Flow of a thixotropic or antithixotropic fluid in a slowly varying channel: the weakly advective regime

A general formulation of the governing equations for the slow, steady, two-dimensional flow of a thixotropic or antithixotropic fluid in a channel of slowly varying width is described. These equations are equivalent to the equations of classical lubrication theory for a Newtonian fluid, but incorporate the evolving microstructure of the fluid, described in terms of a scalar structure parameter. We demonstrate how the lubrication equations can be further simplified in the weakly advective regime in which the advective Deborah number is comparable to the aspect ratio of the flow, and present illustrative analytical and semi-analytical solutions for particular choices of the constitutive and kinetic laws, including a purely viscous Moore–Mewis–Wagner model and a regularised viscoplastic Houska model. The lubrication results also allow the calibration and validation of cross-sectionally averaged, or otherwise reduced, descriptions of thixotropic channel flow which provide a first step towards models of thixotropic flow in porous media, and we employ them to explain why such descriptions may be inadequate.

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title: Thixotropy—a review
abstract: The ensuing mechanical response to stressing or straining a structured liquid results in various viscoelastic phenomena, either in the linear region where the microstructure responds linearly with respect to the stress and strain but does not itself change, or in the nonlinear region where the microstructure does change in response to the imposed stresses and strains, but does so reversibly. The complication of thixotropy arises because this reversible, microstructural change itself takes time to come about due to local spatial rearrangement of the components. This frequently found time-response of a microstructure that is itself changing with time makes thixotropic, viscoelastic behaviour one of the greatest challenges facing rheologists today, in terms of its accurate experimental characterisation and its adequate theoretical description. Here a history of thixotropy is given, together with a description of how it is understood today in various parts of the scientific community. Then a mechanistic description of thixotropy is presented, together with a series of applications where thixotropy is important. A list of different examples of thixotropic systems is then given. Finally the various kinds of theories that have been put forward to describe the phenomenon mathematically are listed.

title: Laminar, unidirectional flow of a thixotropic fluid in a circular pipe
abstract: In this paper we study the unidirectional, axisymmetric flow of a bentonite mud in a circular pipe. Bentonite mud is an inelastic, thixotropic, generalised-Newtonian fluid. We use a rheological model that characterises this behaviour in terms of a single parameter Λ which is a measure of the amount of structure in the fluid. The behaviour of Λ is determined by a single rate equation which models the tendency of fluid structure to increase whilst being limited by the imposed shear rate. We find that, for certain parameter ranges, the model is not structurally stable, but that this problem can be eliminated by including diffusion of fluid structure. A graph of the equilibrium shear stress for a given shear rate (the rheogram) is not monotonic, yet no mechanical instability occurs in pipe flow. We contrast this with recent work on the pipe flow of a Johnson-Segalman-Oldroyd fluid which displays spurting and oscillatory behaviour. The difference lies in the relative magnitude of normal stress effects in the two fluids. There appear to be no grounds for discarding the constitutive model studied here simply because of the non-monotonicity of the equilibrium rheogram.

title: Aging and free surface flow of a thixotropic fluid
abstract: Free surface flows of thixotropic fluids such as paints, self-compacting concrete, or natural mudflows are of noticeable practical interest. Here we study the basic characteristics of the uniform flow of a layer of thixotropic fluid under gravity. A theoretical approach relying on a simple thixotropy constitutive equation shows that after some time at rest over a small slope angle the fluid layer should start to flow rather abruptly beyond a new, larger, critical slope angle. The theory also predicts that the critical time at which the layer velocity should significantly increase is proportional to the duration of the preliminary rest and tends to infinity when the new slope approaches the critical slope. Experiments carried out with different suspensions show that the qualitative trends of the flows are in very good agreement with the theoretical predictions, except that the critical time for flow start appears to be proportional to a power 0.6 of the time of rest whereas the theory predicts a linear dep...

