Showing papers in "Journal of Non-newtonian Fluid Mechanics in 2007"
TL;DR: In this article, a review of the use of the concept of plasticity in geophysical fluid dynamics is presented, focusing on the role of pore pressure and friction in the bulk dynamics.
Abstract: The objective of this review is to examine how the concept of plasticity is used in geophysical fluid dynamics. Rapid mass movements such as snow avalanches or debris flows involve slurries of solid particles (ice, boulder, clay, etc.) within an interstitial fluid (air, water). The bulk behavior of these materials has often been modeled as plastic materials, i.e., a plastic material yields and starts to flow once its stress state has significantly departed from equilibrium. Two plastic theories are of common use in fluid dynamics: Coulomb plasticity and viscoplasticity. These theories have little in common, since ideal Coulomb materials are two-phase materials for which pore pressure and friction play the key role in the bulk dynamics, whereas viscoplastic materials (e.g., Bingham fluids) typically behave as single-phase fluids on the macroscopic scale and exhibit a viscous behavior after yielding. Determining the rheological behavior of geophysical materials remains difficult because they encompass coarse, irregular particles over a very wide range of size. Consequently, the true nature of plastic behavior for geophysical flows is still vigorously debated. In this review, we first set out the continuum-mechanics principles used for describing plastic behavior. The notion of yield surface rather than yield stress is emphasized in order to better understand how tensorial constitutive equations can be derived from experimental data. The notion of single-phase or two-phase behaviors on the macroscopic scale is then examined using a microstructural analysis on idealized suspensions of spheres within a Newtonian fluid; for these suspensions, the single-phase approximation is valid only at very high or low Stokes numbers. Within this framework, the bulk stress tensor can also be constructed, which makes it possible to give a physical interpretation to yield stress. Most of the time, depending on the bulk properties (especially, particle size) and flow features, bulk behavior is either Coulomb-like or viscoplastic in simple-shear experiments. The consequences of the rheological properties on the flow features are also examined. Some remarkable properties of the governing equations describing thin layers flowing down inclined surfaces are discussed. Finally, the question of parameter fitting is tackled: since rheological properties cannot be measured directly in most cases, they must be evaluated from field data. As an example, we show that the Coulomb model successfully captures the main traits of avalanche motion, but statistical analysis demonstrates that the probability distribution of the friction coefficient is not universal.
361 citations
TL;DR: In this article, the authors describe the structure of Carbopol dispersions in terms of polydisperse glasses made of individual swollen hydrophylic elastic sponges, which display elastoplasticity and significant dissipation both below and above their yield stress.
Abstract: This paper gives details of new data on neutralized Carbopol 940 dispersions. Appropriate techniques have been used to characterize the physical properties of the bulk gel and inter-phase slip at the wall. Previously published data are analysed and used wherever possible. Terminology and measurement difficulties are also addressed. The structure of Carbopol dispersions can be described in terms of polydisperse glasses made of individual swollen hydrophylic elastic sponges. They display elastoplasticity, and significant dissipation both below and above their yield stress. Hardly any creep was observed over 10 months of experiments. Scaling laws are proposed for Carbopol mechanical properties as a function of concentration. Carbopol behaviour varies with the value of concentration, and in particular when comparing percolation concentration with close-packing concentration. Constitutive equations for bulk shear stresses and for friction slip at the wall can be deduced from contact mechanics which fit rheometry data and scaling laws. Herschel–Bulkley constitutive equations may be used by ignoring elasticity, normal shear stresses and transients.
340 citations
TL;DR: This work presents a new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids using separated representations and tensor product approximations basis for treating transient models.
Abstract: Kinetic theory models described within the Fokker-Planck formalism involve high-dimensional spaces (including physical and conformation spaces and time). One appealing strategy for treating this kind of problems lies in the use of separated representations and tensor product approximations basis. This technique that was introduced in a former work [A. Ammar, B. Mokdad, E Chinesta, R. Keunings, A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids, J. Non-Newtonian Fluid Mech. 139 (2006) 153-176] for treating steady state kinetic theory models is extended here for treating transient models. (c) 2007 Elsevier B.V. All rights reserved.
323 citations
TL;DR: In this paper, the interplay of fluid inertia and fluid elasticity in planar entry flows was explored by studying the flow of weakly elastic solutions through micro-fabricated planar contraction geometries.
