–10.© 2009 Elsevier B.V. All rights reserved. 1. Introduction Suspension rheology represents a complex subject which, how-ever, is of great importance for practical applications, engineeringpurposes and in material processing. A relatively simple case of asuspensionisthatofadilutesuspensionofnon-deformablespheresin a weakly viscoelastic matrix fluid. Although this problem hasbeen studied by many authors in the literature [1–11], it appearsthat the results, either for the velocity and the pressure field or forthe bulk properties of the suspension often do not agree even upto order . Due to the lack of analytical solutions for suspensions,it is of great importance to confirm which one, if any, of the avail-able analytical solutions for the dilute suspension of rigid spheresis correct.The methodology followed here to derive the analytical solutionis the one used by Greco et al. and described in a series of articles[5,9,12]. The correctness of the final solution given here has beenconfirmed using the “Mathematica” software package and is com-pared to the available solutions in the literature. (We will be glad tosend a copy of the ‘Mathematica’ analysis to any interested party.)The work falls into two parts:(i) In the dilute solution the motion of a single sphere is solved,ignoring any interactions with other spheres. This step is com-mon to all workers [1–11].

We report molecular dynamics simulation of crystallization of model alkane systems conducted under constant pressure conditions. We have studied crystallization of n -eicosane (C 20 H 42 ) and n -hexacontane (C 60 H 122 ) under quiescent and shearing conditions. We find preshearing before subjecting the melt to quiescent crystallization enhances the crystallization of higher molecular weight hexacontane, whereas, for low molecular weight eicosane, no significant change can be detected. For both alkanes applying steady planar shear significantly speeds up the crystallization. The crystal growth rate increases with the shear rate. However, we find that the critical shear rate above which the crystallization is enhanced, is inversely proportional to the size of the chains. In all cases the Weissenberg numbers of the sheared systems are moderate. We estimate them to be in the range of 0.01–10. Our quiescent simulations for eicosane predict crystallization temperature and lattice parameters of the crystalline phase in good agreement with experimental measurements. We have compared an order parameter used in the simulations against one analogous to that used in dilatometry experiments. Using this order parameter as a measure of crystallinity we predict the crystal growth rate of n -eicosane to be a maximum at ∼300 K in good agreement with experiments. Fitting crystallization growth data to Avrami's model we have calculated Avrami growth functions and exponents for many cases. For quiescent crystallization of n -eicosane we found the Avrami exponent calculated using our order parameter for defining degree of crystallinity, agrees well with that obtained in the experiments. For C60 the crystallization process is very slow at quiescent conditions; however preshearing enhances the crystal growth.

By looking back to the formal beginning of rheology (1929) and studying more recent offerings at congresses, we see that there is a drift of content in the subject. In the beginning rheology was seen as an extension of classical continuum mechanics, but more recently one has seen the withering of offerings in solids rheology and the start of some new branches of the subject. Some opinions on possible future directions are given.

A formulation describing the rheology of crystallising polymers is discussed. For some semi-crystalline polymers where spherulites form as part of the crystallisation process, the use of a suspension-type model is appropriate. Whilst it is possible to so describe simple shearing and elongational rheology during on-going crystallisation with such models, the flow through a capillary tube is much more complex and numerical solution is usually necessary. To give some insight into this complex flow, a ‘step function’ or ‘amorphous-frozen’ model of the viscosity changes due to crystallisation has been devised so that a semi-analytical approach is feasible. We use this simple model and compare the results with recently published experiments in tubes and channels at high (O(103 s − 1)) shear rates using poly(butene-1). A direct correlation between simple shear and tube flow crystallisation onset times is found.