title: A study of start-up flow of thixotropic fluids including inertia effects on an inclined plane
abstract: Despite the practical importance of thixotropic fluids, there is no reliable way at present to predict the onset of thixotropic flow. The start-up flow of thixotropic fluids including inertia effects falling down along an inclined plate is studied in this paper. The effects of the unsteady term in the NS equations on the start-up process are clarified and a criterion parameter A is presented to measure this unsteady effect. The parameter A is defined as the ratio of the Reynolds number and generalized Weissenberg number W, where W is the ratio of the characteristic time of microstructure changes and the characteristic time of flow. According to flow characteristics, we classify the motion into three cases. In case 1, avalanche happens and the initial viscosity is big. The start-up process is divided into two stages: creep and flow. Velocity profiles of both stages are discussed. In this case, if A is small enough, the inertia effects could be neglected. Otherwise, the inertial unsteady term will protract ...

title: Avalanche behavior in yield stress fluids.
abstract: We show that, above a critical stress, typical yield stress fluids (gels and clay suspensions) and soft glassy materials (colloidal glasses) start flowing abruptly and subsequently accelerate, leading to avalanches that are remarkably similar to those of granular materials. Rheometrical tests reveal that this is associated with a bifurcation in rheological behavior: for small stresses, the viscosity increases in time; the material eventually stops flowing. For slightly larger stresses the viscosity decreases continuously in time; the flow accelerates. Thus the viscosity jumps discontinuously to infinity at the critical stress. We propose a simple physical model capable of reproducing these effects.

title: Dam Break Wave of Thixotropic Fluid
abstract: Thixotropy is the characteristic of a fluid to form a gelled structure over time when it is not subjected to shearing, and to liquefy when agitated. Thixotropic fluids are commonly used in the construction industry e.g., liquid concrete and drilling fluids, and related applications include some forms of mud flows and debris flows. This paper describes a basic study of dam break wave with thixotropic fluid. Theoretical considerations were developed based upon a kinematic wave approximation of the Saint-Venant equations down a prismatic sloping channel. A very simple thixotropic model, which predicts the basic rheological trends of such fluids, was used. It describes the instantaneous state of fluid structure by a single parameter. The analytical solution of the basic flow motion and rheology equations predicts three basic flow regimes depending upon the fluid properties and flow conditions, including the initial "degree of jamming" of the fluid related to its time of restructuration at rest. These findings were successfully compared with systematic bentonite suspension experiments. The present work is the first theoretical analysis combining the basic principles of unsteady flow motion with a thixotropic fluid model and systematic laboratory experiments.

title: Viscous fingering of a thixotropic fluid in a porous medium or a narrow fracture
abstract: We consider the flow of a fluid with a modified Bautista–Manero rheology injected radially into a porous medium or unidirectionally into a widening fracture. Linear stability analysis of the flows demonstrates that viscous fingering instabilities are possible, supported essentially by the thixotropic nature of the fluid: the growth rate of unstable perturbations scales approximately with the relaxation rate λ ˆ − 1 of the fluid, while their spatial scale is set in the radial case by ( Q ˆ λ ˆ ) 1 / 2 where Q ˆ is the areal injection rate, and in the fracture case by a ‘relaxation lengthscale’ which cannot be deduced by purely dimensional reasoning. It appears likely that the same instability mechanism can occur in other shear-thinning thixotropic fluids.

title: Understanding thixotropic and antithixotropic behavior of viscoelastic micellar solutions and liquid crystalline dispersions. I. The model
abstract: A simple model consisting of the Upper Convected Maxwell constitutive equation and a kinetic equation for destruction and construction of structure, first proposed by Fredrickson in 1970, is used here to reproduce the complex rheological behavior of viscoelastic systems that also exhibit thixotropy and rheopexy under shear flow. The model requires five parameters that have physical significance and that can be estimated from rheological measurements. Several steady and unsteady flow situations were analyzed with the model. The model predicts creep behavior, stress relaxation and the presence of thixotropic loops when the sample is subjected to transient stress cycles. Such behavior has been observed with surfactant-based solutions and dispersions. The role of the characteristic time for structure built up, λ, in the extent and shape of the thixotropic loops is demonstrated.

