J. Non-Newtonian Fluid Mech. 135 (2006) 1–7

Identification of Bingham fluid flow parameters using a simple squeeze test

Nicolas Roussel a,∗, Christophe Lanos b, Zahia Toutou a

a Division B.C.C., Laboratoire central des Ponts et Chaussees, 58, boulevard Lefebvre, 75732 Paris Cedex 15, Franceb Laboratoire Genie Civil Genie Mecanique, I.N.S.A., 20 avenue des Buttes de Coesmes, CS 14315, 35043 Rennes Cedex, France

Received 17 January 2003; received in revised form 7 November 2005; accepted 4 December 2005

Abstract

Squeeze flow of a Bingham fluid between two parallel plates is studied by means of a variational approach [K.J. Zwick, P.S. Ayyaswamy, I.M.Cohen, Variational analysis of the squeezing flow of a yield stress fluid, J. Non-Newtonian Fluid Mech., 63 (1996) 179–199.]. The trial velocityfield consists of a central region of pure extensional flow, mid-way between the plates, and sheared regions adjacent to the plates. The width of theregion of extensional flow is chosen to minimise an energy dissipation functional. The analysis assumes that the separation of the plates is smallcompared to the plate radius, and the predictions are compared with those of previously published analyses.© 2005 Elsevier B.V. All rights reserved.

K

1

ttpqfibaa

ttifpa

2

tn

0d

eywords: Squeeze test; Bingham fluid; Squeezing flow paradox; Variational approach; Extensional flow

. Introduction

Squeeze tests are often used in practice as a straightforwardechnique to determine the flow properties of highly concen-rated suspensions such as concrete, molten polymers, ceramicastes, etc. Most of those materials behave as highly viscous oruasi-plastic fluids and can be described as Bingham fluids as arst approximation. The maximum particle size and the slidingehaviour at the interface of such materials prevent the use ofny traditional viscometric tests (coaxial cylinders or cone-platepparatus for instance).

In this work, several assumptions on the flow pattern allowshe generation of a kinematically admissible velocity field inerms of a parameter β, in respect to which a functional is min-mised, using the fact that the upper and lower bounds of thisunctional are converging for small disc separation. The com-ression load is calculated from the dissipation inside the samplend the obtained solution is compared with the literature.

. Description of the squeeze test

plates, without any rotation. The upper disc can be displaced atcontrolled constant velocity, while the lower one remains sta-tionary. The squeezing of the sample between the two platesinduces a radial and axial flow. R is the radius of the plates,h the height of the sample, F the compression load applied onthe plates and c is the compression speed (Fig. 1). The testedmaterial fills permanently the area between the plates. In thefollowing, we will speak of large and small h/R ratios. However,it has to be noted that both small and large h/R ratios are smallerthan 1. The assumption that the thickness of the sample is smallcompared to its radius is always assumed to be fulfilled.

The roughness of the plates can be also a test parameter. Useof rough plates imposes a sticking flow; the no-slip boundarycondition is assumed to be fulfilled. On the other hand, in the caseof smooth plates, the material can slip along the solid surface.Only rough plates and sticking flows are considered in this work.

3. Behaviour law

The yield criterion used in the Bingham behaviour law inthis study is the von Mises yield criterion [1]. It writes in three

The squeeze test is a simple compression test (plastometerest) carried out on cylindrical samples with reduced slender-ess. This apparatus consists in two coaxial circular parallel

dimensions and in the case of an axi-symmetric flow:When

12 (σ(d)

rr )2 + 1

2 (σ(d)θθ )

2 + 12 (σ(d)

zz )2 + (σrz)2 < K2

i (1)

t

∗ Corresponding author.

E-mail address: nicolas.roussel@lcpc.fr (N. Roussel).

377-0257/$ – see front matter © 2005 Elsevier B.V. All rights reserved.oi:10.1016/j.jnnfm.2005.12.001

here is no velocity gradient in the sample.

