158970944 Axial Couette Flow Bingham Fluid

J. Non-Newtonian Fluid Mech. 165 (2010) 14941504

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Journal of Non-Newtonian Fluid Mechanics

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Axial C s t

Yu-QuanDepartment of

a r t i c l

Article history:Received 25 JuReceived in reAccepted 29 Ju

Keywords:Bingham uidAxial CouetteConcentric annAnalytical solu

uilleof onmisdepennd the numa mat

1. Introdu

Flows ofby the forcare frequencially in oilwuids exhibit is viscoplasticity, which indicates that the strain ratewill retain zero until a yield stress 0 is reached. Therefore, studiesof annular ows utilizing a variety of viscoplasticmodels have beencarried out during the last several decades.

Some relatively simple models of yield stress uids, which arealso widelyCasson

=

1/20 +

HerschelB

=(

0

and Roberts

=

1/n0

CorresponE-mail add

ma0 =lytich simhavely an

Bird et al. [1], includingowsdrivenbya torque applied to theoutercylinder, an axial motion of the inner cylinder or a constant pres-sure gradient in a concentric annulus. Fordham et al. [2] developeda general method for calculating viscoplastic ows in concentricannuli of arbitrary radius ratio, which is worked for models (1a),

0377-0257/$ doi:10.1016/j.applied in researches, are presented here. They are the

(0)1/2

2

(1a)

ulkley

n1 + 0)

(1b)

onStiff models

(n1)/n +

(0)1/n

n

(1c)

ding author. Tel.: +86 10 6277 1654; fax: +86 10 6278 1824.ress: [email protected] (Y.-Q. Liu).

(1b) and (1c) mentioned above.To the best of our knowledge, analytical solutions possibly

derived for Binghamuidowthrough concentric annuli are nearlyall under the following three conditions:

Rotation of the inner/outer cylinder. Axial moving of the inner cylinder. An axial pressure gradient.

Notice that what Bird et al. [1] reviewed referred to individ-ual instances of the conditions above. From then on, solutions forcombinations of these were found in succession. Bittleston andHassager [3] combined the rotation of the inner cylinder and thepressure gradient together, and considered ow of Bingham uidsboth in a slot and in an annulus. Filip and David [4] investigatedRobertsonStiff uid ow produced by the inner cylinder movingalong its axis and by the pressure gradient imposed in the axialdirection, which is supposed to cover the results for Bingham andpower-law uids. Moreover, Peng and Zhu [5] worked on Bing-hamuids in spiral Couette ow,whichmeant TaylorCouette ow

see front matter 2010 Elsevier B.V. All rights reserved.jnnfm.2010.07.013ouettePoiseuille ow of Bingham uid

Liu , Ke-Qin ZhuEngineering Mechanics, Tsinghua University, Beijing 100084, China

e i n f o

ly 2009vised form 21 June 2010ly 2010

sPoiseuille owulitions

a b s t r a c t

In this paper, the axial CouettePoiseAnalytical solutions of different typestwonew types of ow,which have beedifferent forms of the velocity prolethe Bingham, axial Couette numbers ain the parameter plane of axial Couettow regimes are analyzed from both

ction

non-Newtonian uids through annuli, induced eithering pressure gradient or by the motion of boundaries,tly encountered in various industrial applications, espe-ell drilling. Oneof themost important properties these

Onen=1 or

Anathrougannulithe earm/locate / jnnfm

hrough concentric annuli

ow of Bingham uids through concentric annuli is studied.w are derived. Compared to previous studies, we emphasizesed previously, are found in our results. Hence, there are eightding on values of three dimensionless parameters, which aree radius ratio. Distributions of these eight forms are speciedber vs. Bingham number for various radius ratios. These newhematical and physical perspective.

