Direct numerical simulation of a bubble suspension in small amplitude oscillatory shear flow

Bubble suspensions can be found in many different fields and studying their rheology is crucial in order to improve manufacturing processes. When bubbles are added to a liquid, the magnitude of the viscosity changes and the behavior of the material is modified, giving it viscoelastic properties. For the purpose of this work, the suspended bubbles are considered to be monodisperse. It is assumed that Brownian motion and inertia can be neglected and that the fluid of the matrix is Newtonian and incompressible. The suspension is subject to an oscillatory strain while remaining in the linear regime. The resulting equations are solved in 3D with direct numerical simulation using a finite element discretization. Results of an ordered and random distribution of bubbles of volume fractions up to 40% are presented. The presence of bubbles has an opposite effect on the rheology of the suspension depending on the applied frequency. When the frequency is low, bubbles act as rigid fillers giving a rise to viscosity. On the contrary, when the frequency is high, the strain rate is being accommodated by the gaseous phase. Hence, bubbles deform, leading to a decrease of the viscosity.     

Überlegungen zur Scherviskosität von Suspensionen aufgrund einer Dimensionsanalyse

Dimensional analysis of the motion of solid particles suspended in a fluid phase shows that the macroscopic relative shear viscosity of suspensions generally depends not only on the volume concentration and particle shape but also on two Reynolds numbers and a dimensionless sedimentation number. These dimensionless numbers are formed using parameters characterizing the structure and motion of the suspension at the microscopic level. The analysis was based on the assumptions that the dispersed particles are rigid and sufficiently large that Brownian motion may be neglected, that the continuous fluid phase is Newtonian and that the interactions between particles and between particles and fluid phase are only hydrodynamic. The Reynolds numbers describe the influence of the inertial forces at the microscopic level, and the sedimentation number the influence of gravity. The dimensionless numbers can be neglected if their values are much smaller than one. For each of the dimensionless numbers both the shear rate and the particle size influence the shear viscosity. Thus sedimentation number is large for low shear rates, whereas the Reynolds numbers are large for high shear rates. The viscosity function for one suspension can be transformed into the viscosity function for another suspension with geometrically similar particles but of a different size. The scale-up rules are derived from the requirement that the relevant dimensionless numbers must be constant. The influence of non-hydrodynamic effects at the microscopic level on the shear viscosity can be detected by deviations from the derived scale-up rules.     

Rheology of carbon black suspensions. II. Well dispersed system

Linear viscoelastic properties were investigated for the suspensions of carbon black (CB) particles having covalently-fused aggregate structures of an average diameter a=120 nm. The suspending medium, an alkyd resin (AR), had a high affinity toward the CB particles, and the aggregates of these particles were well dispersed to form no higher-order agglomerates. Consequently, the suspensions obeyed the time-temperature superposition and their Arrhenius-type activation energy was identical to that of the medium. From comparison of the zero shear viscosity η₀ for the CB suspensions and hard-sphere silica suspensions, an effective volume fraction φ_eff of the CB particles was found to be 2.7 times larger than the bare volume fraction of the particles. The CB particles exhibited a slow relaxation process, and the terminal relaxation time of this process was close to the Peclet time (Brownian diffusion time) evaluated from the aggregate size a and a high frequency viscosity. Furthermore, the terminal relaxation mode distribution of the CB suspensions was well scaled with an intensity factor H_t that was evaluated from the φ_eff in a way utilized for the hard-sphere silica suspensions. These results demonstrated that the slow relaxation in the CB suspensions was dominated by the Brownian diffusion of the CB aggregates, as similar to the situation for the silica suspensions.     

The effect of pH,ionic strength,and temperature on the rheology and stability of aqueous clay suspensions

The effect of pH level, ionic strength, and temperature on the theology and stability of aqueous suspensions of attapulgite clay was systematically investigated. A Rheometrics Mechanical Spectrometer with cone and plate fixtures was used to measure the steady shear viscosity of the system. The edge charges of the clay particles can be adjusted by changing the pH level of the suspending medium so as to influence the flocculation state and, consequently, the rheological behavior of the suspension. This pH effect may be counteracted by the ionic strength effect at both very high and very low pH levels where the ionic strength is high enough to cause flocculation of the electrostatically stabilized suspension. The temperature effect study indicates that the relative contribution of Brownian motion and shear flow to the viscosity is dependent on the flocculation state of the suspension.     

