The dynamics of a simple model for a thixotropic yield stress fluid

We study the dynamics of a model for thixotropic yield stress fluids which was recently proposed in [1]. This is based on the partially extended convected model (PEC), modified to allow for a non-zero shear stress limit for large shear rates (PECR), and combined with a Newtonian solvent to enable yielding to homogeneous shear flow (PECR-N) under a prescribed shear stress. We define ? to be the ratio of retardation time to relaxation time and focus on the limit ? → 0. Multiple time scales arise and the solutions are investigated with perturbation methods, in conjunction with direct computation of the full set of equations. Both PEC-N and PECR-N capture the experimentally observed phenomenon that the value of yield stress depends on the observation time. For the PECR-N model, we find transient solutions that are not expected from prior work on the PEC-N model, such as blow-up in finite time on the slow manifold, and yielded time-periodic flow.     

Quasi-static bubble in a yield stress fluid: elasto-plastic model

The influence of the yield stress of Carbopol® gel dispersions on the behaviour of quasi-static bubbles was investigated. Many fluids, from many different industrial fields, have yield stress behaviour. Most of them contain gas bubbles. To study bubble behaviour in such suspensions, a transparent model fluid (dispersion of Carbopol® in water) was used. The experimental device allowed to quasi-statically increase bubble internal pressure with small pressure step to reach a maximum target internal pressure and the pressure setpoint was inverted to return to the initial pressure. Hysterical behaviour of the bubbles was highlighted as they did not regain their initial shape because of yield stress. We show that the rheological behaviour is related to the internal pressure, bubble geometry and yield stress in quasi-static conditions. A modification of the Laplace law depending on the yield stress of the fluid and bubble sphericity was proposed and validated.     

Flow of a yield stress fluid over a rotating surface

We study the flow of yield stress fluids over a rotating surface when both the viscoelastic solid behavior below a critical deformation (γ _c) and liquid properties beyond γ _c can play a significant role. We review the detailed characteristics of the flow in the solid regime in the specific case of a pure elongational strain (large height to radius ratio). We, in particular, show that there exists a critical rotation velocity (ω _c) associated with the transition from the solid to the liquid regime. We then consider the specific case of lubricational regime (small height to radius ratio) in the liquid regime. In that case we describe the different possible evolutions of the equilibrium shape of the material as a function of the rotation velocity (ω), from which we extrapolate the transient shape evolutions as ω increases. We show that for a sufficiently large rotation velocity the sample separates into two parts, one remaining at rest around the rotation axis, the other going on moving radially. These predictions are then compared with systematic spin-coating tests under increasing rotation velocity ramps followed by a plateau at ω _f with typical yield stress fluids. It appears that there exists a critical velocity below which the material undergoes a limited elongation and beyond which it starts to spread significantly over the solid surface. For a larger ω _f value the sample forms a thick peripheral roll, leaving behind it a thin layer of fluid at rest relatively to the disc. These characteristics are in qualitative agreement with the theoretical predictions. Beyond a sufficiently large ω _f value this roll eventually spreads radially in the form of thin fingers. Moreover, in agreement with the theory in the lubricational regime, the different curves of deformation vs ω fall along a master curve when the rotation velocity is scaled by ω _c for different accelerations, different sample radii, or different material yield stress. The final thickness of the deposit seems to be mainly governed by the displacement of the roll, the characteristics of which take their origin in the initial stage of the spreading, including the solid–liquid transition.     

Heat transfer and flow of a thermodependent and yield stress fluid

In this paper, we study the flow of a fluid possessing a yield stress, in a cylindrical pipe being the wall heated with a constant flux; experimental and numerical results being presented here. We are interested in the influence of the different parameters on the Nusselt number, and in the pressure variation. We propose a model which enables us to estimate the Nusselt number and pressure variation, which takes into account the variation of the physical properties due to temperature variations. The proposed correlations agree well with experimental measurement.