Abstract: We explore the interplay of fluid inertia and fluid elasticity in planar entry flows by studying the flow of weakly elastic solutions through micro-fabricated planar contraction geometries. The small characteristic lengthscales make it possible to achieve a wide range of Weissenberg numbers (0.4 < Wi < 42) and Reynolds numbers (0.03 < Re < 12), allowing access to a large region of Wi-Re space that is typically unattainable in conventional macroscale entry flow experiments. Experiments are carried out using a series of dilute solutions (0.78 < clc* < 1.09) of a high molecular weight polyethylene oxide, in which the solvent viscosity is varied in order to achieve a range of elasticity numbers, 2.8 < El = WilRe < 68. Fluorescent streak imaging and micro-particle image velocimetry ([L-PIV) are used to characterize the kinematics, which are classified into a number of flow regimes including Newtonian-like flow at low Wi, steady viscoelastic flow, unsteady diverging flow and vortex growth regimes. Progressive changes in the centreline velocity profilt are used to identify each of the flow regimes and to map the resulting stability boundaries in Wi-Re space. The same flow transitions can also be detected through measurements of the enhanced pressure drop across the contraction/expansion which arise from fluid viscoelasticity. The results of this work have significant design implications for lab-on-a-chip devices, which commonly contain complex geometric features and transport complex fluids, such as those containing DNA or proteins. The results also illustrate the potential for using micro-fabricated devices as rheometric tools for measuring the extensional properties of weakly elastic fluids. (C) 2007 Elsevier B.V. All rights reserved.
244 citations
TL;DR: In this article, a new model for elastoviscoplastic fluid flow is presented, which extends the Bingham viscoplastic model and the Oldroyd viscoelastic model.
Abstract: From a thermodynamic theory, a new model for elastoviscoplastic fluid flow is presented It extends the Bingham viscoplastic model and the Oldroyd viscoelastic model Fundamental flows are studied: simple shear flow, uniaxial elongation and large amplitude oscillatory shear The complex moduli (G',G'') are founded to be in qualitative agreement with experimental data for materials that present microscopic network structures and large scale rearrangements Various fluids of practical interest, such as liquid foams, droplet emulsions or blood, present such elastoviscoplastic behavior: at low stress, the material behaves as a viscoelastic solid, whereas at stresses above a yield stress, the material behaves as a fluid
197 citations
TL;DR: The main goal of this article is to review various results and methods concerning the numerical simulation of Bingham visco-plastic flow; these results have been obtained from the early 1970s to now.
Abstract: The main goal of this article is to review various results and methods concerning the numerical simulation of Bingham visco-plastic flow; these results have been obtained from the early 1970s to now. We consider first the case of flow in cylindrical pipes and then flow in multi-dimensional cavities. The methods to be discussed include classical ones relying on regularization, (kind of) Lagrange multipliers and augmented Lagrangian techniques; they include also a duality–penalty method whose implementation relies on a Newton-Conjugate Gradient-Uzawa algorithm which seems to be new (in this context at least). Other issues are addressed; they concern particularly the accelerated calculation of steady state solutions and the time discretization of the unsteady flow models. The results of numerical experiments are presented, including the simulation of the wall driven flow in square cavities.
156 citations
TL;DR: In this article, a Couette cell was designed to allow for high-resolution optical access in a simple shear flow of a surfactant system comprised of cetylpyridinium chloride and sodium salicylate in aqueous sodium chloride.
Abstract: A series of experiments were performed to further investigate the phenomenon of shear-banding in surfactant solutions. Many surfactant solutions, through their unique amphiphilic chemistry, form long wormlike micelle structures which behave like living polymers. These wormlike micelles have interesting viscoelastic properties and have been the subject of a number of recent studies. These water-based surfactant systems are widely used in many commercial and industrial applications; however, many aspects of their complex flow behavior are still not fully understood. In this study, a Couette cell was designed to allow for high-resolution optical access in a simple shear flow of a surfactant system comprised of cetylpyridinium chloride and sodium salicylate in aqueous sodium chloride. Beyond a critical stress, this system is found to enter a non-linear regime in which there is a plateau in the shear stress with increasing shear rate. Within this plateau, the fluid forms distinct bands of varying shear rate. The goal of this study was to obtain high spatial and temporal resolution particle-image velocimetry and flow-induced birefringence results in both steady and transient-startup flows. As a consequence of the high resolution, steady PIV results suggest the existence of multiple-shear bands. In the transient PIV experiments, we observe a propagating damped elastic wave, as well as fluctuations in the shear-band evolution on timescales of less than one relaxation time. Pointwise FIB gap profiles show a diffuse birefringent region prior to the onset of shear-banding in the velocity profiles. These results provide insight on the flow behavior, as well as a full set of experimental data which will drive development of constitutive models capable of predicting shear-banding.
143 citations
TL;DR: In this paper, a network model for wormlike micellar solutions is presented which incorporates scission and reforming of the chains, based on a discrete version of Cates' living polymer theory.
