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Mechanics Research Communications 53 (2013) 36– 42

Contents lists available at ScienceDirect

Mechanics Research Communications

jo ur nal ho me pag e: www.elsev ier .com/ locate /mechrescom

lastic and viscous effects on flow pattern of elasto-viscoplastic fluidsn a cavity

enato da R. Martinsa, Giovanni M. Furtadoa, Daniel D. dos Santosa, Sérgio Freya,ônica F. Naccacheb,∗, Paulo R. de Souza Mendesb

Department of Mechanical Engineering, Federal University of Rio Grande do Sul, Rua Sarmento Leite 425, Porto Alegre, RS 90050-170, BrazilDepartment of Mechanical Engineering, Pontifícia Universidade Católica-RJ, Rua Marquês de São Vicente 225, Rio de Janeiro, RJ 22453-900, Brazil

r t i c l e i n f o

rticle history:eceived 27 December 2012eceived in revised form 20 June 2013ccepted 27 July 2013vailable online 8 August 2013

eywords:

a b s t r a c t

Inertialess flows of elasto-viscoplastic fluids inside a leaky cavity are numerically analyzed using the finiteelement technique, with the goal of understanding the influence of both the elastic and viscous effectson the topology of the yield surfaces of an elasto-viscoplastic material. Assuming that the collapse of thematerial microstructure is instantaneous, a mechanical model is composed of the governing equationsof mass and momentum for incompressible fluids, and associated with a hyperbolic equation for theextra-stress tensor based on the Oldroyd-B model (Nassar et al., 2011). The main feature of the model is

iscoplastic fluidlasto-viscoplastic modelield stressid-driven cavityLS-type method

the consideration of the viscosity and relaxation time as functions of the strain rate to allow the shear-thinning of the viscosity and to restrict the elastic effects to the unyielded regions of the material. Thenumerical simulations are performed through a three-field Galerkin least-squares-type method in termsof the extra-stress tensor and the pressure and velocity fields. The results indicate that the material yieldsurfaces are strongly influenced by the interplay between the elastic and viscous effects, in accordancewith recent experimental visualization of elasto-viscoplastic flows.

© 2013 Elsevier Ltd. All rights reserved.

. Introduction

Numerical results of inertialess flows of elasto-viscoplastic flu-ds inside a lid-driven cavity are presented and discussed. The flowf a viscoplastic fluid is characterized by two distinct regions: annyielded region, where the stress is below the material yield limit,nd the deformation is extremely low, and an yielded region, whichppears where the stresses exceed such a limit. In many fluids, elas-icity is present in regions below the yield stress. Modeling thisype of behavior is not an easy task because there are few constitu-ive equations available in the literature that account for this typef material response. In addition, this behavior forms a complexathematical problem to be solved, because of the non-linearity

hat is inherent to the material and governing equations of theseuids.

Elasto-viscoplastic fluids presenting can be found in several pro-

esses in the oil, cosmetic, and food industries, as well as in otherpplications. Therefore, the proposition of models that lead to a bet-er understanding of the mechanical behavior of these fluids is of

∗ Corresponding author. Tel.: +55 21 3527 1174; fax: +55 21 3527 1165.E-mail addresses: naccache@puc-rio.br, mnaccache@me.com (M.F. Naccache).

093-6413/$ – see front matter © 2013 Elsevier Ltd. All rights reserved.ttp://dx.doi.org/10.1016/j.mechrescom.2013.07.012

extreme industrial relevance. The main goal of the present work isto evaluate the response of a recently proposed constitutive equa-tion for elasto-viscoplastic fluids (Nassar et al., 2011) and to analyzethe viscous and elastic contributions to the flow pattern within thelid-driven cavity.

