Viscometric Functions Research Articles

Rheology of a suspension of deformable spheres in a weakly viscoelastic fluid

In this work, we consider a suspension of weakly deformable solid particles within a weakly viscoelastic fluid. The fluid phase is modelled as a second-order fluid, and particles within the suspended phase are assumed linearly elastic and relatively dilute. We apply a cell model as a proxy for mean field flow, and solve analytically within a cellular fluid layer and its enclosed particle. We use an ensemble averaging process to derive analytical results for the bulk stress in suspension, and evaluate the macroscopic properties in both shear and extensional flow. Our viscometric functions align with existing literature over a surprisingly broad range of fluid and solid elasticities.The suspension behaves macroscopically as a second-order fluid, and we give simple formulae by which the reader can calculate the parameters of this effective fluid, for use in more complex simulations. We additionally calculate the particle shape and orientation, and in simple shear flow show that the leading-order modifications to the angle of inclination ζ act to align the particle towards the flow direction, giving ζ=π/4−3Cae/4+α0Wi/2α1 where Cae is the elastic capillary number, Wi is the Weissenberg number, and αi are material properties of the suspending second-order fluid, for which the ratio α0/α1 is negative.

An analytical study on nonlinear viscoelastic lubrication in journal bearings

This paper presents a novel analytical solution for journal-bearing viscoelastic lubrication using the perturbation method. The nonlinear Giesekus model was used for the constitutive equations to study the effects of fluid elasticity, shear-thinning viscometric functions, and strain-hardening elongational viscosity of viscoelastic lubrication. The investigation focuses on the impact of characteristic parameters such as mobility factor, eccentricity ratio, and Weissenberg number on the fluid film pressure distribution, load capacity, and shear stress. Although distinguishing between the normal stress differences and extensional viscosity in mixed viscoelastic flows is complicated, we investigated the role and contribution of these two factors. By increasing the elasticity of the fluid, the portion of both mentioned parameters increases consequently. Furthermore, analyses and comparisons show the contributions of the first normal stress and elongational viscosity to the load capacity of the bearing through the stress ratio and flow type parameter for the first time. The research findings indicate that fluid elasticity enhances the load capacity of the bearing compared to a Newtonian lubricant with the same effective viscosity. Moreover, the bearing load capacity is divided into two regions. In the linear region, the mobility factor and Weissenberg numbers have minimal effects leading to a linear increase in the load distribution, and in the exponential region, the load capacity changes are considerable. This research provides valuable insights into the behavior of viscoelastic lubrication in journal-bearing systems.

On the rheology and magnetization of dilute magnetic emulsions under small amplitude oscillatory shear

A dilute magnetic emulsion under the combined action of a uniform external magnetic field and a small amplitude oscillatory shear is studied using numerical simulations. We consider a three-dimensional domain with a single ferrofluid droplet suspended in a non-magnetizable Newtonian fluid. We present results of droplet shape and orientation, viscoelastic functions and bulk emulsion magnetization as functions of the shear oscillation frequency, magnetic field intensity and orientation. We also investigate how the magnetic field induces mechanical anisotropy by producing internal torques in oscillatory conditions. We found that, when the magnetic field is parallel to the shear plane, the droplet shape is mostly independent of the shear oscillation frequency. Regarding the viscometric functions, we show how the external magnetic field modifies the storage and loss moduli, especially for a field aligned to the main velocity gradient. The bulk emulsion magnetization is studied in the same fashion as the viscoelastic functions of the oscillatory shear. We show that the in-phase component of the magnetization with respect to the shear rate reaches a saturation magnetization, at the high frequencies limit, dependent on the magnetic field intensity and orientation. On the other hand, we found a non-zero out-of-phase response, which indicates a finite emulsion magnetization relaxation time. Our results indicate that the magnetization relaxation is closely related to the mechanical relaxation for dilute magnetic emulsions under oscillatory shear.