Abstract: In this paper a network model for wormlike micellar solutions is presented which incorporates scission and reforming of the chains, based on a discrete version of Cates’ ‘living polymer’ theory. Specifically we consider two elastically active Hookean species: long chains which can break to form two short chains, which can themselves recombine to form a long chain. The chains undergo rupture at a rate dependent on the local elongation and deformation rate. This two species model, developed ultimately to enable understanding of inhomogeneous flows, is examined in this paper for various deformations; steady-state shear flow, step strain, extension, and linear small amplitude oscillatory flow in homogeneous conditions. We also examine how systematic variations in the model parameters affect the rheological predictions and material functions.
137 citations
TL;DR: In this paper, a new numerical method for incompressible non-Newtonian fluid flows based on the lattice Boltzmann method (LBM) is proposed, which is applied to two representative test case problems.
Abstract: A new numerical method for incompressible non-Newtonian fluid flows based on the lattice Boltzmann method (LBM) is proposed. The essence of the present method lies in the determination of shear-dependent viscosity of the fluid by using a variable parameter related to the local shear rate. Also, the relaxation time in the BGK collision term is kept at unity taking account of numerical stability. The method is applied to two representative test case problems, power-law fluid flows in a reentrant corner geometry and non-Newtonian fluid flows in a three-dimensional porous structure. These simulations indicate that the method can be useful for practical non-Newtonian fluid flows, such as shear-thickening (dilatant) and shear-thinning (pseudoplastic) fluid flows.
120 citations
TL;DR: In this paper, two models, initially developed for colloidal suspensions and fiber suspensions, have been used to describe the observed phenomena, and a modified version has been proposed by adding a molecular diffusivity contribution in the Folgar-Tucker equation.
Abstract: Forward and reverse stress growth experiments have been conducted on polypropylene/organoclay nanocomposites containing the same clay loading but characterized by different microstructures. Stress overshoots have been observed for the initial start-up experiments and for the following reverse start-up experiments after a certain rest time. The amplitude of these overshoots increased with the applied shear rate and rest time, but the overshoots occurred at the same strain of about 1.7. The overshoots are related to the structure of the nanocomposites, in particular the magnitude of the overshoots increased with the degree of the clay exfoliation in the matrix. Two models, initially developed for colloidal suspensions and fiber suspensions, have been used to describe the observed phenomena. The overshoots are fairly well predicted by the first (structure network) model and explained by the competing effects of the structure breakdown under flow and reorganization during rest time. However, the model predicts that the shear stress following the overshoot decreases and reaches steady-state too rapidly. The second model developed for ellipsoid suspensions describes quite well the stress overshoots for the initial forward flow, but no effect of rest time is predicted. A modified version has been proposed by adding a molecular diffusivity contribution in the Folgar-Tucker equation. The effect of the particle disorientation is qualitatively predicted, but the kinetics is too slow compared to that deduced from experiments.
116 citations
TL;DR: Vinay et al. as discussed by the authors presented a one-dimensional model of early transients in pipeline restarts of waxy crude oils, which are modelled as compressible yield stress fluids.
Abstract: We present a one-dimensional model of early transients in pipeline restarts of waxy crude oils, which are modelled as compressible yield stress fluids. We show that restart transients are effectively controlled by three dimensionless numbers: a Reynolds number, Re, a compressibility number X ∗ , and the Bingham number B. The first two of these combine to define three different dimensionless timescales, for compressive pressure diffusion, for the propagation of acoustic waves and for viscous damping. The Bingham number governs whether or not the pipeline restarts. We illustrate each qualitatively different regime computationally, and make favourable comparisons against the fully two-dimensional model in [G. Vinay, A. Wachs, J.-F. Agassant, Numerical simulation of weakly compressible Bingham flows: the restart of pipeline flows of waxy crude oils, J. Non-Newtonian Fluid Mech. 136 (2006) 93–105]. Our model is able to explain certain counter-intuitive observations, such as the fact that a pipeline full of more compressible fluid may in certain circumstances restart earlier than the same pipeline filled with a less compressible fluid, see e.g. [M.G. Cawkwell, M.E. Charles, An improved model for start-up of pipelines containing gelled crude oil, J. Pipelines 7 (1987) 41–52].
TL;DR: In this article, the authors present results of an experimental study of the motion of air bubbles rising in a column of Carbopol solution under gravity, and derive a new empirical prediction of the stopping condition for static bubbles in viscoplastic fluids.
