# Issues in the flow of yield-stress liquids

## Abstract

Yield-stress liquids are materials that are solid below a critical applied stress and flow like mobile liquids at higher stresses. Classical descriptions of yield-stress liquids, which have been the basis for asymptotic and computational studies for five decades, are inadequate to describe many recent experimental observations, and it is clear that the time dependence of microstructure must be taken into account in the description of many real yield-stress liquids.

### Keywords

Yield stress Thixotropy Viscoplastic fluid Microstructure Bingham fluid## Introduction

Yield-stress liquids are broadly defined as materials that are solid below a critical applied stress and flow like mobile liquids at higher stresses. They are typically composed of colloidal or nanoscale constituents, and they are prevalent in consumer products, coatings and paints, industrial fluids, foods, mineral wastes, etc. Understanding bubble motion in yield-stress liquids is sometimes important, as exemplified by the need to remove air bubbles from cement and the emission of flammable gas bubbles from tanks of radioactive colloidal sludge at the US Department of Energy’s Hanford, Washington site.

*τ*

_{y}is the yield stress and

*η*

_{p}is commonly known as the plastic viscosity. The Bingham equation is linear in the shear rate following the onset of flow, but the fluid is in fact highly shear thinning; the viscosity, which is defined as the ratio of the shear stress to the shear rate, is

*y*

_{0}from the centerplane at which the stress equals the yield stress. If we perform a macroscopic force balance on a segment of the plug of length

*L*, where the pressure drop is Δ

*p*, it readily follows that

*y*

_{0}=

*τ*

_{y}

*L*/Δ

*p*. As long as

*y*

_{0}is smaller than the channel half-width, we must have shear banding, in which there is a plug of undeformed material adjacent to the centerplane and a sheared layer between the center plug and the wall, with a discontinuity in the velocity gradient at the interface. We then easily obtain the full velocity and stress distribution by integrating the equation of motion with the appropriate stress constitutive equation (Bingham, Herschel–Bulkley, Casson) between

*y*

_{0}and the wall, requiring continuity of the velocity and shear stress at

*y*=

*y*

_{0}. The requirement of continuity of the tangential velocity is a very strong statement about the material, to which we will return subsequently; classical plasticity permits tangential velocity discontinuities at interior slip planes, whereas slip planes are forbidden in the classical treatment of yield stress liquids. The analysis of channel flow is straightforward because it is possible to carry out an a priori computation of the location of the boundary between yielded and unyielded material.

## Non-viscometric flows

**A**, \(\boldsymbol{\Delta} \equiv \nabla {\bf v} + \nabla {\bf v}^{\rm T}\) is the rate of deformation tensor, and \(\boldsymbol{\gamma}\) is the strain tensor. Equation 5b is rarely employed in applications; it is conventional to assume that the modulus

*G*is infinite, in which case there can be no deformation; the condition in the unyielded region is then

Flows of Bingham fluids are frequently characterized by the dimensionless Bingham number, \(\mbox{Bn}=\tau _{\rm y} R/\eta _{\rm p} V\), where *R* and *V* are a characteristic length and velocity, respectively. *τ*_{y} is a characteristic stress at low deformation rates, while *η*_{p}*V*/*R* characterizes the viscous stress at very high rates, where the yield stress is irrelevant; hence, this dimensionless group has obvious applicability only in the limits of zero or infinity. The true viscosity scales as \(\eta \sim \eta _{\rm p} (1+\mbox{Bn})\), so a comparison of *τ*_{y} to *ηV*/*R* meaningfully reflects the relative importance of the yield stress and the stress from viscous deformation; this comparison suggests that the relevant group for scaling is \(\mbox{Bn}/(1 + \mbox{Bn})\) rather than Bn. It is also important to keep in mind that this type of scaling is the sole determinant of the flow only when there is a single characteristic length scale.

The location of the yield surface is unknown in general flows. It is straightforward to demonstrate from strictly kinematical arguments that continuity of the velocity and stress at the yield surface cannot be satisfied within the context of conventional lubrication theory, and asymptotic methods must be used with delicacy (Lipscomb and Denn 1984); the issues are addressed in recent work by Putz et al. (2009). Variational methods can be used, but these are best for bounding macroscopic quantities (the drag coefficient, for example) and less satisfactory for establishing the details of velocity and stress fields. The most common approach is to remove the discontinuity by regularization, which transforms the computational problem into a conventional one for a purely viscous liquid, and then to vary the regularization parameter to try to obtain convergence to the solution of the discontinuous problem. Three regularizations are in common use:

The Bingham model should be approached in all three formulations in the limit as *ε*→0. There are no universal convergence proofs, and numerical issues usually become important in numerical solutions before *ε* can become sufficiently small to establish convergence of the stress field. Thus, whenever regularization is employed there must be a small amount of flow (apparent creep) in what is interpreted as the unyielded region, since *ε* must always be non-zero. A large number of solutions can be found in Mitsoulis (2008). Convergence of the smooth regularizations is discussed in Frigaard and Nouar (2005). A detailed treatment of convergence for creeping flow around a sphere that is not discussed by Frigaard and Nouar can be found in Liu et al. (2002). The “gold standard” for checking the validity of calculations of flow around a sphere in a Bingham fluid is that of Beris et al. (1985). Some new results demonstrating convergence problems when the yield surface is discontinuous are described in Putz et al. (2009). Overall, the regularization methodology appears to be satisfactory in most instances and is incorporated into commercial CFD codes, although some authors (e.g., Putz and Frigaard 2010) have recently employed an augmented Lagrangian approach that allows more accurate determination of the unyielded regions. The two illustrative calculations shown subsequently in this paper employ the Bercovier–Engelman regularization.

*τ*

_{y}≥ (Δ

*ρ*)

*gR*/6, where Δ

*ρ*is the density difference and

*g*is the gravitational acceleration. (The 6 in the denominator is replaced by 8.33 for a falling drop at zero Reynolds number.) Other computational results for bubbles and drops in Bingham liquids using regularization methods may be found in Potapov et al. (2006) and Tsamopoulos et al. (2008).

## Measurement issues

Direct measurement of the rheological functions of a yield-stress liquid is fraught with difficulty. Extrapolation of measured shear stress data to zero shear rate in order to obtain the yield stress is unreliable. Furthermore, wall slip often occurs; slip may be evident from the shape of the measured rheological functions (e.g., Nguyen and Boger 1983), and it has been observed directly by painting markers on the free surface in a rotational viscometer (e.g., Kalyon 2005) and through magnetic resonance imaging (e.g., Bertola et al. 2003; Wassenius and Callaghan 2004). In contrast to the behavior of polymer melts, where slip may occur at high deformation rates (Denn 2001, 2008), slip in yield-stress liquids is most likely to occur at low rates. Roughened surfaces are routinely employed to minimize slip, but a common means to avoid slip while measuring the shear stress is to use a rotating vane. The vane does not provide a direct measurement of the shear stress and requires a theoretical treatment to extract the shear stress–shear rate relation.

Nguyen et al. (2006) reported the results of a study of the yield stress of 50% and 60% TiO_{2} suspensions carried out at six laboratories using a variety of measurement techniques, as well as flow curve fitting and extrapolation. The reported yield stresses differed by a factor of two, both from laboratory to laboratory and within laboratories that used multiple methods. The overall standard deviation was 49% of the mean for the 50% suspension and 40% of the mean for the 60% suspension. The laboratory-to-laboratory variability is easily explained by the fact that samples were prepared on site, and the preparation methods differed substantially, pointing to the significance of the microstructure in determining the yield stress. The very large deviations with different techniques within several laboratories point to the unreliability of some methods. Three laboratories used the vane method, and the reported standard deviation was approximately 10% of the mean for both concentrations.

*γ*is the shear strain. If failure occurs at a critical yield strain,

*γ*

_{y}, then the static value of the yield stress

*τ*

_{y}is

*G*

*γ*

_{y}. In an experiment at constant shear rate we would then have a dynamic apparent yield stress

*τ*

_{app,y}such that \(\tau _{\rm app,y} -\tau _{\rm y} =\eta \dot{\gamma}\). Nguyen and Boger’s data for the 66% red mud suspension are shown in Fig. 3, where it appears that the behavior expected for the Kelvin–Voigt material is roughly followed. What appears to be Kelvin–Voigt behavior in ketchup prior to yielding was recently reported by Benmouffok-Benbelkacem et al. (2010).

*τ*

_{y}, and what is the origin of its time dependence?

## Shear-dependent structure

^{7}M

_{w}polystyrene in a mixture of decalin and cyclohexyl bromide. At rest (A), the gel exhibits a percolated structure and exhibits a yield stress of about 5 Pa. Just after flow (B), the gel has broken up into individual flocs and there is no measurable yield stress.