The topology of the yielded and unyielded regions of viscoplasticflows have been numerically analyzed in several published works.Most of these works employed regularization strategies for theclassical Bingham and Herschel–Bulkley equations (e.g. Alexandrouet al., 2001a,b, 2003; De Besses et al., 2003; Burgos and Alexandrou,1999; Burgos et al., 1999; Mitsoulis et al., 2006, 1993; Naccacheand Barbosa, 2007; Hammad et al., 2001; Liu et al., 2002; Zisis andMitsoulis, 2002; Roquet and Saramito, 2003). All of these worksneglected inertia effects and use the regularized equations pro-posed by Papanastasiou (1987), Bercovier and Edelman (1980) orDe Souza Mendes and Dutra (2004).

The elastic effects on viscoplastic fluid flows have beenaddressed in some articles in the literature. Warren and Carter(1987) experimentally investigated the die swell flow of mixtures

of nitrocellulose and nitroglycerin. Sikorski et al. (2009) studiedthe velocity and shape of air bubbles rising through yield stressCarbopol dispersions. The form of the bubbles – with round headsand tapered tails – indicate that elasticity plays a significant role

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n the system. De Souza Mendes et al. (2007) performed numer-cal simulations and flow visualizations of Carbopol solutions fornertialess flows through an axisymmetric expansion followed by aontraction. The asymmetric behavior of the flow pattern obtainedxperimentally contrasted with the symmetric numerical patterns

obtained using a purely viscous viscoplastic function, leadingo the conclusion that elasticity cannot be neglected in the flow

odeling.Beverly and Tanner (1989) performed experiments and numeri-

al simulations of the die swell of elastic materials with yield stress.he numerical predictions and experimental observations are onlyn good agreement for low Weissenberg flows; for high Weis-enberg flows, a non-negligible deviation was verified. Saramito2007) presented a new three-dimensional model for elasto-iscoplasticity based on both the Bingham and the Oldroyd-Bodels and obtained good results in the study of simple flows.

ofou et al. (2008) experimentally studied the rheology of breadough employing two equations to model the material. The vis-oplastic Herschel-Bulkley equation was used to describe the yieldtress and the shear-thinning behavior of the flour dough, while the-BKZ model with a yield stress, was employed to represent thetress relaxation and the viscoelastic nature of the material. Nassart al. (2011) employed an elasto-viscoplastic model to simulate anxpansion–contraction axisymmetric flow, comparing the resultsith experimental data from the literature. The yielded regionsresented symmetric shapes in the purely viscous case; however,hen elasticity was incorporated in the modeling, the yielded

egions showed an asymmetric pattern, a trend that is in agree-ent with visualization results found in the literature. De Souzaendes (2009) improved the latter model using a microstructure

arameter that indicates the level of the structural breakage of theiscoplastic material as a function of the stress level applied to theaterial. In this equation, thixotropy is also taken into account, and

he relaxation time and viscosity are functions of the structural levelf the material. In a later work, De Souza Mendes (2011) extendedhis model to an Oldroyd-B-like fluid.

The mechanical model for the problem under analysis is com-osed of the usual governing equations of mass and momentum for

ncompressible fluids, coupled with the constitutive equation forlasto-viscoplastic fluids recently proposed by Nassar et al. (2011).he solution is obtained using a three-field Galerkin least-squaresGLS) finite element formulation (Behr et al., 1993), which accountsor the velocity, pressure and extra-stress fields as primal variables.

ith the addition of mesh-dependent terms in the flow governingquations, the formulation is able to successfully capture the elasticnd viscous effects present in the current model, making use of anqual-order combination of linear Lagrangian finite element inter-olations. The role of elasticity, yield stress, shear-thinning andinematics on the topology of the yielded and unyielded regionsre presented and discussed.