Abstract: We present results of an experimental study of the motion of air bubbles rising in a column of Carbopol solution under gravity. In Dubash and Frigaard [N. Dubash, I.A. Frigaard, Conditions for static bubbles in viscoplastic fluids, Phys. Fluids 16 (12) (2004) 4319–4330] we have predicted conditions under which sufficiently small bubbles will not propagate in a yield stress fluid. The bubbles in our experiments all exceed the bounds in Dubash and Frigaard (2004), which show themselves to be quite conservative. For smaller bubbles we observe a rapid increase in terminal bubble velocity as the ratio of bubble to column radius, R / R c increases. For R / R c ≥ 0.7 the terminal velocity approaches a maximum. By considering a simple control volume model for the energy dissipation rate of a steadily rising bubble, we are able to show that all our data collapse onto a linear relation between the work done by buoyancy and the sum of the contributions from inertia, yield stress, viscous dissipation and surface tension. By extrapolating this linear relation to the case of zero bubble velocity, we derive a new empirical prediction of the stopping condition. This condition is compatible with those previously derived, but is much less conservative.
TL;DR: In this article, a fictitious domain method for the dynamic simulation of particle motion in a Bingham viscoplastic fluid at moderate Bingham numbers is presented, which is built on the framework established by Glowinski and his coworkers, in the sense that they use their formulation and their operator splitting idea to simplify the computation, but differs from their method in both spatial and temporal discretizations of the governing equations.
Abstract: In this study, we present a fictitious domain method for the dynamic simulation of particle motion in a Bingham viscoplastic fluid at moderate Bingham numbers. Our method is built on the framework established by Glowinski and his coworkers, in the sense that we use their formulation and their operator-splitting idea to simplify the computation, but differs from their method in both spatial and temporal discretizations of the governing equations. Concerning the space scheme, we use a finite-difference method to discretize equations and a collocation-element method to enforce the rigid-body motion constraint inside the particle boundaries. Concerning the time scheme, the combined system is decoupled into three sub-systems: a Navier–Stokes problem, a plasticity problem and a rigid-body-motion problem, and we solve the Navier–Stokes problem with the classic projection method. The plasticity and rigid-body-motion multipliers at the previous time-level are retained in the Navier–Stokes problem to reduce the splitting error. In addition, the present study shows that retaining a viscous diffusion term in the plasticity problem is not necessary for the convergence of the iteration. The method is verified by comparing our results on the lid-driven cavity flow, the drag coefficient for a sphere settling in a tube and the hydrodynamic interactions between two spheres translating along the tube axis to the data available in the literature. Our results confirm that the drag coefficient for a sphere settling in a Bingham fluid at non-zero Reynolds numbers can be well correlated with an effective Reynolds number. For two approaching spheres, there exists a critical separation distance above which a drag-reduction is observed and below which a drag-enhancement takes place, compared to the drag for a single sphere. The drag-reduction however does not happen to a sphere falling towards a solid wall at the moderate Bingham numbers we studied. These observations are explained by a consideration of the competition between a shear-thinning plastic force and a repulsive lubrication force on the sphere occurring in the squeezing flow.
TL;DR: In this article, the internal flow of viscoplastic liquids through ducts consisting of an abrupt axisymmetric expansion followed by an abrupt contraction was examined, and steady, inertialess numerical solutions were obtained by solving the conservation equations of mass and momentum via the finite volume method.
Abstract: We examine the internal flow of viscoplastic liquids through ducts consisting of an abrupt axisymmetric expansion followed by an abrupt contraction. Steady, inertialess numerical solutions were obtained by solving the conservation equations of mass and momentum via the finite volume method. The viscoplastic behavior of the liquid was modeled by the generalized Newtonian liquid model with a recently proposed viscosity function. Flow visualization experiments were also conducted with Carbopol aqueous solutions at different concentrations. Yield surfaces delimiting yielded and unyielded regions were observed for different combinations of the governing parameters. The unyielded region is located in the large-diameter portion of the channel, near the wall. The size of the unyielded region is shown to be a strong function of the geometrical, rheological, and flow parameters.
TL;DR: In this article, the authors present a theoretical and experimental analysis of the dam break of a viscoplastic fluid in a horizontal channel, using a shallow, slow fluid model based on the Herschel-Bulkley constitutive law to characterize the early and late stages of flow, the final state and the dependence on yield stress and nonlinear viscosity.
Abstract: We present a theoretical and experimental analysis of the dam break of a viscoplastic fluid in a horizontal channel. A shallow, slow fluid model based on the Herschel-Bulkley constitutive law allows one to characterize the early and late stages of the flow, the final state and the dependence on yield stress and nonlinear viscosity. A particular diagnostic is advanced (time ratios based on the length of time required for the fluid to slump certain distances from the broken dam) that may assist an experimentalist to unravel those dependences. Experiments are conducted with cornsyrup, and aqueous suspensions of xanthan gum, kaolin, carbopol, cornstarch and apple puree. Cornsyrup xanthan gum and kaolin show fair quantitative agreement with theory. Carbopol compares less favourably, due primarily to inertial effects which are missing from the theory. The results for cornstarch confirm that it is shear thickening, but its detailed rheology remains unknown (and unexplored). Apple puree also appears to compare well with theory, although repeating the dam break in a roughened channel leads to substantially different results, suggesting that fluid separation can induce effective wall slip (a problem that also probably plagues the Bostwick device). Finally, theory is compared with Bostwick tests with fruit puree, with limited success.