## Predictive ability of classical models

There is a dearth of quantitative comparisons between the predictions of classical models like Eq. 5 and experiments on yield-stress liquids in non-trivial geometries, but two recent publications using particle imaging velocimetry to obtain detailed velocity data for yield-stress fluids moving past single spheres at very low Reynolds numbers are instructive. Both groups found fore-aft symmetry for Newtonian fluids, as expected, but both observed large deviations from fore-aft symmetry with the yield-stress liquids. Gueslin et al. (2006) studied a Laponite clay suspension, which is a thixotropic material (Abou et al. 2003); the settling velocity depended on the aging time, which in turn depended on the stress. (Aluminum and brass spheres, with buoyant forces differing by a factor of four, were used.) Putz et al. (2008) used a 0.08% Carbopol solution, which, unlike most observations on Carbopol, appeared to exhibit a degree of shear hysteresis. A constitutive description in the form of Eq. 5, or any generalization without a time dependence resulting from structural change, cannot describe these observations even qualitatively. (Topkavi et al. (2009) reported velocity profiles for the flow of a Carbopol gel past a stationary cylinder in which they, too, observed a deviation from fore-aft symmetry, but they provided no data with a Newtonian fluid for comparison.)

There are two other important experimental observations that are inconsistent with the classical rheological models. Shear banding is required for Bingham-like fluids when there is a stress gradient and the stress falls below the critical value in some region of the flow field; shear banding cannot occur in a uniform stress field, however. Magnetic resonance imaging studies of a bentonite clay suspension in a cone-and-plate instrument, where the shear stress should be uniform, do show shear bands above a critical shear rate (Møller et al. 2008). Finally, visual observation of the free surface of a 0.48% Laponite clay suspension in a parallel plate rheometer during transient stress development at a constant rate shows the onset of shear localization at the midplane, reminiscent of a slip plane in a solid, apparently indicating that the material yielded only locally (Pignon et al. 1996).

## Requirements for simulation

Thixotropy

Avalanche behavior

Loss of fore-aft symmetry in flow

Shear banding without a stress gradient

Shear localization

*λ*, varies between zero and unity, where unity corresponds to the equilibrium structure and zero to complete structural breakdown. (Some investigators prefer to permit 0 ≤

*λ*≤ ∞.) If the structure is anisotropic then we would require a structural tensor.

*a*and

*b*are typically taken to have integer values. If the rate of buildup is driven only by the distance from equilibrium then

*b*would be expected to be zero, as is usually done in polymer network theories. For a simple shear flow we would then have a kinetic equation of the form

*d*/

*dt*by the substantial derivative

*D*/

*Dt*and the shear rate \(\dot{\gamma}\) by \(\sqrt {\frac{1}{2}\mbox{II}}_{\boldsymbol{\Delta}}\). There is a steady-state structure in a steady shear flow:

*a*>

*b*to ensure the correct limits

*λ*→1, 0 as \(\dot{\gamma}\to 0\), ∞. (Equations of this type for the structural variable do not appear to admit the possibility of avalanches.) Many such models of structured liquids have been proposed, and they are reviewed in papers by Mujumdar et al. (2002) and Mewis and Wagner (2009); interestingly, the condition

*a*>

*b*is violated by some of these models. Some models include two mechanisms for structure buildup, a Brownian term that depends only on (1 −

*λ*) and a shear-dependent term. Pinder (1964) and Coussot et al. (2002b) assume a constant rate of buildup, which permits

*λ*to become infinite; the latter formulation does admit avalanches.

*G*=

*λG*

_{o}, where

*G*

_{o}is the equilibrium modulus. For a fractal structure the form would be \(G = \lambda^{n}G_{\rm o}\),

*n*> 1. It is likely that the yield stress would have the same dependence on

*λ*as the modulus. The dependence of the dissipative parameters on

*λ*would more than likely be taken initially to be a power law; in fact, most of the models reviewed by Mewis and Wagner take the plastic viscosity to be proportional to

*λ*. A minimal generalization of Eq. 5 would then be of the form

*G*→ ∞, does not introduce new conceptual issues, whereas any attempt to include the viscoelastic deformation of the unyielded region appears to be incompatible with conventional regularization approaches. Viscoelasticity after yielding can be accommodated by extending the structural parameter format or using a memory functional approach like that of White (1979), who proposed incorporating thixotropy through a memory-dependent yield stress (see also Suetsugu and White 1984). Separately, the issue of continuity of the velocity under all circumstances is a major unresolved conceptual issue.

