Simple yield stress fluids

Available online at www.sciencedirect.com

ScienceDirect

Current Opinion in

Colloid & Interface Science

Simple yield stress fluids Ian Frigaard Abstract

Recent years have seen a flurry of research on yield stress fluids, approached from different perspectives (physicochemical, rheological and fluid mechanical), considering different length scales and timescales, using a range of tools: experimental, computational and analytical. In this context, it has been common to denigrate the simple models of yield stress fluids for their various acknowledged deficiencies and proffer improved models. Here, we push back, in defence of the century-old simplicity, exploring what is new and useful and what these models have given in the past few decades. Addresses Departments of Mechanical Engineering and Mathematics, University of British Columbia, Vancouver, BC, Canada Corresponding author: Frigaard, Ian ([email protected])

beyond that, as the concept of a yield stress has been debated continually for 3e4 decades. However, colloidal gels, emulsions, soft glassy materials and jammed noncolloidal suspensions are classes of materials that definitively can exhibit yield stress behaviour albeit stemming from different mechanisms. Thus, here we do not revive the old debate on the existence of a yield stress. The terms simple (or ideal) yield stress (or viscoplastic) fluids have become commonly used recently. The usage of ideal dates from the study by Coussot et al. [15] and was used to distinguish thixotropic and nonthixotropic yield stress fluids. Later, this migrated to simple with the same meaning [16], and this has stuck through the more recent attempts to characterise yield stress fluids in a broader sense [4]. Simple yield stress fluids are meant those for which the shear stress depends only on the imposed shear rate, that is, these are both nonthixotropic and inelastic. These are the topic of this article.

Current Opinion in Colloid & Interface Science 2019, 43:80–93 This review comes from a themed issue on Rheology Edited by Yujun Feng and Erin Koos For a complete overview see the Issue and the Editorial https://doi.org/10.1016/j.cocis.2019.03.002 1359-0294/© 2019 Elsevier Ltd. All rights reserved.

Keywords Yield stress, Bingham fluid, Herschel bulkley fluid, Casson fluid.

Simple viscoplastic fluids Viscoplastic fluids are a category of non-Newtonian fluid that is characterised by having a yield stress. They are synonymously called yield stress fluids. In the current scientific literature, it is fair to say that the terminology means slightly different things to different researchers. After a lack of effective review for w30 years, since the study by Bird et al. [1], there have recently been a number of contributions that cover the breadth of the field [2e4]: texts that provide useful introductions [5,6], the wonderful 100-year Bingham commemorative collection of articles [7e13] that include targeted reviews and finally in this volume the nicely written study by de Souza Mendes and Thompson [14]. There has also been an enormous expansion of the literature that makes a concise summary such as the study by Bird et al. [1] no longer possible. Although there is agreement that viscoplastic fluids are characterised by a yield stress, there is less consensus Current Opinion in Colloid & Interface Science 2019, 43:80–93

The recent revival of interest in yield stress fluids has been largely critical of simple yield stress models, which has some justification. One response has been a spate of new (and older) rheological models that are intended to offer more complete descriptions. One of the author’s favourite pages is the “Rheology Drugstore” in the front of D.D. Joseph’s text [17], which lists candidate viscoelastic models that were current 30 years ago. We can probably expect a similar culling of the new breed of model. There have been some recent attempts to quantify thixo-elasto-visco-plastic (TEVP) effects and characterise their regimes of usage and validity in terms of dimensionless groups [9]. This is a difficult task which so far has been largely rheological in focus, that is, comparing rheometrically separated timescales and/or stress scales, while in application, the fluid mechanics scales often dominate. Figure 1 is a different recent perspective: placing classical viscoplastic fluid models at the centre and capturing many of the interesting applications and deficiencies at the periphery. As in politics, different viewpoints make debate worthwhile. This (slightly satirical ‘pre-Socratean’) world map was constructed during a recent workshop. Similar to the ancient maps, this is as much as an indicator of known comfort (at the core) and where the sea monsters live (at the edge of the world). It helps us to place our own scientific and engineering concepts of models, fundamental mechanical behaviours, measurements and flow types, within the www.sciencedirect.com

Simple yield stress fluids Frigaard

outer realms of applications: existing, emerging and unknown. Hopefully, such constructs encourage humility, helpful and humourous interaction between explorers. Simple yield stress fluids are those models of popular rheology and fluid mechanics: the Bingham, Herschele Bulkley, Casson and other similar fluid models. In the following section, we outline the models and give a brief historical and mechanics-focused perspective on these fluids. In the remainder of the article, we then address some key practical questions: Are these models still useful? Is there anything new with these models and are there still challenges? What are the limitations of these models? The article is thus part review of the past 2e3 decades of research, specifically targeted at these common models, and part a ‘defence’ of the continuing relevance of the simple models and their use in fluid mechanical terms. For a number of reasons, we do not review experimental work, although the author conducts experimental

81

research and highly values it. First, experimental research is reviewed in the study by Coussot [3]. The second point is that simple yield stress fluid models have always been idealisations, and experimental fluids are not. Although we can find many instances where there is a good agreement between experiment and theory [18,19], we can also find disagreement [20]. Apart from experimental error, disagreement is connected to deficiencies of the simple models in describing (often time-dependent) mechanical behaviour at low shear rates, where elasticity and thixotropy conspire to shatter our innocence. Different fluids and experiments expose different deficiencies, but at the same time, the simple models retain value as the common idealisation of all (Figure 1) and allow us to move forward in our understanding of how yield stress fluids should behave. Models

Mathematically, the main simple yield stress models describe the deviatoric stress tensor tij, by the constitutive laws:

Figure 1

Current Opinion in Colloid & Interface Science

A viscoplastic worldview, courtesy of G.H. McKinley. www.sciencedirect.com

Current Opinion in Colloid & Interface Science 2019, 43:80–93

82 Rheology


tij ¼

_ þ hv ðgÞ

 tY g_ ij ⇔t>tY ; g_

g_ ¼ 0⇔t  tY ;

(1)

(2)

where the components of the strain rate tensor g_ ij are: vui vuj þ ; g_ ij ¼ vxj vxi with u ¼ ðu1 ; u2 ; u3 Þ the velocity vector and where 2 31=2 31=2 3 h i2 3   X X 1 1 2 5 t ¼ 4 tij 5 : g_ ¼ 4 g_ 2 i;j ¼ 1 ij 2 i;j ¼ 1 2

(3)

We have denoted the viscous part of the effective vis_ for example, hv ¼ mN for Bingham fluids, cosity hv ðgÞ, for a HerscheleBulkley fluids and hv ¼ kg_ n1pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hv ¼ 1 þ 2 tY =ðmN g_ Þ for a Casson fluid; see Figure 2. _ þ tY =g. _ _ ¼ hv ðgÞ The effective viscosity is simply hðgÞ Here, mN is the plastic viscosity, k is the consistency and n is the power law index. In the flow context, these fluids satisfy the Cauchy momentum equations and flows are typically incompressible. The range of flow configurations and boundary conditions is application-specific and mimics those of Newtonian fluids. The constants tY , mN , k and n are fitted from rheometric tests. Historical perspective

The most popular model for a simple yield stress material is that of the Bingham fluid, which dates back to work carried out by Bingham in the early part of the 20th century. In 1911, Bingham and Durham are credited with the observation that the fluidity of a suspension approaches zero at constant pressure as the solids fraction is increased. However, Bingham’s 1916 article [21] Figure 2

Current Opinion in Colloid & Interface Science

The most common simple yield stress fluid models. Current Opinion in Colloid & Interface Science 2019, 43:80–93

is more widely accepted as the inception of the Bingham fluid, wherein a number of complex fluids were driven under pressure through a range of capillary tubes. When the driving pressure exceeds a friction constant, flow occurs and Bingham fits the flow to a curve that is linear in the differential pressure. This linear relation later evolved into the Bingham constitutive law with the friction constant related to the yield stress. In the 1920s, these observations were put on a firmer footing by Bingham [22] and other nonlinear flow laws also emerged [23], also based on capillary flows using a similar apparatus to that of Bingham. This was also an era where the language of rheology was being developed. Although various viscometric flows were studied in these years, driven either by measurement or fluid transport applications, it was not until the 1930s that Prager [24] wrote the full tensorial equations for an incompressible Bingham fluid. The following decades saw the velocity minimisation and stress maximisation principles derived formally by Prager [25] for Stokes flows and an influential series of articles by Oldroyd [26]. In the Soviet Union, a number of works by Ilyushin appeared independently in this period (reviewed in the study by Georgeievskii [27]) and later the seminal analytical articles of Mosolov and Miasnikov [28]. The late 1970s and early 1980s saw the theoretical background for weak solutions established, using convex analysis and variational inequalities [29], and this formed the basis for the computational theory pioneered by Glowinski et al. [30] and Fortin and Glowinski [31]. The review by Bird et al. [1] provides a concise review of the literature on viscoplastic fluids up until the early 1980s, summarising amongst other things the known analytical solutions to different flows. The 1980s were an interesting era for viscoplastic fluids. First, as rheometry evolved with improved ranges of low shear rates, better precision, control and reliable controlled stress rheometers, Barnes and Walters [32] raised the question of whether the yield stress was purely an artifact of insufficient precision. This initiated a debate that has persisted. The notion of an apparent yield stress and the provocative notion that there was no yield stress at all and that all such fluids would flow, governed by a large zero-shear viscosity, took hold amongst a part of the fluid mechanics community concerned with such fluids. This led to perhaps the first mechanical/mathematical schism in how viscoplastic fluids should be treated. It is worth considering deeper the environment of the time. First, this was an era in which various viscoelastic fluid models were being dismissed as ill-posed or otherwise unrealistic. It was also an era in which the mathematical community was excited by free and moving boundary problems (e.g. the Stefan problem), with related questions of regularity and ill-posedness, and the Bingham fluid was identified as one such problem [33]. www.sciencedirect.com

