scholarly journals Revisiting ignited–quenched transition and the non-Newtonian rheology of a sheared dilute gas–solid suspension

2017 ◽  
Vol 833 ◽  
pp. 206-246 ◽  
Author(s):  
Saikat Saha ◽  
Meheboob Alam
Keyword(s):  
Normal Stress ◽  
Shear Viscosity ◽  
Hard Spheres ◽  
Moment Tensor ◽  
Stokes Number ◽  
Simple Shear Flow ◽  
Restitution Coefficient ◽  
Dilute Gas ◽  
Solid Suspension ◽  
Normal Stress Differences

The hydrodynamics and rheology of a sheared dilute gas–solid suspension, consisting of inelastic hard spheres suspended in a gas, are analysed using an anisotropic Maxwellian as the single particle distribution function. For the simple shear flow, the closed-form solutions for granular temperature and three invariants of the second-moment tensor are obtained as functions of the Stokes number ($St$), the mean density ($\unicode[STIX]{x1D708}$) and the restitution coefficient ($e$). Multiple states of high and low temperatures are found when the Stokes number is small, thus recovering the ‘ignited’ and ‘quenched’ states, respectively, of Tsao & Koch (J. Fluid Mech., vol. 296, 1995, pp. 211–246). The phase diagram is constructed in the three-dimensional ($\unicode[STIX]{x1D708},St,e$)-space that delineates the regions of ignited and quenched states and their coexistence. The particle-phase shear viscosity and the normal-stress differences are analysed, along with related scaling relations on the quenched and ignited states. At any $e$, the shear viscosity undergoes a discontinuous jump with increasing shear rate at the ‘quenched–ignited’ transition. The first (${\mathcal{N}}_{1}$) and second (${\mathcal{N}}_{2}$) normal-stress differences also undergo similar first-order transitions: (i) ${\mathcal{N}}_{1}$ jumps from large to small positive values and (ii) ${\mathcal{N}}_{2}$ from positive to negative values with increasing $St$, with the sign change of ${\mathcal{N}}_{2}$ identified with the system making a transition from the quenched to ignited states. The superior prediction of the present theory over the standard Grad’s method and the Burnett-order Chapman–Enskog solution is demonstrated via comparisons of transport coefficients with simulation data for a range of Stokes number and restitution coefficient.

Computational Modelling of Mesophase Pitches’ Shear Rheology

2001 ◽  
Vol 709 ◽  
Author(s):  
Dana Grecov ◽  
Alejandro D. Rey
Keyword(s):  
Normal Stress ◽  
Shear Viscosity ◽  
Computational Modelling ◽  
Simple Shear Flow ◽  
Shear Rates ◽  
Extra Stress Tensor ◽  
Normal Stress Differences ◽  
Macroscopic Orientation ◽  
Arbitrary Flow ◽  
Extra Stress

ABSTRACTFlow modelling of mesophase pitches is performed using a previously formulated mesoscopic viscoelastic rheological theory [1] that takes into account flow-induced texture transformations. A complete extra stress tensor equation is developed from first principles for liquid crystal materials under non-homogeneous arbitrary flow. Predictions for a given simple shear flow, under non-homogeneous conditions, for the apparent shear viscosity and first normal stress differences are presented. The rheological functions are explained using macroscopic orientation effects, which predominate at low shear rates. The predicted normal stress differences and apparent shear viscosity are in agreement with experimental measurements.

Microstructural theory and the rheology of concentrated colloidal suspensions

2012 ◽  
Vol 713 ◽  
pp. 420-452 ◽  
Author(s):  
Ehssan Nazockdast ◽  
Jeffrey F. Morris
Keyword(s):  
Normal Stress ◽  
Volume Fraction ◽  
Solid Volume Fraction ◽  
Perturbation Expansion ◽  
Colloidal Suspensions ◽  
Simple Shear Flow ◽  
Fluid Viscosity ◽  
Analytical Prediction ◽  
Regular Perturbation ◽  
Normal Stress Differences