TL;DR: In this article, the authors reported the results of a systematic numerical investigation, using the upper-convected Maxwell model, of viscoelastic flow through smooth planar contractions of various contraction ratios with particular emphasis placed on the divergent flow regime.
Abstract: This study reports the results of a systematic numerical investigation, using the upper-convected Maxwell (UCM) model, of viscoelastic flow through ‘smooth’ planar contractions of various contraction ratios with particular emphasis placed on the ‘divergent flow’ regime. It is shown that both inertia and/or shear-thinning are not required for divergent flow to be predicted in contrast to the existing results in the literature where inertia has always been present when the phenomenon has been observed. Guided by the numerical results a simple explanation is presented for the occurrence of divergent flow and the conditions under which it arises. In addition, above a critical Deborah number, the flow becomes unsteady and we use an analysis based on the scaling laws of McKinley et al. [G.H. McKinley, P. Pakdel, A. Oztekin, Rheological and geometric scaling of purely elastic flow instabilities, J. Non-Newtonian Fluid Mech. 67 (1996) 19–47] for purely elastic instabilities to show that the square of this critical Deborah number varies linearly with contraction ratio in excellent agreement with the numerical results obtained in this study.
TL;DR: In this article, the authors present an alternative reasoning for the choice of characteristic quantities to be employed in the non-dimensionalization of the governing equations of non-Newtonian fluid flow problems.
Abstract: I present an alternative reasoning for the choice of characteristic quantities to be employed in the non-dimensionalization of the governing equations of non-Newtonian fluid flow problems The usual non-dimensionalization procedure generates well known dimensionless groups such as the Reynolds number, Deborah or Weissenberg number, Carreau number, Bingham number, and capillary number The groups that represent dimensionless rheological properties (eg Deborah number, Carreau number and Bingham number) involve flow quantities such as characteristic velocities or deformation rates Consequently, for a fixed flowing material, the values of these groups change with the flow rate In the alternative procedure, the resulting dimensionless rheological groups are actually dimensionless rheological properties, and thus remain fixed for a given flowing material Moreover, each set of values of these dimensionless rheological properties defines a class of rheologically equivalent materials The proposed non-dimensionalization procedure is physically more sound, and renders simpler both the application of dimensionless results to engineering situations and the comparisons between numerical and experimental results in scientific investigations
TL;DR: In this paper, the displacement flow of a weakly compressible waxy crude oil from a pipeline, in the case that the displacing fluid is incompressible and less viscous, is considered.
Abstract: We consider the displacement flow of a weakly compressible waxy crude oil from a pipeline, in the case that the displacing fluid is incompressible and less viscous. We show that fluid compressibility only has a significant effect on the timescale over which all residual fluid is drained from the pipeline, but no noticeable effect on the initial breakthrough of new fluid. We derive analytic estimates for this drainage time, for the cases where either the pressure drop or the displacement rate is fixed. In the case of the fixed displacement rate, the drainage time may be infinite since it is possible for there to be a residual static layer of waxy crude left behind on the walls of the pipe, see also [M. Allouche, I.A. Frigaard, G. Sona, Static wall layers in the displacement of two visco-plastic fluids in a plane channel, J. Fluid Mech. 424 (2000) 243–277]. We provide estimates for the maximal residual wax fraction, which is attained by our simplified displacement model. However, fully two-dimensional displacement simulations typically produce a significantly thinner static layer, due to the occurrence of fully two-dimensional velocity and stress fields close to the displacement front.
TL;DR: In this paper, a numerical method for simulating viscoelastic axisymmetric free surface flow of an Oldroyd B fluid was developed for the computation of the non-Newtonian extra-stress components on rigid boundaries and on the symmetry axis.
Abstract: This work deals with the development of a numerical method for simulating viscoelastic axisymmetric free surface flow of an Oldroyd B fluid. A novel formulation is developed for the computation of the non-Newtonian extra-stress components on rigid boundaries and on the symmetry axis. The full free surface stress conditions are employed. The resulting governing equations are solved by finite differences on a Marker-and-cell (MAC) type grid. Validation is provided by simulating a pipe flow problem. The classical die-swell problem is solved and swelling ratios are provided. The height of the splash caused by a falling liquid drop for various Reynolds and Weissenberg numbers is then studied, and the height of the splash is shown to diminish with increasing viscoelasticity.
TL;DR: In this paper, a comprehensive numerical study of the effects of the contraction ratio upon viscoelastic flow through axisymmetric contractions was carried out, and the results enabled the construction of vortex pattern maps, with CR and De as independent parameters, elucidating the role of these dimensionless groups in controlling vortex growth, vortex type (lip or corner vortices), and pressure drop characteristics.