2 N. Roussel et al. / J. Non-Newtonian Fluid Mech. 135 (2006) 1–7

Nomenclature

c compression speedDij strain rate tensorF compression forceF* reduced force, F* = −Fh/πR3

g global Bingham number, g = ηcR/Kih2

h sample heightI2 second invariant of the strain rate tensorJv functional associated to the flowKi plastic yield valueR plates radiusV sample volumez0 boundary between the two flowing zones

z0 = βh/2

Greek symbolsβ non-dimensional parameter describing the

position of the boundary between the two flowingzones

Γ energy function linked to the behaviour law(cf. Eq. (7))

η plastic viscosityσ

(d)ij stress tensor deviatoric part

When

12 (σ(d)

rr )2 + 1

2 (σ(d)θθ )

2 + 12 (σ(d)

zz )2 + (σrz)2 ≥ K2

i (2)

there is a velocity gradient. Here Ki is the yield stress. σ(d)rr , σ(d)

θθ ,σ(d)

zz and σrz are the components of the stress tensor deviatoricpart.

If (2) is true, the following general form of the behaviour lawis then used, under the assumption of an incompressible body.It links the stress tensor deviatoric part and the strain rate tensorvia a scalar function of the second invariant of the rate of straintensor:

σ(d)ij = 2f (I2)Dij (3)

where in the case of a Bingham viscoplastic fluid P,

2f (I2) = Ki√I2 + 2η

(4)

here Dij is the rate of strain tensor and I2 the second invariant ofthe rate of strain tensor. η is the plastic viscosity.

Lipscomb and Denn [2] argue that, in general, complex flowscannot admit unyielded zones and that, in the present case, thewhole of the fluid has to yield. It has been shown recently (Smyr-naios and Tsamopoulos [3], Matsoukas and Mitsoulis [4]) thatsome unyielded zones can exist around stagnation points. In thepresent work, the von Mises yield criterion is assumed to be ful-filled everywhere in the sample. As a consequence, there is noplug flow and the unyielded zones are neglected. We will comeback to this point in Section 6.

4. Three-dimensional yield criterion

As noted by Wilson [5], there is an immediate difficulty con-cerning the existence of yield surfaces in the material itself,which has caused some disagreement in the literature. A com-mon assumption made while studying a squeezing flow is thath/R is small. This led numerous authors (Scott [6] for powerlaw fluids, Covey and Stanmore [7] for viscoplastic fluids orSherwood et al. [8] in his earlier work on non-homogeneousyctc

|

|

w

bBtatytoftvpaabaiBcc

Fig. 1. Squeeze test geometry.

ield stress fluids) to make the usual approximations of lubri-ation theory. The predominant stress tensor component is thenhe shear stress σrz. This approach is mono-dimensional and theonstitutive behaviour law becomes:

σrz| > Ki → σrz = 2η∂Vr

∂z+ Ki (5a)

σrz| < Ki → ∂Vr

∂z= 0 (5b)

here Vr is the radial flow velocity.This simplification eases the analytic treatment of the flow

ut, as a direct consequence, a paradox appears on the mid-plane.y symmetry, the shear stress falls to zero at z = 0. According to

he previous behaviour law, the plastic criterion is not fulfillednd the fluid must move as a rigid plug. But, at the same time,he gap between the plates is being narrowed and the fluid has toield even on the mid-plane. The plug must therefore deform andhe solution is inconsistent. It should be noted that, even if thebtained flow field is kinematically inconsistent, the expressionor the compression force in terms of the geometric configura-ion of the test agrees well with the experimental values on aariety of yield stress fluids. In fact, while integrating the com-ression force, any error in the calculated flow field is averagednd the overall energy dissipation stays in reasonable bounds. Asn alternative, Wilson [5] approximates the behaviour law by aiviscosity equation. The Bingham model is then approached aslimit of the biviscosity model. However, this alternative method

s only a way to avoid the paradox in the particular case of theingham fluid. In the more general case, the inconsistency in thealculated velocity fields is explained by the fact that the lubri-ation analysis only allows a one-dimensional yield criterion to