Direct simulation of flexible fibers

A numerical simulation of multiple flexible fibers in suspension in Newtonian simple shear flow is presented. The method used is similar to those of previous recent simulation works by Fan et al. [J. Non-Newtonian Fluid Mech. 74 (1998) 113] and Yamane et al. [J. Non-Newtonian Fluid Mech. 54 (1994) 405], however, the method has been modified to allow a small amount of bending and torsion in the fibers. A restoring moment acts to straighten the fibers as they interact in the flow.It is demonstrated that this simulation can be used to extract basic rheological information about the suspension including fiber orientations and suspension viscosity. The viscosity of semi-concentrated to concentrated flexible fiber suspensions are shown to increase by a magnitude of the order 7–10% greater than the equivalent rigid fiber suspension tested. This is in qualitative agreement with previous experimental work by Goto et al. [Rheologica Acta 25 (1986) 119] and Blakeney [J. Colloid Interface Sci. 22 (1966) 324]. The implication is that any constitutive relation involving particulate suspensions described by orientation vectors may quantitatively underestimate suspension viscosity, particularly for fibers of large aspect ratio, or low Young’s modulus, whereby the tendency to flex is greater [Rheologica Acta 25 (1986) 119]. If particulate deformation were accounted for (by whatever means) in the existing constitutive relationship, predictions of bulk suspension parameters such as viscosity should be noticeably improved. A method is developed to modify an existing rigid-fiber viscosity to an equivalent flexible fiber viscosity, hence improving viscosity prediction ability.     

Effect of temperature on rheological properties of suspensions

A parameter, the product of viscosity of the suspension medium and applied shear rate, is introduced to describe the effect of temperature on the rheological properties of suspensions, considering the mechanism of agglomeration in suspensions. It is found that, by plotting rheograms of the shear stress versus this parameter, a single master curve can be obtained independently of temperature. The dependence of viscosity on temperature and the flow of suspensions are shown to be well described.     

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.     

A model of the steady-state motion of suspensions

The authors examine the steady-state one-dimensional motions of suspensions whose particles have a density equal to that of the corresponding dispersion medium. As a whole, the mechanical behavior of such suspensions is described by equations of motion that coincide in form with the Navier-Stokes equations for a certain incompressible fluid whose viscosity is a known function of the particle concentration in the suspensions. To close these equations, the authors postulate a principle of minimum energy dissipation for steady-state motion, which plays the paxt of an equation of state for the suspension. This new equation permits the determination of the spatial distribution in the concentration of solids. Exact solutions are presented for certain variational problems associated with the Poiseuille flow of a fluid of this kind in circular tubes and Couette flows between concentric cylinders and parallel planes. It is shown that in most cases separation of the suspension takes place.     

Observations on the rheological response of alkali activated fly ash suspensions: the role of activator type and concentration

This paper reports the influence of activator type and concentration on the rheological properties of alkali-activated fly ash suspensions. A thorough investigation of the rheological influences (yield stress and plastic viscosity) of several activator parameters, including: (i) the cation type and concentration of alkali hydroxide and (ii) the alkali-to-binder ratio (n) and silica modulus (M_s), and (iii) the volume of the activation solution, on the suspension rheology is presented. The results indicate a strong dependence on the cation and its concentration in the activation solution. The viscosity of the activation solution and the volumetric solution-to-powder ratio are shown to most strongly influence the plastic viscosity of the suspension. The suspension yield stress is predominantly influenced by the changes in fly ash particle surface charge and the ionic species in the activator. A shift from non-Newtonian to Newtonian flow behavior is noted in the case of silicate-based suspensions for M_s?≤?1.5. This behavior, which is not observed at higher M_S values, or when the fly ash is dispersed in hydroxide solutions or pure water, is hypothesized to be caused by colloidal siliceous species present in this system, or surface charge effects on the fly ash particles. Comparisons of the rheological response of alkali-activated suspensions to that of portland cement-water suspensions are also reported.     

Dynamics and rheology of a dilute suspension of dipolar nonspherical particles in an external field: Part 1. Steady shear flows

Suspensions of small nonspherical particles having dipolar moments exhibit non-Newtonian behavior under the influence of shear and external fields. Numerical methods are presented for calculating the rheological and rheo-optical properties of dilute suspensions of Brownian particles having permanent dipoles subject to time-dependent shear and external fields. The numerical methods employ the Galerkin method of weighted residuals to solve the differential equation for the particle orientation distribution function. The steady-state shear flow intrinsic viscosity of suspensions of particles with sufficiently extreme aspect ratio is predicted to exhibit a maximum value attained for intermediate shear rates at selected field orientations. These numerical results provide valuable insight into the coupling which occurs between the effects of rotary Brownian motion, the hydrodynamic resistance of nonspherical particles, and the external torque exerted on dipolar particles. The results are applicable to both suspensions of magnetic particles and electrically dipolar particles.     