Simple yield stress fluids Frigaard

The Bingham model carried with it the complications of rigid behaviour at subyield stresses, meaning that in plug regions, (i) the stress tensor was generally indeterminate and (ii) the effective viscosity was infinite. Worse still, there was no equation to explicitly describe evolution of the yield surface. These conceptual difficulties led to the suspicion that there was maybe something wrong with the Bingham model (and its relatives). In this context, the rheological assertion that very viscous behaviour occurred at low shear proved convenient for some mechanicians. The biviscous model was advocated as a way of resolving various analytical difficulties. For example, the so-called lubrication paradox [34] was cast into further confusion by its nonresolution using a method of distinguished limits [35], which blended geometric and rheological small parameters. Strangely, the doubts of the 1980s and 1990s seemed to ignore the decade-old results of Duvaut and Lions [29] that had established existence and uniqueness of solutions to twodimensional (2D) and three-dimensional (3D) inertial flows (and various continuity results) under broadly similar conditions to those for Newtonian fluids. This was also the advent of computational solutions of viscoplastic problems. Although it was known how to compute Bingham fluid flows using the augmented Lagrangian method [30,31], these methods were not widely known outside of the mathematical community for another 20 years and did not evolve beyond benchmark calculations in the 1980s and early 1990s. The majority of computations in the 1980s and 1990s used methods of viscosity regularisation [36,37] which were convenient in turning viscoplastic fluid flow problems into nonlinearly viscous computations for which more standard and familiar iterative methods were used. Many computational studies of this era motivated the use of regularisation method by statements to the effect that the Bingham model was discontinuous or that real fluids behaved as very viscous fluids at low shear; the study by Barnes and Walters [32] provided a convenient truth. The assertions of Barnes and Walters [32] were countered by a number of authors shortly thereafter, as is summarised in the study by Coussot [8], with the commonly accepted view that the yield stress was at minimum an engineering reality, that is, if there was no discernible flow on the timescale of interest, then a fluid might as well be considered as unyielded. This pragmatic approach has however never fully been adopted as a protocol for the interpretation of flows that are computed using viscosity regularisation, which remain the most popular computational methods today. Some lessons could be learned from our experiences with viscosity regularisation as we look to model more complex behaviours. (a) Rheometric advances and studies may reveal new insights about the mechanical behaviour of complex www.sciencedirect.com

83

fluids, but rarely is it correct to assert that all materials will behave in a certain way. (b) The scientific response to mathematical difficulties in understanding a rheological model, or how to compute it, should be to try harder and succeed rather than assume that a mathematical difficulty is a physical reality. (c) Innocuous changes to models made for ease have both physical and mathematical consequences.

Regarding (b), it is now widely accepted that there is nothing wrong mathematically with the simple models: harder work may admittedly be needed to solve specific analytical problems. We can today resolve flows in 2D and 3D using either the exact or regularised versions of these models [13,38]. Regarding (c), regularisation methods are examined in the study by Frigaard and Nouar [39] wherein it is shown that (i) certain flows may be linearly unstable, due only the regularisation; (ii) for slowly varying geometries, these methods can produce large discrepancies when compared with the nonregularised models.

Are these models still useful? Rheology is a complex field that is both a material and a mechanical science, but a key factor in its development is that it has been driven by application as much as theory. Part of the applicability comes from the design of materials with specific properties, and these designs are increasingly performed at the colloidal, polymeric or other microscale. The other part of the applicability comes from the process/product and its mechanical (or other) behaviour. To assess this requires mechanical understanding at whatever length scales and timescales that are relevant. Often this is at continuum scales, and hence, continuum (non-Newtonian) fluid mechanics remains one of the most important disciplines for the application of rheology. The first reason then that models such as the Bingham fluid persist to this day is that they reflect the simplest description of phenomena of practical engineering relevance. As with the pragmatic response to the yield stress myth of Barnes and Walters [32], the same is true of the more current yield stress introspection. It does not matter if the yield stress has colloidal, granular/jamming, polymeric or other origins. It does not matter if different materials show time dependency, elastic creep, shearbanding and viscoelastic responses, etc. If the phenomenon you care about is predominantly a yield stress phenomenon, you must weigh the complexity of the model against your need for answers in moving ahead and then decide which model to use. Simple yield stress fluid models are often the winners in this contest. Essentially, these deliver answers and transfer the burden to interpreting results: did the missing physics matter? Current Opinion in Colloid & Interface Science 2019, 43:80–93

84 Rheology

Sometimes yes, sometimes no and sometimes it depends on the question asked. Second, in many interdisciplinary and industrial contexts, simple yield stress fluid models are also those first adopted. Continuum mechanics is anyway complicated, and the physical intuition of nonexperts is limited to a few notions. For example, asking a lay person to comment on the viscosity of the soup in front of them might elicit a range of responses related to taste, texture or thickness. Only on probing their understanding do we implant a different more physical intuition, that is, regarding resistance to stress. Similarly, a yield stress is intuitively simple to explain in a lay person’s terms and perhaps shear thinning too. It is not therefore surprising that industries evolve by adopting minimalist and simple rheological models that their personnel can use, share in a common language and develop intuition about. As an example, most oilfield rheometry is performed on a simple concentric viscometer that runs at 6 (or 12) standard angular speeds but is used to characterise a bewildering range of polymeric fluids, emulsions and colloidal suspensions. Before the 1990s, the power law model and the Bingham fluid were the standard industry models, slowly replaced over 2 decades by the HerscheleBulkley model. It is the aforementioned combination of simplicity of concept and the commonality of intuition gained from using simple models in complicated processes that has preserved the utility of simple viscoplastic models until now and, in all likelihood, will do so into the future. It would be ludicrous to throw away a simple rheological model that, for example, gives a reasonable prediction of the pressure drop of a dairy product through a rectangular duct, including the minimal pressure drop needed to mobilise fluids in the corners, simply because we understand that there are other rheological features not predicted by this model. This third reason for usage is is there a better model that can be used today? Some of the flow phenomena that occur with yield stress fluids and for which simple yield stress models are commonly used include the following. (a) Flows that are predominantly steady and shear dominated. These include pipe and duct flows, viscometric flows, slowly moving thin film and Hele-Shaw type flows, some tribological flows. (b) Interior or exterior flows in which some part of the flow is stuck, that is, not moving. Typically, this coincides with a geometric irregularity/ nonuniformity, for example, a narrow part of a duct or a corner. (c) Flows that encapsulate particles, droplets or bubbles. In these flows, there are typically two types of problem: a transport problem, whereby the dispersed phase is to be carried in the yield stress fluid, and a settling/rising problem, in which the yield stress fluid is static and the dispersed phase migrates. (d) Geophysical/industrial flows where precise rheological fitting presents difficulties, but the effect of a yield Current Opinion in Colloid & Interface Science 2019, 43:80–93

stress is of interest. Examples include predicting basaltic lava flows, unrestrained Corium flow after a nuclear meltdown, onset of mudslides and stability of mined tailings dams/structures.