## Concluding remarks

Many yield-stress liquids do not correspond to the classical description, which fails to take stress-dependent structure into account. Modification of the classical regularized continuum description to incorporate a structural variable is conceptually straightforward as long as one does not seek to include the deformation of the unyielded material, but it appears that new solution techniques would be required if one wished to incorporate a strain-based yield criterion and possible viscoelastic behavior of the unyielded solid. The elementary structure formulation described here does not appear to be able to describe avalanche behavior, and one of the most challenging unresolved issues is whether tangential velocity continuity is always appropriate in a yield-stress liquid.

## Notes

### References

- Abou B, Bonn D, Meunier J (2003) Nonlinear rheology of Laponite suspensions under an external drive. J Rheol 47:979–988CrossRefGoogle Scholar
- Barnes HA (1999) The yield stress—a review or ‘
*πα**ντ**α**ρε**ι*”—everything flows? J Non-Newton Fluid Mech 81:133–178CrossRefGoogle Scholar - Barnes HA, Walters K (1985) The yield stress myth. Rheol Acta 24:323–326CrossRefGoogle Scholar
- Benmouffok-Benbelkacem G, Caton F, Bavarian C, Skali-Lami S (2010) Non-linear viscoelasticity and temporal behavior of typical yield stress fluids: carbopol, xanthan, and ketchup. Rheol Acta 49:305–314CrossRefGoogle Scholar
- Bercovier M, Engelman M (1980) A finite-element method for incompressible non-Newtonian flows. J Comput Phys 36:313–326CrossRefGoogle Scholar
- Beris AN, Tsamopoulos JA, Brown RA, Armstrong RC (1985) Creeping flow around a sphere in a Bingham plastic. J Fluid Mech 158:219–244CrossRefGoogle Scholar
- Bertola V, Bertrand F, Tabuteau H, Bonn D, Coussot P (2003) Wall slip and yielding in pasty materials. J Rheol 47:1211–1226CrossRefGoogle Scholar
- Bonn D, Denn MM (2009) Yield stress fluids slowly yield to analysis. Science 324:1401–1402CrossRefGoogle Scholar
- Coussot P, Nguyen QD, Huynh HT, Bonn D (2002a) Avalanche behavior in yield stress fluids. Phys Rev Lett 88:175501CrossRefGoogle Scholar
- Coussot P, Nguyen QD, Huynh HT, Bonn D (2002b) Viscosity bifurcation in thixotropic, yielding fluids. J Rheol 46:573–589CrossRefGoogle Scholar
- Denn MM (2001) Extrusion instabilities and wall slip. Annu Rev Fluid Mech 33:265–287CrossRefGoogle Scholar
- Denn MM (2008) Polymer melt processing: foundations in fluid mechanics and heat transfer. Cambridge University Press, New YorkGoogle Scholar
- Divoux T, Tamarii D, Barentin C, Manneville S (2010) Transient shear banding in a simple yield stress fluid. Phys Rev Lett 104:208301CrossRefGoogle Scholar
- Frigaard IA, Nouar C (2005) On the usage of viscosity regularization methods for visco-plastic fluid flow computation. J Non-Newton Fluid Mech 127:1–26CrossRefGoogle Scholar
- Gartling DK, Phan-Thien N (1984) A numerical simulation of a plastic flow in a parallel plate plastometer. J Non-Newton Fluid Mech 14:347–360CrossRefGoogle Scholar
- Gueslin B, Talini L, Herzhaft B, Peysson Y, Allain C (2006) Flow induced by a sphere settling in an aging yield-stress fluid. Phys Fluids 18:103101CrossRefGoogle Scholar
- Kalyon DM (2005) Apparent slip and viscoplasticity of concentrated suspensions. J Rheol 49:621–640CrossRefGoogle Scholar
- Lipscomb GG, Denn MM (1984) Flow of Bingham fluids in complex geometries. J Non-Newton Fluid Mech 14:337–346CrossRefGoogle Scholar
- Liu BT, Muller SJ, Denn MM (2002) Convergence of a regularization method for creeping flow of a Bingham material about a rigid sphere. J Non-Newton Fluid Mech 102:179–191CrossRefGoogle Scholar
- Mewis J, Denn MM (1983) Constitutive equations based on the transient network concept. J Non-Newton Fluid Mech 12:69–83CrossRefGoogle Scholar
- Mewis J, Wagner NJ (2009) Thixotropy. Adv Colloid Interface Sci 147–148:214–227CrossRefGoogle Scholar