. The mechanical model

Consider � ⊂ R2 an open domain bounded by a regular polygo-al boundary. In accordance with Nassar et al. (2011), the inertialessteady flow of an elasto-viscoplastic material can be modeled by theollowing governing equations,

· u = 0 in � (1)

P − ∇� = 0 in � (2)

here u is velocity vector, P = p + �� is the modified pressure, and

= − ∇ � is the gravitational force per unit mass. The extra-stressensor � is described by a modified Oldroyd-B equation,

+ �1( �)� = 2�( �)[D + �2( �)D

]in � (3)

Communications 53 (2013) 36– 42 37

where D is the strain rate tensor, � =√

2trD2 is the magnitude ofthe strain rate tensor, and D and � are the upper-convected timederivative of tensors � and D, respectively:

� = (∇�)u − (∇u)� − �(∇u)T and D = (∇D)u − (∇u)D − D(∇u)T

(4)

The viscoplastic viscosity function is a modified version of themodel introduced in De Souza Mendes and Dutra (2004);

�( �) =[

1 − exp

(−�0

�y�

)](�y

�+ K �n−1

)+ �∞ (5)

where �0 is the low shear rate viscosity plateau, �y is the yield stress,K is the consistency index, n is the power-law index, and �∞ is thehigh shear rate viscosity plateau. The relaxation and retardationtimes of the fluid are given, respectively, by the following functions:

�1( �) = (�01 − �∞1 )exp

(−�0

�y�

)+ �∞1

�2( �) = (�02 − �∞2 )exp

(−�0

�y�

)+ �∞2 (6)

where �01 and �∞1 are the below-yield and above-yield relaxationtimes, respectively; and analogously �02 and �∞2 are, respectively,the below-yield and above-yield retardation times. In this workelasticity is only considered below the yield limit so that �∞1 =�∞2 = 0. According to Eqs. (5) and (6), in the limit � → 0 (unyieldedregions), �( �) → �0, �1( �) → �01 , and �2( �) → �02 – i.e. the modeltends to the classical Oldroyd-B model. On the other hand, when� > �0 (yielded regions), �1( �) → �∞1 , and �2( �) → �∞2 , and weobtain an Oldroyd-B type equation, with a variable (viscoplas-tic) viscosity function – note also that for the particular situationanalyzed in the present work, �∞1 = �∞2 = 0, the Generalized New-tonian Liquid model is recovered, and the fluid behaves as aninelastic viscoplastic fluid.

3. The numerical solution

To approximate the mechanical model described above, a multi-field Galerkin least-squares formulation, in terms of velocity,pressure and extra-stress, is employed. This formulation may beviewed as a direct extension of the model introduced by Behr et al.(1993) for constant viscosity fluids, to flows of elasto-viscoplasticmaterials. Proposed by Hughes et al. (1986) for Stokes flow, andlater extended by Franca and Frey (1992) for Navier–Stokes flow,this formulation has been successfully applied to many engineeringapplications.

This model overcomes the shortcomings present in classicalGalerkin approximations for fluid problems of interest, primar-ily, the need to satisfy functional compatibility conditions amongthe finite element subspaces of its primal variables. The modelproduces stable and meaningful approximations for fluid prob-lems of interest that are exempt from numerical pathologies, evenemploying equal-order combinations of Lagrangean finite elements(for details, see Franca and Frey (1992) and Behr et al. (1993) andreferences therein). Exploiting such a feature in all of the compu-tations shown in the upcoming numerical section, an equal-orderbi-linear (Q1) finite element interpolation is used. More details ofthe numerical formulation are found in Dos Santos et al. (2011).The geometry considered in this article, which is depicted in Fig. 1,

includes a square cavity of length L, with a tap wall subjectedto a horizontal velocity uc from left to right, along with no-slipand impermeability conditions (u = 0) on the remaining walls. Amesh independence procedure to evaluate the relative error of the

38 R.d.R. Martins et al. / Mechanics Research Communications 53 (2013) 36– 42

etmw

Yield surfaces are herein computed as the locus of point in which

Fig. 1. The geometry and boundary conditions.

xtra-stress magnitude is performed, as shown in Fig. 2. Althoughhe results are similar for the meshes investigated, the more refined

esh tested, with 104 Q1 finite elements and 10,201 nodal points,hich produces the smallest value of its non-dimensional element

Fig. 3. Yield surfaces: effect of the jump number for �∗01

= 250, n = 0.5 a

Fig. 2. Vertical profile of the stress modulus (� =√

1/2tr�2), for three differentmeshes.

size h∗Kmin

= hK /L equals to 2 × 10−2, is selected to guarantee moreaccurate approximations close to the cavity corners.