AbstractA theory for the analytical prediction of microstructure of concentrated Brownian suspensions of spheres in simple-shear flow is developed. The computed microstructure is used in a prediction of the suspension rheology. A near-hard-sphere suspension is studied for solid volume fraction $\phi \leq 0. 55$ and Péclet number $Pe= 6\lrm{\pi} \eta \dot {\gamma } {a}^{3} / {k}_{b} T\leq 100$; $a$ is the particle radius, $\eta $ is the suspending Newtonian fluid viscosity, $\dot {\gamma } $ is the shear rate, ${k}_{b} $ is the Boltzmann constant and $T$ is absolute temperature. The method developed determines the steady pair distribution function $g(\mathbi{r})$, where $\mathbi{r}$ is the pair separation vector, from a solution of the Smoluchowski equation (SE) reduced to pair level. To account for the influence of the surrounding bath of particles on the interaction of a pair, an integro-differential form of the pair SE is developed; the integral portion represents the forces due to the bath which drive the pair interaction. Hydrodynamic interactions are accounted for in a pairwise fashion, based on the dominant influence of pair lubrication interactions for concentrated suspensions. The SE is modified to include the influence of shear-induced relative diffusion, and this is found to be crucial for success of the theory; a simple model based on understanding of the shear-induced self-diffusivity is used for this property. The computation of the microstructure is split into two parts, one specific to near-equilibrium ($Pe\ll 1$), where a regular perturbation expansion of $g$ in $Pe$ is applied, and a general-$Pe$ solution of the full SE. The predicted microstructure at low $Pe$ agrees with prior theory for dilute conditions, and becomes increasingly distorted from the equilibrium isotropic state as $\phi $ increases at fixed $Pe\lt 1$. Normal stress differences are predicted and the zero-shear viscosity predicted agrees with simulation results obtained using a Green–Kubo formulation (Foss & Brady, J. Fluid Mech., vol. 407, 2000, pp. 167–200). At $Pe\geq O(1)$, the influence of convection results in a progressively more anisotropic microstructure, with the contact values increasing with $Pe$ to yield a boundary layer and a wake. Agreement of the predicted microstructure with observations from simulations is generally good and discrepancies are clearly noted. The predicted rheology captures shear thinning and shear thickening as well as normal stress differences in good agreement with simulation; quantitative agreement is best at large $\phi $.

scholarly journals Normal stress differences in dense suspensions

2018 ◽  
Vol 857 ◽  
pp. 200-215 ◽  
Author(s):  
Ryohei Seto ◽  
Giulio G. Giusteri
Keyword(s):  
Normal Stress ◽  
Simple Shear ◽  
Volume Fraction ◽  
Hydrodynamic Lubrication ◽  
Contact Forces ◽  
Simple Shear Flow ◽  
First Normal Stress Difference ◽  
Dense Suspensions ◽  
Normal Stress Differences ◽  
Dynamics Simulations

The presence and the microscopic origin of normal stress differences in dense suspensions under simple shear flows are investigated by means of inertialess particle dynamics simulations, taking into account hydrodynamic lubrication and frictional contact forces. The synergic action of hydrodynamic and contact forces between the suspended particles is found to be the origin of negative contributions to the first normal stress difference $N_{1}$ , whereas positive values of $N_{1}$ observed at higher volume fractions near jamming are due to effects that cannot be accounted for in the hard-sphere limit. Furthermore, we found that the stress anisotropy induced by the planarity of the simple shear flow vanishes as the volume fraction approaches the jamming point for frictionless particles, while it remains finite for the case of frictional particles.

Fundamental Characteristics of Viscoelastic Fluids Simulated by a Bead-Spring-Damper Macro Model With Interaction

2000 ◽  
Author(s):  
Takuji Ishikawa ◽  
Nobuyoshi Kawabata ◽  
Katsushi Fujita ◽  
Yutaka Miyake
Keyword(s):  
Normal Stress ◽  
Constitutive Equations ◽  
Viscoelastic Fluids ◽  
Polymer Chains ◽  
Viscoelastic Flow ◽  
Simple Shear Flow ◽  
Tetrahedral Structure ◽  
Macro Model ◽  
Normal Stress Differences ◽  
Basic Characteristics

Abstract The flow field of viscoelastic fluids is commonly analyzed by using constitutive equations. In this paper, a bead-spring-damper macro model with interaction is proposed as an alternative to analyze a viscoelastic flow. A tetrahedral structure of beads and springs models a gathering of intertwined polymer chains. Behavior of the macro model and the cluster is computed under a simple shear flow condition. Shear-thinning of viscosity, the mechanism of generation of normal stress differences and the effect of slip in the interaction are investigated. The results show that the elongation of clusters to the x direction is the mechanism of the normal stress differences generation, and that the slip in the interaction weakens the stresses. Consequently, it is found that the bead-spring-damper macro model can express the behavior of polymer chains in viscoelastic fluids and basic characteristics of viscoelastic fluids without using constitutive equations.