N. Roussel et al. / J. Non-Newtonian Fluid Mech. 135 (2006) 1–7 3

be specified. Adams et al. [9] can be quoted: “A comprehensiveyield criterion is one which is based upon a combination of allof the acting components of the stress”. Let us come back nowto the squeeze flow paradox but, this time, let us use the three-dimensional expression of the von Mises yield criterion (2). Thesymmetry imposes that there is a zone around the mid-planewhere the shear stress falls to zero. Eq. (2) then becomes:

If

12 (σ(d)

rr )2 + 1

2 (σ(d)θθ )

2 + 12 (σ(d)

zz )2 ≥ K2

i

there is a velocity gradient.The previously neglected stress tensor components σ(d)

rr , σ(d)θθ

and σ(d)zz allow the criterion to be fulfilled even on the axis. There

is no rigid plug. The fluid is yielding everywhere in the sampleas assumed in the present study. More recently, Sherwood andDurban [10] have carried out a rigorous three-dimensional stressanalysis for generalised non-Newtonian fluids, including a Bing-ham fluid, under slip wall boundary conditions. The present workalso follows a three-dimensional approach to a Bingham fluidsqueeze flow but under no-slip wall boundary conditions. A vari-ational principle is used to obtain an appropriate approximatedsolution.

5. Variational analysis

etwantfl

iap

O

O

I

a

∇

fvoud

J

where Γ is an energy function defined by the following relation:

σ(d)ij = ∂Γ

∂Dij

(8)

In the squeeze flow geometry studied here, St is the free surfaceat the periphery of the squeezing test. Boundary conditions aret = 0 on St. For a Bingham fluid, the Γ function can be writtenusing the behaviour laws (3) and (4) in terms of the secondinvariant of the strain rates tensor:

Γ = 2ηI2 + 2Ki

√I2 (9)

Zwick et al. [14] found that the lower and upper bounds of thefunctional Jv associated with the flow of an Herschel–Bulkleyfluid converge (minimum principle for strain and maximum prin-ciple for stress) for large values of a non-dimensional parameterthat reduces in the case of a Bingham fluid to:

h = 3Fh

2πKiR3 (10)

At the end of the test, for small h/R, the strain rate increases.The viscous stress terms dominate the yield stress terms and themagnitude of Eq. (10) becomes increasingly great. As the twobounds of the functional converge for small h/R, it is not of anyinterest from a practical point of view to determine both of themin order to analyse the flow when the film thickness is smallcfittfpfii

6

•

•

ttanaflsctwv

Using a variational principle which focuses on the rate ofnergy dissipation within the control volume bounds a func-ional, which is related to the energy of the flow. This principleas initially applied by Prager [11] to slow Bingham fluids flow

nd was then extended by Johnson [12] to creeping flows ofon-Newtonian fluids. Yoshioka and Adachi [13] showed thathis principle was applicable for fully yielded flow of yield stressuids.

Let us consider a control volume V, with a surface S dividednto a velocity boundary Sv and a traction boundary St. A bound-ry value problem for the steady, inertialess flow of an incom-ressible non-Newtonian fluid can be written [14]:

n Sv, U = Vsv (6a)

n St, n · (σ(d)ij − pI) = t (6b)

n V, ∇ · (σ(d)ij − pI) − f = 0 (6c)

nd

· U = 0 (6d)

is the body force per unit volume, p the pressure, U the flowelocity, Vsv the velocity imposed on Sv, t the imposed tractionn St and n is the unit outward normal on S. There exists strictpper and lower bounds for the following functional, which isirectly linked to the total volumetric dissipation rate:

v =∫

V

Γ dV −∫

St

(t · U)dS (7)

ompared to the radius of the plates. Knowing the difficulty ofnding a stress field that satisfies both the stress boundaries and

he equilibrium equations, only the upper bound of the func-ional is calculated in the present study. The minimum principleor strain is used to obtain an appropriate approximation of theroblem solution. A necessary step is then to find a velocityeld that satisfies the boundary conditions and the equation of

ncompressibility.