An experimental study of multimodal glass suspension rheology to test and validate a polydisperse suspension viscosity model

Experimental measurements of non-colloidal multimodal suspension viscosities are performed over a wide range of mixing ratios and used to test the robustness and predictive capability of a recent viscosity model (Mwasame et al. in Phys Fluids 28:061701, 2016b), subsequently referred to as the MWB model. Three unimodally distributed particle suspensions with narrow size distributions are blended to make the bimodal and trimodal suspensions used in the rheological experiments. We demonstrate how predictions for mixture viscosities can be made using the MWB model only requiring the volume-weighted average particle sizes and viscosity correlations of the individual unimodal suspensions comprising the blend. The resultant model predictions are found to be in good agreement with measured bimodal and trimodal viscosity data to within expected experimental uncertainty. The datasets provided here can be used to validate future modeling efforts, and the MWB model can be used to optimize the viscosity of multimodal suspension mixtures for specific performance criteria.     

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.     

Solutions of xanthan gum/guar gum mixtures: shear rheology,porous media flow,and solids transport in annular flow

Mixtures of xanthan and guar gum in aqueous solution were studied in two flow situations: simple shear and porous media. In addition, solids transport in vertical annular flow of sand suspensions was explored. The zero shear rate viscosity of the solutions displayed a pronounced synergy: the viscosity of the mixture is higher than that of the polymer solutions in a wide range of relative concentrations of the two polymers, in agreement with previous literature. However, at relatively high shear rates, the viscosity approaches the value of the more viscous xanthan gum solutions at mass fractions of xanthan gum between 0.1 and 0.15, and the degree of synergy substantially decreases. Stress relaxation experiments in simple shear indicate that the polymer mixtures exhibit a well-defined yield stress after relaxation that is absent in solutions of pure polymers. In porous media flow experiments, a synergistic behavior mimicking the shear flow results was obtained for the polymer mixtures at low shear rates. However, at a critical shear rate, the apparent viscosity in porous media flows exceeds the shear viscosity due to the elongational nature of flow in the pores. The solids transport capacity in annular flows is well-represented by trends in shear viscosity and stress relaxation behavior. However, the lack of viscosity synergy at high shear rates limits the applicability of the mixtures as a way to improve solids suspension capacity in annular flows.     

Fibre suspension rheology: effect of concentration, aspect ratio and fibre size.

Viscosity data for fibre suspensions are produced using cone-and-plate geometry of enhanced dimensions for the reduced influence of fibre-wall interactions. Semi-concentrated suspensions of monodisperse polyamide fibres in silicone oil, with a variety of fibre concentrations (2, 5 and 8%), lengths and diameters, were studied. The suspension viscosity was measured in a range of shear stress in order to study the stress dependence. The study here focuses on the nature of the forces and interactions that contribute to the suspension viscosity. The results show that at sufficiently high stress levels, the suspension viscosity tends to reach a steady-state. At very low stress levels the suspension viscosity increases over time, most likely due to structures formed by adhesive forces. At higher concentrations, the viscosity depends on the absolute size of the fibres, again indicating the presence of non-hydrodynamic interactions.     

The rheology of suspensions of porous zeolite particles in polymer solutions

We present data and predictive models for the shear rheology of suspended zeolite particles in polymer solutions. It was found experimentally that suspensions of zeolite particles in polymer solutions have relative viscosities that dramatically exceed the Krieger–Dougherty predictions for hard sphere suspensions. Our investigations show that the major origin of this discrepancy is due to the selective absorption of solvent molecules from the suspending polymer solution into zeolite pores. The effect raises both the polymer concentration in the suspending medium and the particle volume fraction in the suspension. Consequently, both the viscosity of the polymer solution and the particle contribution to the suspension viscosity are increased. We propose a predictive model for the viscosity of porous zeolite suspensions by incorporating a solvent absorption parameter, α, into the Krieger–Dougherty model. We experimentally determined the solvent absorption parameter by comparing viscosity data for suspensions of porous and nonporous MFI zeolite particles. Our results are in good agreement with the theoretical pore volume of MFI particles.     