Inertial effects on the rheology of a dilute emulsion

2010 ◽  
Vol 646 ◽  
pp. 255-296 ◽  
Author(s):  
R. VIVEK RAJA ◽  
GANESH SUBRAMANIAN ◽  
DONALD L. KOCH
Keyword(s):  
Shear Flow ◽  
Normal Stress ◽  
Simple Shear ◽  
Reynolds Numbers ◽  
Reciprocal Theorem ◽  
Simple Shear Flow ◽  
Ohnesorge Number ◽  
Linear Flow ◽  
Bulk Stress ◽  
Normal Stress Differences

The behaviour of an isolated nearly spherical drop in an ambient linear flow is examined analytically at small but finite Reynolds numbers, and thereby the first effects of inertia on the bulk stress in a dilute emulsion of neutrally buoyant drops are calculated. The Reynolds numbers, Re = a2ρ/μ and $\hat{\Rey} \,{=}\, \dot{\gamma}a^2\rho/\hat{\mu}$, are the relevant dimensionless measures of inertia in the continuous and disperse(drop) phases, respectively. Here, a is the drop radius, is the shear rate, ρ is the common density and and μ are, respectively, the viscosities of the drop and the suspending fluid. The assumption of nearly spherical drops implies the dominance of surface tension, and the analysis therefore corresponds to the limit of the capillary number(Ca) based on the viscosity of the suspending fluid being small but finite; in other words, Ca ≪ 1, where Ca = μa/T, T being the coefficient of interfacial tension. The bulk stress is determined to O(φRe) via two approaches. The first one is the familiar direct approach based on determining the force density associated with the disturbance velocity field on the surface of the drop; the latter is determined to O(Re) from a regular perturbation analysis. The second approach is based on a novel reciprocal theorem formulation and allows the calculation, to O(Re), of the drop stresslet, and hence the emulsion bulk stress, with knowledge of only the leading-order Stokes fields. The first approach is used to determine the bulk stress for linear flows without vortex stretching, while the reciprocal theorem approach allows one to generalize this result to any linear flow. For the case of simple shear flow, the inertial contributions to the bulk stress lead to normal stress differences(N1, N2) at O(φRe), where φ(≪1) is the volume fraction of the disperse phase. Inertia leads to negative and positive contributions, respectively, to N1 and N2 at O(φRe). The signs of the inertial contributions to the normal stress differences may be related to the O(ReCa) tilting of the drop towards the velocity gradient direction. These signs are, however, opposite to that of the normal stress differences in the creeping flow limit. The latter are O(φCa) and result from an O(Ca2) deformation of the drop acting to tilt it towards the flow axis. As a result, even a modest amount of inertia has a significant effect on the rheology of a dilute emulsion. In particular, both normal stress differences reverse sign at critical Reynolds numbers(Rec) of O(Ca) in the limit Ca ≪ 1. This criterion for the reversal in the signs of N1 and N2 is more conveniently expressed in terms of a critical Ohnesorge number(Oh) based on the viscosity of the suspending fluid, where Oh = μ/(ρaT)1/2. The critical Ohnesorge number for a sign reversal in N1 is found to be lower than that for N2, and the precise numerical value is a function of λ. In uniaxial extensional flow, the Trouton viscosity remains unaltered at O(φRe), the first effects of inertia now being restricted to O(φRe3/2). The analytical results for simple shear flow compare favourably with the recent numerical simulations of Li & Sarkar (J. Rheol., vol. 49, 2005, p. 1377).

Normal stresses in concentrated non-Brownian suspensions

2013 ◽  
Vol 715 ◽  
pp. 239-272 ◽  
Author(s):  
T. Dbouk ◽  
L. Lobry ◽  
E. Lemaire
Keyword(s):  
Normal Stress ◽  
Pore Pressure ◽  
Volume Fraction ◽  
Hard Spheres ◽  
Particle Volume Fraction ◽  
Radial Profile ◽  
Normal Stresses ◽  
Particle Volume ◽  
Wide Range ◽  
Normal Stress Differences

AbstractWe present an experimental approach used to measure both normal stress differences and the particle phase contribution to the normal stresses in suspensions of non-Brownian hard spheres. The methodology consists of measuring the radial profile of the normal stress along the velocity gradient direction in a torsional flow between two parallel discs. The values of the first and the second normal stress differences, ${N}_{1} $ and ${N}_{2} $, are deduced from the measurement of the slope and of the origin ordinate. The measurements are carried out for a wide range of particle volume fractions (between 0.2 and 0.5). As expected, ${N}_{2} $ is measured to be negative but ${N}_{1} $ is found to be positive. We discuss the validity of the method and present numerous tests that have been carried out in order to validate our results. The experimental setup also allows the pore pressure to be measured. Then, subtracting the pore pressure from the total stress, ${\mbrm{\Sigma} }_{\mathbf{22} } $, the contribution of the particles to the normal stress ${ \mbrm{\Sigma} }_{\mathbf{22} }^{\mathbi{p}} $ is obtained. Most of our results compare well with the different experimental and numerical data present in the literature. In particular, our results show that the magnitude of the particle stress tensor component and their dependence on the particle volume fraction used in the suspension model balance proposed by Morris & Boulay (J. Rheol., vol. 43, 1999, p. 1213) are suitable.

