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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 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.

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.

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.

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%.

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 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 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.

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 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 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.

The rheological behaviour of a 58 vol.% dispersion of styrene/acrylate particles in ethylene glycol was investigated using a plate-on-plate rheometer. Experimental results showed that the concentrated polymer dispersion exhibited a strong shear-thickening transition under both steady shear and dynamic oscillatory conditions. The low-frequency dynamic oscillatory behaviour could be reasonably interpreted in terms of the steady shear behaviour. Accordingly, the critical dynamic shear rate \( \dot{\gamma }_{{{\text{c\_d}}}} , \) agreed well with the critical shear rate obtained in steady flow \( \dot{\gamma }_{{{\text{c\_s}}}} , \) where \( \dot{\gamma }_{{{\text{c\_d}}}} \) was calculated as the maximum shear rate by the critical dynamic shear strain γc and the frequency ω, i.e. \( \dot{\gamma }_{{{\text{c\_d}}}} = \omega \gamma_{\text{c}} . \) However, during high-frequency dynamic oscillation, it was observed that the shear thickening occurred only when an apparent critical shear strain was reached, which could not be fully explained by the wall-slipping effect. Based on freeze fracture microscopic observations, the effect of the micro-sized flocculation of particles on the rheology of concentrated dispersions was also discussed.

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.

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 new set of experiments on a bread dough includes small-strain oscillatory behaviour, larger-strain oscillatory behaviour, simple shearing beginning from rest, uniaxial elongation beginning from rest, relaxation after sudden shear and recoil from elongation. We believe this is the most complete set of rheological data yet reported for a bread dough. Analysis of these soft-solid experiments proceeds from a Lodge-type rubberlike material with a power-law memory function. The model suggests that the response to steady shear and elongational flows may be described as a product of (strain rate)p times a function of strain; the exponent p is found to be about 0.2–0.3 from small-strain oscillatory measurements. Experiments confirm this finding. The model overestimates stresses, and in order to improve predictions, the use of a KBKZ model and a damage function model are investigated. Due to the eventual fracture of the soft-solid material, the idea of a “damage function” was adopted to produce a simple accurate, integral-type constitutive model for small-strain oscillations, simple shearing and elongation. Further analysis of reversing strains, for example, larger-strain oscillatory flows and recoil, is needed.

There are many curious phenomena when very thin lubricant films interact with sliding surfaces. It is difficult to do experiments on these nano-thickness films, and here we describe the use of non-equilibrium molecular dynamics (MD) to investigate friction in alkane lubricant layers. The pronounced increase of viscosity in layers of thickness seven molecules and less enables an increased load capacity to be sustained in such films. We also demonstrate the existence of a newly-discovered low-friction regime.

The effective viscosity of confined lubricant films less than 6–7 molecular layers is usually enhanced by many orders of magnitude. For dodecane the high friction film has a strong in-plane order with “mosaic-like” structures that extend across the film and effectively form “crystalline bridges” [Jabbarzadeh A, Harrowell P, Tanner RI. Crystal bridge formation marks the transition to rigidity in a thin lubrication film. Phys Rev Lett 2006; 96: 206102-1/4] resulting in high friction. Using molecular dynamics simulations, we have identified three routes to lower the friction. We show that the structure of confined films and their response to shearing are affected by atomic in-plane order and smoothness of the confining surfaces, the relative orientation of two crystalline surfaces and the direction of shear. We show a small increase in surface roughness in going from crystalline to amorphous surfaces can lead to a much lower friction. We demonstrate that misaligning (twisting) one surface with respect to the other by 45° results in a much lower effective viscosity. Application of shear for extended times induces alignment of lubricant molecules into a nematic-like order with ultra-low effective viscosity. The magnitude of reduction in the friction and the physical process through which it happens varies for each of these three routes. Depending on the method used, destruction of crystalline bridges, multilayer or fault plane slip provides a route for dramatic reduction in friction.

Purpose – To present a new collocation method for numerically solving partial differential equations (PDEs) in rectangular domains.Design/methodology/approach – The proposed method is based on a Cartesian grid and a 1D integrated‐radial‐basis‐function scheme. The employment of integration to construct the RBF approximations representing the field variables facilitates a fast convergence rate, while the use of a 1D interpolation scheme leads to considerable economy in forming the system matrix and improvement in the condition number of RBF matrices over a 2D interpolation scheme.Findings – The proposed method is verified by considering several test problems governed by second‐ and fourth‐order PDEs; very accurate solutions are achieved using relatively coarse grids.Research limitations/implications – Only 1D and 2D formulations are presented, but we believe that extension to 3D problems can be carried out straightforwardly. Further, development is needed for the case of non‐rectangular domains.Originality/...