Is there anything new? Over the past 2 decades, there has been a significant increase in the number of published articles and other studies using the models outlined in Models section. From a utilitarian perspective, the main advances have been (i) methodological; (ii) in solving classical fluid mechanics problems, not solved previously; (iii) in multiphase flows and (iv) in applications. Methodologies

As in other branches of non-Newtonian fluid mechanics, the main methodological advance of the past 2 decades is computational. Flows that formerly required researchlevel computational methods can today be solved with relative ease using both open source and commercially available computational fluid dynamics tools. Typically, these use a form of regularisation, and care is needed to interpret results in the correct physical way, but otherwise, these are useful tools that produce reasonable quality results for applications. A number of research groups use augmented Lagrangian methods, and there is a wide range of finite element or finite volume implementations, meshing strategies, etc. Some of these more advanced numerical methods are needed to study certain limiting flows, for example, flow onset/stopping and some boundary layers, where the yield surfaces have interesting geometric features. The second area for notable fluid mechanical advance has been in asymptotic methods or more specifically, the application of these methods to broad classes of longthin problems: shear flows, extensional flows, viscoplastic boundary layers and flows in slowly varying geometries. The lubrication paradox unleashed in the study by Lipscomb and Denn [34] was resolved for antiplane shear flows a few years later [40] and then for 2D flows Ref. [41]. Other recent studies by Balmforth and co-authors [42e 46] have led the way in understanding extensional flows, sheets and boundary layers, so that there is a veritable toolkit to use. An instructive introduction and overview of these methods can be found in chapter 3 of [6]. The third area of viscoplastic fluid mechanics that has developed significantly is that of hydrodynamic stability. In the context of linear stability, the only difficulty to overcome is that of the yield surface and plug regions. In a typical shear flow, for example, Poiseuille flow, the shear stress varies linearly and thus lies a finite amount below the yield stress in the interior of plug regions. Infinitesimal stress perturbations cannot therefore break the plug, and instead only a linear perturbation of the yield surface needs be considered. Linearisation and www.sciencedirect.com

Simple yield stress fluids Frigaard

derivation of the yield surface boundary conditions was carried out in Ref. [47] and has been applied to many classical linear stability problems [48e53]. Owing to the linearisation of the effective viscosity about a simple shear flow, the viscous part of the linear stability operator is anisotropic, with the consequence that simplifying transformations such as Squire’s theorem are not directly applicable. There have also been a number of interfacial flows studied using linear methods. Three qualitatively different categories of interfacial flows have been addressed: (i) multilayer shear flows in which the fluids are yielded on each side of the interface; (ii) multilayer shear flows in which one or more fluids are unyielded at the interface and (iii) free surface flows. Under (i), the flow stability problems are not much different from Newtonian viscouseviscous flows, except fed by the local shear-dependent values of the effective viscosity and tangent viscosity. Category (ii) leads to suppression of interfacial instabilities and so-called VPL flows, (viscoplastic lubrication) reviewed later in the following sections. Under (iii), it has been noted that if the flow develops in the streamwise direction, for example, a thin film, then the plug regions also experience extensional stress and following [41] may be just yielded, requiring a different analysis [54]. While linear stability analyses have developed to the point of straightforward application, it is with nonlinear stability that the significant fundamental results have come. The ability to resist a shear stress at rest is the key qualitative feature of a yield stress fluid. Thus, for example, it is intuitive that for sufficiently large yield stress, dense objects are statically suspended in fluid, a layer of yield stress fluid on a vertical surface does not flow, a stress-controlled vane viscometer cannot turn (neglecting slip and elastic effects), etc. These are properties of a static flow and physically represent the dimensionless balance Y, between the yield stress and whatever is the driving force (stress) of the flow, for example, pressure-driven flow along a pipe, settling of a particle under gravity, or bubble rise (see Table 1). In the past 10e15 years, the static limit has been defined in terms of a critical dimensionless yield number Yc and given a general and formal mathematical definition [57]. The roots of this go back to the 1965 analysis of Mossolov and Miasnikov [28] for antiplane shear flows (meaning flows in one direction and 2 dimensions, e.g. flow along a rectangular duct) [29,55]. It has recently become more widely appreciated that the critical limit Y  Yc means not only that the steady flow has zero velocity but also that the associated unsteady flows converge to the static steady state. In many cases, this stability is irrespective of the size of the perturbation, that is, the zero flow states are globally energy stable [57]. Furthermore, the convergence to the static state occurs in www.sciencedirect.com

85

Table 1 Physical meaning of dimensionless numbers in the text. Name Bingham number Yield number Reynolds number Archimedes number Bond number

Description Yield stress Viscous stress Yield stress Driving stress of flow Inertial stress Viscous stress   Buoyancy stress 2 Viscous stress Buoyancy stress Surface tension ðstressÞ

a finite time, which for large yield stress is inversely proportional to the yield stress. This type of result, originally established by Bristeau [56] for channel flow has been used extensively in the context of one-dimensional (1D) shear flows by Georgiou and co-workers e.g. Ref. [58]. However, it is the generality of the result and its link to the static flow limit Yc that leads to novel flow applications, as we shall outline. Classical flows

The advances described previously have enabled the study of many classical multidimensional flows in the past 2 decades. With reliable computational methods, we have no longer been restricted to limiting geometries or small parameters allowing simplification, but also computation has combined with these methods for deeper understanding. Some examples are given in the following paragraphs. We have seen extensive calculations of flows around axisymmetric and 2D particles. An early work here by Beris et al. [59] was groundbreaking in both numerical quality and in framing the theoretical questions. Contributions have been both pragmatic (e.g. with the target of fitting drag coefficients) and more analytical (e.g. with the target of understanding the flow structure in the limit of zero flow) [60e69]. In heat transfer, computational advances have enabled extensive research on natural convection problems, whereas previously, only problems of forced convection had been studied. These include both those focused at computation of Nusselt numbers to fit engineering correlations [70e72] and those targeting onset and stability, for ´nard and heated cavities [73e example, in RayleigheBe 75], which are examples of energy stable flows. Linear and weakly nonlinear stability of some mixed convection flows has also been studied, for example, the plane ´nardePoiseuille flow [76,77]. RayleigheBe Squeeze flows (between parallel or axisymmetric plates) have been studied both analytically and computationally, exposing the structure of the flow at the central Current Opinion in Colloid & Interface Science 2019, 43:80–93

86 Rheology

symmetry plane/point and near the edges of the plates [78e83]. This flow was considered in the context of the lubrication paradox [34] but in particular [82,83] have largely resolved these asymptotic issues. In the case of finite plates, these flows represent particleeparticle contacts within yield stress suspensions. An interesting feature of these flows in the particleeparticle context is that unlike Newtonian fluids, it may be necessary to consider the entire flow around the particle to understand contact. The point here is that although large stresses are generated in the interparticle gap, far from the minimal gap, the pressure gradients decline and resistance to contact can come from the far field; see the discussion in chapter 3 of [6] and also the study by Koblitz et al. [84]. A final area is that of flows along channels with uneven walls. First, these studies include many computed flows through expansions and/or contractions, of direct relevance to polymer processing [85e89]. At the other end of the spectrum, the studies by Frigaard and Ryan [90], Putz et al. [91], Roustaei and Frigaard [92] and Roustaei et al [93], study wavy walled channel flows as amplitude and frequency of the wall perturbation varies. Thus, initially the plug withstands the extensional stresses generated [90] but breaks at critical amplitude. The broken plugs separate regions of pseudoplugs and true unyielded plugs (confirmed by asymptotics and computation) [91]. At larger amplitudes still, static fouled regions form within the deepest part of the channel [92]. At very large yield stresses, these vuggy wall regions fill progressively with unyielded fluid and the central plug regions fill out to approach the minimal channel thickness [93]. This leaves asymptotically thin shear layers that correspond to viscoplastic boundary layers [45]. Multiphase flows

In the context of gaseliquid flows, with the exception of the study by Stein and Buggisch [94] on spherical bubbles, we know no analytic flow solutions. However, in a yield stress fluid, bubbles held in suspension are not necessarily spherical; see Figure 3. The limit of zero flow for bubbles has been studied mathematically in the study by Dubash and Frigaard [95] wherein a number of estimates for critical Y values are made. Because the interface deforms on motion however, shape dependency becomes important. The question of what is the smallest yield stress to prevent onset of motion of a given static bubble is well defined (if the shape is fixed), whereas understanding if a moving bubble keeps moving (given a fixed yield stress) is somewhat different. The latter problem is addressed by Tsamopoulos and coworkers [96,97], who solved the steady incompressible bubble propagation flow, for varying rheology, density, surface tension and bubble volume. They iteratively adapt the bubble shape for each computation, finding Current Opinion in Colloid & Interface Science 2019, 43:80–93

Figure 3

Current Opinion in Colloid & Interface Science

Air bubbles trapped in a Carbopol solution in a tube 50 mm in diameter (tY z10Pa); width of images, 30 mm. The tube was shaken and banged on the author’s desk a few times to create the shapes, all of which remained static.

the shape and rise velocity of a steadily moving bubble for each set of dimensionless parameters (Archimedes, Bond and Bingham numbers). Panoramas of the shapes are given, to the limits of their computations, which largely represent where bubble rise becomes unsteady. An interesting observation is that the steady rise problem at nonzero velocity appears to have a unique bubble shape (within the ranges computed), but static bubbles have no unique shape. For steadily rising bubbles in the studies by Tsamopoulos et al. [96] and Dimakopoulos et al. [97], for increasing yield stress, a critical value (Yc ) is found, which varies with the Bond number but only slightly with the Archimedes number. The second area where significant advances have been made is fluidefluid flows in pipes and ducts. First, there have been studies that extend the classical works of Newtonian fluid mechanics. Transient displacement of capillary tubes by air is considered in the studies by Dimakopoulos et al. [98], de Sousa et al [99] and Thompson et al [100]. A key feature of displacing yield stress fluids is that residual layers of fluid may remain behind on the walls of the pipe/duct after the displacement front has passed, as first observed in the study by Allouche et al. [101]. These residual layers are often the motivation for studying the flow, and much of the research effort has gone into predicting either the residual layer thickness or its maximum [102e106]. While mainly these studies are computational, others www.sciencedirect.com

