Inelastic Fluids Research Articles

New contributions on laminar flow of inelastic non-Newtonian fluid in the two-dimensional symmetric expansion: Creeping and slowly moving flow conditions

The steady flow of generalized Newtonian fluid in a two-dimensional 1:3 sudden expansion was studied numerically. Finite volume method was applied to solve the momentum equations along with the continuity equation and the Power law rheological model within the laminar flow regime for a range of Reynolds number and Power law index values. The values of generalized Reynolds number, based on physical and rheological properties, upstream channel height and bulk velocity, were varied between 0.0001 ≤ Re gen ≤ 10, while the Power law index values mapped the 0.60 ≤ n ≤ 1.40 range, allowing for the investigation of both shear-thinning and shear-thickening effects at creeping as well as slowly moving fluid flow conditions. We report accurate results of a systematic study with a focus on most important characteristics of recirculating fluid flow in the downstream section of sudden expansion geometry. It is shown that for the creeping flow regime there exist finite sized redevelopment length, extra pressure drop (Couette correction) and recirculation zones (also called as Moffatt vortices) that are influenced by the non-Newtonian viscous behaviour.

Numerical simulation of a bubble rising in shear-thinning fluids

The motion of a single bubble rising freely in quiescent non-Newtonian viscous fluids was investigated experimentally and computationally. The non-Newtonian effects in the flow of viscous inelastic fluids are modeled by the Carreau rheological model. An improved level set approach for computing the incompressible two-phase flow with deformable free interface is used. The control volume formulation with the SIMPLEC algorithm incorporated is used to solve the governing equations on a staggered Eulerian grid. The simulation results demonstrate that the algorithm is robust for shear-thinning liquids with large density ( ρ 1/ ρ g up to 10 3) and high viscosity ( η 1/ η g up to 10 4). The comparison of the experimental measurements of terminal bubble shape and velocity with the computational results is satisfactory. It is shown that the local change in viscosity around a bubble greatly depends on the bubble shape and the zero-shear viscosity of non-Newtonian shear-thinning liquids. The shear-rate distribution and velocity fields are used to elucidate the formation of a region of large viscosity at the rear of a bubble as a result of the rather stagnant flow behind the bubble. The numerical results provide the basis for further investigations, such as the numerical simulation of viscoelastic fluids.

Dynamics of a dilute sheared inelastic fluid. I. Hydrodynamic modes and velocity correlation functions

The hydrodynamic modes and the velocity autocorrelation functions for a dilute sheared inelastic fluid are analyzed using an expansion in the parameter =(1-e);{12} , where e is the coefficient of restitution. It is shown that the hydrodynamic modes for a sheared inelastic fluid are very different from those for an elastic fluid in the long-wave limit, since energy is not a conserved variable when the wavelength of perturbations is larger than the "conduction length." In an inelastic fluid under shear, there are three coupled modes, the mass and the momenta in the plane of shear, which have a decay rate proportional to k;{23} in the limit k-->0 , if the wave vector has a component along the flow direction. When the wave vector is aligned along the gradient-vorticity plane, we find that the scaling of the growth rate is similar to that for an elastic fluid. The Fourier transforms of the velocity autocorrelation functions are calculated for a steady shear flow correct to leading order in an expansion in . The time dependence of the autocorrelation function in the long-time limit is obtained by estimating the integral of the Fourier transform over wave number space. It is found that the autocorrelation functions for the velocity in the flow and gradient directions decay proportional to t;{-52} in two dimensions and t;{-154} in three dimensions. In the vorticity direction, the decay of the autocorrelation function is proportional to t;{-3} in two dimensions and t;{-72} in three dimensions.