Nearly 30 years ago the PTT constitutive equation for molten polymers and solutions was published, and since then it has been widely used. In the intervening time much progress has been made in polymer science and mechanics, especially via network and tube theories. It is natural to unite these points of view in a revisitation of PTT-type models. We show that a new improved form of the PTT model, called the PTT-X model, can closely describe, in particular, the rheological behaviour of low-density polyethylene in (steady and transient) shear and elongational flow and in recoil from an elongational stretch. The results of an attempt to reduce the number of parameters are also described and we place the new model within the context of general network theory and tube theory.

We consider the drawing of slender optical fibres containing an axial hole, which is initially circular and is not necessarily coaxial with the fibre. A Newtonian creeping flow analysis is used to show that changes in hole shape from circular occur when the hole is not on the fibre axis, but these changes only appear in proportion to the square of the slenderness ratio of the fibre. The final oval shape predicted compares qualitatively with experiments on polymethyl-methacrylate fibres.

International audience; A numerical method to simulate fibre suspensions in transient and steady state shear flows for concentrated solutions is reported, which takes into account short-range hydrodynamic interaction via lubrication forces, contact forces and hydrodynamic forces. Fibres are assumed to have varying lengths as observed in industrial composites and this can be described with a fibre distribution length function. Stresses in the composite have been calculated from a constitutive equation or from an approximate expression for the effective stress of fibre suspensions. This simulation will assist in modelling of short fibre filled industrial composites. Our first results shown here qualitatively agree with experiments.

A numerical study on the transient flow of a viscoelastic fluid is presented. The numerical framework is that of Smoothed Particle Hydrodynamics (SPH) already used by Ellero et al. in previous simulations of Non-Newtonian flows [J. Non-Newtonian Fluid. Mech. 105 (2002) 35–51]. In particular, the start-up flow between parallel plates is simulated for an Oldroyd-B and UCM fluid at low Reynolds number. Results for a Newtonian fluid are also shown for comparison. The numerical results are presented and compared with available theoretical solutions, showing a very good agreement. In particular, the simulations of an Oldroyd-B fluid have been found to be stable and accurate for a wide range of the Weissenberg number. In the case of a UCM fluid, the absence of a viscous term in the momentum equation makes its numerical modelling harder. Namely, the process is characterised by a travelling damped wave, which, if not accurately resolved, can lead to the rapid growth of small oscillations in time eventually causing divergence. On the other hand, if specifically dealing with transient flow problems, stabilising techniques such as BSD, EVSS or AVSS can not be used either; whenever used in conjunction with decoupled solution algorithms, they give an excessive oversmoothing in the results which deteriorates the final accuracy. In this work, we consider an exact SPH discretisation of the hyperbolic equation characterising the UCM model. SPH simulations are finally performed for different Weissenberg numbers showing very promising results. Finally, a discussion on the SPH treatment of the boundary conditions for general hydrodynamics problems is also outlined following the approach of ‘SPH boundary particles’ introduced by Morris for the simulations of low Reynolds number flows.

The flow past a cylinder in a channel with the aspect ratio of 2:1 for the upper convected Maxwell (UCM) fluid and the Oldroyd-B fluid with the viscosity ratio of 0.59 is studied by using the Galerkin/Least-square finite element method and a p-adaptive refinement algorithm. A posteriori error estimation indicates that the stress-gradient error dominates the total error. As the Deborah number, De, approaches 0.8 for the UCM fluid and 0.9 for the Oldroyd-B fluid, strong stress boundary layers near the rear stagnation point are forming, which are characterized by jumps of the stress-profiles on the cylinder wall and plane of symmetry, huge stress gradients and rapid decay of the gradients across narrow thicknesses. The origin of the huge stress-gradients can be traced to the purely elongational flow behind the rear stagnation point, where the position at which the elongation rate is of 1/2De approaches the rear stagnation point as the Deborah number approaches the critical values. These observations imply that the cylinder problem for the UCM and Oldroyd-B fluids may have physical limiting Deborah numbers of 0.8 and 0.9, respectively.

There is now in the literature a number of mathematical models of the effects of flow on crystallization of polymers, and here we compare the performance of various models in slow simple shearing and sinusoidal strain (oscillatory) flows. We compare the model predictions to the experiments of Wassner and Maier [E. Wassner, R.-D. Maier, Shear-induced crystallization of polypropylene melts. in: D.M. Binding, et al. (Ed.), Proceedings of the XIII International Congress on Rheology, Cambridge, 2000, pp. 1–183] on isotactic polypropylene. Both the Kolmogorov [A.N. Kolmogorov, On the statistics of the crystallization process on metals. Bull. Akad. Sci. USSR, Class Sci., Math. Nat. 1 (1937) 355–359] and Nakamura et al. [K. Nakamura, T. Watanabe, K. Katayama, T. Amano, Some aspects of non-isothermal crystallization of polymers I, J. Appl. Polym. Sci. 16 (1972) 1077–1091] crystallinity formulations are discussed, and the advantages of the former are emphasized. For the cases considered, we see that strain-based crystallinity enhancement formulations appear to give results close to the experiments at the low flow rates which were studied, although they have some drawbacks. Several other ideas also lead to acceptable results; a new formulation based on both strain and strain-rate appears to give the best overall fit.