4. Dimensionless parameters

the magnitude of strain rate is below to the lowest strain rate valuefor which the viscosity equals the higher plateau �0 – namely, when

nd U* = 0.01, and (a) J = 500; (b) J = 1000; (c) J = 5000; (d) J =10,000.

R.d.R. Martins et al. / Mechanics Research Communications 53 (2013) 36– 42 39

0, J = 1

�bsi

x

wab

tf

∇

∇

�

�

�

Tp

Fig. 4. Yield surfaces: effect of the power-law index for �∗01

= 25

˙ ≤ �0 (see Dos Santos et al., 2011 for details). In order to analyzeoth viscous and elastic effects on the flow pattern, the dimen-ionless parameters that govern the problem are obtained with thentroduction of the following set of dimensionless quantities:

∗ = xL

; u∗ = u�1L

; �∗ = �

� 1; P∗ = P

�y; �∗ = �

�y; �∗ = �

�y/ �1(7)

here the superscript * indicates that the variable is dimensionless,nd �1 = (�y/K)1/n denotes the strain rate at which the viscosityegins to have a power-law behavior.

Using the above definitions, the dimensionless governing equa-ions and boundary conditions presented previously, assume theollowing form:

∗ · u∗ = 0 in �∗ (8)

∗P∗ − ∇∗�∗ = 0 in �∗ (9)

∗ + �∗1( �∗)�∗ = 2�∗( �∗)

[D∗ + �∗

2( �∗)D∗]

in� (10)

∗( �∗) = [1 − exp(−(J + 1) �∗)](

1�∗ + �∗(n−1)

)+ �∗

∞ (11)

∗1( �) = �∗

01exp (−(J + 1) �∗) �∗

2( �) = �∗02

exp (−(J + 1) �∗) (12)

herefore, from the above equations, the following governingarameters may be identified:

000 and U* = 0.1, and (a) n = 0.4; (b) n = 0.7; (c) n = 0.9; (d) n = 1.0.

U∗ = uc

�1L; �∗

01= �01 �1; �∗

02= �02 �1;

J ≡ �1 − �o

�o= �0

(�1/n−1

y

K1/n

)− 1 (13)

The dimensionless velocity U* arises from the nondimensionaliza-tion of the velocity boundary condition at the cavity lid, which givesthe flow intensity inside the cavity. The parameter J gives a rela-tive measure of the colapse of the material microstructure – see DeSouza Mendes et al. (2007) for more details.

Remark. A comment could be added concerning the scaling usedto obtain Eqs. (9)–(12). Unlike the usual nondimensionalization ofthe relaxation and retardation times, which scales the equationsby a characteristic strain rate of the flow, giving rise to the clas-sical Weissenberg number, in Eq. (13), the time is scaled usinga rheological quantity of the material, �1. As a consequence, thenon-dimensional elastic parameters, �∗

1 and �∗2, which arise, are not

sensitive to increasing flow kinematics. This feature is exploited inthe physical interpretations of the upcoming results.

5. Results and discussion

Numerical simulations of inertialess flows of elasto-viscoplasticfluids are undertaken to evaluate the elasticity, yield stress, shear-thinning, and kinematic effects on the flow pattern. This taskis achieved by varying the flow intensity, U*, the dimensionless

40 R.d.R. Martins et al. / Mechanics Research Communications 53 (2013) 36– 42

F1

= 20

rJ�

ownflW

ozte

ig. 5. Yield surfaces: effect of elasticity for n = 0.5, J = 1000 and U* = 0.01, and (a) �∗0

elaxation time, �∗01

, the power-law index, n, and the jump number,. All of the results are obtained for negligible retardation times,∗02

