Shear viscosity behavior of highly concentrated suspensions at low and high shear-rates

A method is proposed for determining the shear viscosity behavior of highly concentrated suspensions at low and high shear-rates through the use of a formulation that is a function of three parameters signifying the effects of particle size distribution. These parameters are the intrinsic viscosity [], a parametern that reflects the level of particle association at the initiation of motion and the maximum packing concentration _m. The formulation reduces to the modified Eilers equation withn = 2 for high shear rates. An analytical method was used for the calculation of maximum packing concentration which was subsequently correlated with the experimental values to account for the surface induced interaction of particles with the fluid. The calculated values of viscosities at low and high shear-rates were found to be in good agreement with various experimental data reported in literature. A brief discussion is also offered on the reliability of the methods of measuring the maximum packing concentration. _r = /₀ relative viscosity of the suspension - volumetric concentration of solids - k _n coefficient which characterizes a specific effect of particle interactions - _m maximum packing concentration - _r,0 relative viscosity at low shear-rates - [] intrinsic viscosity - n, n parameter that reflects the level of particle interactions at low and high shear-rates, respectively - _r, relative viscosity at high shear-rates - (_m)_s, (_m)_i, (_m)_l packing factors for small, intermediate and large diameter classes - v _s, v_i, v_l volume fractions of small, intermediate and large diameter classes, respectively - _si, _sl coefficient to be used in relating a smaller to an intermediate and larger particle group, respectively - _is, _il coefficient to be used in relating an intermediate to a smaller and larger particle group, respectively - _ls, _li coefficient to be used in relating a larger to a smaller and intermediate particle group, respectively - _m0 maximum packing concentration for binary mixtures - _m,e measured maximum packing concentration - _m,c calculated maximum packing concentration     

Experiments on the effective viscosity of concentrated suspensions of solid spheres

The effective viscosity of concentrated suspensions of solid spherical particles was determined experimentally, under zero shear conditions, by means of a convection experiment. For solid concentrations in the range 30–50 per cent the results verify extrapolations to low shear rates of previously reported conventional viscosity measurements, and suggest that, in this range of concentrations and at low shear, a suspension of neutrally bouyant spheres behaves effectively as an equivalent Newtonian fluid.     

Random close packing and relative viscosity of multimodal suspensions

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.     

Time periodic viscosity of concentrated kaolin suspensions at constant shear rates

Knowledge of the viscosity of concentrated suspensions is required for several technical applications, e.g. process control in mechanical engineering, casting of ceramics and pipeline transport of solids. Our previous viscometric investigations of concentrated suspensions showed, under particular shear conditions, an apparent viscosity that was periodic in time for a constant shear rate and temperature. These results were obtained with rotational viscometers with a set coaxial geometry. The inner cylinder was rigidly coupled to the viscometer driving axis. In this paper we describe a viscosity time behavior which was found using another type of coupling. Measurements were performed with rotational viscometers with a non-rigidly linked inner cylinder (small sample adapter supplied by Brookfield). Using kaolin suspensions of 30% solid mass content, viscosity oscillations appear. They show a regular time pattern at certain intervals of low shear rates. The amplitudes reach up to 20% of the viscosity mean value. In addition a motion of the inner cylinder away from the coaxial position is observed. This dislocation is followed by a relocation into the coaxial position. A maximum in the viscosity value is correlated with a maximum of the dislocation position. The process of dislocation and relocation of the inner cylinder is assumed to be caused by local anisotropically distributed inhomogeneities, which originate from shear-induced agglomeration and deglomeration of suspended particles. The motion of the inner cylinder is described by introducing a perturbation term into the equation of motion. The parameters of the perturbation term are fitted to the experimental data. Received: 10 September 1998 Accepted: 28 April 1999     

On viscous flow and effective viscosity of concentrated suspensions and emulsions

Summary Previous analysis byHappel (3) of viscous flow in concentrated solid suspensions has been extended to include concentrated emulsions of slightly deformable fluid particles in the presence or absence of surfactant impurities.General expressions were obtained for viscous flow in multi-particle systems when arbitrary shear fields are imposed. Specific relations were then derived for uniform,Couette and hyperbolic fields. The behavior is found to be strongly dependent upon particle concentration and surfactant concentration. The theoretical expressions obtained for effective viscosity of emulsions compare favorably with experimental data ofNeogy andGhosh (18),Sibree (15),Sherman (17), andBroughton andWindebank (16). These results support other studies on ensemble velocities [(10), (12), and in particular (22)], which strongly indicate the practical value and factual reliability of cell models in predicting the behavior of suspensions and emulsions.     