Structural-Phenomenological Rheological Model for Engineering Calculations of Polymeric Media Flows

Studying the behavior of flows of solutions and polymer melts in the field of nonlinear viscoelasticity allows describing the rheological properties in more details and more accurately assess the adequacy of rheological models. A new structural-phenomenological model is proposed to describe the rheological behavior of melts of branched polymers. This model can be recommended for engineering calculations of flows of polymeric media. The model is obtained from the modified Vinogradov-Pokrovsky model which is based on the microstructur-al approach and describes the dynamics of a polymer fluid. Stationary viscometric functions for simple shear and uniaxial tension, as well as stationary shear viscosity, the first-difference coefficient of normal stresses, stationary viscosity at uniaxial tension, are calculated using the obtained model. Also, the influence of model parameters on the form of the functions has been studied. It is shown that the model describes with good accuracy the nonlinear viscoelastic behavior of flowing polymer systems: an anomaly in viscosity, a drop in the coefficient of the first difference of normal stresses, and a nonmonotonic nature of the dependence of the steady-state elongation viscosity on the tensile rate. The viscometric functions data are compared with the experimental data for an industrial polyethylene melt sample.

A conservative lubrication dynamics method for the simulation of dense non-colloidal suspensions with particle spin

In this paper, a novel semi-implicit lubrication dynamics method that can efficiently simulate dense non-colloidal suspensions is proposed. To reduce the computational cost in the presented methodology, inter-particle lubrication-based forces and torques alone are considered together with a short-range repulsion to enforce finite inter-particle separation due to surface roughness, Brownian forces or other excluded volume effects. Given that the lubrication forces are singular, i.e. scaling inversely with the inter-particle gap, the strategy to expedite the calculations is severely compromised if explicit integration schemes are used, especially at high concentrations. To overcome this issue, an efficient semi-implicit splitting integration scheme to solve for the particles translational and rotational velocities is presented. To validate the proposed methodology, a suspension under simple shear test is simulated in three dimensions and its rheology is compared against benchmark results. To demonstrate the stability/speed-up in the calculations, performance of the proposed semi-implicit scheme is compared against a classical explicit Velocity-Verlet scheme. The predicted viscometric functions for a non-colloidal suspension with a Newtonian matrix are in excellent agreement with the reference data from the literature. Moreover, the presented semi-implicit algorithm is found to be significantly faster than the classical lubrication dynamics methods with Velocity-Verlet integration schemes.

Viscometric functions and rheo-optical properties of dilute polymer solutions: Comparison of FENE-Fraenkel dumbbells with rodlike models

Rigid macromolecules or polymer chains with persistence length on the order of the contour length (or greater) have traditionally been modelled as rods or very stiff springs. The FENE-Fraenkel-spring dumbbell, which is finitely extensible about a non-zero natural length with tunable harmonic stiffness, is one such model which has previously been shown to reproduce bead–rod behaviour in the absence of hydrodynamic interactions. The force law for the FENE-Fraenkel spring reduces to the Hookean or FENE spring force law for appropriately chosen values of the spring parameters. It is consequently possible to explore the crossover region between the limits of bead–spring and bead–rod behaviour by varying the parameters suitably. In this study, using a semi-implicit predictor–corrector Brownian dynamics algorithm, the FENE-Fraenkel spring is shown to imitate a rod with hydrodynamic interactions when spring stiffness, extensibility and simulation timestep are chosen carefully. By relaxing the spring stiffness and extensibility, the FENE-Fraenkel spring can also reproduce spring-like behaviour, such as a crossover from −1∕3 to −2∕3 power-law scaling in the viscosity with shear rate, and a change from positive to negative second normal stress difference. Furthermore, comparisons with experimental data on the viscosity and linear dichroism of high aspect ratio, rigid macromolecules shows that the extensibility and stiffness of the FENE-Fraenkel spring allows for equal or improved accuracy in modelling inflexible molecules compared to rodlike models.

Investigation into the rheology of a solid sphere suspension in second-order fluid using a cell model

The rheology of suspensions of solid spheres in viscoelastic background media is analytically calculated for dilute values of solid volume fraction. A mean-field cell model is used to express the influence of the solid particles in a suspending fluid, which obeys the second-order fluid model. A semi-analytical constitutive equation is created for the whole suspension, itself a second-order fluid with modified material parameters. The resulting rheology of the whole suspension is presented in simple shear, which shows predictions for the behavior of viscometric functions past explicitly dilute concentrations.