Dynamics of a dilute sheared inelastic fluid. II. The effect of correlations

The effect of correlations on the viscosity of a dilute sheared inelastic fluid is analyzed using the ring-kinetic equation for the two-particle correlation function. The leading-order contribution to the stress in an expansion in =(1-e);{12} is calculated, and it is shown that the leading-order viscosity is identical to that obtained from the Green-Kubo formula, provided the stress autocorrelation function in a sheared steady state is used in the Green-Kubo formula. A systemmatic extension of this to higher orders is also formulated, and the higher-order contributions to the stress from the ring-kinetic equation are determined in terms of the terms in the Chapman-Enskog solution for the Boltzmann equation. The series is resummed analytically to obtain a renormalized stress equation. The most dominant contributions to the two-particle correlation function are products of the eigenvectors of the conserved hydrodynamic modes of the two correlated particles. In Part I, it was shown that the long-time tails of the velocity autocorrelation function are not present in a sheared fluid. Using those results, we show that correlations do not cause a divergence in the transport coefficients; the viscosity is not divergent in two dimensions, and the Burnett coefficients are not divergent in three dimensions. The equations for three-particle and higher correlations are analyzed diagrammatically. It is found that the contributions due to the three-particle and higher correlation functions to the renormalized viscosity are smaller than those due to the two-particle distribution function in the limit -->0 . This implies that the most dominant correlation effects are due to the two-particle correlations.

Combined effect of variable fluid properties on nonisothermal flow of an inelastic fluid with viscous heating

This study investigates the effect of variable thermal conductivity and variable viscosity on the flow characteristics and flow rates of an inelastic fluid, including viscous heating terms in the analysis. The exponential dependence of viscosity on temperature is modeled through Arrhenius law and shear rate dependence is modeled through Carreau rheological equation while it is assumed that the thermal conductivity varies linearly with temperature. Flow governing motion and energy balance equations are coupled and the solution is found iteratively facilitating a pseudospectral method based on the Chebyshev polynomial expansions. Effect of variable thermal conductivity and viscosity as well as the effect of power law index, material time constant, Brinkman and Peclet numbers on the velocity and temperature profiles are presented. Influence of the above parameters on the flow rate is also calculated.

Viscoelastic flow in a 3D square/square contraction: Visualizations and simulations

The inertialess three-dimensional (3D) flow of viscoelastic shear-thinning fluids in a 4:1 sudden square-square contraction was investigated experimentally and numerically and compared with the flow of inelastic fluids. Whereas for a Newtonian fluid the vortex length remains unchanged at low Reynolds numbers, with the non-Newtonian fluid there is a large increase in vortex length with fluid elasticity leading to unstable periodic flow at higher flow rates. In the steady flow regime the vortices are 3D and fluid particles enter the vortex at the middle plane, rotate towards its eye, drift sideways to the corner-plane vortex, rotate to its periphery, and exit to the downstream duct. Such dynamic process is reverse of that observed and predicted with Newtonian fluids. Numerical predictions using a multimode Phan-Thien–Tanner viscoelastic model are found to match the visualizations accurately and in particular are able to replicate the observed flow reversal. The effect of fluid rheology on flow reversal, vor...

Editorial

Editorial

Galerkin Least-Squares Multifield Approximations for Flows of Inelastic Non-Newtonian Fluids

The aim of this work is to investigate a Galerkin least-squares (GLS) multifield formulation for inelastic non-Newtonian fluid flows. We present the mechanical modeling of isochoric flows combining mass and momentum balance laws in continuum mechanics with an inelastic constitutive equation for the stress tensor. For the latter, we use the generalized Newtonian liquid model, which may predict either shear-thinning or shear-thickening. We employ a finite element formulation stabilized via a GLS scheme in three primal variables: extra stress, velocity, and pressure. This formulation keeps the inertial terms and has the capability of predicting viscosity dependency on the strain rate. The GLS method circumvents the compatibility conditions that arise in mixed formulations between the approximation functions of pressure and velocity and, in the multifield case, of extra stress and velocity. The GLS terms are added elementwise, as functions of the grid Reynolds number, so as to add artificial diffusivity selectively to diffusion and advection dominant flow regions—an important feature in the case of variable viscosity fluids. We present numerical results for the lid-driven cavity flow of shear-thinning and shear-thickening fluids, using the power-law viscosity function for Reynolds numbers between 50 and 500 and power-law exponents from 0.25 to 1.5. We also present results concerning flows of shear-thinning Carreau fluids through abrupt planar and axisymmetric contractions. We study ranges of Carreau numbers from 1 to 100, Reynolds numbers from 1 to 100, and power-law exponents equal to 0.1 and 0.5. Besides accounting for inertia effects in the flow, the GLS method captures some interesting features of shear-thinning flows, such as the reduction of the fluid stresses, the flattening of the velocity profile in the contraction plane, and the separation of the boundary layer downstream the contraction.