- Mitsoulis E (2008) Flows of viscoplastic fluids: models and computations. In: Binding DM, Hudson NE, Keunings R (eds) Rheology reviews 2007. British Society of Rheology, GlasgowGoogle Scholar
- Møller PCF, Rodts S, Michels MAJ, Bonn D (2008) Shear banding and yield stress in soft glassy materials. Phys Rev E 77:041507CrossRefGoogle Scholar
- Møller PCF, Fall A, Bonn D (2009a) Origin of apparent viscosity in yield stress fluids below yielding. Europhys Lett 87:38004CrossRefGoogle Scholar
- Møller PCF, Fall A, Chikkadi V, Derks D, Bonn D (2009b) An attempt to categorize yield stress fluid behaviour. Phil Trans Roy Soc A 367:5139–5155CrossRefGoogle Scholar
- Mujumdar A, Beris AN, Metzner AB (2002) Transient phenomena in thixotropic systems. J Non-Newton Fluid Mech 102:157–178CrossRefGoogle Scholar
- Nguyen QD, Boger DV (1983) Yield stress measurements for concentrated suspensions. J Rheol 27:321–349CrossRefGoogle Scholar
- Nguyen QD, Akroyd T, De Kee DC, Zhu LX (2006) Yield stress measurements in suspensions: an inter-laboratory study. Korea-Australia Rheol J 18:15–24Google Scholar
- Oldroyd JG (1947) A rational formulation of the equations of plastic flow for a Bingham solid. Proc Camb Philos Soc 43:100–105CrossRefGoogle Scholar
- Papanastasiou TC (1987) Flow of materials with yield stress. J Rheol 31:385–404CrossRefGoogle Scholar
- Pignon F, Magnin A, Piau JM (1996) Thixotropic colloidal suspensions and flow curves with minimum: identification of flow regimes and rheometric consequences. J Rheol 40:573–587CrossRefGoogle Scholar
- Pinder KL (1964) Time dependent rheology of tetrahydrofuran-hydrogen sulfide gas hydrate slurry. Can J Chem Eng 42:132–138CrossRefGoogle Scholar
- Potapov A, Spivak R, Lavrenteva OM, Nir A (2006) Motion and deformation of drops in Bingham fluid. Ind Eng Chem Res 45:6985–6995CrossRefGoogle Scholar
- Prager W (1961) Introduction to mechanics of continua. Ginn, BostonGoogle Scholar
- Putz A, Frigaard IA (2010) Creeping flow around particles in a Bingham fluid. J Non-Newton Fluid Mech 165:263–280CrossRefGoogle Scholar
- Putz AMV, Burghelea TI, Frigaard IA, Martinez DM (2008) Settling of an isolated spherical particle in a yield stress shear thinning fluid. Phys Fluids 20:033102CrossRefGoogle Scholar
- Putz A, Frigaard IA, Martinez DM (2009) On the lubrication paradox and the use of regularisation methods for lubrication flows. J Non-Newton Fluid Mech 163:62–77CrossRefGoogle Scholar
- Saramito P (2007) A new constitutive equation for elastoviscoplastic fluid flows. J Non-Newton Fluid Mech 145:1–14CrossRefGoogle Scholar
- Singh JP, Denn MM (2008) Interacting two-dimensional bubbles and droplets in a yield-stress fluid. Phys Fluids 20:040901CrossRefGoogle Scholar
- Suetsugu Y, White JL (1984) A theory of thixotropic plastic viscoelastic fluids with a time-dependent yield surface and its comparison to transient and steady state experiments on small particle filled polymer melts. J Non-Newton Fluid Mech 14:121–140CrossRefGoogle Scholar
- Topkavi DL, Jay P, Magnin A, Jossic L (2009) Experimental study of the very slow flow of a yield stress fluid around a circular cylinder. J Non-Newton Fluid Mech 164:35–44CrossRefGoogle Scholar
- Tsamopoulos J, Dimakopoulos Y, Chatzidai N, Karapetsas G, Pavlidis M (2008) Steady bubble rise and deformation in Newtonian and viscoplastic fluids and conditions for bubble entrapment. J Fluid Mech 601:123–164CrossRefGoogle Scholar
- Wassenius H, Callaghan PT (2004) Nanoscale NMR velocimetry by means of slowly diffusing tracer particles. J Magn Reson 169:250–256CrossRefGoogle Scholar
- White JL (1979) A plastic-viscoelastic constitutive equation to represent the rheological behavior of concentrated suspensions of small particles in polymer melts. J Non-Newton Fluid Mech 5:177–190CrossRefGoogle Scholar