Simple yield stress fluids Frigaard

have developed thin-film style models that lead to some semianalytical results and qualitative understanding [107,108]. Both miscible and immiscible flows have been considered, although the miscible flows are usually ´clet number limit. in the large Pe In a different limit, of miscible displacement, Taylor dispersion occurs and has been studied for both laminar and turbulent flows [109,110]. The large body of work in this area is driven by the enormous applicability, that is, in many industries (food, oil and gas, mining, dairy, cosmetics), we transport yield stress material and inevitably these materials get stuck (e.g. in corners) and/or need to be cleaned out. Second, considering rheological differences, flow regimes, density differences, etc., easily results in large sets of dimensionless parameters that need to be explored. A third area of advance is towards (idealised) suspension flows, that is, particles in a simple yield stress fluid. Industrially used suspensions often fall into this category, and although greatly simplified, it is of some value to understand how, for example, a hard sphere suspension in a simple yield stress fluid (Bingham?) should behave, even in Stokes flow. There are a number of computational studies that deal with multiple particles [84,111e115] in both steady and transient situations. Ostensibly, these studies are focused at the microscale hydrodynamics, and they do uncover details of yielding, bridging between particles and contact. However, it is not clear what the next step is for such studies: unlike Newtonian fluids, the viscoplastic Stokes equations are not linear in the velocity, so superposition principles fail and methods such as Stokesian dynamics are not valid. A pure computational approach has been followed in the studies by Yu and Wachs [116] and Wachs and Frigaard [117] in which a fictitious domain approach is used. This approach models the full fluidesolid domain and is implemented within the augmented Lagrangian framework. Although suited to suspensions, the computational algorithms are relatively slow to converge at each time step and transient calculations have thus been limited to small numbers of particles in two dimensions. Recent work in Ref. [118] has instead focused on the steady Stokes flow of 2D particles arranged in random configurations to represent a given solid fraction. These calculations are interesting in that, by repeating with successively large yield stress, we begin to estimate the critical yield limit Yc of a suspension, that is, at which point, the entire suspension is static. Also, the relative economy of the Stokes flow calculations means that a good level of mesh refinement is feasible. We are beginning to uncover some of the true complexity of the stress field in suspensions. www.sciencedirect.com

87

Applications and opportunities

Of the many areas of application of yield stress fluid mechanics in the past 2 decades, we highlight only a few. First, there has been a proliferation of thin film flows, both with and without the pseudoplug corrections that explained the lubrication paradox. These include flows over topography [119e121], those combining with heat transfer [122], models of flow through obstacles [123], onset of avalanches [124], a number of studies of dambreak flows [125,126], some of steady film motion [19], etc . In addition, other free surface flows such as the slump test have been studied in detail [126e128]. Related to these flows, we can also find novel applications of adhesion [129]. Second, our better understanding of critical limits and the associated energy stability has led to a few studies in which these ideas are exploited. Finite stopping time of a particle in a flow implies a finite distance travelled [117], and these ideas have been used in context of particle fractionation using yield stress fluids [130,131]. Here, a fluid-filled centrifuge is used to adjust acceleration, and hence, particles experience a different yield number at different radii: large particles move while small particles are held. This idea extends to continuous fractionation operations [132]. In the study by Karimfazli et al. [75], the idea of a thermal switch/filter is advanced: a high yield stress arrests motion leaving only conduction, but for reduced yield stress, motion ensues and convective heat transfer dominates. In Ref. [75], yield stress variation via an electrorheological and magnetorheological (ER/MR) fluid is proposed and estimates indicate that a relatively weak ER/MR effect would be sufficient, for example, yield stresses w102 Pa. More broadly in the ER/MR context, there are numerous applications in vibration damping, valves and automotive clutches. These are all technologically interesting but use the ER/MR effect in an oneoff mode. We feel there are other areas where less extreme ER/MR effects can be applied with fluid-mechanic interest [133]. Third, there have been many attempts to model porous media flows at microscales and macroscales. It has long been known that in porous media, viscoplastic fluids are modelled qualitatively by an extension of Darcy’s law into nonlinear filtration/seepage regimes. In these flows, there is no flow at all if the pressure gradient lies below a limiting pressure gradient. Limiting pressure gradient models have been common since the 1960s [134,135], and their analysis leads to porous media flows with dead zones and flowing zones, as might be expected. Much of the earlier literature is summarised in Ref. [136]. The newer research is complementary to these continuum porous media descriptions, in trying to predict either scaling laws of the flow or the limiting pressure gradient. Some of the research is experimental, for example, flows though beds of packed beads, and some computational. Current Opinion in Colloid & Interface Science 2019, 43:80–93

88 Rheology

Computational studies, using the Stokes equations and either regular or irregular geometries, have been conducted by various authors [93,137e140]. Most of these computations are limited to 2D. We note that for Newtonian fluids, fully 3D computation flows through digitised porous media geometries (scanned from real core samples) have been feasible for the past decade and are giving promising results. Perhaps this will come for viscoplastic fluids too. In the meantime, macroscale poreethroat network models have been developed [137]. Other interesting work in this direction, using networks or lattices, includes the studies by Chen et al. [141], Sochi et al. [142], Balhoff et al. [143], Talon et al. [144], Talon and Bauer [145] and Chevalier and Talon [146]. These approaches are interesting in that statistical properties of the network can be varied, and we are able to analyse macroscale dynamics of the porous media flow. For example, in Ref. [144], we see unexpected transitions in the scaling laws of pressure drop vs. flow rate through a porous sample: the initial flow is along a single critical path, then the number of paths increases and then we recover the more macroscopic features we would expect from the rheological flow law. Finally, fluidefluid flows have found wide application. First, there are a number of applications in the oil and gas industry: in cementing and in the restart of heavy oil pipelines, as reviewed in Ref. [10]. Second, we have seen exciting new biomedical applications such as in the studies by Zamankhan et al. [147,148], wherein the authors model the propagation of a mucus plug along pulmonary airways using yield stress models. Third, there are a number of flows proposed that exploit the stabilising effect of retaining an unyielded plug at a fluidefluid interface. This method has allowed the establishment of stable multilayer flows in configurations that would be unstable for Newtonian fluids [149e 151]. Slight variations allow controlled droplet transport [152], sculpting of the interface [153] and stable coreannular flows for oil pipelining applications [154]. Figure 1 captures some of the aforementioned and other application directions being pursued. The question of rheological design of fit-for-purpose soft solids has been recently considered by Ewoldt [155], which is a useful step in bridging between a fluid-mechanic effect, desired rheological properties and a colloidal approach to designing the right fluid.

of analytical challenges, many are flow dependent, that is, in applying methods or adapting them to a problem at hand. Stress indeterminacy has hampered progress, although it is part of these models. Understanding decay of the stress field around a settling particle, for example, would allow estimates to be made in general of the extent of yielded flow region and could also lead to generic information regarding yield stress suspensions. Recent work, for example [68,126,127], has explored the connections with perfect plasticity, which certainly captures some features of flows at the yield limit. However, less has been made of connections with elasticity, which can be viewed as a different regularisation of simple yield stress models and may lead to at least qualitative results [156]. Regarding computation, although reliable computations are now routinely performed, they are significantly slower than with Newtonian fluids, and more so when the augmented Lagrangian method is used. This can make 3D computations painful in situations for which other generalised Newtonian fluid models might be quickly used. Recent advances offer some improvement [13,157,158]. As with other branches of fluid mechanics, recent years have seen a number of applications of Lattice Boltzmann (LB) methods to these flows [159e 161]. It is hard to evaluate potential advantages of these methods in comparison to more conventional finite element or finite volume (FEM/FV) NaviereStokes discretisation because we have not seen comprehensive comparisons over different meaningful flow types. Many recent LB studies effectively model yield stress fluids by converging to regularised versions of the effective viscosity. This is probably due to familiarity of the authors with other generalised Newtonian fluid models, not realising that the yield stress behaviour is physically different. In fact, this regularisation is anyway not needed in the LB framework. Earlier work of Vikhansky [162] showed that the true yield stress behaviour could be included implicitly. An interesting recent article [163] makes comparisons between these methods. Also, in Ref. [163], it is shown that some care is needed in the choice of lattice representation to avoid the generation of spurious flows within plug regions. Will LB methods save the day? It is probably too early to tell. While they offer scalability properties, mesh adaptivity options are not as advanced with these methods as with FEM/FV approaches. Thus, problems such as finding yield limits, where an adaptive mesh is helpful, are less attractive with LB currently.

Challenges

As we have seen previously, a wide range of flows are currently studied. The last decades have given us new tools and methods with which to study these flows. It is fair to say that the study of simple yield stress fluid models has reached a high level of maturity in which many practical problems are tractable using a combination of analytical and computational methods. In terms Current Opinion in Colloid & Interface Science 2019, 43:80–93

With respect to multiphase flows, in some respects, progress has been good, but in comparison with the scale of problems that are being solved at the leading edge of Newtonian flows, for example, with either suspensions or bubbly flows, it is apparent that viscoplastic fluid mechanics lags far behind. Partly this is due to the computational challenges discussed previously. www.sciencedirect.com

Simple yield stress fluids Frigaard

Partly the underlying nonlinearity prevents generalisation of techniques that can accelerate study of specific regimes, for example, Stokesian dynamics. Partly too there are closure-type problems that are not fully solved for these fluids, for example, lubrication/drag closures, needed to resolve particle motion in some computational approaches. These are areas of active research.