Effect of Matrix Elasticity on the Continuous Foaming of Food Models

The aim is to understand the effect of matrix elasticity on continuous foaming using food models based on glucose syrup. This was modified by adding polyacrylamide (PAA) with 2% whey protein isolate (WPI) or Tween 80 as foaming agents. Foaming was conducted in a stirred column. Rotation speed N and gas-to-liquid flow ratio (G/L) were varied. Overrun, average bubble size d (32), texture and stability were measured using densimetry, image analysis, and rheometry, respectively. Experimental results showed that 0.01% PAA did not modify the viscosity of 2% WPI models, but conferred low elastic behavior. PAA (0.05%) doubled matrix viscosity and drastically increased elasticity. The increase of elasticity became slower for further PAA addition. Foaming experiments demonstrated that theoretical overrun could not be achieved for inelastic WPI models in two cases: for high viscosity and low N, as dispersion effectiveness was reduced; for high G/L and N because of enhanced coalescence. Matrix elasticity was shown to increase overrun at constant viscosity for high G/L by enhancing interface stabilization. However, in elastic models, gas dispersion was more difficult and d (32) was higher than in inelastic fluids of similar viscosity. Finally, when the limiting step was dispersion, foaming was shown to be negatively affected by matrix elasticity.

CFD analysis of several design parameters affecting the performance of the Maxblend impeller

A CFD characterization of the hydrodynamics of the Maxblend impeller with experimental validations has been carried out with viscous Newtonian and non-Newtonian inelastic fluids. The mixing cases investigated were the non-baffled configuration with Newtonian and shear-thinning fluids, and the baffled configuration with only Newtonian fluids. The study focused on the effect of the impeller bottom clearance and the Reynolds number on the power characteristics, the distribution of shear rates and the overall flow conditions in the vessel. It was found that the bottom clearance plays a significant role on the power consumption, and that the value of the Reynolds number and the power law index strongly affect the axial pumping efficiency and the shear rate profile. The best performance was obtained when the impeller Reynolds number is superior to 10.

CFD analysis of viscous non-Newtonian flow under the influence of a superimposed rotational vibration

The effects of a superimposed sinusoidal rotational vibration on the flow of non-Newtonian fluids in a tube are studied numerically by computational fluid dynamics (CFD). Inelastic time-independent fluids of the power law, Herschel–Bulkley, Bingham plastic, and Newtonian types are investigated. Newtonian flow is unchanged by any superimposed oscillations but the flow of non-Newtonian fluids is greatly affected. The flow of shear-thinning fluids and viscoplastic fluids is enhanced, whilst the flow of shear-thickening fluids is retarded. The effects of the various rheological as well as vibration parameters are studied in detail. Flow is affected by both vibration frequency and amplitude, but different amplitude–frequency combinations which correspond to the same peak acceleration result in the same effect. Mechanical vibration in the sonic range generates substantial flow enhancements in low to moderately viscous fluids, but has limited scope for highly viscous fluids. Mechanical vibration in the ultrasound range, however, has a good potential for the processing of highly viscous materials, being able to generate orders of magnitude enhancement in flow. The extent of flow enhancement achieved is also dependent on the nature of the superimposed vibration: a rotational oscillation produces more flow enhancement than a transversal oscillation, but less than a longitudinal oscillation.

Development-Length Requirements for Fully Developed Laminar Pipe Flow of Inelastic Non-Newtonian Liquids

In the current study, we report the results of a detailed and systematic numerical investigation of developing pipe flow of inelastic non-Newtonian fluids obeying the power-law model. We are able to demonstrate that a judicious choice of the Reynolds number allows the development length at high Reynolds number to collapse onto a single curve (i.e., independent of the power-law index n). Moreover, at low Reynolds numbers, we show that the development length is, in contrast to existing results in the literature, a function of power-law index. Using a simple modification to the recently proposed correlation for Newtonian fluid flows (Durst, F. et al., 2005, “The Development Lengths of Laminar Pipe and Channel Flows,” J. Fluids Eng., 127, pp. 1154–1160) to account for this low Re behavior, we propose a unified correlation for XD∕D, which is valid in the range 0.4<n<1.5 and 0<Re<1000.