2010 Elsevier B.V. All rights reserved.

y obtain the Bingham or power-law model by setting0 in (1b) and (1c), respectively.al or semi-analytical solutions of viscoplastic uid owsple geometries, such as pipes, slots and concentricbeen quite thoroughly studied and published. Some ofalytical solutions for Bingham uids were reviewed by

Y.-Q. Liu, K.-Q. Zhu / J. Non-Newtonian Fluid Mech. 165 (2010) 14941504 1495

between two concentric rotating cylinders with the sliding of theinner cylinder.

Due to the different position of the plug zone, different typesof the velocity prole may be observed. Assuming the maximumvelocity ofas the chardivide the pBingham none specicformed thisit more preplane Coueporous para

Neverthfor the axiand they dow upon tfor their seRobertsoncan be recofully analytWe have doand Chen aDavid [4], Tthe same af

As we whave somedresults origto errors inthere are twwhich the p

The rest2, a brief inboundary ccorrespondour results,conclusion

2. Geomet

Considehamplasticand R2 as tder moves wstationary.imposed inin Fig. 1.

The goveequations

U = 0

U

t+ (U

where , Uvector, precylindrical

U = Urer +

where Ur, Uazimuthal aof these thr

. Sch

conbelo

0 + = 0 anhile

ijiiji

thermsion

+ (U)T (6)ording to the axisymmetric property of the problem, wee, as Filip and David [4] did, that the only non-zero veloc-ponent is Uz, which is a function only of r. So the continuity

on and the momentum equations for Ur and U are naturallyd, while the momentum equation in the axial direction isd to the following one

= dpdz

(7)

ere rz = rz (r) is the only non-zero component of . Then (4)ced to

0 +0dUz/dr

)dUzdr

|rz | > 0

dUzdr

= 0 |rz | 0

(8)

boundary conditions include the non-slip conditions at the

1= Uc (9a)

2= 0 (9b)Poiseuille ow for Newtonian uids in a circular pipeacteristic velocity for non-dimensionalization, one canarameter plane of rotational/axial Couette number vs.umber into a couple of regions, in any of which onlytype of ow appears. Bittleston and Hassager [3] per-task, while Tsangaris et al. [6] and Chen and Zhu [7] didcisely in their respective work, which considered thettePoiseuille ow of Bingham uids with two equallyllel walls.eless, Filip and David [4] only gave out the restrictionsal Couette number except for the Bingham number,id not outline the distribution of different types ofhe parameter plane. Frankly, this is totally understoodmi-analytical results caused by the complexity of theStiff model. But as we mentioned, the Bingham modelvered by setting n=1 in (1c), which makes their resultsical. Then it is possible to map out the parameter space.ne so, and found it similar to that in Tsangaris et al. [6]nd Zhu [7]. Actually, ow cases presented in Filip andsangaris et al. [6] and Chen and Zhu [7] are essentiallyter comparison.ill show in this paper, ow cases in annular channelsistinctions fromthose inplane channels, andextendinginated from Filip and David [4] to Bingham uids leadsthe parameter plane. More importantly, we stress thato new types of ow, missed in Filip and David [4], witharameter plane is nally able to be fullled.of this paper will be organized as follows. In Sectiontroduction to the geometry, governing equations andonditions is given. In Section 3, the analytical solutionsing todifferent typesofowarederived.Wewill discussand compare them with former works in Section 4. Awill be reached in Section 5.

ry, governing equations and boundary conditions

r the steady, laminar and incompressible ow of a Bing-uid between two innite concentric cylinders with R1he inner and outer radii, respectively. The inner cylin-ith a constant velocity Uc while the outer one remains

Also, a constant negative pressure gradient dp/dz 0

0(4)

d represent the deviatoric stress and strain rate ten-their second invariants and are

j/2

j/2(5)

ore, 0 is the yield stress and 0 is the viscosity. Theof is

1496 Y.-Q. Liu, K.-Q. Zhu / J. Non-Newtonian Fluid Mech. 165 (2010) 14941504

together with two continuity conditions for the velocity compo-nents and stress across the yield surface