Rheological equations of state of weakly concentrated suspensions of rigid ellipsoidal particles

Rheological equations of state of dilute suspensions of rigid ellipsoidal particles (ellipsoids of revolution) are derived [1–4] from the vantage point of the structural-continuum approach, with attention given both to rotational Brownian motion of particles and to their inertia and the outer force fields. Interaction between particles is ignored in those treatments given the low concentration of the suspended particles. In this paper, the earlier findings [1–4] are generalized to higher concentrations. The effect of hydrodynamical interaction between particles on the rheological behavior of the suspension is treated in the light of the Simha approach [5].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 141–145, January–February, 1973.     

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

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

A simple semiempirical model for the effective viscosity of multicomponent suspensions

We propose a methodology to approximate the viscosity of multicomponent suspensions. The procedure consists of successive applications of expressions for the viscosity of binary mixtures, originally written as the product of monomodal stiffening functions. First, the viscosity of a binary mixture made of the two smallest components is calculated. This allows to extract a volume fraction that will be used, together with the volume fraction of the third component, to feed the next iteration of the procedure to calculate the viscosity of a trimodal mixture and so on. The application of this approach to arbitrary mixtures requires the detailed knowledge of the geometry of the system in the form of size ratios and compositions. When this information is unknown, an approximation of the model can still be used as a fitting tool. With that purpose, the final expression for the viscosity is written in terms of an effective volume fraction that is further approximated by the use of a (1,2) Padé approximant. This approximation allows to incorporate the crowding effects due to different species in a volume fraction-dependent crowding factor that can be used as a fitting parameter to match experimental or simulation data. We have applied the model to mixtures of particles with different sizes and tested its accuracy comparing with experimental results obtaining very good agreement.     

Rheological behavior of concentrated suspensions as affected by the dynamics of the mixing process

The rheological characterizations of concentrated suspensions are generally carried out assuming “well-mixed” suspensions. However, the variation of the concentration distributions of the ingredients of the formulation, i.e., the “goodness of mixing”, the size and shape distributions of the particle clusters and the rheological behavior of the suspension all depend on the thermo-mechanical history that the suspension is exposed to during the mixing process. Here, various experimental tools are used for the characterization of the degree of mixedness (concentration distributions) of various ingredients along with the characterization of rheological material functions, wall slip behavior and the maximum packing fraction of a graphite/elastomer suspension. The degree of mixedness values of the ingredients of the suspensions processed using batch and continuous processes and under differing operating conditions were characterized quantitatively using wide-angle X-ray diffraction and thermo gravimetric analysis and were elucidated under the light of the electrical properties of the suspension as affected by the mixing process. Upon achieving better homogeneity of the graphite particles and the binder and decreases in the size and breadth of the size distributions of particle clusters (as inferred from electrical measurements and maximum packing fraction values), the elasticity (storage modulus) and the shear viscosity (magnitude of the complex viscosity from small-amplitude oscillatory shear and shear viscosity from steady torsional and capillary rheometry) of the suspension decreased significantly and the wall slip velocity values increased. These findings demonstrate the intimate relationships that exist between the rheological behavior of concentrated suspensions and the thermo-mechanical history that they are exposed to during the processing stage and suggest that the preparation conditions for suspensions should be carefully selected and well documented to achieve reproducible characterization of rheological material functions.     