Limitations and final words The limitations of simple yield stress models rheologically are well known. First, even supposing their validity for a particular flow, it is necessary to measure the yield stress and other properties (parameters). Here, we refer to Refs. [8,164] as examples that illustrate some of the methods and difficulties. Second, there are limitations which are specific to the different category of yield stress fluid considered, as reviewed in Ref. [4]. Some of these are limitations of scale, and others relate to the continuum mechanics description of macroscopic behaviour. In this vein, there are shearthinning fluids for which there is a large zero-shear viscosity, such that yield stress behaviour is apparent, that is, the assertion of Barnes and Walters [32]. For those fluids with a yield stress, it is also common that they exhibit some combination of thixotropy and elasticity at low shear rates. Regarding elastic effects, a 1D Maxwell-type visco-elasto-plastic model was proposed by Schwedoff [165], long before Bingham’s work. An elastic strain was also accounted for by Oldroyd [26] to describe subyield behaviour, and Saramito [166,167] has developed this approach further. The recent study [168] introduces a new elasto-visco-plastic model, incorporating kinematic hardening and including thixotropy (TEVP model). Another attractive TEVP model is the BMP model [169], which is approaching 20 years old. This model is mostly analysed in the thixo-visco-elastic fluid context. There are also many interesting and recent developments with thixotropic viscoplastic fluids, reviewed within this volume [14]. The rheology of TEVP fluids is a hot topic these days. Without wishing to dampen my colleagues’ enthusiasm, a little caution is needed. There is little mathematical analysis of any of these models to date, addressing fundamental questions about well-posedness and other topics. Much of the work is still largely focused at modelling rheometric effects in 1D (or even 0D). Although there have been implementations of some models into computational codes, these too are not common. Therefore, although there are limitations to simple yield stress models, at least these are limitations that are known and understood. The more complex models also have limitations, but we have perhaps not yet encountered them. At present, this makes things a little impractical to use TEVP models widely. We are in a

www.sciencedirect.com

89

decade (?) of evolution: a period of constitutive model generation, then a thinning of the field and hopefully we will arrive at a shortlist of TEVP models suitable for different classes of yield stress fluid. On the way, hopefully we can move from rheometry to fluid mechanics. Having said this, recent computational work [170] has shown the utility of TEVP models in explaining longstanding paradoxical behaviour, such as the loss in fore-aft symmetry of Stokes flow around a sphere [20], and we can hope for more success with TEVP. As presented in this article, the study of simple yield stress fluids is alive and well from the fluid mechanics perspective. There have been significant advances in fundamental understanding, in the methods of analysis and computation and in the range of applications that are being effectively studied. These include new applications thrust upon the field, where there is a need to understand the consequences of the fluids in a particular flow having a yield stress, for example, mucus plugs, oil well cementing displacements. These also include new applications where better understanding of the dynamics allows control of either instabilities or motion. From the perspective of colloidal science, fluid mechanics delivers understanding on the macroscale. The opportunities are to be able to engineer complex liquids of specific properties. In the current context, being able to design the yield stress, with a degree of control over elastic and thixotropic timescales of the fluids, would be useful, both practically and from a fundamental perspective. Equally, smart fluids that can adjust yield stress (and other properties) in response to an applied field are of interest from the perspective on control applications. These exist of course, in many ER/MR fluids. We observe however that most ER/MR applications exploit on/off flows in narrow gaps, where generating a high yield stress (e.g. tY >104 Pa) rapidly is a key e objective. If the flow control aimed for is instead say interfacial (temporarily stopping growth of an instability), or is to dampen convective transport, then significantly smaller yield stresses can probably be effective. There are certainly new products and processes to be discovered and invented. A critical rheologist might say, “but these models don’t capture the elastic behaviour, or .”. Perhaps correct, but innovation and applications are built initially on ideas; simple yield stress models allow us to study the fluid mechanics at a sufficient depth that generates new ideas.

Conflict of interest statement Nothing declared.

Current Opinion in Colloid & Interface Science 2019, 43:80–93

90 Rheology

Acknowledgement

21. Bingham EC: An investigation of the laws of plastic flow. Bull. Bur. Stand. 1916, 13:309–353.

This work has been supported by the Natural Sciences and Engineering Research Council of Canada via their Discovery Grants programme (Grant No. RGPIN-2015-06398).

22. Bingham EC: Fluidity and plasticity. New York: McGraw-Hill; 1922.

References

23. Herschel WH, Bulkley R: Konsistenzmessungen von Gummibenzollösungen. Koll Z 1926, 39:291–300.

Papers of particular interest, published within the period of review, have been highlighted as: * of special interest * * of outstanding interest 1.

Bird RB, Dai GC, Yarusso BJ: The rheology and flow of viscoplastic materials. Rev Chem Eng 1983, 1:2–70.

2. *

Balmforth N, Frigaard IA, Ovarlez G: Yielding to stress: recent developments in viscoplastic fluid mechanics. Annu Rev Fluid Mech 2014, 46:121146. Good recent general view of fluid mechanical aspects of yield stress fluids 3. Coussot P: Yield stress fluid flows: a review of experimental * data. J Non-Newtonian Fluid Mech 2014, 211:31–49. Current summary of experimental work in the area 4. *

Bonn D, Denn M, Berthier B, Divoux B, Manneville S: Yield stress materials in soft condensed matter. Rev Mod Phys 2017, 89, 035005. Up to date review of attempts to classify vield stress fluids. Puts “simple” fluids in context

5.

Huilgol RR: Fluid mechanics of viscoplasticity. Springer; 2015.

6.

Ovarlez G, Hormozi S. Lectures on visco-plastic fluid mechanics. Springer; 2019.

7.

Cloitre M, Bonnecaze RT: A review on wall slip in high solid dispersions. Rheol Acta 2017, 56:283–305.

8.

Coussot P: Binghams heritage. Rheol Acta 2017, 56:163–176.

9. Ewoldt R, McKinley G: Mapping thixo-elasto-visco-plastic * behavior. Rheol Acta 2017, 56:195–210. A more mechanical attempt to position yield stress behaviour rheologically

24. Prager W: Mécanique des solides isotropes au delà du domaine élastique. Mem Sci Math 1937, 87. 25. Prager W: On slow visco-plastic flow. Report A11-84, Brown University, graduate Division of Applied Mathematics. In Also available in Studies in Mathematics and mechanics, a volume Presented to Richard von Mises by friends, colleagues, and pupils. New York: Academic Press Inc.; 1952: 208 – 216. 1954. 26. Oldroyd JG: A rational formulation of the equations of plastic flow for a Bingham solid. Math Proc Camb Phil Soc 1947, 43: 100–105. 27. Georgeievskii D: Stability of Bingham flows: from the earliest works of A.A. Ilyushin to the present. J Eng Math 2011, https:// doi.org/10.1007/s10665-011-9471-7. 28. Mossolov PP, Miasnikov VP: Variational Methods in the theory * * of the fluidity of a viscous plastic medium. J Mech Appl Math 1965, 29:468–492. With successive papers in 1966 & 1967, these are a must read for anyone with an interest in qualitative behaviour 29. Duvaut G, Lions JL. Inequalities in mechanics and physics, vol. * 219. Springer-Verlag; 1976:279–327. Theoretical basis 30. Glowinski R, Lions JL, Tremolieres R: Numerical analysis of variational inequalities. 1981 [Amsterdam: North-Holland]. 31. Fortin M, Glowinski R: Augmented Lagrangian methods. 1983 [Amsterdam: North-Holland]. 32. Barnes HA, Walters K: The yield stress myth? Rheol Acta 1985, 24:323–326. 33. Comparini E: A one-dimensional Bingham flow. J Math Anal Appl 1992, 169:127–139.

10. Frigaard I, Paso G, de Souza Mendes PR: Binghams model in the oil and gas industry. Rheol Acta 2017, 56:259–282.

34. Lipscomb GG, Denn MM: Flow of Bingham fluids in complex geometries. J Non-Newtonian Fluid Mech 1984, 14:337–346.

11. Malkin A, Kulichikhin V, Ilyin S: A modern look on yield stress fluids. Rheol Acta 2017, 56:177–188.

35. Wilson SDR: A note on thin-layer theory for Bingham plastics. J Non-Newtonian Fluid Mech 1999, 85:29–33.

12. Mitsoulis E, Tsamopoulos J: Numerical simulations of complex * yield-stress fluid flows. Rheol Acta 2017, 56:231–258. A recent review on computational applications and methods

36. Bercovier M, Engleman M: A finite-element method for incompressible non-Newtonian flows. J Comp Physiol 1980, 36:313–326.