Uzr=y = Up (10a)

dUzdr

r=y

=

where y is tof the plug

For owmotion ofmomentum

( n)

where =mal to . Sthis problemconsidered

So far, tdimensiona

r = rR2

Uz

where U0 =HagenPoisthe sake ofless forms o

d(rrz)r dr

=

the constitu

rz =(

1 +

d

d

and the bou

Uzr= = Co

Uzr=1 = 0

In thisters: the rB= 0R2/0can be eithe

3. Analytic

No mattrz from (12

rz = r + C

whereC is auid has yione shear of Uz would

Uz = 12 r2

with an extNow we

3.1. Cases corresponding to those in Filip and David [4]

In this part, all ve cases are already studied in Filip and David[4], but they only gave out the restrictions for the axial Couette

r exte reouetts in tay b

Casewi

2.his cylinderfacform

12

Up

12

lyinyiel

or sebe ansion

= Br de

Co +r20 l

r20 l

hichlowin

r2(r

tionsordinould

1 reove

(2r2

ce, a

o

Cor2

Cor2

Casewi

3.his cathe pe ha0 (10b)

he position of the yield surface, and Up is the velocityzone.with a plug zone detached from both cylinders, the

the plug zone, , is determined by conservation of, which is represented by Frigaard et al. [8]

ds =

(

dU

dt

)d (10c)

p+ , is the unit tensor, and n is the outward nor-ince no rotational motions of the plug zone occur in, the conservation of angular momentum will not be

here.he problem is established mathematically. To non-lize it, the following dimensionless variables are used:

= UzU0

rz = rzR20U0(11)

(R22/20)dp/dz is twice of the max velocity ofeuille ow. From here on, the tildes are eliminated forconvenience. Therefore, we can obtain the dimension-f the momentum equation (7)

2 (12)

tive equations (8)

BdUz/dr)

dUzdr

|rz | > B

Uzr

= 0 |rz | B(13)

ndary condition (9a) and (9b)

(14a)

(14b)

problem we have three dimensionless parame-adius ratio =R1/R2 (0, 1), the Bingham numberU0 0, and the axial Couette number Co=Uc/U0, whichr a positive number or a negative one.

al solutions

er what the velocity prole is like, a general solution of) exists:

r(15)

n integral constant. According to (13), once theBinghamelded, the strain rate dUz/dr would not change sign inow zone. Substituting (15) into (13), a general solutionbe

+ C ln r B sgn(

dUzdr

)r + D (16)

ra integral constant D.analyze various cases of the velocity prole in detail.

numbeaccuraaxial Cregionow, m

3.1.1.Flow

in Fig.In t

outer cthe int

The

Uz =

Appat twotions fshoulda dime

r2 r1Afte

Uz =

In wthe fol

2Co + 2

RestricAcc

case sh

0 B Mo

dCo

dr2=

Hen

Co2 Cwhere

Co2 =

Co3 =

3.1.2.Flow

in Fig.In t

whiler= r1, wcept for the Bingham number. We will determine thestrictions of every individual case concerning both theenumberCo and theBinghamnumberB, so that speciche parameter plane CoB, related to specic types ofe recognized.

Ith a plug zone detached from both cylinders, as shown

ase, two shear ow zones are attached to the inner anders respectively with the plug zone between them. Ate r= r1 and r= r2, we have rz =B, where r1 r2 1.of the velocity prole is

r2 + C1 ln r Br + C2 ( r r1)(r1 r r2)

r2 + C3 ln r + Br + C4 (r2 r 1)(17)

g the conditions on both boundaries (14a) and (14b) andd surfaces (10a) and (10b) leads to a set of six equa-ven unknowns: C1, C2, C3, C4, r1, r2 and Up. Eq. (10c)dded due to the equilibrium of the plug zone, which inless form is

(18)

tailed analyses, we nd the velocity prole is

r20 lnr

B(r ) 1

2(r2 2) ( r r1)

n r2 B(1 r2) +12(1 r22) (r1 r r2)

n r B(1 r) + 12(1 r2) (r2 r 1)

(19)

r20 = r1r2 is the position where rz =0, and r2 satisesg implicit expression:

2 B) lnr2 Br2

+ 2B(1 r2) 1 + (B + )2 = 0 (20)

g to r1 r2 1 and (18), the rst restriction of thisbe

(21)

r, we have +B r2 1, and can obtain from (20) that

B) ln r2r2 B

0 (22)

nother restriction for Co should be

Co3 (23)

=1 = (1 B) ln

1 B +12[1 ( + B)2] (24)

=+B = ( + B) ln( + B) +12[(1 B)2 2] (25)

IIth a plug zone attached to the outer cylinder, as shown

se, the shear ow zone is attached to the inner cylinderlug zone is attached to the outer one. At the interfaceve rz =B, where r1 1.


Fig. 2. Schematic of case I: ow with a plug zone detache

The form of the velocity prole is

Uz ={

12

r2 + C5 ln r Br + C6 ( r r1)0 (r1 r 1)

(26)

Fig. 3. Schem

Applyinyield surfacthree unkn

After de

Uz ={

Co +0

satiwith r1atic of case II: ow with a plug zone attached to the outer cylinder.

2Co + 2r1(r

RestrictionsThe dist

rz = r +

In thiswith r10B1 that

dCo

dr1= (2r

So the o{Co1 CoCo1 Co

where

Co1 = Cor1

Notice tsatised, wd from both cylinders.

g the conditions on the inner boundary (14a) and at thee (10a) and (10b) leads to a set of three equations forowns: C5, C6 and r1.tailed analyses, we nd the velocity prole is

r1(r1 + B) lnr

B(r ) 1

2(r2 2) ( r r1)

(r1 r 1)(27)

sfying the following implicit expression:1 + B) lnr1

2B(r1 ) (r21 2) = 0 (28)

ribution of rz in the annulus is

r1(r1 + B)r

(29)

case we need |rz |r=1| B. Substituting (29) into it1, we get r1 1B, which means 1B r1 1 when, or r1 1 when B>1 . We can obtain from (28)

1 + B) lnr1

0 (30)

nly restriction of this case for Co is

Co2 (0 B 1 ) 0 (B > 1 ) (31)

=1 = (ln + 1 )B +[ln + 1

2(1 2)

](32)

hat for any B0, case II would appear as long as (31) ishich indicates no extra restriction for B in case II.


Fig. 4. Schema

3.1.3. CaseFlow wit

in Fig. 4.In this ca

while the pr= r2, we ha

The form

Uz ={

Co

12

Applyingyield surfacthree unkno

After de

Uz ={

Co

r2(r

with r2 sati

2Co 2r2(rRestrictions

The distr

rz = r + r

In this(36) into itensure r2 e[B , B + case:

0 B 1 +

chem

rwa,1] aed fo

13tic of case III: ow with a plug zone attached to the inner cylinder.

IIIh a plug zone attached to the inner cylinder, as shown

se, the shear ow zone is attached to the outer cylinderlug zone is attached to the inner one. At the interfaceve rz =B, where r2 1.

Fig. 5. S

Afteand [identi

0 < of the velocity prole is

( r r2)r2 + C7 ln r + Br + C8 (r2 r 1)

(33)

the conditions on the outer boundary (14b) and at thee (10a) and (10b) leads to a set of three equations forwns: C7, C8 and r2.


( r r2)2 B) ln r B(1 r) +

12(1 r2) (r2 r 1)

(34)

sfying the following implicit expression:

2 B) ln r2 + 2B(1 r2) (1 r22) = 0 (35)

ibution of rz in the annulus is

2(r2 B)r

(36)

case,rz |r= B should be satised. Substituting

with r2 1, we have B r2 B+ . In order toxists, there must be an intersection between intervals] and [,1], which induces the restriction for B in this

(37)

13

< < 1

We can

dCo

dr2= (2r2

which is vaCo:

0 < 13

13

< < 1

where

Co4 = Cor2

Co5 = Cor2

3.1.4. CaseShear o

in Fig. 5.In this ca

is dependenthat in planatic of case IV: shear ow dominant throughout the annulus (Co