= 0. For the class of liquids investigated here, elasticity playsnly a non-negligible role in the unyielded regions (i.e. regionshere � < �0). In these unyielded regions, the local Weissenbergumber, classically defined as the ratio between a characteristicuid time and a local flow strain rate, is extremely low, becausei ≡ �01 � ≈ �/ �1 1 in these regions.Fig. 3 shows the influence of the jump number, J, on the topology

f the yield surfaces, for U* = 0.1, n = 0.5, and �∗01

= 250. The blackones in the figures represent the unyielded regions ( � < �0), whilehe white zones denote the yielded regions. The unyielded regionsxperience very low strain rates, as described by Eq. (5), where

0; (b) �∗01

= 800; (c) �∗01

= 1000; (d) �∗01

= 1300; (e) �∗01

= 1500; (f) �∗01

= 2000.

elasticity can play a relevant role. According to these figures, twodistinct unyielded regions exist inside the cavity: an almost stag-nant region adhered to the cavity bottom, and another region closeto the cavity’s lid. The latter region is associated with the flow recir-culation induced by the lid motion and consequently, is subjectedto higher strain rates than the quiescent regions close to the bot-tom of the cavity. From the definition of the jump number, J, Eq.(13), higher values of J indicate that the viscoplastic behavior of thefluid is more pronounced. Therefore, both unyielded regions grow

monotonically as J increases. For the lowest value of J, shown inFig. 3a, the material behaves almost as a shear-thinning fluid, withonly small portions of unyielded fluid being attached to both lowercorners of the cavity. Concerning their shapes, it is evident that the

R.d.R. Martins et al. / Mechanics Research Communications 53 (2013) 36– 42 41

Fig. 6. Yield surfaces: effect of flow intensity for �∗ = 250, n = 0.5 and J = 1000: (a) U* = 0.001; (b) U* = 0.005; (c) U* = 0.10; (d) U* = 0.20; (e) U* = 0.40; (f) U* = 1.00.

yboe

iustTc

01

ield surfaces are asymmetric. This deviation from the symmetricehavior of a purely viscoplastic model is caused by the presencef elasticity within the unyielded regions, a phenomenon that isxplained in more detail in Fig. 5.

The effect of the power-law index on the flow pattern is shownn Fig. 4, for U* = 0.1, J = 1000, and �∗

01= 250. While the stagnant

nyielded regions at the bottom of the cavity are almost insen-

itive to an increase in n, the unyielded regions associated withhe flow recirculation are visibly reduced by increasing values of n.his trend is related to the shear-thinning of the viscoplastic vis-osity function, which causes higher stress levels as n increases.

As the unyielded zones at the cavity bottom are quiescent,presenting an almost null strain rate, the shear-thinning of viscos-ity could not visibly alter the morphology of these regions in thecurrent simulations.

The effect of elasticity is addressed in Fig. 5. As in the previousset of figures, the presence of elasticity disturbs the usual symmet-ric pattern of the yield surfaces prescribed by purely viscoplastic

models. The yield surfaces become increasingly asymmetric withincreasing elasticity levels, an effect that acts on both the upperand lower unyielded regions. This result is consistent with relatedexperimental works (e.g. De Souza Mendes et al., 2007), which

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Tcvtrstivczdsbwti

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2 R.d.R. Martins et al. / Mechanics Re

how the same asymmetric behavior in inertialess flows of vis-oplastic fluids, a trend that is explained by elasticity. Anothernfluence of the elasticity on the morphology of unyielded regionss related to the regions’ sizes. As illustrated in these figures, anncrease in the elastic levels significantly decreases the regions’izes, another behavior that is associated with the higher stressevels generated by the viscoelastic nature of the stress field.

The influence of the flow intensity on the flow pattern is pre-ented in Fig. 6. According to the definition given by Eq. (13), U* isorrelated with the inverse of the Herschel-Bulkley number,

∗ =(

KUn

�0Ln

)1/n

= HB−1/n (14)

herefore, a monotonic decrease of the unyielded regions inside theavity is to be expected with an increase in U*. This trend is clearlyerified in Fig. 6, both in the upper and lower unyielded regions ofhe cavity. Located in the main vortex of cavity, the upper unyieldedegion is surrounded by regions that are subjected to increasingtress levels as U* increases. The influence of the flow intensity inhe unyielded regions at the bottom of the cavity is less than thenfluence on the regions of the upper cavity. Even for the largestalue of U* the unyielded regions are still present, near the cavityorners. After a critical value of U*, between U* = 0.20 and 0.40, theseones diverge, with the corner regions separating by an increasingistance as U* increases. This divergence indicates that the highertress levels associated with the main vortex are reaching the cavityottom. The kinematic effects seem to overlap the elastic effectsith increasing values of U*, because the asymmetric shapes of both

he upper and lower unyielded regions tend to disappear as the flowntensity increases.