13. Saramito P, Wachs A: Progress in numerical simulation of * yield stress fluid flows. Rheol Acta 2017, 56:211–230. A recent review on computational applications and methods

37. Papanastasiou TC: Flows of materials with yield. J Rheol 1987, 31:385–404.

14. de Souza Mendes P, Thompson R: Time-dependent yield stress materials. Curr Opin Colloid Interface Sci 2019, 43: 15–25.

38. Glowinski R, Wachs A: On the numerical simulation of viscoplastic fluid flow. Handb Numer Anal 2011, 16:483–717.

15. Coussot P, Nguyen Q, Huynh H, Bonn D: Viscosity bifurcation in thixotropic, yielding fluids. J Rheol 2002, 46:573–590.

39. Frigaard IA, Nouar C: On the use of viscosity regularisation * methods for visco-plastic fluid flow computation. J NonNewtonian Fluid Mech 2005, 127:1–26. Critical review on regularization methods

16. Moller P, Fall A, Chikkadi V, Derks D, Bonn D: An attempt to categorize yield stress fluid behaviour. Phil. Trans. R. Soc. A. 2009, 367:5139–5155.

40. Walton IC, Bittleston SH: The axial flow of a Bingham plastic in * a narrow eccentric annulus. J Fluid Mech 1991, 222:39–60. First resolution of the lubrication paradox

17. Joseph DD: Fluid dynamics of viscoelastic liquids. Springer; 1990.

41. Balmforth NJ, Craster RV: A consistent thin-layer theory for Bingham fluids. J Non-Newtonian Fluid Mech 1999, 84:65–81.

18. Tabuteau H, Coussot P, de Bruyn J: Drag force on a sphere in steady motion through a yield-stress fluid. J Rheol 2007, 51: 125–137.

42. Balmforth NJ, Dubash N, Slim A: Extensional dynamics of viscoplastic filaments: I. Long-wave approximation and the Rayleigh instability. J Non-Newtonian Fluid Mech 2010, 165: 1139–1146.

19. Freydier P, Chambon G, Naaim M: Experimental characterization of velocity fields within the front of viscoplastic surges down an incline. J Non-Newtonian Fluid Mech 2017, 240:56–69. 20. Putz A, Burghelea T, Martinez DM, Frigaard IA: Settling of an isolated spherical particle in a yield stress shear thinning fluid. Phys Fluids 2008, 20, 033102.

Current Opinion in Colloid & Interface Science 2019, 43:80–93

43. Balmforth NJ, Dubash N, Slim A: Extensional dynamics of viscoplastic filaments II. Drips and bridges. J Non-Newtonian Fluid Mech 2010, 165:1147–1160. 44. Balmforth NJ, Hewitt IJ: Viscoplastic sheets and threads. J Non-Newtonian Fluid Mech 2013, 193:28–42.

www.sciencedirect.com

Simple yield stress fluids Frigaard

45. Balmforth NJ, Craster RV, Hewitt DR, Hormozi S, Maleki A: * Viscoplastic boundary layers. J Fluid Mech 2017, 813: 929–954. Answering questions dating back to the 1940s 46. Hewitt DR, Balmforth NJ: Viscoplastic slender-body theory. J Fluid Mech 2018, 856:870–897. 47. Frigaard IA, Howison SD, Sobey IJ: On the stability of Pois* euille flow of a Bingham fluid. J Fluid Mech 1994, 263: 133–150. Linear stability basics and discussion on how to treat the yield surfaces 48. Kabouya N, Nouar C: On the stability of a Bingham fluid flow in an annular channel. Compt Rendus Mec 2003, 331: 149 –156. 49. Peng J, Zhu KQ: Linear stability of Bingham fluids in spiral Couette flow. J Fluid Mech 2004, 512:21–46. 50. Landry MP, Frigaard IA, Martinez DM: Stability and instability of Taylor-Couette flows of a Bingham fluid. J Fluid Mech 2006, 560:321–353. 51. Nouar C, Kabouya N, Dusek J, Mamou M: Modal and nonmodal linear stability of the plane Bingham-Poiseuille flow. J Fluid Mech 2007, 577:211–239. 52. Liu R, Liu QS: Non-modal stability in Hagen-Poiseuille flow of a Bingham fluid. Phys Fluids 2014, 26, 014102. 53. Liu R, Ding Z, Hu K: Stabilities in plane Poiseuille flow of Herschel–Bulkley fluid. J Non-Newtonian Fluid Mech 2018, 251:132–141. 54. Balmforth NJ, Liu JJ: Roll waves in mud. J Fluid Mech 2004, 519:33–54. 55. Temam R, Strang G: Functions of bounded deformation. Arch Ration Mech Anal 1980, 75:7–21. 56. Bristeau M M: Application de la méthode des éléments finis à la résolution numérique d’inéquations variationnelles de type Bingham. These de 3ème cycle, Universite de Paris VI; 1975. 57. Karimfazli I, Frigaard I: Flow, onset and stability: qualitative * * analysis of yield stress fluid flow in enclosures. J Non-Newtonian Fluid Mech 2016, 238:224–232. Yield numbers and stability 58. Chatzimina M, Georgiou G, Argyropaidas I, Mitsoulis E, Huilgol R: Cessation of Couette and Poiseuille flows of a Bingham plastic and finite stopping times. J Non-Newtonian Fluid Mech 2005, 129:117–127. 59. Beris AN, Tsamopoulos JA, Armstrong R, Brown R: Creeping * motion of a sphere through a Bingham plastic. J Fluid Mech 1985, 158:219–244. Old, but an important paper 60. Beaulne M, Mitsoulis E: Creeping motion of a sphere in tubes filled with Herschel-Bulkley fluids. J Non-Newtonian Fluid Mech 1997, 72:55–71. 61. Zisis T, Mitsoulis E: Viscoplastic flow around a cylinder kept between parallel plates. J Non-Newtonian Fluid Mech 2002, 105:1–20. 62. Deglo de Besses B, Magnin A, Jay P: Viscoplastic flow around a cylinder in an infinite medium. J Non-Newtonian Fluid Mech 2003, 115:27–49. 63. Deglo de Besses B, Magnin A, Jay P: Sphere drag in a viscoplastic fluid. AIChE J 2004, 50:2627–2629. 64. Mitsoulis E: On creeping drag flow of a viscoplastic fluid past a circular cylinder: wall effects. Chem Eng Sci 2004, 59: 789–800. 65. Tokpavi D, Magnin A, Jay P: Very slow flow of Bingham * viscoplastic fluid around a circular cylinder. J Non-Newtonian Fluid Mech 2008, 154:65–76. Thorough and detailed study using a regularization method 66. Jossic L, Magnin A: Drag of an isolated cylinder and interactions between two cylinders in yield stress fluid. J NonNewtonian Fluid Mech 2009, 164:9–16.

www.sciencedirect.com

91

67. Putz A, Frigaard IA: Creeping flow around particles in a Bingham fluid. J Non-Newtonian Fluid Mech 2010, 165: 263–280. 68. Chaparian E, Frigaard IA: Yield limit analysis of particle motion in a yield-stress fluid. J Fluid Mech 2017, 819:311–351. 69. Chaparian E, Frigaard IA: Cloaking: particles in a yield-stress * fluid. J Non-Newtonian Fluid Mech 2017, 243:47–55. With 67 & 68, gives a clear over view of static flow limit for symmetric particles and associated phenomena 70. Turan O, Poole RJ, Chakraborty N: Aspect ratio effects in laminar natural convection of Bingham fluids in rectangular enclosures with differentially heated side walls. J Non-Newtonian Fluid Mech 2011, 166:208–230. 71. Turan O, Poole RJ, Chakraborty N: Laminar Rayleigh-Bénard convection of yield stress fluids in a square enclosure. J NonNewtonian Fluid Mech 2012:83–96. 171172. 72. Nirmalkar N, Bose A, Chhabra R: Free convection from a heated circular cylinder in Bingham plastic fluids. Int J Therm Sci 2014, 83:33–44. 73. Zhang JY, Frigaard IA, Vola D: Yield stress effects on Rayleigh* Bénard convection. J Fluid Mech 2006, 566:389–419. One of the first examples of energy stability for a simple yield stress fluid 74. Balmforth NJ, Rust AC: Weakly nonlinear viscoplastic convection. J Non-Newtonian Fluid Mech 2009, 158:36–45. 75. Karimfazli I, Frigaard I, Wachs A: A novel heat transfer switch using the yield stress. J Fluid Mech 2015, 783:526–566. 76. Métivier C, Nouar C: On linear stability of Rayleigh-Bénard Poiseuille flow of viscoplastic fluids. Phys Fluids 2008, 17: 104106. 77. Métivier C, Nouar C, Brancher JP: Weakly nonlinear dynamics of thermoconvective instability involving viscoplastic fluids. J Fluid Mech 2010, 660:316–353. 78. Smyrnaios D, Tsamopoulos J: Squeeze flow of Bingham plastics. J Non-Newtonian Fluid Mech 2001, 100:165–189. 79. Matsoukas A, Mitsoulis E: Geometry effects in squeeze flow of Bingham plastics. J Non-Newtonian Fluid Mech 2003, 109: 231–240. 80. Karapetsas G, Tsamopoulos J: Transient squeeze flow of viscoplastic materials. J Non-Newtonian Fluid Mech 2006, 133: 35–56. 81. Fusi L, Farina A, Rosso F: Planar squeeze flow of a Bingham fluid. J Non-Newtonian Fluid Mech 2015, 225:1–9. 82. Muravleva L: Axisymmetric squeeze flow of a viscoplastic Bingham medium. J Non-Newtonian Fluid Mech 2017, 249: 97–120. 83. Singeetham PK, Puttanna VK: Viscoplastic fluids in 2D plane * squeeze flow: a matched asymptotics analysis. J Non-Newtonian Fluid Mech 2019, 263:154–175. Detailed explanation 84. Koblitz A, Lovett S, Nikiforakis N: Viscoplastic squeeze flow between two identical infinite circular cylinders. Phys. Rev. Fluids 2018, 3, 023301. 85. Mitsoulis E, Huilgol R: Entry flows of Bingham plastics in expansions. J Non-Newtonian Fluid Mech 2004, 122:45–54. 86. de Souza Mendes P, Naccache M, Varges P, Marchesini F: Flow * of viscoplastic liquids through axisymmetric expansion– contractions. J Non-Newtonian Fluid Mech 2007, 142:207–217. Nice visualization of the yield surfaces 87. Naccache M, Barbosa R: Creeping flow of viscoplastic materials through a planar expansion followed by a contraction. Mech Res Commun 2007, 34:423–431. 88. Hermany L, DallOnder dos Santos D, Frey S, Naccache M, de Souza Mendes P: Flow of yield-stress liquids through an axisymmetric abrupt expansion-contraction. J Non-Newtonian Fluid Mech 2013, 201:35–52.

