Issues in the flow of yieldstress liquids
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Abstract
Yieldstress liquids are materials that are solid below a critical applied stress and flow like mobile liquids at higher stresses. Classical descriptions of yieldstress 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 yieldstress liquids.
Keywords
Yield stress Thixotropy Viscoplastic fluid Microstructure Bingham fluidIntroduction
Yieldstress 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 yieldstress 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.
Nonviscometric flows
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 nonzero. 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.
Measurement issues
Direct measurement of the rheological functions of a yieldstress 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 yieldstress 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 laboratorytolaboratory 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.
Sheardependent structure
Predictive ability of classical models
There is a dearth of quantitative comparisons between the predictions of classical models like Eq. 5 and experiments on yieldstress liquids in nontrivial geometries, but two recent publications using particle imaging velocimetry to obtain detailed velocity data for yieldstress fluids moving past single spheres at very low Reynolds numbers are instructive. Both groups found foreaft symmetry for Newtonian fluids, as expected, but both observed large deviations from foreaft symmetry with the yieldstress 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 foreaft 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 Binghamlike 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 coneandplate 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 foreaft symmetry in flow

Shear banding without a stress gradient

Shear localization
Concluding remarks
Many yieldstress liquids do not correspond to the classical description, which fails to take stressdependent 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 strainbased 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 yieldstress liquid.
Notes
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