Abstract: A comprehensive numerical study of the effects of the contraction ratio upon viscoelastic flow through axisymmetric contractions was carried out. Six contraction ratios were examined (CR = 2, 4, 10, 20, 40 and 100) using the Oldroyd-B and Phan–Thien–Tanner (PTT) constitutive equations, under creeping-flow conditions and for a wide range of Deborah numbers (De). The results enabled the construction of vortex pattern maps, with CR and De as independent parameters, elucidating the role of these dimensionless groups in controlling vortex growth, vortex type (lip or corner vortices), and pressure-drop characteristics. The extensional parameter of the PTT model was also varied (e = 0–0.5) and it was found that for small values of e the Couette correction is a monotonic decreasing function of De, while for high e values it is a monotonic increasing function. © 2007 Elsevier B.V. All rights reserved.
TL;DR: In this article, the shape of the extrudate, and in particular the thickness and diameter swells, as a function of the dimensionless power-law index (in the case of pseud-plasticity) and the dimensions of the yield stress (in case of viscoplasticity).
Abstract: Numerical simulations have been undertaken for the benchmark problem of annular extrudate swell present in pipe extrusion and parison formation in blow molding. The finite element method (FEM) is used to provide numerical results for different inner/outer diameter ratios κ under steady-state conditions. The Herschel-Bulkley model of viscoplasticity is used with the Papanastasiou regularization, which reduces with appropriate parameter choices to the Bingham–Papanastasiou, power-law and Newtonian models. The present results provide the shape of the extrudate, and in particular the thickness and diameter swells, as a function of the dimensionless power-law index (in the case of pseudoplasticity) and the dimensionless yield stress (in the case of viscoplasticity). The pressures from the simulations have been used to compute the excess pressure losses in the system (exit correction). While shear-thinning leads to reduced swelling relative to the Newtonian values for all κ-values, the opposite is true for shear-thickening fluids, which exhibit considerable swelling. Viscoplasticity leads to decreased extrudate swell as the dimensionless yield stress goes from zero (Newtonian behaviour) to an asymptotic value of fully plastic behaviour. The exit correction decreases to zero with a decrease in the power-law index to zero and an increase in the dimensionless yield stress to its asymptotic limit. However, the decrease is not monotonic: for power-law fluids it has maxima in the range of power-law indices between 0.8 and 0.6, while for viscoplastic fluids it has maxima around Bingham number values of 5.
TL;DR: Hulsen et al. as mentioned in this paper used the log-conformation formulation to solve the high Weissenberg number problem for viscoelastic flow problems, which can be used to solve all the governing equations (continuity, conservation of momentum and constitutive equation) in a strongly coupled way.
Abstract: The log-conformation formulation has alleviated the long-standing high Weissenberg number problem associated with the viscoelastic fluid flows [R. Fattal, R. Kupferman, Constitutive laws for the matrix-logarithm of the conformation tensor, J. Non-Newtonian Fluid Mech. 123 (2004) 281–285]. This formulation ensures that solutions of viscoelastic flow problems are physically admissible, and it is able to capture sharp elastic stress layers. However, the implementations presented in literature thus far require changing the evolution equation for the conformation tensor into an equation for its logarithm, and are based on loosely coupled (partitioned) solution procedures [M.A. Hulsen, et al., Flow of viscoelastic fluids past a cylinder at high Weissenberg number: stabilized simulations using matrix logarithms, J. Non-Newtonian Fluid Mech. 127 (2005) 27–39]. A simple alternate form of the log-conformation formulation is presented in this article, and an implementation is demonstrated in the DEVSS-TG/SUPG finite element method. Besides its straightforward implementation, the new log-conformation formulation can be used to solve all the governing equations (continuity, conservation of momentum and constitutive equation) in a strongly coupled way by Newton’s method. The method can be applied to any conformation tensor model. The flows of Oldroyd-B and Larson-type fluids are tested in the benchmark problem of a flow past a cylinder in a channel. The accuracy of the method is assessed by comparing solutions with published results. The benefits of this new implementation and the pending issues are discussed.
TL;DR: In this article, a series of novel flow structures comprised of highly interconnected filaments created by the growth of multiple internal holes that develop within the fluid sheet was observed. But, the results were limited to the case of flat-fan and hollow-cone spray nozzles.