. Bingham fluid flow

The volume between the plates can be divided in two zones:

Zone 1: the plastic criterion is reached and overcome. Shearflow occurs in this zone.Zone 2: the plastic criterion is reached and the shear stressequals zero. This zone must include the mid-plane. The flowin this zone is extensional.

Analysing the flow pattern, zone 1 appears to be located closeo the plates where the shear stress is higher. It can be noted herehat, although plug flow cannot exist on the plane of symmetrys shown in Section 4, some unyielded zones such as the onesumerically predicted by Smyrnaios and Tsamopoulos [3] mayppear. These zones exist around the two stagnation points ofow at the center of disks and cover a fraction of the axis ofymmetry. In order to simplify our chosen velocity field, wehoose here to neglect these unyielded zones and assume thathe von Mises yield criterion is fulfilled everywhere. However,e will come back on this point in Section 9 to show that theelocity field obtained by Smyrnaios and Tsamopoulos taking

4 N. Roussel et al. / J. Non-Newtonian Fluid Mech. 135 (2006) 1–7

into account these unyielded zones is probably closer to the realvelocity field. In the central zones, a predominantly extensionalflow is noted by Lanos [15].

In zone 2, the following boundary conditions and equationsmust be fulfilled.

Vz(r, z = 0) = 0 (11)

The mass balance equation under the assumption of an incom-pressible body becomes;

∂Vz

∂z= −∂Vr

∂r− Vr

r(12)

and the shear stress and the associated strain rate equal zero:

1

2

(∂Vz

∂r+ ∂Vr

∂z

)= 0 (13)

One trivial solution for this type of extensional flow is, Sherwoodand Durban [10], Lanos [15], Roussel [16]:⎛⎜⎝

Vr(r, z)

Vθ(r, z)

Vz(r, z)

⎞⎟⎠ =

⎛⎜⎝

A1r

0

−2A1z

⎞⎟⎠ (14)

A1 is a parameter depending on the test geometry, the plasticviscosity and the plastic yield value.

In zone 1, close to the plates, the following boundary condi-t

V

V

ao[g⎛⎜⎝

ApflF

cmt

Fig. 2. Zones separation in the proposed velocity field.

In zone 1, for 0 ≤ z ≤ βh/2,

Vr(r, z) = 3cr

2h(β + 2)(18a)

Vz(r, z) = − 3cz

h(β + 2)(18b)

In zone 2, for βh/2 ≤ z ≤ h/2,

Vr(r, z) = −3cr(4z2 − 4βhz + h2(2β − 1))

2h3(β3 − 2β + 2)(19a)

Vz(r, z) = c(8z3 − 12βhz2 + 6h2z(2β − 1) − β3h3)

2h3(β3 − 2β + 2)(19b)

This velocity field is of course not the exact solution of theproblem. It is a kinematically admissible velocity field lessinconsistent than the ones obtained using a one-dimensionalanalysis but it is still an approximation of the exact flow field.This speed field is however similar to the one obtained numer-ically by Adams et al. [9]. It presents a central zone with a flatradial velocity profile surrounded by two zones where the dom-inant strain rate is the component Drz.

7. Minimum principle for strain

At this stage, the parameter β is still unknown and a familyofTSwπ

J

Wtcb

J

Tn

ions and equations must be fulfilled:

r

(r, z = h

2

)= 0 (15)

z

(r, z = h

2

)= c

2(16)

The mass balance equation is identical to Eq. (12). By annalogy with a simple Newtonian fluid flow, an approximationf the solution of the mass balance equation is given by Lanos15] for a Bingham fluid. This solution can be considered as aeneralised Stefan [17] solution:

Vr(r, z)

Vθ(r, z)

Vz(r, z)

⎞⎟⎠

=

⎛⎜⎜⎜⎜⎝

(A2

(z2 − h2

4

)+ A3

(z − h

2

))r

0

−A2

6(h3 − 3h2z + 4z3) + A3

4(h2 − 4hz + 4z2) − c

2

⎞⎟⎟⎟⎟⎠

(17)

2 and A3 are parameters that depend on the test geometry, thelastic viscosity and the plastic yield value. Assuming such aow pattern, zone 1 and zone 2 are then separated as shown inig. 2.