Effects of the Injection Period on the Rise Velocity and Shape of a Bubble in a Non-Newtonian Fluid

The present work aims at studying the influence of the injection period on the bubble rise velocity and their corresponding shape. Three different fluids were employed: a Newtonian fluid (99.5% glycerol); an inelastic shear thinning fluid (2% carboxymethylcellulose—CMC in water) and an elastic shear thinning fluid (0.5% polyacrylamide— PAAm in water). Five different injection periods ranging from 0.3 to 60 s were applied to bubble volumes from 20 to 1000 × 10−9 m3. Except for the case of the injection period of 0.3 s, the rise velocity in the glycerol and CMC solutions does not depend on the injection period. However, the rise velocity decreases significantly with the injection period in the PAAm solution. In the glycerol solution, bubbles take an oblate spheroid shape, and are flattened vertically. On the other side, in the CMC and PAAm solutions, bubbles are vertically elongated and have a teardrop shape with a tail. However, the recent flow field results around a rising bubble obtained by PIV tend to show a disconnection between the bubble shape and surrounding flow field.

Filament-breaking length—a measure of elasticity in extension

A technique has been developed to assess elasticity in extension of fluids which are weakly elastic. The technique is based on stretching a fluid sample held between two small closely spaced co-axial disks until the fluid bridge or filament breaks. The distance between the disks on breakup, ‘the breaking length’, is measured. When the fluid is elastic, this length is greater than that of an equivalent inelastic fluid. An inelastic baseline was established by measuring the breaking lengths of a wide variety of Newtonian fluids. Measurements with weakly elastic fluids reveal that the extra breaking length increases with polymer type and concentration and with the rate of stretching, the expected behavior for elastic liquids. The breaking lengths of model paper coatings were measured and found to correlate with the degree of misting in a roll coating machine, indicating that droplet formation is related to extensional elasticity.

Linear stability of circular Couette flow of inelastic viscoplastic fluids

Results for the fully non-axisymetric, wide gap, linear stability of the circular Couette flow for two models of inelastic viscoplastic fluids are presented. 1 1 This work was presented at the 2005 AERC conference in Grenoble. For a given rheological behaviour, an analytical technique is used to obtain the matrix problem associated with the linear stability. This problem is then solved using a standard collocation technique. For power law fluids, as long as the index n is larger than 0.2, the first unstable regime is axisymetric, and the non newtonian behaviour of the fluid has a stabilizing influence on the flow. For indexes smaller than 0.2, however, a new non-axisymmetric regime is obtained, consisting of a large number of columnar vortices. This flow is similar to those observed in some granular and fiber suspension flows. The same stability analysis is performed for a regularized Bingham model, with similar findings. Indeed, when the Bingham number is not too large, the first unstable regime is again axisymetric. When the Bingham becomes large enough, the columnar vortices regime is obtained as in the power law case. More generally, the stability of the circular Couette flow is little dependent on the details of the rheological laws. The stability results using a regularized Bingham case not only compare well to the results using a non-regularized Bingham model, but they also compare qualitatively well to the flow stability of a power law fluid with small indexes. Finally, a scaling analysis of the linear stability in the Bingham case indicates that for these fluids, the instability occurs when the supplied mechanical energy is larger than the energy necessary to “fluidize” the material.

Progressive Collapse Analysis of RC Frames Subjected to Blast Loading

Progressive collapse of a building can be triggered by damage to certain critical load-carrying elements immediately following a blast. The initial damage can be followed by a chain of events which progressively destabilize the building. The modelling of progressive collapse is very complex and must take into account the nature and extent of damage inflicted to the individual elements, the behaviour resulted from the damage and the structural interaction between the elements. Importantly, the chain of events to be modelled is dynamic in nature. This paper critically compares some analysis methods available for progressive collapse design from the very simple linear elastic static analysis to the most rigorous and time consuming nonlinear dynamic analysis which takes into accounts both the primary and secondary effects of the blast loading. Inelastic non-linear dynamic finite element analyses and computational fluid dynamics have been employed in the evaluation which revealed certain deficiencies with the simplified methods including the non-linear dynamic analysis method which does not take into account the direct impact of the blast loading.

