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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.

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.

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.

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.

Technologies that use optical force to actively control particles in microchannels are a significant area of research interest in various fields. An optical force is generated by the momentum change caused by the refraction and reflection of light, which changes the particle surface as a function of the angle of incidence of light and which in turn feeds back and modifies the force on the particle. Simulating this phenomenon is a complex task. The deformation of a particle, the interaction between the surrounding fluid and the particle, and the reflection and refraction of light should be analyzed simultaneously. Herein, a deformable particle in a microchannel subjected to optical interactions is simulated using the three-dimensional lattice Boltzmann immersed-boundary method. The laser from the optical source is analyzed by dividing it into individual rays. To calculate the optical forces exerted on the particle, the intensity, momentum, and ray direction are calculated. The optical-separator problem with one optical source is analyzed by measuring the distance traveled because of the optical force. The optical-stretcher problem with two optical sources is then studied by analyzing the relation between the intensity of the optical source and particle deformation. This simulation will help the design of sorting and measuring by optical force.

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.

The analytical solution of the Stokes-Darcy equations is derived for a Newtonian, statistically homogenous, suspension with non-Brownian (non-colloidal) rigid spherical particles. Navier-type linear slip over the surface of the particles is applied. The mathematical model is based on Brinkman's (1949) original idea for the viscous force exerted by a flowing fluid on a dense swarm of spherical particles. First, the equations are solved analytically for the pressure-driven and uniform flow of the suspension. Self-consistency of the model allows for the evaluation of the resistance parameter as function of the volume fraction of the solid phase and the dimensionless slip coefficient. The results show that for fixed solid concentration, the drag force on each particle decreases due to the slippage of the fluid over its surface. Consequently, the resistance parameter decreases compared to the classic no-slip case. However, in all cases, divergence of the resistance parameter occurs as the solid volume fraction approaches the value 2/3 which is very close to the maximum volume fraction 0.637( ≈ 2/π)for random close packing of spherical particles. The analysis is also performed for steady shear and steady uniaxial elongational flows and exact analytical solutions for the velocity and the pressure are derived. Then, a volume average of the total stress tensor in the suspension allows for the analytical evaluation of the shear and elongational viscosities of the complex flow system. Limiting expressions for the no-slip and perfect slip cases are also found and discussed.

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%.

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.

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.

Two fundamental test systems were used to evaluate the visco-elastic properties of doughs from wheat samples of three varieties grown at four distinct sites. For comparison, tests were also performed with traditional equipment, namely the Mixograph, an extension tester and a Farinograph-type small-scale recording mixer. Uniaxial dough elongation (with an Instron) produced results similar to the conventional extension tester, except that results were provided in fundamental units (Pascals), the critical value recorded being the elongational stress at maximum strain. Stress relaxation measurements were performed following a small initial shear strain. With this method, it was possible to distinguish between the viscosity and the elastic components of dough visco-elasticity. In all the tests the extra dough-strength properties were evident for the variety (Guardian) that had the 5 + 10 glutenin subunits, in contrast to the other two with the 2 + 12 combination of subunits.

The present Dissipative Particles Dynamics (DPD) simulation study aims at extracting some correlations between velocity and temperature profiles, and dynamic viscosity with externally applied shear velocity via Lees-Edwards boundary condition. System physical and rheological behaviours are studied under changes made to shear velocity, cutoff radii and weight function exponent in the definition of conservative, dissipative and random forces. Two cutoff radii are altered up to the level where system of particles shows anomalous behaviour. Radial distribution function and temperature (T) are invoked to rule out invalid cutoff radii combinations. Calculated viscosity in a range of weight functions exponents (S) is compared against theory in a variety of shear velocities, where reasonable agreement with respect to T control is achieved.

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...

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.

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.

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.

Shear thinning-a reduction in suspension viscosity with increasing shear rates-is understood to arise in colloidal systems from a decrease in the relative contribution of entropic forces. The shear-thinning phenomenon has also been often reported in experiments with noncolloidal systems at high volume fractions. However its origin is an open theoretical question and the behavior is difficult to reproduce in numerical simulations where shear thickening is typically observed instead. In this letter we propose a non-Newtonian model of interparticle lubrication forces to explain shear thinning in noncolloidal suspensions. We show that hidden shear-thinning effects of the suspending medium, which occur at shear rates orders of magnitude larger than the range investigated experimentally, lead to significant shear thinning of the overall suspension at much smaller shear rates. At high particle volume fractions the local shear rates experienced by the fluid situated in the narrow gaps between particles are much larger than the averaged shear rate of the whole suspension. This allows the suspending medium to probe its high-shear non-Newtonian regime and it means that the matrix fluid rheology must be considered over a wide range of shear rates.

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.

A viscosity-oriented, flow-induced crystallization model is used to predict the rate of crystal layer growth in a tube at high shear rates. A combined strain and strain rate dependence of the enhancement of crystallization kinetics was proposed that showed excellent agreement with viscosity measurements at low deformation rates in simple shear, and it is here considered in the more complex Poiseuille flow. Suspension mechanics is used to link the relative crystalline volume fraction to the viscosity of the semicrystalline polymer. The microstructure is directly related to the thermomechanical histories and this was accounted for in the total volume fraction using the Avrami-Kolmogorov model. The key characteristic of our model is the coupling of the flow history to induced crystallization and the linkage of the flow-enhanced nucleation with viscosity. In this way, the flow is described in terms of changes in crystallization due to changes in rheological behavior. A finite volume numerical treatment was employed using the ANSYS CFX software to model the layer growth. The model is further tested with the presence of an organic nucleating agent in which the sensitivity of the rheological properties of the pigment–polymer blend to stress and temperature was evident. Reasonable agreement with experiments was observed. POLYM. ENG. SCI., 2012. © 2012 Society of Plastics Engineers

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.

We regard gluten dough as a mixture of gluten, starch and water. We show that stress intensification around the starch particles enables one to describe the rapid strain softening of dough at low strains. The starch is in the form of a combination of A-particles (close to oblate spheroids) and B-particles (almost spherical). This suggests that a suspension theory should be able to account for the linear viscoelastic properties of doughs. We develop a new representation for the prediction of the linear viscoelastic properties of a viscoelastic matrix (gluten) with embedded oblate spheroids and spherical particles. The calculations are compared with experiments on gluten mixes derived from an Australian Baker’s flour. We note that the non-sphericity of the A-particles is very important in stiffening the gluten matrix and also that the effective volume fraction of the starch is greater than that calculated by assuming a starch density of 1.4 g/ml.

In this paper we report molecular dynamics simulation results of lubricant films only a few nanometres thick confined between atomically smooth gold or model mica surfaces. We have studied dodecane (C 12 H 26 ) of various film thicknesses. We show below a critical film thickness structural transitions take place with the formation of crystal bridges. We demonstrate this critical film thickness is larger with Au(1 0 0) surface. Below this critical film thickness we observe a large enhancement of apparent shear viscosity. This enhancement, however, is much lower for films confined by gold. We find the extrapolated zero shear viscosity of a ∼2.4-nm-thick film confined by gold is almost two orders of magnitude lower than that of a film, of the same thickness, confined by model mica. We find the source of this difference is the weaker pinning of the layers next to the surface of the gold. This leads to stronger slip, at the lubricant–gold interface, which persists at solid-like and liquid-like states. For mica a larger part of shearing takes place inside the film. The solid-like structure of the films confined by gold shows stronger resilience to shear and melt at considerably larger shear rates than those confined by model mica surfaces. These differences could be important in developing low friction devices at microscale.

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.