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Wärmeübertragungs- und Strömungsverhalten eines Fluids mit temperatur- und scherspannungsabhängigem Fließvermögen

Zusammenfassung In dieser Arbeit wird das Strömungsverhalten eines Fluids mit temperatur- und scherspannungsabhängigem Fließvermögen in einem mit konstanter Wärmestromdichte beaufschlagten Kreisrohr untersucht. Hierbei fallen experimentell und numerisch gefundene Ergebnisse an. Besonders interessiert der Einfluß verschiedener Parameter auf die Nusselt-Zahl und den Druckverlust. Es wird ein Modell zur Berücksichtigung der Auswirkung temperaturabhängiger Stoffwerte auf beide Intensitätsparameter vorgeschlagen. Die hieraus resultierenden Korrelationen stimmen gut mit den experimentellen Befunden überein.

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Nomenclature a, b K =a e ^–bT - C f friction factor - C f* reduced friction factor - C _p specific heat capacity (J/Kg/°C) - D internal diameter (m) - h heat transfer coefficient (W/m² °C) - H free energy - Hb Herschel-Bulkley number - K fluid consistency (Pa · sⁿ) - L axial length (m) - mass flow (Kg · m^–3) - n flow behaviour index - Nu Nusselt number - p pressure (Pa) - Pe Peclet numberRe ·Pr=Re _g ·Pr _g - Pr Prandtl numberPr _g =kC _p / - Pr _g generalised Prandtl numberPr _g =kC _p /(U ₀/D ^n–1 - r radial coordinate (m) - R pipe radius (m) - R _c plug radius - Re Reynolds number - Re Metzner Reynolds number (Ostwald fluid) - Re _g generalised Reynolds number - Re _s Metzner Reynolds number (Herschel-Bulkley fluid) - t time (s) - T temperature (°C) - T _e entrance temperature (°C) - T _m mean temperature (°C) - T _p wall temperature (°C) - U ₀ mean velocity (m/s) - u axial velocity (m/s) - v radial velocity (m/s) - X ⁺ Cameron number - y, r radial coordinate (m) - z axial coordinate (m) Greek symbols R _c /R - exponent used forNu - shear rate (s^–1) - (3n + 1)/4n - * /((1 –a)) - thermal conductivity (W/m °C) - _a apparent viscosity (Pa · s) - density (kg/m³) - shear stress (Pa) - ₀ wall shear stress (Pa) - =b _p D/2 - _p heat flux (W/m²) - =1+2n/2n+1+2n ²/(2n+1)(n+1) ² - p*/z p*/z=p/z/p/z (isothermal) Indices p wall - i axial index - j radial index     

Explosively driven particle fields imaged using a high speed framing camera and particle image velocimetry

A high speed framing camera and a particle image velocimetry instrument were used to determine the properties of explosively driven particle fields in early microsecond and later millisecond times. Test items were configured in a two inch long cylindrical shape with a half inch diameter core of organic explosive. The core was surrounded by a particle bed of aluminum or tungsten powder of a specific particle size distribution. Position data from the leading edge of the particle fronts for each charge was recorded with a high speed framing camera at early time and with a particle image velocimetry (PIV) instrument at later time to determine particle velocity. Using a PIV image, a velocity gradient along the length of the particle field was established by using the mean particle velocity value determined from three separate horizontal bands that transverse the particle field. The results showed slower particles at the beginning of the particle field closest to the source and faster ones at the end. Differences in particle dispersal, luminescence, and agglomeration were seen when changes in the initial particle size and material type were made. The aluminum powders showed extensive luminescence with agglomeration forming large particle structures while the tungsten powder showed little luminescence, agglomeration and no particle structures. Combining velocity data from the high speed framing camera and PIV, the average drag coefficient for each powder type was determined. The particle field velocities and drag coefficients at one meter showed good agreement with the numerical data produced from a computational fluid dynamics code that takes advantage of both Eulerian and Lagrangian solvers to track individual particles after a set post detonation time interval.     