Abstract: This article reports experimental observations of the flow kinematics and stability of thin fluid sheets produced by a series of commercially available flat-fan and hollow-cone spray nozzles for a series of viscoelastic wormlike micelle solutions. As the flow rate through the nozzle is increased, the sheets of viscoelastic fluid grow larger and eventually becoming unstable and atomizing into drops. For the sheets of water produced by the flat-fan nozzles, the fluid rims of the sheets were found to destabilize first. The addition of viscoelasticity was found to stabilize the rim while simultaneously destabilizing the internal fluid sheet. What results is a series of novel flow structures comprised of highly interconnected filaments created by the growth of multiple internal holes that develop within the fluid sheet. Increasing viscoelasticity of the test fluid was found to stabilize the thin films produced by both the flat-fan and hollow-cone spray nozzles, thereby shifting the break-up of the sheets to larger flow rates. However, beyond the critical flow rate for sheet rupture, increases to the fluid elasticity were found to alter the dynamics of the atomization of the viscoelastic fluid sheets by increasing the number and growth rate of holes in the sheet while simultaneously reducing the initiation time for sheet rupture.
TL;DR: In this article, a jump discontinuity of the rise velocity of a single air bubble in viscoelastic liquids was investigated. But the authors focused on the non-dimensional properties of the polymer solutions.
Abstract: Bubbles rising in viscoelastic liquids may exhibit a jump discontinuity of the rise velocity as a critical bubble volume is exceeded. We carried out detailed experiments to investigate the occurrence of this discontinuity with single air bubble rising in various polymer solutions without influence of surfactants. The polymer solutions were characterized thoroughly by means of shear and elongational rheometry, as well as tensiometry. The experiments showed that a jump discontinuity can exist only if the non-dimensional group λE(g3ρ1/σ)1/4, found by dimensional analysis, exceeds a critical value. A universal correlation of non-dimensional numbers for the non-dimensional critical bubble volume at the jump discontinuity was found. The non-dimensional numbers represent the relevant rheological and dynamic liquid properties. This is the first time that the prediction of the critical bubble volume as well as the potential of the solution to exhibit a bubble rise velocity discontinuity becomes possible based on liquid properties only. In the correlation found, the relaxation time of the polymer solutions in elongational flow of the viscoelastic liquid was found to play a key role.
TL;DR: Wu et al. as mentioned in this paper studied the surface instability of a confined viscoelastic liquid film under the influence of an applied electric field using the Maxwell and Jeffreys models for the liquid and showed that inclusion of fluid inertia removes the singularity and leads to finite but large growth rates for all values of Deborah number.
Abstract: We study the surface instability of a confined viscoelastic liquid film under the influence of an applied electric field using the Maxwell and Jeffreys models for the liquid. It was shown recently for a Maxwell fluid in the absence of inertia that the growth rate of the electrohydrodynamic instability diverges above a critical value of Deborah number [L. Wu, S.Y. Chou, Electrohydrodynamic instability of a thin film of viscoelastic polymer underneath a lithographically manufactured mask, J. Non-Newtonian Fluid Mech. 125 (2005) 91] and the problem of pattern length selection becomes ill-defined. We show here that inclusion of fluid inertia removes the singularity and leads to finite but large growth rates for all values of Deborah number. The dominant wavelength of instability is thus identified. Our results show that the limit of small inertia is not the same as the limit of zero inertia for the correct description of the dynamics and wavelength of instability in a polymer melt. In the absence of inertia, we show that the presence of a very small amount of solvent viscosity (in the Jeffreys model) also removes the non-physical singularity in the growth rate for arbitrary Deborah numbers. Our linear stability analysis offers a plausible explanation for the highly regular length scales of the electric field induced patterns obtained in experiments for polymer melts. Further, the dominant length scale of the instability is found to be independent of bulk rheological properties such as the relaxation time and solvent viscosity.
TL;DR: In this article, the motion of micron-sized fluorescent tracer particles in a gel of Carbopol ETD 2050 was studied using fluorescence microscopy, which indicated that the material is inhomogeneous, with different particles sampling different microrheological environments.
Abstract: The motion of micron-sized fluorescent tracer particles in a gel of Carbopol ETD 2050 is studied using fluorescence microscopy. For a Carbopol concentration at which the material shows yield-stress behavior on the bulk scale, the tracer particles display a range of behavior: Some of the particles show slightly subdiffusive behavior, while others are almost completely immobilized. This indicates that the material is inhomogeneous, with different particles sampling different microrheological environments. From our results we calculate the range of microrheological viscous and elastic moduli of the material, which we compare with bulk values of the moduli as determined by conventional shear rheometry.
TL;DR: Dam break flows of viscoplastic fluids are studied theoretically using a Herschel-Bulkley constitutive law and a lubrication model of the motion in this paper, and the evolution of these flows from initiation to arrest is studied by integrating the equations of motion numerically.