Current Opinion in Colloid & Interface Science 2019, 43:80–93

92 Rheology

89. Marly A, Vigneaux P: Augmented Lagrangian simulations study of yield-stress fluid flows in expansion-contraction and comparisons with physical experiments. J Non-Newtonian Fluid Mech 2017, 239:1–9. 90. Frigaard IA, Ryan D: Flow of a visco-plastic fluid in a channel of slowly varying width. J Non-Newtonian Fluid Mech 2004, 123:67–83. 91. Putz A, Frigaard IA, Martinez DM: On the lubrication paradox * and the use of regularisation methods for lubrication flows. J Non-Newtonian Fluid Mech 2009, 163:62–77. Detailed review of lubrication paradox and is resolution in a channel flow 92. Roustaei A, Frigaard IA: The occurrence of fouling layers in the flow of a yield stress fluid along a wavy-walled channel. J Non-Newtonian Fluid Mech 2013, 198:109–124. 93. Roustaei A, Chevalier T, Talon L, Frigaard IA: Non-Darcy effects in fracture flows of a yield stress fluid. J Fluid Mech 2016, 805: 222–261. 94. Stein S, Buggisch H: Rise of pulsating bubbles in fluids with a yield stress. Z Angew Math Mech 2000, 80:827–834. 95. Dubash N, Frigaard IA: Conditions for static bubbles in viscoplastic fluids. Phys Fluids 2004, 16:4319–4330. 96. Tsamopoulos J, Dimakopoulos Y, Chatzidai N, Karapetsas G, Pavlidis M: Steady bubble rise and deformation in Newtonian and viscoplastic fluids and conditions for bubble entrapment. J Fluid Mech 2008, 601:123–164. 97. Dimakopoulos Y, Pavlidis M, Tsamopoulos J: Steady bubble rise in Herschel-Bulkley fluids and comparison of predictions via the Augmented Lagrangian Method with those via the Papanastasiou model. J Non-Newtonian Fluid Mech 2012, 200: 35–56. 98. Dimakopoulos Y, Tsamopoulos J: Transient displacement of Newtonian and viscoplastic liquids by air in complex tubes. J Non-Newtonian Fluid Mech 2007, 142:162–182. 99. de Sousa DA, Soares EJ, de Queiroz RS, Thompson RL: Numerical investigation on gas-displacement of a shear-thinning liquid and a visco-plastic material in capillary tubes. J Non-Newtonian Fluid Mech 2007, 144:149–159. 100.Thompson RL, Soares EJ, Bacchi R: Further remarks on numerical investigation on gas displacement of a shear-thinning liquid and a visco-plastic material in capillary tubes. J NonNewtonian Fluid Mech 2010, 165:448–452. 101. Allouche M, Frigaard IA, Sona G: Static wall layers in the displacement of two visco-plastic fluids in a plane channel. J Fluid Mech 2000, 424:243–277. 102. Frigaard IA, Leimgruber S, Scherzer O: Variational methods and maximal residual wall layers. J Fluid Mech 2003, 483: 37–65. 103. Wielage-Burchard K, Frigaard IA: Static wall layers in plane channel displacement flows. J Non-Newtonian Fluid Mech 2011, 166:245–261. 104. Freitas J, Soares E, Thompson R: Viscoplasticviscoplastic displacement in a plane channel with interfacial tension effects. Chem Eng Sci 2013, 91:54–60. 105. Zare M, Roustaei A, Frigaard IA: Buoyancy effects on microannulus formation: density stable displacement of NewtonianBingham fluids. J Non-Newtonian Fluid Mech 2017, 247: 22–40. 106. Eslami A, Frigaard IA, Taghavi SM: Viscoplastic fluid displacement flows in horizontal channels: numerical simulations. J Non-Newtonian Fluid Mech 2017, 249:79–96. 107. Frigaard IA: Stratified exchange flows of two Bingham fluids in an inclined slot. J Non-Newtonian Fluid Mech 1998, 78: 61–87. 108. Taghavi SM, Seon T, Martinez DM, Frigaard IA: Buoyancydominated displacement flows in near-horizontal channels: the viscous limit. J Fluid Mech 2009, 639:1–35.

Current Opinion in Colloid & Interface Science 2019, 43:80–93

109. Zhang J, Frigaard IA: Dispersion effects in the miscible displacement of two fluids in a duct of large aspect ratio. J Fluid Mech 2006, 549:225–251. 110. Maleki A, Frigaard IA: Axial dispersion in weakly turbulent flows of yield stress fluids. J Non-Newtonian Fluid Mech 2016, 235:1–19. 111. Liu BT, Muller SJ, Denn M: Interactions of two rigid spheres translating collinearly in creeping flow in a Bingham material. J Non-Newtonian Fluid Mech 2003, 113:49–67. 112. Tokpavi DL, Jay P, Magnin A: Interaction between two circular cylinders in slow flow of Bingham viscoplastic fluid. J NonNewtonian Fluid Mech 2009, 157:175–187. 113. Prashant S, Derksen J: Direct simulations of spherical particle motion in Bingham liquids. Comput Chem Eng 2011, 35: 1200–1214. 114. Chaparian E, Wachs A, Frigaard IA: Inline motion and hydrodynamic interaction of 2D particles in a viscoplastic fluid. Phys Fluids 2018, 30, 033101. 115. Fahs H, Ovarlez G, Chateau X: Pair-particle trajectories in a shear flow of a Bingham fluid. J Non-Newtonian Fluid Mech 2018, 261:171–187. 116. Yu Z, Wachs A: A fictitious domain method for dynamic simulation of particle sedimentation in Bingham fluids. J Non-Newtonian Fluid Mech 2007, 145:78–91. 117. Wachs A, Frigaard IA: Particle settling in yield stress fluids: limiting time, distance and applications. J Non-Newtonian Fluid Mech 2016, 238:189–204. 118. Koblitz A, Lovett S, Nikiforakis N: Direct numerical simulation of * particle sedimentation in a Bingham fluid. Phys. Rev. Fluids 2018, 3, 093301. The first clear indication of the complexity of a yield stress fluid suspension, as it ceases to flow. 119. Wilson SK, Duffy BR, Ross AB: On the gravity-driven draining of a rivulet of a viscoplastic material down a slowly varying substrate. Phys Fluids 2002, 14:555–571. 120. Balmforth NJ, Craster RV, Rust AC, Sassi R: Viscoplastic flow over an inclined surface. J Non-Newtonian Fluid Mech 2006, 139:103–127. 121. Ionescu IR: Viscoplastic shallow flow equations with topography. J Non-Newtonian Fluid Mech 2013, 193:111–128. 122. Balmforth NJ, Craster RV, Sassi R: Dynamics of cooling viscoplastic domes. J Fluid Mech 2004, 499:149–182. 123. Bernabeu N, Saramito P, Harris A: Laminar shallow viscoplastic fluid flowing through an array of vertical obstacles. J Non-Newtonian Fluid Mech 2018, 257:59–70. 124. Ionescu IR, Lupascu: Modeling shallow avalanche onset over complex basal topography. Adv Comput Math 2016, 42:5–26. Cite as. 125. Liu Y, Balmforth NJ, Hormozi S, Hewitt D: Two–dimensional viscoplastic dambreaks. J Non-Newtonian Fluid Mech 2016, 238:65–79. 126. Liu Y, Balmforth NJ, Hormozi S: Axisymmetric viscoplastic dambreaks and the slump test. J Non-Newtonian Fluid Mech 2018, 258:45–57. 127. Dubash N, Balmforth N, Slim A, Cochard S: What is the final * shape of a viscoplastic slump? J Non-Newtonian Fluid Mech 2009, 158:91–100. Insights into plasticity theory and simple yield stress fluids 128. Hogg A, Matson G: Slumps of viscoplastic fluids on slopes. J Non-Newtonian Fluid Mech 2009, 158:101–112. 129. Ewoldt R, Clasen C, Hosoi AE, McKinley GH: Rheological fingerprinting of gastropod pedal mucus and synthetic complex fluids for biomimicking adhesive locomotion. Soft Matter 2007, 3:634–643. 130. Madani A, Storey S, Olson JA, Frigaard IA, Salmela J, Martinez DM: Fractionation of non-Brownian rod-like particle

www.sciencedirect.com

Simple yield stress fluids Frigaard

suspensions in a viscoplastic fluid. Chem Eng Sci 2010, 65: 1762–1772.