Free-surface flow of concentrated suspensions

Free-surface flows of concentrated suspensions exhibit many interesting phenomena such as particle segregation and surface corrugation. In this work the flow structures associated with free-surface has been studied experimentally. The free-surface velocity for neutrally buoyant suspension of uniform spheres in a gravity driven inclined channel flow was determined by particle imaging velocimetry (PIV) technique. Experiments were carried out for concentrated suspensions with particle fractions ? ranging from 0.40 to 0.50. The measured velocities show blunted profile in the channel. The blunting of the velocity profile increases with the particle concentration. The rms velocity fluctuations measured at the free-surface progressively increase with particle fraction ? and are linear in shear rate γ. The surface roughness were characterized by analyzing the power spectral density of the refracted light from the free-surface. The characteristics observed are in support of earlier findings.     

Heat conductivity of concentrated suspensions

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 94–97, November–December, 1990.     

Stability of some shear flows for concentrated suspensions

In this paper we investigate the stability of some viscometric flows for a concentrated suspension model which allows for the effects of shear-induced migration, including plane and circular Couette and Poiseulle flows, and unbounded and bounded torsional flows. In the bounded torsional flow, where its radial outer boundary is assumed frictionless, an exact closeform solution is given. With the exception of torsional flows, we find that a limit point for all the steady-state solutions can exist for certain range in the parameter values. In all cases, disturbances can persist for a long time, O (H ²/a ²), where H is a dimension of the flow field, and a is the particles' radius.     

Shear viscosity of settling suspensions

Summary A new apparatus is presented which allows the determination of the viscosity of suspensions of high density particles in low viscosity media. Results obtained on suspensions of low density particles compare well with those obtained on a Weissenberg Rheogoniometer equipped with a cone-and-plate and a bob-and-cup. Also, a very interesting observation is that the Eilers equation, conveniently modified to take into account the shape of the particles by means of the intrinsic viscosity, can well correlate all the data.

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Zusammenfassung Es wird ein neues Gerät beschrieben, das die Bestimmung der Viskosität von Suspensionen mit Teilchen hoher Dichte in niedrig viskosen Flüssigkeiten ermöglicht. Die an Suspensionen von Teilchen mit niedriger Dichte erhaltenen Ergebnisse stimmen gut mit den mittels eines Weissenberg-Rheogoniometers gewonnenen überein, das entweder mit einer Kegel-Platte- oder einer Koaxial-Zylinder-Meßeinrichtung ausgerüstet war. Als ein überraschendes Ergebnis stellt sich heraus, daß die Eilers-Gleichung, zum Zweck der Erfassung der Teilchenform mit Hilfe des Staudinger-Index in bequemer Weise modifiziert, alle experimentellen Daten gut zu korrelieren imstande ist.

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Nomenclature shear rate (sec^–1) - viscosity (Poise) - _r relative viscosity - [] intrinsic viscosity - volume concentration - _max maximum volume concentration With 7 figures and 3 tables     

A model for the shear viscosity of non-colloidal suspensions with Newtonian matrix fluids

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.     

On the effective viscosity of pseudoplastic suspensions

Summary An expression is developed that predicts the concentration dependence of the apparent viscosity of a concentrated pseudoplastic suspension. The result, which is an extension of the Frankel and Acrivos Newtonian suspension analysis, shows that the influence of particles concentration on the effective viscosity of pseudoplastic suspensions increases as the power law index of the suspension increases. Experimental data shows good agreement with the theoretical predictions.With 8 figures     

Viscosity of concentrated suspensions of spheres

Difficulties associated with the viscosity measurement of concentrated suspensions of particulate solids in a liquid solvent can effectively be overcome with the falling needle technique reported here. The comparison of the settling (terminal) velocity of a given needle in a Newtonian solvent, with its terminal velocity in a suspension, yields the suspension viscosity ratio directly. The van den Brule and Jongschaap constitutive model describes our high concentration data best. Falling sphere data (diameter of sphere/diameter of suspended particle 10) agree well with the falling needle data over the whole range (up to 40%) of solids concentrations used in our tests.In the opaque suspensions used, the passage of sedimenting needles and spheres was initially observed radiographically. Later tests used a more convenient technique using an inductance coil particle detector driven by a Colpitts oscillator.     

A differential model for the rheological properties of concentrated suspensions with weakly viscoelastic matrices

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