Rheology of semi-dilute suspensions with a viscoelastic matrix

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

A hydrodynamic model for discontinuous shear-thickening in dense suspensions

Restricted sliding or rotational motion of colloidal particles plays a key role in the emergence of discontinuous shear thickening (DST). From viscometric functions to the number of contacting neighbors under an applied deformation, a hindrance to sliding motion significantly changes the behavior of dense suspensions on all scales. In this work, implicitly by using a modified hydrodynamic model based on Stokesian dynamics and explicitly by solving for the hydrodynamics of nonsmooth colloids, we show that lubrication forces that arise from surface asperities effectively provide such constraints to tangential particle motion. A transition from continuous shear thickening to DST is observed as the surface roughness of the particles is systematically increased. In this hydrodynamic model for DST, normal stress differences remain negative in the shear-thickened state (STS). Study of the spatial stress distribution indicates the onset of DST to be a highly localized event; however, particle self-diffusivity and the microstructural network suggest a rather uniform structure in the STS.

Numerical simulation of the dynamics and calculation of the rheological characteristics of the dispersed systems using BEM

Dispersed systems of various types occupy a significant place in nature, technology, and everyday life. Unfailing interest in this field is shown by researchers from the physics and mechanics, colloid chemistry, micro-manufacturing, and biology, which is due to the variety of phenomena and effects associated with dispersed systems of different nature. High-efficient computational techniques for direct modeling of the dispersed system are required to more accurately determine the rheological parameters of such systems, based on the calculated properties of its components. The present work is dedicated to the numerical investigation of the dispersed system features in a shear flow at low Reynolds numbers using the boundary element method. The results of the simulations and the method details are discussed. Calculations are presented for different types of dispersed inclusions. Viscous droplets and rigid particles of different shapes in a volume of carrier viscous liquid are considered. Furthermore, the standard viscometric functions that characterize the behaviour of an emulsion or suspensions, regarded as a homogeneous medium, are calculated and studied.

Forced Convection Heat Transfer of a Giesekus Fluid in Circular Micro-Channels Subjected to a Constant Wall Temperature

Abstract In this paper, an exact analytical solution for forced convective heat transfer of nonlinear viscoelastic fluid in isothermal circular micro-channel is presented. The nonlinear Giesekus constitutive equation is used to model the Giesekus fluid heat transfer in micro-channel with constant wall temperature, which is the main innovative aspect of the current study. This constitutive equation is a powerful tool and able to model the fractional viscometric functions, extensional viscosity, and elastic property. The solution of temperature profile and Nusselt number is obtained based on the Frobenius method. The effects of Weissenberg number, mobility factor, slip coefficient, and Navier index on temperature distribution, velocity profile, and Nusselt number are investigated in detail. The results show that the increases in both slip coefficient and Navier index cause the increases in slip velocity and maximum dimensionless temperature at the wall and the micro-channel center, respectively. Moreover, the Nusselt number has an upward trend with increases in slip coefficient and Navier index parameters. The results are indicated that the flow and temperature fields have a complex relation with mobility factor which controls the level of the nonlinearity of the Giesekus model. Additionally, three correlations for Nusselt number of Giesekus flow in micro-channel are presented.

Mechanism of shear thickening in suspensions of rigid spheres in Boger fluids. Part I: Dilute suspensions

The mechanism for shear thickening of the viscometric functions of rigid non-Brownian sphere suspensions in highly elastic fluids is studied numerically. Particular focus is placed on the particle-induced fluid stress, which is the dominant contribution to the shear thickening behavior, especially at moderate to high Weissenberg numbers. We determine the Weissenberg number scaling for both the magnitude of the particle-induced fluid stress and the volume containing most of the particle-induced fluid stress to understand the overall suspension behavior at moderate Weissenberg numbers. Furthermore, we analyze the flow type in the regions of significant polymer-induced fluid stress and find that the stretch of polymers in strain-dominated flow within closed streamlines around the particles generates the large stresses that contribute to the thickening behavior. Thus, understanding the response of the suspending fluid to extensional deformation is important for predicting the shear rheology of the bulk suspension. Lastly, we explore the effect of polymer viscosity and polymer extensibility on the behavior of the particle-induced fluid stress.