Progress in numerical simulation of yield stress fluid flows

Numerical simulations of viscoplastic fluid flows have provided a better understanding of fundamental properties of yield stress fluids in many applications relevant to natural and engineering sciences. In the first part of this paper, we review the classical numerical methods for the solution of the non-smooth viscoplastic mathematical models, highlight their advantages and drawbacks, and discuss more recent numerical methods that show promises for fast algorithms and accurate solutions. In the second part, we present and analyze a variety of applications and extensions involving viscoplastic flow simulations: yield slip at the wall, heat transfer, thixotropy, granular materials, and combining elasticity, with multiple phases and shallow flow approximations. We illustrate from a physical viewpoint how fascinating the corresponding rich phenomena pointed out by these simulations are.     

Power spectra of fluid velocities measured by laser Doppler velocimetry

The power spectrum and the correlation of the laser Doppler velocimeter velocity signal obtained by sampling and holding the velocity at each new Doppler burst are studied. Theory valid for low fluctuation intensity flows shows that the measured spectrum is filtered at the mean sample rate and that it contains a filtered white noise spectrum caused by the steps in the sample and hold signal. In the limit of high data density, the step noise vanishes and the sample and hold signal is statistically unbiased for any turbulence intensity.List of symbols A cross-section of the LDV measurement volume, m² - A empirical constant - B bandwidth of velocity spectrum, Hz - C concentration of particles that produce valid signals, number/m³ - d _m diameter of LDV measurement volume, m - f(₁, ₂ | u) probability density of t _i; and t _j given (t) for all t, Hz² - probability density for t _j-t_i, Hz - n (t, t) number of valid bursts in (t, t) = N + n - N (t, t) mean number of valid bursts in (t, t) - N _e mean number of particles in LDV measurement volume - valid signal arrival rate, Hz - mean valid signal arrival rate, Hz - R _uu time delayed autocorrelation of velocity, m²/s² - S _u power spectrum of velocity, m²/s²/Hz - t ₁, t ₂ times at which velocity is correlated, s - t _i, t _j arrival times of the bursts that immediately precede t ₁ and t ₂, respectively, s - t _ij t _j–t _i s - T averaging time for spectral estimator, s - T _u integral time scale of u (t), s - T Taylor's microscale for u (t), s - u velocity vector = U + u, m/s - u fluctuating component of velocity, m/s - U mean velocity, m/s - u _m sampled and held signal, m/s Greek symbols (t) noise signal, m/s - _m (t) sampled and held noise signal, m/s - bandwidth of spectral estimator window, radians/s - time between arrivals in pdf, s - Taylor's microscale of length = UT m - kinematic viscosity - ₁, ₂ arrival times in pdf, s - root mean square of noise signal, m/s - _u root mean square of u, m/s - delay time = t ₂ - t ₁ s - B duration of a Doppler burst, s - circular frequency, radians/s - _c low pass frequency of signal spectrum radians/s Other symbols ensemble average - conditional average - ^ estimate     

Radial flow velocity profiles of a yield stress fluid between smooth parallel disks

In rock grouting, idealized 2D-radial laminar flow of yield stress fluids (YSF) is a fundamental flow configuration that is used for cement grout spread estimation. A limited amount of works have presented analytical and numerical solutions on the radial velocity profiles between parallel disks. However, to the best of our knowledge, there has been no experimental work that has presented measured velocity profiles for this geometry. In this paper, we present velocity profiles of Carbopol (a simple YSF), measured by pulsed ultrasound velocimetry within a radial flow model. We describe the design of the physical model and then present the measured velocity profiles while highlighting the plug-flow region and slip effects observed for three different apertures and volumetric flow rates. Although the measured velocity profiles exhibited wall slip, there was a reasonably good agreement with the analytical solution. We then discuss the major implications of our work on radial flow.