Fig. 6. Schem

The form

Uz = 12 r2

Applyingleads to a sedetailed an

Uz =[Co


rz = r + C

Case IV d

Co B(1

As ln 0), as shown

se, theplug zonedoesnot exist, either. In Tsangaris et al.n and Zhu [7], cases IV and V are organized together andas Pure shear ow. Here we split them apart becauseof the restriction for Co, which will be shown below.of the velocity prole is

+ C11 ln r + Br + C12 (48)

cases. Howparameterinto three r

(1) Co5 (2) 0 Co(3) 0 Co

Apparenarea, whichin annular c

3.2.1. CaseAnother

shown in FiIn this c

outer cylindthe interfacPay attentiotrated in thand dUz/dr

The form

Uz =

12

Up

12B(1 ) 12(1 2) ln r

ln B(1 r) + 1

2(1 r2) (49)

ribution of rz in the annulus is

Co + B(1 ) 1/2(1 2)r ln

(50)

emands rz B throughout r1, which gives) + 1/2(1 2) + r(r B) ln (51)imum of the right side of (51) depends on the valuef 0B2, the maximum is reached when r= , result-

(52a)

2, the maximum is reached when r=B/2, resulting in(52b)

>2, the maximum is reached when r=1, resulting in

(52c)

) + 12(1 2) B

2

4ln (53)

+ 1 )B +[ln + 1

2(1 2)

](54)

ses discovered by us

ouettePoiseuille owof Binghamuids in plane chan-es of ow consist of the ve cases discussed above andlug ow case, where no ow actually occurs due to thebetween the yield stress and the applied pressure gra-

he parameter plane CoB is fully lled with these sixever, for that in annular channels, a blank area in theplane CoB remains to be identied, which is dividedegions:

Co Co6 2 B 1 + Co6 1 + B 2 Co7 B 2tly none of the cases given above can occupy this blankimplies an essential requirement for new types of owhannels besides those already presented.

VIowwith a plug zone detached from both cylinders, asg. 7.ase, two shear ow zones are attached to the inner anders respectively with the plug zone between them. Ate r= r1 and r= r2, we have rz =B, where r1 r2 1.n to the differences between case VI and case I concen-e inner shear ow zone, where dUz/dr < 0 in case VI>0 in case I.of the velocity prole is

r2 + C13 ln r + Br + C14 ( r r1)(r1 r r2)

r2 + C15 ln r + Br + C16 (r2 r 1)(55)


Fig. 7. Schemcylinders.

Applyingat two yieldfor seven uadded duesionless for

r1 + r2 = BAfter de

Uz =

Co r1r1

with r2 sati

2Co + 2r2(r

RestrictionsAccordin

case should

2 B 2Consequ

r1 B 1 rMoreove

dCo

dr2= (r1

chem.

ce, a

Co Coatic of case VI: another ow with a plug zone detached from both

the conditions on both boundaries (14a) and (14b) andsurfaces (10a) and (10b) leads to a set of six equationsnknowns: C13, C14, C15, C16, r1, r2 and Up. Eq. (10c) isto the equilibrium of the plug zone, which in a dimen-

Fig. 8. Scylinder

Hen{Co5Co8m is

(56)


r1r2 lnr

+ B(r ) 1

2(r2 2) ( r r1)

r2 ln r2 B(1 r2) +12(1 r22) (r1 r r2)

r2 ln r B(1 r) +12(1 r2) (r2 r 1)

(57)

sfying the following implicit expression:

2 B) lnB r2r2

1 + (B )2 + 2B(1 r2) = 0 (58)

g to r1 r2 1 and (56), the rst restriction of thisbe

(59)

ently, the value ranges for r1 and r2 are

B

2 r2 B (2 B 1 + )

1 B

2 r2 1 (1 + < B 2)