Birefringence and stress growth in uniaxial extension of polymer solutions

Simultaneous measurements of extensional stresses and birefringence are rare, especially for polymer solutions. This paper reports such measurements using the filament stretch rheometer and a phase modulated birefringence system. Both the extensional viscosity and the birefringence increase monotonically with strain and reach a plateau. Estimates of this saturation value for birefringence, using Peterlin’s formula for birefringence of a fully extended polymer chain are in agreement with the experimental results. However, estimates of the saturation value of the extensional viscosity using Batchelor’s formula for suspensions of elongated fibres are much higher than observed. Reasons for the inability of the flow field to fully unravel the polymer chain are examined using published Brownian dynamics simulations. It is tentatively concluded that the polymer chain forms a folded structure. Such folded chains can exhibit saturation in birefringence even though the stress is less than that expected for a fully extended molecule.Simultaneous measurements of stress and birefringence during relaxation indicate that the birefringence decays much more slowly than the stress. The stress-birefringence data show a pronounced hysteresis as predicted by bead-rod models. The failure of the stress optic coefficient in strong flows is noted.Experiments were also performed wherein the strain was increased linearly with time, then held constant for a short period before being increased again. The response of the stress and birefringence in such experiments is dramatically different and can be traced to the different configurations obtained during stretching and relaxation. The results cast doubt on the appropriateness of pre-averaging the non-linear terms in constitutive equations.     

A self-similar behavior for the relative viscosity of concentrated suspensions of rigid spheroids

A viscosity model for suspensions of rigid particles with predictive capability over a wide range of particle volume fraction and shear conditions is of interest to quantify the transport of suspensions in fluid flow models. We study the shear viscosity of suspensions and focus on the effect of particle aspect ratio and shear conditions on the rheological behavior of suspensions of rigid bi-axially symmetric ellipsoids (spheroids). We propose a framework that forms the basis to microscopically parameterize the evolution of the suspension microstructures and its effect on the shear viscosity of suspensions. We find that two state variables, the intrinsic viscosity in concentrated limit and the self-crowding factor, control the state of dispersion of the suspension. A combination of these two variables is shown to be invariant with the imposed shear stress (or shear rate) and depends only on the particle aspect ratio. This self-similar behavior, tested against available experimental and numerical data, allows us to derive a predictive model for the relative viscosity of concentrated suspensions of spheroids subjected to low (near zero) strain rates. At higher imposed strain rates, one needs to constrain one of the state variables independently to constrain the state of dispersion of the suspension and its shear dynamic viscosity. Alternatively, the obtained self-similar behavior provides the means to estimate the state variables from the viscosity measurements made in the laboratory, and to relate them to microstructure rearrangements and evolution occurring during deformation.     

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.     

Strain-induced low-frequency relaxation in colloidal DGEBA/SiO2 suspensions

Diglycidyl ether of bisphenol A (DGEBA) is widely exploited as an epoxy resin in adhesives and coatings. In this paper, it is used as an oligomer matrix for silica-filled nanocomposites. Rheological measurements show that the pure matrix obeys power-law relaxation dynamics in the vicinity of the dynamic glass transition of this low-molecular-weight glass former. In the filled systems, a low-frequency relaxation appears additionally to the structural α-process of the matrix. Considering the nanocomposites as Newtonian hard-sphere suspensions at low angular frequencies (or high temperatures), the modified terminal regime behavior of the matrix can be linked to strain-induced perturbations of the isotropic filler distributions. While in the low-frequency regime hydrodynamic stresses relax instantaneously, the Brownian stress relaxation is viscoelastic and can be evidenced by dynamic rheological measurements. At higher angular frequencies, the α-process of the matrix superimposes on the Brownian stress relaxation. In particular, we were able to depict the low-frequency anomaly for concentrated, semi-dilute, and even for dilute suspensions.     

Rheology of brown coal-water suspensions

Techniques for measuring the fundamental flow properties of as-mined Victorian brown coal suspensions are developed. Flow properties are presented for Morwell, Yallourn and Loy Yang coals as a function of concentration for fixed particle size distribution. Even at the relatively low solids concentration of 20 to 30 percent by weight, the suspensions exhibit complex non-Newtonian characteristics. Generally at high concentration, the suspensions are thixotropic with a shear rate dependent viscosity and exhibit a yield stress. The inherent thixotropy of the coals can be exploited and it is technically but perhaps not economically feasible to transport the as-mined coals in a pipeline. The power requirement to pump the coals in a pipeline is found to be 10 to 30 times that required to pump black coal in the Black Mesa pipeline in the U.S.A. The differences in the rheological properties of the three coals is related to the surface properties of the coal particle — expressed in terms of carboxylate and inorganic cation content, and to differences in the pore volume of the coals. The flow characteristics of one coal can be converted to that of another simply by varying either the carboxylate or ionic content of the suspension.