     

Magnetorheology in an aging, yield stress matrix fluid

Field-induced static and dynamic yield stresses are explored for magnetorheological (MR) suspensions in an aging, yield stress matrix fluid composed of an aqueous dispersion of Laponite? clay. Using a custom-built magnetorheometry fixture, the MR response is studied for magnetic field strengths up to 1?T and magnetic particle concentrations up to 30?v%. The yield stress of the matrix fluid, which serves to inhibit sedimentation of dispersed carbonyl iron magnetic microparticles, is found to have a negligible effect on the field-induced static yield stress for sufficient applied fields, and good agreement is observed between field-induced static and dynamic yield stresses for all but the lowest field strengths and particle concentrations. These results, which generally imply a dominance of inter-particle dipolar interactions over the matrix fluid yield stress, are analyzed by considering a dimensionless magnetic yield parameter that quantifies the balance of stresses on particles. By characterizing the applied magnetic field in terms of the average particle magnetization, a rheological master curve is generated for the field-induced static yield stress that indicates a concentration–magnetization superposition. The results presented herein will provide guidance to formulators of MR fluids and designers of MR devices who require a field-induced static yield stress and a dispersion that is essentially indefinitely stable to sedimentation.     

Influence of yield stress on the fluid droplet impact control

The impact and the spreading of a drop of the yield stress fluid on a solid surface have been experimentally investigated. A yield stress fluid chosen as a model fluid can shed some light on the manner in which it is possible to control the impacted drop's profile. Several gels based on polymer concentration were prepared to obtain different levels of yield stress. Their shear rheological behaviours were characterized and their flow behaviours were modeled using Herschell–Bulkley equation. Droplets were impacted in a wide range of velocities upon a dry and smooth polymethylmetacrylate substrate. Their dynamics on the impacted surface were captured using a high-speed camera. The spreading and recoil of drops are studied and their behaviour was compared to that of a Newtonian fluid at each impact velocity.Influence of the yield stress level and intensity of inertia on the transient and final stages of drops impact were studied. It was shown how the increasing yield stress dictates the drop formation and also led to an emphasis of the inhibition of spreading and the weakening of retraction in the case of high inertial impacts. It was also noticed that the magnitude of the gravitational subsidence observed for the low impact velocities, is governed by the initial non-spherical shape of droplets. Dimensionless numbers were defined in the case of yield stress fluids, allowing us to compare the effects of forces present in the process and better understand the phenomena observed. Wall slip of gels on the PMMA substrate was characterized. Its influence on the drop spreading has been discussed.     

Determination of yield stress fluid behaviour from inclined plane test

The aim of this paper is to determine precisely under which conditions an inclined plane can be used as a rheometer, which could represent a practical and rapid technique for various types of industrial or natural viscoplastic coarse suspensions. We first examine its efficiency and relevancy for determining fluid yield stress in a straight way by measuring the deepest fluid layer able to stay on the inclined plane. We have made experiments with different materials (clay-water suspensions) whose yield stress ranged from 35 to 90 Pa, using 1 m long open rectangular channels with a slope ranging from 10 to 30° and a width ranging from 5 to 25 cm. Our procedure involved measuring the final fluid depth far from edges a long time after the end of the slow gravity-induced emptying of a dam placed upstream. The fluid yield stress was also estimated independently by fitting a Herschel-Bulkley model to simple shear rheometry data obtained within a relatively wide shear rate range. A good agreement between inclined rectangular channel tests and independent usual rheometrical tests is obtained even for aspect ratios (flow depth to channel width ratio) as large as 1 when one assumes that, when the fluid has stopped, the side and bottom wall shear stresses are equal to the fluid yield stress. These results prove the efficiency of the inclined plane test for determining yield stress when appropriate experimental precautions are taken for both tests. In addition we examine the possibility of determining the simple shear flow curve of a mud suspension from fluid depth, velocity and discharge measurements of different steady flows in a wide open channel (8 m long; 60 cm wide) equipped with a recirculating system. The results obtained from inclined plane tests are in good agreement with independent rheometrical data (with torsional geometries). However it is technically difficult to cover a wide shear rate range from the inclined plane technique since this requires a rather wide channel flow rate range.     