At the boundary z = z0 = βh/2 with 0 ≤ β ≤ 1, the continuityondition on axial and radial velocities and on their derivativesust be fulfilled. By solving the continuity condition equations,

he velocity field then reduces to:

f velocity fields may be generated by varying β. Amongst thisamily, one velocity field is the closest to the actual flow field.his particular flow field minimises the functional defined inection 5. β is then the free parameter with respect to whiche will minimise the functional Jv calculated over the volumeR2h:

v =∫

zone 12ηI2 + 2Ki

√I2 dV +

∫zone 2

2ηI2 + 2Ki

√I2 dV

(20)

ithout going through all the minutiae, the value of the func-ional is expressed in terms of β. If it is assumed that h � R, then,onserving only the predominant terms R4 and R3, the functionalecomes:

v =(

3ηcR2

h3(1 − β)(β + 2)2 + 2KiR

h(β + 2)

)πcR2 (21)

his functional can be expressed in terms of the global Binghamumber g = ηcR/Kih2 which compares the magnitude of viscous

N. Roussel et al. / J. Non-Newtonian Fluid Mech. 135 (2006) 1–7 5

and plastic forces [7]. The functional becomes:

Jv =(

3g

2(1 − β)(β + 2)2 + 1

(β + 2)

)2πcKiR

3

h(22)

Differentiating (22), the β value that minimises Jv is one of thethree roots of the polynomial equation:

2β3 − 3β(3g + 2) + 4 = 0 (23)

As β must fulfil the geometrical condition 0 ≤ β ≤ 1, the solutionof the minimisation problem is

β =√

2√

(3g + 2)sin

×⎛⎝1

3tan−1

⎛⎝2

√2(3g + 2)3/2

√g

(3g+2)3

3g√

(3g2 + 6g + 4)

⎞⎠⎞⎠ (24)

The perfect plastic fluid is a particular case of the Bingham fluidwith η → 0. In this case, the Bingham number g → 0 and β → 1.The shearing flow zone disappears and the flow in the sampleis entirely extensional and sliding at the interface with a shearstress equal to the yield stress. On the other hand, the Newtonianviscous fluid is another particular case of the Bingham fluidwith Ki → 0. In this case, the Bingham number g → +∝ andβ → 0. The extensional flow zone disappears and the velocityfififl

8

tmitphttzutc

bi

−

On

F

Fig. 3. Fractures in the outcoming sample (clay paste).

or, using Eq. (23) and the definition of the Bingham number:

F = −2πKiR2(3

√3β2h + βR + 2R)

3βh(β + 2)(27)

The load also depends on the yield stress and on the plasticviscosity via the β parameter.

In the case of a perfect plastic fluid, β = 1 and η = 0, the forceexpression then becomes:

F = −2πKiR2

√3

− 2πKiR3

3h(28)

The first term is linked to the dissipation due to the extensionalflow whereas the second term is linked to the dissipation associ-ated with the shear at the plate surface. The first term is generallyignored by previous authors (Covey [18] or Lanos [19]) but istaken in account by Adams et al. [9]. For perfect plastic fluidflow, this term corresponds to p(r = R, z) = 0 and σ(d)

rr = 3−1/2Ki

(effect of the traction in the outcoming sample). The load doesnot depend on the compression speed. This was expected froman analysis based upon plasticity theory. In the case of a plasticfluid, Eq. (28) can be compared to the one obtained by Sher-wood and Durban [10]. This expression is obtained imposing anaverage boundary condition:∫ h/2

F

ws(rs

eld calculated from (19a) and (19b) equals the theoretical floweld originally given by Stefan [17] for a Newtonian viscousuid.

. Compression load

Once the flow field is obtained, the constitutive law allowshe identification of the deviatoric part of the stress tensor. The

omentum equation gives the pressure gradient at the platenterface. The compression load can then be integrated fromhe pressure on the plates which is itself integrated from theressure gradient. But, as it will be shown in Section 9, thereas been disagreement about the stress boundary conditions athe edge of sample (r = R). As a consequence, the integration ofhe pressure gradient is not obvious. Most of the time, p(r = R,= h/2) = 0 is assumed but the pressure for r = R and z ≤ h/2 isnknown. The stress in the outcoming sample that generateshe fractures shown in Fig. 3 is not taken into account in thealculation.