title: Leveling of thixotropic liquids
abstract: In this paper the effect of thixotropy in the hydrodynamic behavior of thin films is studied. The simple problem of leveling on a horizontal substrate is considered. The rheological properties of the material are assumed to evolve over time due purely to changes in its internal structure. These changes are modeled in terms of a single structural variable. Neither elastic nor yielding effects are taken into account. More specifically, two distinct rheological models are considered: the simple model proposed by Moore and the more complex model proposed by Baravanian et al. These models exhibit a large range of variation for the liquid viscosity across the film thickness. After deriving the hydrodynamic equations governing leveling flows with the standard assumptions required by the lubrication approximation and running time-dependent numerical simulations, the nonlinear leveling history of the liquid can be predicted as a function of the initial microstructural state, rheological parameters, and initial disturbance of the liquid free surface. The main effort of this work is devoted to devising approximation schemes which lead to significant simplifications of the governing equations and their numerical computations. By approximating the inverse of viscosity as a monotonic function between its substrate and free-surface values, excellent agreement is found for the film amplitude, irrespective of the values of the rheological parameters of both models. Finally, a linear analysis yields a generalization of the Orchard’s law of leveling for Newtonian liquids to take into account the effect of thixotropy.

title: Mathematical modeling of thixotropic drilling mud and crude oil flow in wells and pipelines—A review
abstract: Many drilling muds and crude oils are known to be thixotropic. Under a wide range of pressures, temperatures and flow regimes, they display unusual complex flow properties when flowing through wells (crude oils and drilling muds) and during storage and pipeline transportation (crude oils). Understanding and modeling the deviation from Newtonian behavior of drilling muds and crude oils are essential in accurately and optimally designing the flow systems associated with these fluids. Despite an impressive amount of experimental and rheological modeling studies concerning the non-Newtonian drilling mud and crude oil behavior, mathematical modeling studies taking into account their thixotropic properties are rare. In addition, there was no literature review of the knowledge gained to date. Thus, a review paper on studies addressing the mathematical modeling of thixotropic drilling mud and crude oil flow in wells and pipelines will pinpoint the challenges and limitations encountered in such studies. This will hopefully trigger further development and new research topics. This review paper focuses mainly on mathematical modeling studies concerning the well and pipeline flow of thixotropic drilling muds and crude oils. After describing how thixotropy is understood today inside and outside of the petroleum industry community, several mathematical models available in the literature are examined. Finally, challenges, limitations, and potential areas for the development of these models are presented.

title: Thixotropic gravity currents
abstract: We present a model for thixotropic gravity currents flowing down an inclined plane that combines lubrication theory for shallow flow with a rheological constitutive law describing the degree of microscopic structure. The model is solved numerically for a finite volume of fluid in both two and three dimensions. The results illustrate the importance of the degree of initial ageing and the spatio-temporal variations of the microstructure during flow. The fluid does not flow unless the plane is inclined beyond a critical angle that depends on the ageing time. Above that critical angle and for relatively long ageing times, the fluid dramatically avalanches downslope, with the current becoming characterized by a structured horseshoe-shaped remnant of fluid at the back and a raised nose at the advancing front. The flow is prone to a weak interfacial instability that occurs along the border between structured and de-structured fluid. Experiments with bentonite clay show broadly similar phenomenological behaviour to that predicted by the model. Differences between the experiments and the model are discussed.

title: Perspectives on shear banding in complex fluids
abstract: In this review, I present an idiosyncratic view of the current state of shear banding in complex fluids. Particular attention is paid to some of the outstanding issues and questions facing the field, including the applicability of models that have “traditionally” been used to model experiments; future directions and challenges for experimentalists; and some of the issues surrounding vorticity banding, which has been discussed theoretically and whose experiments are fewer in number yet, in many ways, more varied in character.

title: Phenomenology and physical origin of shear localization and shear banding in complex fluids
abstract: We review and compare the phenomenological aspects and physical origin of shear localization and shear banding in various material types, namely emulsions, suspensions, colloids, granular materials, and micellar systems. It appears that shear banding, which must be distinguished from the simple effect of coexisting static-flowing regions in yield stress fluids, occurs in the form of a progressive evolution of the local viscosity toward two significantly different values in two adjoining regions of the fluids in which the stress takes slightly different values. This suggests that from a global point of view, shear banding in these systems has a common physical origin: Two physical phenomena (for example, in colloids, destructuration due to flow and restructuration due to aging) are in competition, and depending on the flow conditions, one of them becomes dominant and makes the system evolve in a specific direction.