Effects of surface properties on the impact process of a yield stress fluid drop

The impact of a yield stress fluid drop onto a solid surface with diversified interface properties has been experimentally investigated. Two smooth substrates with distinct surface energies and three similar substrates with different roughnesses have been used. The bulk shear rheological behaviour of Carbopol gels, concentrated suspensions of swollen micro-gels, has been measured. Wall friction has also been characterized on each substrate. Slip effects of gels proved to be greater on a more hydrophobic substrate. They decreased with an increase in roughness. The drop hydrodynamics during the impact was correlated with the wall friction of the gels on all substrates and with the ratio of surface roughness to size of the swollen micro-gels. At very low impact velocities, the gravitational subsidence amplitude depends greatly on surface properties. At higher impact velocities, no significant difference is observed during the spreading phase. The drop behaviour differs during the retraction depending on the substrate. Interface effects during the retraction stage proved to diminish when the yield stress value increases.     

Effect of yield stress on the flow of a Casson fluid in a homogeneous porous medium bounded by a circular tube

The effect of yield stress on the flow characteristics of a Casson fluid in a homogeneous porous medium bounded by a circular tube is investigated by employing the Brinkman model to account for the Darcy resistance offered by the porous medium. The non-linear coupled implicit system of differential equations governing the flow is first transformed into suitable integral equations and are solved numerically. Analytical solution is obtained for a Newtonian fluid in the case of constant permeability, and the numerical solution is verified with that of the analytic solution. The effect of yield stress of the fluid and permeability of the porous medium on shear stress and velocity distributions, plug flow radius and flow rate are examined. The minimum pressure gradient required to start the flow is found to be independent of the permeability of the porous medium and is equal to the yield stress of the fluid.     

A correlation for yield stress fluid flow through packed beds

Relatively few correlations are available for non-Newtonian fluid flows through packed beds, even though such fluids are frequently used in industry. In this paper, a correlation is presented for yield stress fluid flow through packed beds. The correlation is developed by introducing the yield stress model in place of the Newtonian model used in deriving Erguns equation. The resulting model has three parameters that are functions of the geometry and roughness of the particle surfaces. Two of the parameters can be deduced in the limit as the yield stress becomes negligible and the model reduces to Erguns equation for Newtonian fluids. The third model parameter is determined from experimental data. The correlation relates a defined friction factor to the dimensionless Reynolds and Hedstrom numbers and can be used to predict pressure drop for flow of a yield stress fluid through a packed bed of spherical particles. Conditions for flow or no-flow are also determined in the correlation. Comparison of model calculations, between a Newtonian and a yield stress fluid for flow penetration into a packed bed of spheres, shows the yield stress fluid initially performs similar to the Newtonian fluid, at large Reynolds numbers. At lower Reynolds numbers the yield stress effect becomes important and the flow rate significantly decreases when compared to the Newtonian fluid.     

Turbulent shear stress profiles in a bubbly channel flow assessed by particle tracking velocimetry

Particle tracking velocimetry (PTV) is applied to a bubbly two-phase turbulent flow in a horizontal channel at Re = 2 × 10⁴ to investigate the turbulent shear stress profile which had been altered by the presence of bubbles. Streamwise and vertical velocity components of liquid phase are obtained using a shallow focus imaging method under backlight photography. The size of bubbles injected through a porous plate in the channel ranged from 0.3 to 1.5 mm diameter, and the bubbles show a significant backward slip velocity relative to liquid flow. After bubbles and tracer particles are identified by binarizing the image, velocity of each phase and void fraction are profiled in a downstream region. The turbulent shear stress, which consists of three components in the bubbly two-phase flow, is computed by analysis of PTV data. The result shows that the fluctuation correlation between local void fraction and vertical liquid velocity provides a negative shear stress component which promotes frictional drag reduction in the bubbly two-phase layer. The paper also deals with the source of the negative shear stress considering bubble’s relative motion to liquid.     