scholarly journals Structure, diffusion and rheology of Brownian suspensions by Stokesian Dynamics simulation

2000 ◽  
Vol 407 ◽  
pp. 167-200 ◽  
Author(s):  
DAVID R. FOSS ◽  
JOHN F. BRADY
Keyword(s):  
Normal Stress ◽  
Volume Fraction ◽  
Hard Spheres ◽  
Normal Stress Difference ◽  
Dynamics Simulation ◽  
Simple Shear Flow ◽  
Stress Difference ◽  
Stokesian Dynamics ◽  
First Normal Stress Difference ◽  
Monodisperse Suspension

The non-equilibrium behaviour of concentrated colloidal dispersions is studied using Stokesian Dynamics, a molecular-dynamics-like simulation technique for analysing suspensions of particles immersed in a Newtonian fluid. The simulations are of a monodisperse suspension of Brownian hard spheres in simple shear flow as a function of the Péclet number, Pe, which measures the relative importance of hydrodynamic and Brownian forces, over a range of volume fraction 0.316 [les ] ϕ [les ] 0.49. For Pe < 10, Brownian motion dominates the behaviour, the suspension remains well-dispersed, and the viscosity shear thins. The first normal stress difference is positive and the second negative. At higher Pe, hydrodynamics dominate resulting in an increase in the long-time self-diffusivity and the viscosity. The first normal stress difference changes sign when hydrodynamics dominate. Simulation results are shown to agree well with both theory and experiment.

Normal stress differences, their origin and constitutive relations for a sheared granular fluid

2016 ◽  
Vol 795 ◽  
pp. 549-580 ◽  
Author(s):  
Saikat Saha ◽  
Meheboob Alam
Keyword(s):  
Normal Stress ◽  
Constitutive Relations ◽  
Moment Tensor ◽  
Normal Stress Difference ◽  
Stress Difference ◽  
Uniform Shear ◽  
Uniform Shear Flow ◽  
Sign Change ◽  
Normal Stress Differences ◽  
Dilute Limit

The rheology of the steady uniform shear flow of smooth inelastic spheres is analysed by choosing the anisotropic/triaxial Gaussian as the single-particle distribution function. An exact solution of the balance equation for the second-moment tensor of velocity fluctuations, truncated at the ‘Burnett order’ (second order in the shear rate), is derived, leading to analytical expressions for the first and second ($\unicode[STIX]{x1D615}_{1}$ and $\unicode[STIX]{x1D615}_{2}$) normal stress differences and other transport coefficients as functions of density (i.e. the volume fraction of particles), restitution coefficient and other control parameters. Moreover, the perturbation solution at fourth order in the shear rate is obtained which helped to assess the range of validity of Burnett-order constitutive relations. Theoretical expressions for both $\unicode[STIX]{x1D615}_{1}$ and $\unicode[STIX]{x1D615}_{2}$ and those for pressure and shear viscosity agree well with particle simulation data for the uniform shear flow of inelastic hard spheres for a large range of volume fractions spanning from the dilute regime to close to the freezing-point density (${\it\nu}\sim 0.5$). While the first normal stress difference $\unicode[STIX]{x1D615}_{1}$ is found to be positive in the dilute limit and decreases monotonically to zero in the dense limit, the second normal stress difference $\unicode[STIX]{x1D615}_{2}$ is negative and positive in the dilute and dense limits, respectively, and undergoes a sign change at a finite density due to the sign change of its kinetic component. It is shown that the origin of $\unicode[STIX]{x1D615}_{1}$ is tied to the non-coaxiality (${\it\phi}\neq 0$) between the eigendirections of the second-moment tensor $\unicode[STIX]{x1D648}$ and those of the shear tensor $\unicode[STIX]{x1D63F}$. In contrast, the origin of $\unicode[STIX]{x1D615}_{2}$ in the dilute limit is tied to the ‘excess’ temperature ($T_{z}^{ex}=T-T_{z}$, where $T_{z}$ and $T$ are the $z$-component and the average of the granular temperature, respectively) along the mean vorticity ($z$) direction, whereas its origin in the dense limit is tied to the imposed shear field.

Investigation of stresses and normal stress differences behavior on symmetric and asymmetric polymeric fluid flow through planar gradual expansions

Meccanica ◽  
2016 ◽  
Vol 52 (8) ◽  
pp. 1889-1909 ◽  
Author(s):  
M. Norouzi ◽  
A. Shahbani Zahiri ◽  
M. M. Shahmardan ◽  
H. Hassanzadeh ◽  
M. Davoodi
Keyword(s):  
Fluid Flow ◽  
Normal Stress ◽  
Polymeric Fluid ◽  
Flow Through ◽  
Normal Stress Differences
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