93

viscoelastic fluids. J Non-Newtonian Fluid Mech 2010, 165: 1578–1587.

131. Madani A, Kiiskinen H, Olson JA, Martinez DM: Fractionation of microfibrillated cellulose and its effects on tensile index and elongation of paper. Nord Pulp Pap Res J 2011, 26:306–311.

151. Hormozi S, Wielage-Burchard K, Frigaard IA: Entry, start up and stability effects in visco-plastically lubricated pipe flows. J Fluid Mech 2011, 673:432–467.

132. Martinez D.M., Olson J.A., Madani A., Frigaard, I.A., Farajisarir D., Lockhart J., 2014. Method and apparatus for continuously fractionating particles contained within a viscoplastic fluid US patent US20140296052 A1, published 02/10/14.

152. Maleki A, Hormozi S, Roustaei A, Frigaard IA: Macro-size drop encapsulation. J Fluid Mech 2015, 769:482–521.

133. Ewoldt R, Tourkine P, McKinley GH, Hosoi AE: Controllable adhesion using field-activated fluids. Phys Fluids 2011, 23, 073104.

153. Hormozi S, Dunbrack G, Frigaard IA: Visco-plastic sculpting. Phys Fluids 2014, 26, 093101. 154. Sarmadi P, Hormozi S, Frigaard IA: Novel triple-layer configuration for stable high-speed lubricated pipeline transport. Phys Rev Fluids 2017, 2, 044302.

134. Sultanov BI: Filtration of visco-plastic fluids in a porous medium. Isrv Akad Nauk AzSSR Ser fiz mati tekh. nauk 1960, 5: 820–833.

155. Ewoldt R: Extremely soft: design with rheologically complex fluids. Soft Robot 2013, 1:12–20.

135. Entov VM: On some two-dimensional problems of the theory of filtration with a limiting gradient. Prikl Mat Mekh 1967, 31: 820–833.

156. Chateau X, Ovarlez G, Luu Trung K: Homogenization approach * to the behavior of suspensions of noncolloidal particles in yield stress fluids. J Rheol 2008, 52:489–506. Theoretical under-pinning of a successful empirical theory

136. Barenblatt GI, Entov VM, Ryzhik VM: Theory of fluid flows through natural rocks. Kluwer Academic Publishers; 1989. 137. Balhoff MT, Thompson KE: Modeling the steady flow of yield* stress fluids in packed beds. AIChE J 2004, 50:3034–3048. Before its time – an approach followed much recently 138. Bleyer J, Coussot P: Breakage of non-Newtonian character in flow through a porous medium: evidence from numerical simulation. Phys Rev E 2014, 89, 063018. 139. Roustaei A, Gosselin A, Frigaard I: Residual drilling mud during conditioning of uneven boreholes in primary cementing. Part 1: rheology and geometry effects in non-inertial flows. J NonNewtonian Fluid Mech 2015, 220:87–98.

157. Treskatis T, Moyers-Gonzalez MA, Price C: An accelerated dual proximal gradient method for applications in viscoplasticity. J Non-Newtonian Fluid Mech 2016, 238:115130. 158. Dimakopoulos Y, Makrigiorgos G, Georgiou G, Tsamopoulos J: The PAL (Penalized Augmented Lagrangian) method for computing viscoplastic flows: a new fast converging scheme. J Non-Newtonian Fluid Mech 2018, 256:23–41. 159. Derksen J: Solid particle mobility in agitated bingham liquids. Ind Eng Chem Res 2009, 48:2266–2274. 160. Derksen J: Simulations of mobilization of Bingham layers in a turbulently agitated tank. J Non-Newtonian Fluid Mech 2013, 191:25–34.

140. Shahsavari S, McKinley G: Mobility and pore-scale fluid dynamics of rate-dependent yield-stress fluids flowing through fibrous porous media. J Non-Newtonian Fluid Mech 2016, 235: 76–82.

161. Huilgol R, Kefayati G: From mesoscopic models to continuum mechanics: Newtonian and non-Newtonian fluids. J NonNewtonian Fluid Mech 2016, 233:146–154.

141. Chen M, Rossen W, Yortsos YC: The flow and displacement in porous media of fluids with yield stress. Chem Eng Sci 2005, 60:4183–4202.

162. Vikhansky A: Lattice-Boltzmann method for yield-stress liq* uids. J Non-Newtonian Fluid Mech 2008, 155:95–100. Don’t do LBM without this

142. Sochi T, Blunt MJ: Pore-scale network modeling of Ellis and Herschel-Bulkley fluids. J Pet Sci Eng 2008, 60:105–124.

163. Regulski W, Leonardi CR, Szumbarski J: On the spatial conver* gence and transient behaviour of lattice Boltzmann methods for modelling fluids with yield stress. 2016. arXiv:1610.01388 [physics.flu-dyn]; under review. This should be published somewhere

143. Balhoff M, Sanchez-Rivera D, Kwok A, Mehmani Y, Prodanovi c M: Numerical algorithms for network modeling of yield stress and other non-Newtonian fluids in porous media. Transport Porous Media 2012, 93:363–379. 144. Talon L, Auradou H, Pessel M, Hansen A: Geometry of optimal path hierarchies. EPL (Europhysics Letters) 2013, 103:30003. 145. Talon L, Bauer D: On the determination of a generalized * Darcy equation for yield-stress fluid in porous media using a Lattice-Boltzmann TRT scheme. Eur Phys J E 2013, 36: 1– 10. Network & lattice flow models are more complex 146. Chevalier T, Talon L: Generalization of Darcy’s law for Bingham fluids in porous media: from flow-field statistics to the flow-rate regimes. Phys Rev E 2015, 91, 023011. 147. Zamankhan P, Helenbrook BT, Takayama S, Grotberg JB: Steady motion of Bingham liquid plugs in two-dimensional channels. J Fluid Mech 2012, 705:258–279. 148. Zamankhan P, Takayama S, Grotberg JB: Steady displacement of long gas bubbles in channels and tubes filled by a Bingham fluid. Phys Rev Fluids 2018, 3, 013302. 149. Moyers-Gonzalez MA, Frigaard IA, Nouar C: Nonlinear stability * of a visco-plastically lubricated viscous shear flow. J Fluid Mech 2004, 506:117–146. Nonlinearly stable multi-phase flows: pretty rare! 150. Moyers-Gonzalez MA, Frigaard IA, Nouar C: Stable two-layer flows at all Re; visco-plastic lubrication of shear-thinning and

www.sciencedirect.com

164. Dinkgreve M, Paredes J, Denn M, Bonn D: On different ways of measuring the yield stress. J Non-Newtonian Fluid Mech 2016, 238:233–241. 165. Schwedoff T: La rigidite des fluides. . Rapports du Congrés intern. de Physique, vol. 1; 1900:478–486. 166. Saramito P: A new constitutive equation for elastoviscoplastic fluid flows. J Non-Newtonian Fluid Mech 2008, 145: 1–14. 167. Saramito P: A new elastoviscoplastic model based on the * Herschel–Bulkley viscoplastic model. J Non-Newtonian Fluid Mech 2009, 158:154–161. An EVP extension that should be more widely explored 168. Dimitriou C, McKinley GH: A canonical framework for modeling elasto-viscoplasticity in complex fluids. J Non-Newtonian Fluid Mech 2019, 265:116–132. 169. Calderas F, Herrera-Valencia E, Sanchez-Solis A, Manero O, Medina-Torres L, Renteria A, Sanchez-Olivares G: On the yield stress of complex materials. Korea Aust Rheol J 2013, 25: 233–242. 170. Fraggedakis D, Dimakopoulos Y, Tsamopoulos J: Yielding the * yield stress analysis: a thorough comparison of recently proposed elasto-visco-plastic (EVP) fluid models. J NonNewtonian Fluid Mech 2016, 238:170–188. Resolves long-standing paradoxical behaviour using an EVP model

Current Opinion in Colloid & Interface Science 2019, 43:80–93