In order to avoid any additional assumptions on the stressoundary conditions, the value of the force acting on the platess calculated from the energy dissipation in the volume πR2h.

cF =∫

πR2h

σ(d)ij Dij dv =

∫πR2h

4η(I2)I2 dv (25)

nce again, it is assumed that h � R. The terms in h2 areeglected and we obtain:

= − 6πηcR4

h3(1 − β)(2 + β)2 − 2πKiR2(

√3βh + R)

h(β + 2)(26)

−h/2σrr dz = 0 at r = R (29)

= −31/2KiπR2

2[(1 − m2)

1/2 + m−1A sin(m)] − 2πmKiR3

3h(30)

ith m → 1 when the shear stress at the wall equals the yieldtress. The ratio between the first terms in Eqs. (28), (30) and30′) is 1.178. However, it has to be noted that this result cor-esponds to a material, which slips at the wall, with the sheartress equal to the yield stress of the material and not to a no-slip

6 N. Roussel et al. / J. Non-Newtonian Fluid Mech. 135 (2006) 1–7

boundary condition, which is what “sticking” is usually takento mean.

On the other hand, using Eq. (26), in the case of a Newtonianviscous fluid, β = 0 and Ki = 0 Pa, the load expression is equal tothe exact solution originally found by Stefan [17] with p(r = R,z = h/2) = 0.

F = −3πηcR4

2h3 (30′)

9. Comparison with previous solutions

The following reduced parameters are more convenient torepresent the test results [19]; the reduced force F* = −Fh/πR3

and the geometrical ratio h/R. Using this representation, theresult for a perfect plastic fluid (Eq. (28) is a linear functionof the geometrical ratio h/R:

F∗ = 2Ki√3

(h

R

)+ 2Ki

3(31)

The lubrication theory, in the case of a Bingham fluid flow witha no-slip condition, gives for the compression force (Sherwoodand Durban [10]):

F = 2πKiR3

3h+ 4πKiR

3

7h(2g)1/2 + O(g) (32)

a

F

te

term includes a correction due to the traction effect at r = R dur-ing plastic flow. With p(r = R) = 0, the correction term must be2Kih/

√3R.

Starting from the fixed plates case, Petrov [20] derived a solu-tion suitable for Bingham number values g � 1 (larger h/R):

F = Fplast

(1 + 4

5

√4Rηc

Kih2

)(34)

where Fplast is the compression force if the plastic viscosityequals zero. This can be calculated using (27).

The solution obtained herein can be compared to these solu-tions. As the compression force is calculated from the dissipationand not from the pressure integration on the plates, there is noneed for a pressure boundary correction for the largest h/R (butstill smaller than 1). The proposed solution is in good agreementwith the corrected lubrication theory (Fig. 4) in the larger h/Rrange.

The numerical simulations done by Matsoukas and Mitsoulis[4] taking into account the unyielded regions confined near thecenter of the disks are also plotted. In the [0.01;0.1] h/R range,they gave the following approximation of their results:

F = ηcR2

h

(1.282

(h

R

)1.981(

1 + 1.238g

(h

R

)1.026))

(35)

asaaatr

F Fh/πf

nd the reduced force is

∗ = 2Ki

3+ 4Ki

7h

(2ηcR

Ki

)1/2

for g � 1 (33)

Some corrections were made by Adams et al. [9] in ordero take into account the pressure boundary conditions at thedge. An additive term of order 31/2Kih/R appears in F*. This

ig. 4. Comparison of various theoretical predictions of the reduced force F* = −or h/R = 0.3.