Abstract: Dam break flows of viscoplastic fluids are studied theoretically using a Herschel–Bulkley constitutive law and a lubrication model of the motion. Following initiation these fluids are gravitationally driven out of the lock in which they had resided. Their motion is eventually arrested because they exhibit a yield stress and they attain a stationary state in which the gravitational forces are in equilibrium with the yield stress. We study the evolution of these flows from initiation to arrest by integrating the equations of motion numerically. We demonstrate that the final arrested state is approached asymptotically and find analytically that the perturbations to the final state decay algebraically with time as 1 / t n , where n is the power index of the Herschel–Bulkley model.
TL;DR: In this paper, the authors deal with the numerical simulation of the uniaxial steady flow of a viscoplastic fluid through an eccentric annular cross-section, where the inner cylinder is treated as a fictitious domain (FD) in which a zero velocity constraint is imposed.
Abstract: This paper deals with the numerical simulation of the uniaxial steady flow of a viscoplastic fluid through an eccentric annular cross-section. The situation considered here refers to mud flow in drilling operations or cement slurry flow in cementing process in the oil and gas industry. The rheological model employed is the classical isothermal Bingham model. The steady solution of the governing equations is obtained through a transient algorithm that allows to compute solutions at an imposed flow rate. Mathematical difficulties related to the Bingham model are overcome thanks to the use of a Lagrange multiplier and an augmented Lagrangian/Uzawa method. The inner cylinder is treated as a fictitious domain (FD) in which a zero velocity constraint is imposed. The velocity constraint is relaxed by the introduction of a distributed Lagrange multiplier (DLM). The governing equations are discretized using a finite element method with triangular elements. The resulting numerical method highlights strong and robust convergence properties. The use of the DLM/FD method enables to study the influence of the eccentricity very conveniently. In particular, we show the effect of the mesh refinement on the solution accuracy in relation with the use of the DLM/FD method. Results obtained underline the effect of the eccentricity on the flow pattern. Finally, we propose an engineering response surface methodology (RSM) approach to predict the pressure drop.
TL;DR: In this article, the free surface of the two-phase flow problem of a gas displacing a non-Newtonian material in a capillary tube is computed using an elliptic mesh generation technique, with the Galerkin Finite Element Method.
Abstract: An elliptic mesh generation technique, with the Galerkin Finite Element Method is used to compute the free surface of the two-phase flow problem of a gas displacing a non-Newtonian material in a capillary tube. Two classes of non-Newtonian materials were investigated: a power-law shear-thinning liquid and a visco-plastic material with the viscosity function proposed by Papanastasiou [T.C. Papanastasiou, Flows of materials with yield-stress, J. Rheol. 31 (1987) 385–404]. The results were given as a function of a non-Newtonian capillary number and a rheological dimensionless parameter: the behavior index in the shear-thinning liquid case and a dimensionless yield-stress (equivalent to Bingham number) in the visco-plastic material. The goal of the present work is to study flow patterns, configuration of the interface between the two phases, and fraction of the mass of non-Newtonian material deposited at the wall, as functions of the dimensionless numbers cited. Some general results of experimental and numerical works found in literature were reproduced with a quite good agreement and a wider range of the dimensionless numbers was investigated. In both classes of materials studied, as the displaced fluid departs from Newtonian behavior, the fraction of the mass deposited on the tube wall decreases and the shape of the interface becomes flatter. It was offered a plausible explanation for this counter-intuitive result related to visco-plastic fluid, i.e. an apparent increase of its viscosity (increasing the dimensionless yield-stress number), induces a decrease of the layer thickness left behind. Concerning flow patterns, it was possible to identify ranges, dependent on rheological properties and capillary number, where the transition between bypass flows and fully recirculating flows occurs. In the Newtonian case all the flow patterns predicted by Taylor [G.I. Taylor, Deposition of a viscous fluid on the wall a tube, J. Fluid Mech. 10 (1961) 161–165] were well captured. In the visco-plastic and pseudo-plastic case it was found an interesting type of intermediate flow regime which is not present in the Newtonian transition.
TL;DR: In this paper, a set of constitutive relationships arising from the coupling of stress with concentration, derived using the extended irreversible thermodynamics formalism, are derived using a simple model, the Bautista-Manero-Puig (BMP) model.
Abstract: In this paper, a set of constitutive relationships arising from the coupling of stress with concentration, are derived using the extended irreversible thermodynamics formalism. In complex fluids that present flow-induced changes in their internal structure, the stress constitutive equation is coupled to an evolution equation of a scalar representing the flow-induced modifications in the internal structure of the fluid, and to the constitutive equation for the mass flux. In particular, the generalization of a simple model, the Bautista–Manero–Puig (BMP) model, which has been used to reproduce the nonlinear viscoelastic behavior of polymer-like micellar solutions and other complex fluids, is derived. This model predicts rather well steady and unsteady nonlinear features of semi-dilute solutions that exhibit shear-banding flow and the shear-thickening transition observed in dilute surfactant solutions.