Finally, the results are compared with the work by Dos Santost al. (2011), where the numerical simulation of a yield stress fluids performed in the same geometry, to analyze the effect of iner-ia in the flow pattern. Both results show that the flow pattern isymmetric if there is no inertia and no elasticity. Both inertia andlasticity lead to an asymmetric flow pattern, but the asymmetryaused by inertia occurs in the opposite direction than that causedy elasticity.

. Final remarks

Numerical simulations of inertialess flows of elasto-viscoplasticuids have been performed, using the constitutive model

ntroduced by Nassar et al. (2011). The constitutive equation isased on the Oldroyd-B model, as modified to account for theependence of the strain rate on the viscosity, relaxation andetardation times. This capability of the model allows the elas-icity to be considered only within the unyielded regions of theow. The mechanical model was approximated by a four-fieldalerkin least-squares method in extra-stress, pressure and veloc-

ty, which afforded the use of an equal-order of linear Lagrangiannterpolations. The numerical results indicate that the size andocation of the yield surfaces are strongly influenced by the relax-tion time (elasticity), shear-thinning, and viscoplastic nature ofhe material. The elasticity was found to be crucial to characterizehe mechanical response of certain viscoplastic materials better.he results demonstrated a qualitatively good agreement with theelated elasto-viscoplastic experimental results available in theiterature.

cknowledgments

The authors acknowledge the brazilian funding agencies CAPESnd CNPq for financial support.

h Communications 53 (2013) 36– 42

References

Alexandrou, A.N., McGilvreay, T.M., Burgos, G., 2001a. Steady Herschel-Bulkley fluidflow in three-dimensional expansions. Journal of Non-Newtonian Fluid Mechan-ics 100, 77–96.

Alexandrou, A.N., Duc, E., Entov, V., 2001b. Inertial, viscous and yield stress effects inBingham fluid filling of a 2-D cavity. Journal of Non-Newtonian Fluid Mechanics96, 383–403.

Alexandrou, A.N., LeMenn, P., Georgiou, G., Entov, V., 2003. Flow instabilitiesof Herschel-Bulkley fluids. Journal of Non-Newtonian Fluid Mechanics 116,19–32.

Behr, M., Franca, L.P., Tezduyar, T.E., 1993. Stabilized Finite Element Methods forthe Velocity-Pressure-Stress Formulation of Incompressible Flows. ComputerMethods in Applied Mechanics and Engineering 104, 31–48.

Bercovier, M., Engelman, M., 1980. A finite element method for incom-pressible non Newtonian flows. Journal of Computational Physics 36,313–326.

Beverly, C.R., Tanner, R.I., 1989. Numerical analysis of extrudate swellin viscoelastic materials with yield stress. Journal of Rheology 33/6,989–1009.

Burgos, G.R., Alexandrou, A.N., 1999. Flow development of Herschel-Bulkley fluidsin a sudden 3-D expansion. Journal of Rheology 43/3, 485–489.

Burgos, G.R., Alexandrou, A.N., Entov, V., 1999. On the determination ofyield surfaces in Herschel-Bulkley fluids. Journal of Rheology 43/3,463–483.

De Souza Mendes, 2009. Modeling the thixotropic behavior of structured fluids.Journal of Non-Newtonian Fluid Mechanics 164, 66–75.

De Souza Mendes, 2011. Thixotropic elasto-viscoplastic model for structured fluids.Soft Matter 7, 2471–2483.