Calculation of turbulent fluid flow and heat transfer in ducts by a full Reynolds stress model

A computational method has been developed to predict the turbulent Reynolds stresses and turbulent heat fluxes in ducts by different turbulence models. The turbulent Reynolds stresses and other turbulent flow quantities are predicted with a full Reynolds stress model (RSM). The turbulent heat fluxes are modelled by a SED concept, the GGDH and the WET methods. Two wall functions are used, one for the velocity field and one for the temperature field. All the models are implemented for an arbitrary three‐dimensional channel. Fully developed condition is achieved by imposing cyclic boundary conditions in the main flow direction. The numerical approach is based on the finite volume technique with a non‐staggered grid arrangement. The pressure–velocity coupling is handled by using the SIMPLEC‐algorithm. The convective terms are treated by the van Leer scheme while the diffusive terms are handled by the central‐difference scheme. The hybrid scheme is used for solving the ε equation. The secondary flow generation using the RSM model is compared with a non‐linear k–ε model (non‐linear eddy viscosity model). The overall comparison between the models is presented in terms of the friction factor and Nusselt number. Copyright © 2003 John Wiley & Sons, Ltd.     

The yield stress myth?

New experimental data obtained from constant stress rheometers are used to show that the yield stress concept is an idealization, and that, given accurate measurements, no yield stress exists. The simple Cross model is shown to be a useful empiricism for many non-Newtonian fluids, including those which have hitherto been thought to possess a yield stress.Paper presented at the 9th International Congress on Rheology, Acapulco, Mexico, October 1984.     

An augmented Lagrangian approach to simulating yield stress fluid flows around a spherical gas bubble

We simulated the flow of a yield stress fluid around a gas bubble using an augmented Lagrange approach. The piecewise linear equal‐order finite elements for both the velocity and the pressure approximations proposed and analyzed by Latché and Vola in 2004 were applied. A mesh adaptive strategy based on this element‐pair choice was also proposed to render the yield surfaces of desired resolution. The corresponding numerical scheme was formulated for general Herschel–Bulkley fluids. Numerical results on Bingham fluid flows around a slowly rising spherical gas bubble were provided to validate the proposed algorithm. Copyright © 2011 John Wiley & Sons, Ltd.     

Inertial,viscous and yield stress effects in Bingham fluid filling of a 2-D cavity

The present study concentrates on Bingham fluid filling of a 2-D cavity and examines the relative importance of inertial, viscous and yield stress effects on the filling profile. The results presented are obtained using PAM-CAST/SIMULOR, which was modified by introducing a regularized Bingham fluid constitutive relation. The results identify five different flow patterns: “mound,” “disk,” “shell,” “bubble,” and a “transition flow” between that of mound and bubble patterns. A complete map of the flow patterns as a function of the Reynolds and Bingham numbers is also presented and discussed using dimensional and physical arguments within a simplified theoretical framework.     

Uniaxial extensional flow for a viscoelastic model that displays thixotropic yield stress behavior

Homogeneous uniaxial extensional flow of a viscoelastic fluid, namely, the partially extending strand convection model combined with a Newtonian solvent, is investigated for large relaxation time. Initial value problems are addressed, for prescribed constant tensile stress. The limit of large relaxation time introduces a slow time scale of evolution, in addition to a fast time scale for flow. Numerical solutions of the original equations show distinct stages of evolution, which are mathematically analyzed with asymptotic analyses for multiple time scales. We discuss the stages of evolution from equilibrium, as well as unloading the applied stress from a yielded solution. The overall picture which emerges captures a number of features which are usually associated with thixotropic yield stress fluids, such as delayed yielding, and hysteresis for up and down stress ramping. Even at large applied tensile stress, there is persistence of an interval of parameters where the deformation rate increases quickly, only after a delayed response.