This paper is concerned with the application of radial basis function networks (RBFNs) for solving non-Newtonian fluid flow problems. Indirect RBFNs, which are based on an integration process, are employed to represent the solution variables; the governing differential equations are discretized by means of point collocation. To enhance numerical stability, stress-splitting techniques are utilized. The proposed method is verified through the computation of the rectilinear and non-rectilinear flows in a straight duct and the axisymmetric flow in an undulating tube using Newtonian, power-law, Criminale-Ericksen-Filbey (CEF) and Oldroyd-B models. The obtained results are in good agreement with the analytic and benchmark solutions.

In this paper, the sedimentation of a sphere and its radial migration in a Poiseuille flow in a vertical tube filled with a Newtonian fluid are simulated with a finite-difference-based distributed Lagrange multiplier (DLM) method. The flow features, the settling velocities, the trajectories and the angular velocities of the spheres sedimenting in a tube at different Reynolds numbers are presented. The results show that at relatively low Reynolds numbers, the sphere approaches the tube axis monotonically, whereas in a high-Reynolds-number regime where shedding of vortices takes place, the sphere takes up a spiral trajectory that is closer to the tube wall than the tube axis. The rotation motion and the lateral motion of the sphere are highly correlated through the Magnus effect, which is verified to be an important (but not the only) driving force for the lateral migration of the sphere at relatively high Reynolds numbers. The standard vortex structures in the wake of a sphere, for Reynolds number higher than 400, are composed of a loop mainly located in a plane perpendicular to the streamwise direction and two streamwise vortex pairs. When moving downstream, the legs of the hairpin vortex retract and at the same time a streamwise vortex pair with rotation opposite to that of the legs forms between the loops. For Reynolds number around 400, the wake structures shed during the impact of the sphere on the wall typically form into streamwise vortex structures or else into hairpin vortices when the sphere spirals down. The radial, angular and axial velocities of both neutrally buoyant and non-neutrally buoyant spheres in a circular Poiseuille flow are reported. The results are in remarkably good agreement with the available experimental data. It is shown that suppresion of the sphere rotation produces significant large additional lift forces pointing towards the tube axis on the spheres in the neutrally buoyant and more-dense-downflow cases, whereas it has a negligible effect on the migration of the more dense sphere in upflow.

It is shown that several (single-mode) models (PTT, XPP) are basically special cases of the general network model. The XPP model [J. Rheol. 45 (2001) 823] is shown to give an identical response to a PTT model at high elongational rates, but the models differ in shear flow at high shear rates. The Giesekus term in the XPP model has practically no effect at any deformation rate except the generation of a second normal stress difference at small shear rates. The possibility of tailoring the shear flow response at high shear rates is explored. The results for the models are also given to show the effect of branching on the response in both transient and steady flows—elongation, shear, planar elongation and biaxial deformation. Finally, the fitting of data with multi-mode models is investigated.

An alternative view of the rheology of semi-crystalline polymers is presented, based on suspension theory. The results in the linear viscoelastic strain regime are in accordance with experimental measurements, spanning the properties of the amorphous and crystalline phases as crystallization proceeds. Unlike some other theories, no change in the relaxation times with crystallization is predicted, especially at low crystallinity, which is in reasonable accord with experimental measurements. Slow shearing and oscillation experiments are shown to be compatible with the idea that strain is a crucial factor in increasing crystallization rates. Some non-linear generalizations are indicated.

A conceptually simple and computationally inexpensive implementation of higher-order convection interpolation schemes for stable and faster convergence of the implicit finite volume method (FVM) is evaluated for multi-dimensional viscoelastic flow calculations as well as convection-dominated Newtonian flow calculations. Based on the deferred-correction method, this effective formulation consistently satisfies the four rules that guarantee bounded numerical solutions. In addition, artificial numerical diffusion, commonly found in numerical solutions with the first-order upwinding convection scheme, can be effectively controlled by consistently using a third-order quadratic upstream interpolation over the entire computational domain. Extensive tests performed for the steady-state wall-driven square enclosure flow of a Newtonian fluid at Reynolds number (Re) up to 5000 demonstrate that the formulation is stable, accurate, and converges well. The accuracy of the formulation for multi-dimensional viscoelastic flow calculations is evaluated by solving the Oldroyd-B equations for the problem of a fully two-dimensional steady frictionless plane flow, and comparing the numerical results with the exact solution available. With the first-order upwinding scheme, the results show that, for hyperbolic constitutive models, regardless of the grid size, there always exists artificial diffusion whenever the flow streamlines are not closely aligned with the grid lines. With the new implementation, artificial numerical diffusion is effectively controlled and the results demonstrate third-order accuracy.