For small h/R, when the viscous effects become predominant,suitable solution should converge towards the purely viscous

olution. This is the case for the solution obtained in this worknd for Covey’s solution and the numerical results of Matsoukasnd Mitsoulis but not for the lubrication theory, Adams solutionnd Petrov solution. Among the three solutions that tend towardshe viscous theoretical solution, it can be noted that the numericalesults obtained by Matsoukas and Mitsoulis give the lowest

R3. Compression speed = 2 mm/s, R = 0.05 m, Ki = 6.5 kPa, initial g value = 0.17

N. Roussel et al. / J. Non-Newtonian Fluid Mech. 135 (2006) 1–7 7

value of the compression force. As the compression speed isconstant, this means that this solution corresponds to the lowestdissipation rate (Eq. (25)). We believe that this means that theirsolution is the closest to the real velocity field.

10. Conclusion

An analytical solution of a Bingham fluid flow between twoparallel moving plates was determined. The solution obtainedis expressed in terms of compression load and plate separationand is suitable for rapid identification of Bingham parametersusing the standard assumptions: the studied flow zone is limitedto the volume between the plates and the height of the sample isassumed to be small compared to the plate radius.

This solution avoids the squeezing flow paradox by usinga three-dimensional approach and a three-dimensional formu-lation of the yield criterion. This demonstrates the need tostudy this type of flow using three-dimensional criteria evenif the problem symmetry and geometry tend to suggest a one-dimensional approach. The final proposed flow field is a com-bination of an extensional flow zone and a shear flow zone. Thethickness of the region of extensional flow is chosen to minimisean energy dissipation functional.

Finally, the solution obtained was compared to the literatureand seems to be suitable to small and higher h/R ratios. It fulfilstwo requirements: it tends towards the perfect plastic solutionff

R

[3] D.N. Smyrnaios, J.A. Tsamopoulos, Squeeze flow of Bingham plastics,J. Non-Newton. Fluid Mech. 100 (2001) 165–190.

[4] A. Matsoukas, E. Mitsoulis, Geometry effects in squeeze flow of Bing-ham Plastics, J. Non-Newton. Fluid Mech. 109 (2003) 231–240.

[5] S.D.R. Wilson, Squeezing flow of a Bingham material, J. Non-Newton.Fluid Mech. 47 (1993) 211–219.

[6] J.R. Scott, Theory and application of the parallel-plate plastimeter, Trans.Inst. Rubber Ind. 7 (1931) 169.

[7] G.H. Covey, B.R. Stanmore, Use of the parallel plate plastometer forthe characterisation of viscous fluids with a yield stress, J. Non-Newton.Fluid Mech. 8 (1981) 249–260.

[8] J.D. Sherwood, G.H. Meeten, C.A. Farrow, N.J. Alderman, Squeeze-filmrheometry of non-uniform mudcakes, J. Non-Newton. Fluid Mech. 39(1991) 311–334.

[9] M.J. Adams, I. Aydin, B.J. Briscoe, S.K. Sinha, A finite element analysisof the squeeze flow of an elasto-viscoplastic paste material, J. Non-Newton. Fluid Mech. 71 (1997) 41.

[10] J.D. Sherwood, D. Durban, Squeeze flow of a power-law viscoplasticfluid, J. Non-Newton. Fluid Mech. 62 (1996) 35.

[11] W. Prager, Studies in Mathematics and Mechanics, R. von Mises Pre-sentation Volume, Academic Press, New York, 1954, pp. 208–216.

[12] M.W.J.R. Johnson, On variational principles for non-Newtonians fluids,Trans. Soc. Rheologie V (1961).

[13] N. Yoshioka, K. Adachi, On variational principles for a non Newtonianfluids, J. Chem. Eng. Jpn. 4 (1971) 217.

[14] K.J. Zwick, P.S. Ayyaswamy, I.M. Cohen, Variational analysis of thesqueezing flow of a yield stress fluid, J. Non-Newton. Fluid Mech. 63(1996) 179–199.

[15] C. Lanos, Reverse identification method associate to compression test,in: Proceedings of the XIIIth Int. Cong. on Rheol., vol. 2, Cambridge,2000, pp. 312–314.

[

[[

[

[

or large h/R ratios and towards the Newtonian viscous solutionor small h/R ratios.

eferences

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