title: On the existence of a simple yield stress fluid behavior
abstract: Materials such as foams, concentrated emulsions, dense suspensions or colloidal gels, are yield stress fluids. Their steady flow behavior, characterized by standard rheometric techniques, is usually modeled by a Herschel–Bulkley law. The emergence of techniques that allow the measurement of their local flow properties (velocity and volume fraction fields) has led to observe new complex behaviors. It was shown that many of these materials exhibit shear banding in a homogeneous shear stress field, which cannot be accounted for by the standard steady-state constitutive laws of simple yield stress fluids. In some cases, it was also observed that the velocity fields under various conditions cannot be modeled with a single constitutive law and that nonlocal models are needed to describe the flows. Doubt may then be cast on any macroscopic characterization of such systems, and one may wonder if any material behaves in some conditions as a Herschel–Bulkley material. In this paper, we address the question of the existence of a simple yield stress fluid behavior. We first review experimental results from the literature and we point out the main factors (physical properties, experimental procedure) at the origin of flow inhomogeneities and nonlocal effects. It leads us to propose a well-defined procedure to ensure that steady-state bulk properties of the materials are studied. We use this procedure to investigate yield stress fluid flows with MRI techniques. We focus on nonthixotropic dense suspensions of soft particles (foams, concentrated emulsions, Carbopol gels). We show that, as long as they are studied in a wide (as compared to the size of the material mesoscopic elements) gap geometry, these materials behave as ‘simple yield stress fluids’: they are homogeneous, they do not exhibit steady-state shear banding, and their steady flow behavior in simple shear can be modeled by a local continuous monotonic constitutive equation which accounts for flows in various conditions and matches the macroscopic response.

title: Constitutive equations for thixotropic fluids
abstract: To distinguish it clearly from nonlinear viscoelasticity, we define “ideal thixotropy” as “a time-dependent viscous response to the history of the strain rate, with fading memory of that history,” endowing such fluids with memory but no elasticity. An “ideal thixotropic fluid” has instantaneous stress relaxation upon cessation of flow and no elastic recoil on removal of stress. We describe “nonideal thixotropic” fluids as those whose viscoelastic time scales governing stress relaxation are much shorter than those governing the thixotropic response. This ensures that a clear distinction can be maintained between “thixotropy” and “nonlinear viscoelasticity.” The stress tensor for an ideal thixotropic fluid can in general be expressed as a contraction product of a fourth rank viscosity tensor with the velocity gradient tensor, in which the viscosity tensor depends on the history of the flow. We show examples of constitutive equations that meet the definitions of ideal thixotropy or nonideal thixotropy. We al...

title: Flow of a thixotropic or antithixotropic fluid in a slowly varying channel: the weakly advective regime
abstract: A general formulation of the governing equations for the slow, steady, two-dimensional flow of a thixotropic or antithixotropic fluid in a channel of slowly varying width is described. These equations are equivalent to the equations of classical lubrication theory for a Newtonian fluid, but incorporate the evolving microstructure of the fluid, described in terms of a scalar structure parameter. We demonstrate how the lubrication equations can be further simplified in the weakly advective regime in which the advective Deborah number is comparable to the aspect ratio of the flow, and present illustrative analytical and semi-analytical solutions for particular choices of the constitutive and kinetic laws, including a purely viscous Moore–Mewis–Wagner model and a regularised viscoplastic Houska model. The lubrication results also allow the calibration and validation of cross-sectionally averaged, or otherwise reduced, descriptions of thixotropic channel flow which provide a first step towards models of thixotropic flow in porous media, and we employ them to explain why such descriptions may be inadequate.

title: On the lubrication paradox and the use of regularisation methods for lubrication flows
abstract: In this paper we complete the analysis of the flow of a Bingham fluid along a wavy-walled channel. We confirm numerically the results of [I. Frigaard, D. Ryan, Flow of a visco-plastic fluid in a channel of slowly varying width,   (2004) 67–83], that there is a true rigid plug for a sufficiently small amplitude long wavelength perturbation of the uniform plane channel. For larger amplitude width perturbations the plug breaks and we provide a comparison between analytical estimates for the breaking amplitude and numerical computations using the augmented Lagrangian method. Plug breaking arises due to the creation of significant extensional stresses within the unyielded part of the fluid. When the plug breaks the extensional stresses govern the structure of the flow within the pseudo-plug. We provide a complete asymptotic solution for the flow in this situation, and compare against numerical computations. In the second part of the paper, we study the efficacity of viscosity regularisation methods for computing this type of flow. We show that accurate resolution of yield surface positions requires proper control of the numerical residuals and having a favourable ratio of regularisation error to critical strain rate. We show that failure to manage these errors means that regularisation methods can produce completely erroneous predictions of the yield surfaces, e.g. predicting broken plug regions when they are in fact intact. For general lubrication-type flows one does not have an estimate of these errors.