3-D particle image velocimetry of the flow field around a sphere sedimenting near a wall: Part 2. Effects of distance from the wall

Through the use of stereoscopic 3-D particle image velocimetry (3-D PIV), the flow field surrounding a sphere sedimenting at a distance of one, two and three ball diameters from the wall is measured. A Newtonian (constant shear viscosity, inelastic) fluid, a Boger (constant shear viscosity, elastic) fluid and a shear-thinning (variable shear viscosity, elastic) fluid are used to study the effects of shear-thinning and elasticity. There are significant differences in the flow fields that can be observed for the different types of fluids and the different distances from the wall from which the sphere is dropped. We find that the structure of the flow field and the motion perpendicular to the wall are dependent on the distance from the wall at which the sphere sediments.

Viscous heating effects on the linear stability of Poiseuille flow of an inelastic fluid

In this paper, the effect of viscous heating on the stability of a non-Newtonian fluid flowing between two parallel plates under the effect of a constant pressure gradient is investigated. The viscosity of the fluid depends on both temperature and shear rate. Exponential dependence of viscosity on temperature is modeled through Arrhenius law. Non-Newtonian behavior of the fluid is modeled according to the Carreau rheological model. Motion and energy balance equations that govern the base flow and the stability of the flow are coupled and the solution to the problem is found iteratively using a pseudospectral method based on the Chebyshev polynomials. In the presence of viscous heating, the effect of activation energy parameter, Prandtl and Brinkman numbers, material time and power-law constants on the stability of the flow is presented in terms of neutral stability curves.

Numerical simulations of non-Newtonian flows between eccentric cylinders by domain decomposition and stream-tube method

In this paper, we study the influence of rheological properties of inelastic and viscoelastic fluids in flows occurring in the annulus of rotating eccentric cylinders. To compute this flow where vortex regions are encountered, the physical domain is split up into a finite number of sub-domains that are mapped into domains where open and closed streamlines are parallel and straight by means of local mapping functions to be determined numerically. Such simplicity for the mapped streamlines makes it easy to handle time-dependent constitutive models in viscoelasticity. The governing non-linear equations are solved by means of an optimization algorithm. The results point out the influence of non-Newtonian properties of the fluids in annular flows between eccentric cylinders as also in particular cases of journal bearings, notably the role of elasticity. Results with Newtonian inertial flows with large gaps underline the possibility of the approach to investigate flows with multiple vortices.

Flows of inelastic non-Newtonian fluids through arrays of aligned cylinders. Part 2. Inertial effects for square arrays

Numerical simulations are presented for flows of inelastic non-Newtonian fluids through periodic arrays of aligned cylinders. The truncated power-law fluid model is used for the relationship between the viscous stress and the rate-of-strain tensor. Results for the drag coefficient for creeping flows of such fluids have been presented in a companion paper [1]. In this second part the effects of finite fluid inertia are investigated for flows through square arrays. It is shown that the Reynolds-number dependence of the drag coefficient of a cylinder in the array is of the form C d≡ F/(ηU) = k 0 + k 2 Re2+ .. for small values of the Reynolds number Re ≡ ρaU/η, where F is the drag force, U is the averaged velocity in the array, η = K (U/a)n-1 is a viscosity scale with K and n the power-law coefficient and index and a the cylinder radius, and k 0 is the drag coefficient for creeping flows. The proportionality constant k 2 depends on the way the drag coefficient and the Reynolds number are defined. It is shown that the observed strong dependence of k 2 on n can almost be eliminated by using length scales different from a in the viscosity scales η used in the definition of Re and in the definition of the drag coefficient. Numerical simulation results are also presented for the velocity variance components. Results for flows at moderate Reynolds number, of order 100, are also presented; these are qualitatively similar to those for Newtonian fluids. The value of the Reynolds number beyond which the flow becomes unsteady was related to the Newtonian fluid case by rescaling. These results for moderate-Reynolds-number flow are compared against previously published experimental data.