Rheological consequences of wet and dry friction in a dumbbell model with hydrodynamic interactions and internal viscosity.

The effect of fluctuating internal viscosity and hydrodynamic interactions on a range of rheological properties of dilute polymer solutions is examined using a finitely extensible dumbbell model for a polymer. Brownian dynamics simulations are used to compute both transient and steady state viscometric functions in shear flow. The results enable a careful differentiation of the influence, on rheological properties, of solvent-mediated friction from that of a dissipative mechanism that is independent of solvent viscosity. In particular, hydrodynamic interactions have a significant influence on the magnitude of the stress jump at the inception of shear flow, and on the transient viscometric functions, but a negligible effect on the steady state viscometric functions at high shear rates. Zero-shear rate viscometric functions of free-draining dumbbells remain essentially independent of the internal viscosity parameter, as predicted by the Gaussian approximation, but the inclusion of hydrodynamic interactions induces a dependence on both the hydrodynamic interaction and the internal viscosity parameter. Large values of the internal viscosity parameter lead to linear viscoelastic predictions that mimic the behavior of rigid dumbbell solutions. On the other hand, steady-shear viscometric functions at high shear rates differ in general from those for rigid dumbbells, depending crucially on the finite extensibility of the dumbbell spring.

A theoretical framework for steady-state rheometry in generic flow conditions

We introduce a general decomposition of the stress tensor for incompressible fluids in terms of its components on a tensorial basis adapted to the local flow conditions, which include extensional flows, simple shear flows, and any type of mixed flows. Such a basis is determined solely by the symmetric part of the velocity gradient and allows for a straightforward interpretation of the non-Newtonian response in any local flow conditions. In steady homogeneous flows, the material functions that represent the components of the stress on the adapted basis generalize and complete the classical set of viscometric functions used to characterize the response in simple shear flows. Such a general decomposition of the stress is effective in coherently organizing and interpreting rheological data from laboratory measurements and computational studies in non-viscometric steady flows of great importance for practical applications. The decomposition of the stress in terms with clearly distinct roles is also useful in developing constitutive models.

Measuring and assessing first and second normal stress differences of polymeric fluids with a modular cone-partitioned plate geometry

We propose a simple, robust method to measure both the first and second normal stress differences of polymers, hence obtaining the full set of viscometric material functions in nonlinear shear flow. The method is based on the use of a modular cone-partitioned plate (CPP) setup with two different diameters of the inner plate, mounted on a rotational strain-controlled rheometer. The use of CPP allows extending the measured range of shear rates without edge fracture problems. The main advantage of such a protocol is that it overcomes limitations of previous approaches based on CPP (moderate temperatures not exceeding 120 °C, multiple measurements of samples with different volume) and yields data over a wide temperature range by performing a two-step measurement on two different samples with the same volume. The method was tested with two entangled polystyrene solutions at elevated temperatures, and the results were favorably compared with both the limited literature data on the second normal stress difference and the predictions obtained with a recent tube-based model of entangled polymers accounting for shear flow-induced molecular tumbling. Limitations and possible improvements of the proposed simple experimental protocol are also discussed.

Understanding the rheological behavior of polymer nanocomposites: Non-equilibrium thermodynamics modeling coupled with detailed atomistic non-equilibrium molecular dynamics simulations

We describe a methodology for parameterizing and validating a continuum model for the rheological behaviour of polymer nanocomposites derived on principles of nonequilibrium thermodynamics based on numerical data collected from large-scale, fully-atomistic equilibrium (MD) and nonequilibrium molecular dynamics (NEMD) simulations. As a model system, we have chosen nanocomposites of polyethylene glycol (PEG) filled with functionalized silica (SiO2) nanospheres. The parameterized continuum model provides a very satisfactory description of the simulation data for the conformational properties of the studied PEO-SiO2 nanocomposite melts for a wide range of shear rates and SiO2 concentration, as well and as with the viscometric functions in steady shear. It also fits quite accurately measured NEMD data for their viscosity as a function of shear rate except from the very high shear rates, also observed in the case of pure PEO melts. Possible ways to remedy this disagreement are proposed.