title: Models for flow of non-Newtonian and complex fluids through porous media
abstract: This article first provides a brief and simple account of continuum models for transport in porous media, and of the role of length scales in passing from pore-scale phenomena to “Darcy” continuum scale representations using averaged variables. It then examines the influence of non-Newtonian rheology on the single- and multi-phase transport parameters, i.e. Darcy viscosity, dispersion lengths and relative permeabilities. The aim is to deduce functional forms and values for these parameters given the rheological properties of the fluid or fluids in question, and the porosity, permeability, dispersion lengths and relative permeabilities (based on Newtonian fluids and equivalent capillary pressures) of the porous medium. It is concluded that micro-models, typically composed of capillary networks, applied at a sub-Darcy-scale, parameterised using data for flows of a well-characterised set of non-Newtonian fluids, are likely to provide the most reliable means.

title: A 1.5D numerical model for the start up of weakly compressible flow of a viscoplastic and thixotropic fluid in pipelines
abstract: We investigate the problem of the start up of a compressible flow in a pipeline filled with a viscoplastic and thixotropic material. The objective of this study is to examine the possibility that the flow can restart for a pressure drop below the value predicted by the conservative relation Δ p ˆ = 4 τ ˆ y L ˆ / D ˆ , where τ ˆ y denotes the yield stress, L ˆ the pipe length and D ˆ the pipe diameter, thanks to the combined effects of compressibility and thixotropy. Our numerical model is a compromise between a fully 2D model and a fully 1D model, i.e., a 1.5D model. Only the velocity component in the direction of the pipe axis is assumed non-zero but it is allowed to vary both in axial and radial directions. We show that this intermediate model yields accurate results which are consistent with the predictions of the fully 2D model. The gain in computing time while keeping an equivalent reliability in the numerical predictions enables us to investigate the effects of the compressibility and thixotropy dimensionless numbers at a reasonable cost.

title: Thixotropy : Build-up and breakdown curves during flow
abstract: Thixotropic systems are usually characterized by measuring the stress transients resulting from sudden changes in shear rate. By using a procedure that takes into account the transfer function of the transducer on a strain-controlled rheometer, the initial part of the transients could be accessed. In this manner complete transient stress curves and accurate constant-structure curves could be recorded on a model system consisting of fumed silica particles dispersed in a Newtonian medium. In agreement with previously obtained stress jump data [Dullaert and Mewis (2005b)], an initial viscoelastic response is detected that precedes the structural time effects in either build-up or breakdown experiments. The effect of flow history on the characteristic times and on the shape of the transient curves is investigated. It is shown that common assumptions in thixotropic models, such as a structure that can be characterized by a single parameter or the existence of exponential stress transients, do not generally hol...

title: Constitutive equations for extensional flow of wormlike micelles: stability analysis of the Bautista–Manero model
abstract: We carry out a stability analysis of the Bautista–Manero (B–M) constitutive equations for extensional flow of wormlike micelles. We show that all solutions for the steady-state extensional viscosity η E are unstable when the elongational rates ɛ ˙ exceed some critical value. In some cases the only real solution for the extensional viscosity is negative at intermediate values of the elongational rate. This critical elongational rate is not unfeasibly large, e.g., 250 s−1 for a typical EHAC solution. We note that the extension rates at which enhanced pressure drop is observed experimentally in porous media flow is generally well below the onset of the instability in the B–M model. However, the extension rates presented here are only average values for the porous media in question and at the pore scale a wide range of values will be encountered. For this reason, we require an improved rheological equation of state. We first separate the contributions to the viscosity of the solution and the wormlike micelles. Then, we remove the term containing the retardation time λ J . We now find that the extensional flow behaviour is well defined for both the transient and steady-state cases.