De Besses, D.D., Magnin, A., Jay, P., 2003. Viscoplastic flow around a cylin-der in an infinite medium. Journal of Non-Newtonian Fluid Mechanics 115,27–49.

De Souza Mendes, P.R., Dutra, E.S.S., 2004. Viscosity function for yield-stress liquids.Applied Rheology 14, 296–302.

De Souza Mendes, P.R., Naccache, M.F., Varges, P.R., Marchesini, F.H., 2007. Flow ofviscoplastic liquids through axisymmetric expansions-contractions. Journal ofNon-Newtonian Fluid Mechanics 142, 207–217.

Dos Santos, D.D., Frey, S., Naccache, M.F., de Souza Mendes, P.R., 2011. Numericalapproximations for flow of viscoplastic fluids in a lid-driven cavity. Journal ofNon-Newtonian Fluid Mechanics 166, 667–679.

Franca, L.P., Frey, S., 1992. Stabilized Finite Element Methods: II. The Incompress-ible Navier–Stokes Equations. Computer Methods in Applied Mechanics andEngineering 99, 209–233.

Hammad, K.J., Vradis, G.C., Otugen, M.V., 2001. Laminar flow of a Herschel-Bulkleyfluid over an axisymmetric sudden expansion. Journal of Fluids Engineering 123,588–594.

Hughes, T.J.R., Franca, L.P., Balusters, M., 1986. A new finite element formulation forcomputational fluid dynamics: V. Circumventing the Babuska-Brezzi condition:A stable Petrov-Galerkin formulation of the Stokes problem accommodatingequal-order interpolations. Computer Methods in Applied Mechanics and Engi-neering 59, 85–99.

Liu, B.T., Muller, S.J., Denn, M.M., 2002. Convergence of regularization method forcreeping flow of a Bingham material about a rigid sphere. Journal of Non-Newtonian Fluid Mechanics 102, 179–191.

Mitsoulis, E., Abdali, S.S., Markatos, N.C., 1993. Flow simulation of Herschel-Bulkleyfluids through extrusion dies. Journal of Non-Newtonian Fluid Mechanics 71,147–160.

Mitsoulis, E., Marangoudakis, S., Spyratos, M., Zizis, Th, Malamataris, N.A., 2006.Pressure-driven flows of Bingham plastics over a square cavity. Journal of FluidsEngineering 128, 993–1003.

Naccache, M.F., Barbosa, R.S., 2007. Creeping flow of viscoplastic materials througha planar expansion followed by a contraction. Mechanics Research Communi-cations 34, 423–431.

Nassar, B., de Souza Mendes, M.F., Naccache, 2011. Flow of elasto-viscoplastic liquidsthrough an axisymmetric expansion-contraction. Journal of Non-NewtonianFluid Mechanics 166, 386–394.

Papanastasiou, T.C., 1987. Flows of materials with yield. Journal of Rheology 31,385–404.

Roquet, N., Saramito, P., 2003. An adaptive finite element method for Bingham fluidflows around a cylinder. Computer Methods in Applied Mechanics and Engi-neering 192, 3317–3341.

Saramito, P., 2007. A new constitutive equation for elastoviscoplastic fluid flows.Journal of Non-Newtonian Fluid Mechanics 145, 1–14.

Sikorski, D., Tabuteau, H., de Bruyn, J.R., 2009. Motion and shape of bubbles risingthrough a yield-stress fluid. Journal of Non-Newtonian Fluid Mechanics 159,10–16.

Sofou, S., Muliawan, E.B., Hatzikiriakos, S.G., Mitsoulis, E., 2008. Rheological char-acterization and constitutive modeling of bread dough. Rheologica Acta 47,369–381.

Warren, R.C., Carter, R.E., 1987. Extrusion stresses, die swell, and vis-cous heating effects in double-based propellants. Journal of Rheology 31,151–173.

Zisis, Th., Mitsoulis, E., 2002. Viscoplastic flow around a cylinder kept between par-allel plates. Journal of Non-Newtonian Fluid Mechanics 105, 1–20.