Bulk fluidity and apparent wall slip of aqueous kaolin suspensions studied using the cone-cone (KK) sensor: The effect of concentration and temperature

The material viscometric functions of aqueous kaolin suspensions were measured using rotational viscometry with an unconventional cone-cone (KK) sensor. The fluidity φ[σ] and slip χ[σ] functions were obtained for various concentrations (30÷40wt.%) and temperatures (10÷40°C), applying different gap thicknesses (0.6÷1.6mm) of a smooth stainless sensor. Both the fluidity and slip functions decrease with increasing kaolin concentration. The fluidity of suspensions decreases also with rising temperature, which is a highly unusual trend. On the contrary, the slip coefficient follows a temperature rise of the fluidity of continuous phase, which supports the hypothesis on existence of a totally depleted slip layer. The relative slip contribution to overall kinematic effect, A, was found to vary over a broad interval. The apparent wall slip dominates at low shear stresses (A≅90%), whereas the bulk fluidity prevails at high shear stresses (A≅10%).

Numerical simulations of the rheology of suspensions of rigid spheres at low volume fraction in a viscoelastic fluid under shear

We present a 3D numerical investigation of the viscometric functions for suspensions in viscoelastic fluids under steady shear including comparisons with theoretical predictions in the dilute regime and experimental results for non-colloidal suspensions in a Boger fluid. The comparison with dilute suspension theoretical predictions resolves a discrepancy in the literature regarding the second normal stress difference calculated using the Landau–Lifshitz averaging procedure versus using the ensemble averaging procedure. We also determine that the particle contribution to the stress in a dilute suspension in viscoelastic fluids shear-thickens in all viscometric material properties and we present the scaling of the viscometric functions with the Weissenberg number (Wi). This shear-thickening behavior is fundamentally different from that shown in 2D numerical simulations which are attributed to elongation flow effects between particles. Comparisons with experimental results at volume fraction ϕ=0.05, which is the lowest volume fraction available in existing experimental measurements of bulk viscometric functions, highlight the important role of long range hydrodynamic interactions between particles to fully describe the suspension rheology even at low volume fractions. We investigate the effect of hydrodynamic interactions in the flow, vorticity, and gradient directions and show that these interactions can greatly enhance or decrease the viscometric functions as well as change the shear rate dependent behavior of the suspension.

Rheological Properties of Aqueous Acid Solutions of Chitosan: Experiment and Calculations of the Viscometric Functions on the Basis of a Mesoscopic Model

Rheological Properties of Aqueous Acid Solutions of Chitosan: Experiment and Calculations of the Viscometric Functions on the Basis of a Mesoscopic Model

Numerical simulations of the rheology of suspensions of rigid spheres at low volume fraction in a viscoelastic fluid under shear

We present a 3D numerical investigation of the viscometric functions for suspensions in viscoelastic fluids under steady shear including comparisons with theoretical predictions in the dilute regime and experimental results for non-colloidal suspensions in a Boger fluid. The comparison with dilute suspension theoretical predictions resolves a discrepancy in the literature regarding the second normal stress difference calculated using the Landau–Lifshitz averaging procedure versus using the ensemble averaging procedure. We also determine that the particle contribution to the stress in a dilute suspension in viscoelastic fluids shear-thickens in all viscometric material properties and we present the scaling of the viscometric functions with Weissenberg number (Wi). This shear-thickening behavior is fundamentally different from that shown in 2D numerical simulations which are attributed to elongation flow effects between particles. Comparisons with experimental results at volume fraction ϕ=0.05, which is the lowest volume fraction available in existing experimental measurements of bulk viscometric functions, highlight the important role of long range hydrodynamic interactions between particles to fully describe the suspension rheology even at low volume fractions. We investigate the effect of hydrodynamic interactions in the flow, vorticity, and gradient directions and show that these interactions can greatly enhance or decrease the viscometric functions as well as change the shear rate dependent behavior of the suspension.