title: Thixotropic flow of toothpaste through extrusion dies
abstract: A commercial toothpaste is investigated in this work as a model paste system to study its processing characteristics in capillary flow using various dies. Its rheological behaviour has been determined as that of a yield-stress, thixotropic material with a time-dependent behaviour, and severe slip at the wall. The rheological data obtained from a parallel-plate rheometer were used to formulate a constitutive equation with a structural parameter which obeys a kinetic equation, typically used to model thixotropy. The predictive capabilities of this model are tested against capillary data for a variety of capillary dies having different length-to-diameter ratios (L/D), contraction angles (2a), and contraction ratios (Db/D) 2 , where Db is the diameter of the barrel of the capillary rheometer. The major trends are well captured by the thixotropic model and show that slip is the essential parameter in predicting the flow behaviour of toothpaste.

title: An exact solution for laminar, unidirectional flow of Houska thixotropic fluids in a circular pipe
abstract: Laminar flow of an incompressible thixotropic liquid which obeys the Houska model as its constitutive equation is investigated theoretically in a circular pipe under equilibrium (steady) conditions. Having derived the single differential equation governing fully-developed, unidirectional, pipe flow of this fluid model, it is shown that an analytical solution is possible for the linear case of the Houska model. The existence and uniqueness of the analytical solution so-obtained is examined theoretically. For the more general nonlinear case of the Houska model, the trust region algorithm is used to obtain numerical results for the flow characteristics. The effects of thixotropic parameters such as breakdown-to-rebuild ratio, viscosity ratio, and Bingham number are investigated on the structural parameter and velocity profiles. As a special case, pipe flow of a thixotropic fluid obeying the Moore model is addressed and a correct boundary condition is proposed at the pipe centerline for this particular fluid model.

title: The Stokes boundary layer for a thixotropic or antithixotropic fluid
abstract: We present a mathematical investigation of the oscillatory boundary layer (‘Stokes layer’) in a semi-infinite fluid bounded by an oscillating wall (the socalled ‘Stokes problem’), when the fluid has a thixotropic or antithixotropic rheology. We obtain asymptotic solutions in the limit of small-amplitude oscillations, and we use numerical integration to validate the asymptotic solutions and to explore the behaviour of the system for larger-amplitude oscillations. The solutions that we obtain differ significantly from the classical solution for a Newtonian fluid. In particular, for antithixotropic fluids the velocity reaches zero at a finite distance from the wall, in contrast to the exponential decay for a thixotropic or a Newtonian fluid. For small amplitudes of oscillation, three regimes of behaviour are possible: the structure parameter may take values defined instantaneously by the shear rate, or by a long-term average; or it may behave hysteretically. The regime boundaries depend on the precise specification of structure build-up and breakdown rates in the rheological model, illustrating the subtleties of complex fluid models in non-rheometric settings. For larger amplitudes of oscillation the dominant behaviour is hysteretic. We discuss in particular the relationship between the shear stress and the shear rate at the oscillating wall.

title: Plasticity and geophysical flows: A review
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.

title: The effect of a variable plastic viscosity on the restart problem of pipelines filled with gelled waxy crude oils
abstract: The effect of a variable plastic viscosity is numerically investigated on the success of the restart operation for a pipeline filled with fully-gelled waxy crude oil. To investigate the separate effects of structure- and shear-dependent viscosity, waxy crude oil is assumed to obey the Houska rheological model. In order to precisely capture the shape and position of the yielding surface, a variational approach is used to formulate the restart problem for this particular fluid model. The numerical results show that a variable plastic viscosity has a significant effect on the restart operation. For certain set of parameters the restart operation is shown to fail if the plastic viscosity is constant but is successful if the plastic viscosity is (moderately) structure-dependent. However, the time needed by the liquefied gel to discharge from the pipe outlet section is increased if the plastic viscosity is structure-dependent. Surprisingly, the shear-thinning behavior of waxy crude oil is predicted to lower the (steady) flow rate even when the restart is successful.

title: Flows of Materials with Yield
abstract: Steady, two‐dimensional flows of Bingham fluids are analyzed by means of a modified constitutive relation that applies everywhere in the flow field, in both yielded and practically unyielded regions. The conservation equations and the constitutive relation are solved simultaneously by Galerkin finite element and Newton iteration. This combination eliminates the necessity for tracking yield surfaces in the flow field. The analysis is applied to a one‐dimensional channel flow, a two‐dimensional boundary layer flow, and a two‐dimensional extrusion flow. The finite element predictions compare well with available analytic solutions for limiting cases.

title: Flow of a visco-plastic fluid in a channel of slowly varying width
abstract: We develop an asymptotic solution for the Poiseuille flow of a Bingham fluid along a two-dimensional channel of slowly varying width, for the case of small amplitude width variation, h . We show that an asymptotic solution with an intact unyielded plug region can be found for h less than a critical value h c ∼ O ( δ ) , where δ is the aspect ratio (channel half-width to wavelength of the variation). Our method involves no form of relaxation of the exact Bingham model. We provide upper and lower estimates for h c . The function h c / δ shows little variation with δ ≪ 1 , and exhibits a maximum in Bingham number, B , at around B ≈ 5 . There is a small variation in the true plug speed from that of the uniform channel flow. At small B the perturbed plug moves slightly slower than that in the uniform channel, whereas at large B the true plug moves faster than the uniform channel flow. Finally, we consider the structure of the flow for h ∼ O ( 1 ) , for which the plug has broken.

title: Advanced Transport Phenomena: Fluid Mechanics and Convective Transport Processes
abstract: 1. A preview 2. Basic principles 3. Unidirectional and one-dimensional flow and heat transfer processes 4. An introduction to asymptotic approximations 5. The thin gap approximation - lubrication problems 6. The thin gap approximation - films with a free surface 7. Creeping flow - general properties and solutions for 2D and axisymmetric problems 8. Creeping flow - 3D problems 9. Convection effects and heat transfer for viscous flows 10. Boundary layer theory for laminar flows 11. Heat and mass transfer at large Reynolds number 12.  Governing equations and vector operations in Cartesian, cylindrical and spherical coordinate systems Appendix B. Cartesian component notation.

title: Squeeze flow behavior of (soft glassy) thixotropic material
abstract: We study the flow behavior of a model soft glassy material − an aqueous suspension of Laponite − when it is squeezed between two circular parallel plates of different roughness. Aqueous suspension of Laponite shows a time dependent aging behavior as reflected in increased elastic modulus as well as yield stress, both of which however also decrease with an increase in the strength of deformation field thereby demonstrating typical thixotropic character. In a squeeze flow situation, under both force as well as velocity controlled modes; we find the behavior to be independent of the initial gap between the plates. In a constant force mode, the gap between the plates decreases until it reaches a finite limiting value, which is found to increase with an increase in age of the material as well as with a decrease in the applied force. In constant velocity experiments, at large gaps between the plates, normal force varies inversely with plate separation. The normal force is higher for a sample aged for a longer time as well as for a larger velocity of the top plate. We observe that the experimental behavior follows prediction of Herschel–Bulkley model solved for the squeeze flow (with different friction coefficients at the two plates) reasonably well under weak deformation fields. However, under strong deformation fields, experimental behavior deviates significantly from the prediction of Herschel–Bulkley model. This deviation arises due to melting or partial yielding of Laponite suspension under large deformation fields causing decrease in the viscosity, elastic modulus and the yield stress.

title: Squeeze Flow of Semi-Solid Slurries
abstract: A standard method used to determine material properties of semi-solid slurries is the squeeze flow experiment in which a fixed amount of material is squeezed under constant force or velocity. The relation between the force and the displacement provides information about the rheology of the slurry. The thixotropy and the time response of the sample however is rarely, if ever, taken into consideration. In this work we study how thixotropy affects the flow characteristics and consequently the predicted material properties. We show that depending on the method of compression and the thixotropic constants the flow can be significantly different. Therefore the predicted material constants can vary and hence cannot be unique.

title: Shallow flows of generalised Newtonian fluids on an inclined plane
abstract: We derive a general evolution equation for a shallow layer of a generalised Newtonian fluid undergoing two-dimensional gravity-driven flow on an inclined plane. The flux term appearing in this equation is expressed in terms of an integral involving the prescribed constitutive relation and, crucially, does not require explicit knowledge of the velocity profile of the flow; this allows the equation to be formulated for any generalised Newtonian fluid. In particular, we develop general solutions for travelling waves on a mild slope and for kinematic waves on a moderately steep slope; these results provide simple and accessible models of, for example, the propagation of non-Newtonian mud and debris flows.