(60)

r, we can obtain form (58) that

r2) lnr1r2

0 (61)

where

Co8 = Cor2

3.2.2. CaseAnother

shown in FiAs we ca

with case Vbelong to o

In this cawhile the pr= r1, we haences betwow zone, w

The form

Uz ={

12

0

Applyinyield surfacthree unkn

After de

Uz ={

Co +0atic of case VII: another ow with a plug zone attached to the outer

nother restriction for Co should be

Co6 (2 B 1 + ) Co6 (1 + B 2) (62)=1 = (B 1) lnB 1

+ 1

2[1 (B )2] (63)

VIIowwith a plug zone attached to the outer cylinder, asg. 8.n see, blank regions (1) and part of (2) have been lledI.Wewill demonstrate that the rest of the blank regionsnly one new case.se, the shear ow zone is attached to the inner cylinderlug zone is attached to the outer one. At the interfaceve rz =B, where r1 1. Pay attention to the differ-een case VII and case II concentrated in the inner shearhere dUz/dr0 in case II.of the velocity prole is

r2 + C17 ln r + Br + C18 ( r r1)(r1 r 1)

(64)

g the conditions on the inner boundary (14a) and at thee (10a) and (10b) leads to a set of three equations forowns: C17, C18 and r1.tailed analyses, we nd the velocity prole is

r1(r1 B) lnr

+ B(r ) 1

2(r2 2) ( r r1)

(r1 r 1)(65)


with r1 satisfying the following implicit expression:

2Co + 2r1(r1 B) lnr1

+ 2B(r1 ) (r21 2) = 0 (66)


rz = r + r

In this c r1 1, wthis case:

B 1 + Afterwa

follows:{ r1 B r1 1

We can

dCo

dr1= (2r

which is va{0 Co 0 Co

So far, th

3.2.3. CasePure pluIn this c

In fact, it hlimit situatiwhich reve

4. Results

4.1. Divisio

For the aconcentric alytically. Wwith corres

Co1 = (ln

Co2 = (1

Co3 = ( +Co4 = ( l

Co5 = (B

Co6 = B2

4Co7 = (ln

Co8 = (B

Accordinobserved wDue to distIII, we illuswith B [0Figs. 9(a) a

Fig. 9. Phase diagram with =0.25: (a) full map and (b) upper half.

cation points of at least three ow cases, from the left corre-ng consecutively to B=2, B=1 , B=1+ , B=2 in Fig. 9(b)1 , B=2, B=1+ , B=2 in Fig. 10(b).

ce the radius ratio appears in three out of four demarca-ints, it has signicant effects upon regions of ow cases. Asases, regions of cases I, VI and VII become narrower in width. 9(b) and 10(b), for the value ranges for B are [0,1 ],and, [1 + ,2.5] respectively. More importantly, deter-whether cases I and VI have an intersection between theiranges for B. When [0, 1/3], an intersection B [2,1 ]and cases I and VI are possible to occur under same B, asrestrictions for Co are satised. When [1/3, 1], no such

menon exists.setting =0.9, we illustrate the variation of the phase dia-hen 1, as shown in Fig. 11. It seems that cases I and

greatly depressed. Considering the longitudinal coordinate,er, all cases are greatly depressed except for cases IV and V.

mparison to results of Filip and David [4]

ip and David [4] studied the axial CouettePoiseuille ow ofsonStiff uids (1c), which is thought to include the resultsgham uids, as (1c) reduces to (4) by setting n=1. Natu-comparison between our results and theirs is supposed to

formed.ribution of rz in the annulus is

1(r1 B)r

(67)

ase we need |rz |r=1| B. Substituting (67) into it withe get r1 B1, which induces the restriction for B in

(68)

rds, the value range for r1 is identied for different B as

1 (1 + B 2)(B > 2)

(69)

obtain from (66) that

1 B) lnr1

0 (70)

lid for (69), and leads to the restriction for Co:

Co8 (1 + B 2)Co7 (B > 2)

(71)

e parameter plane CoB has been lled up.

VIIIg owase, the Bingham uid stays stationary in the annulus.as been involved in all the cases except IV and V as aon. As long as Co /= 0, shear ow will absolutely occur,als Co=0 and B1 as the restrictions of this case.

and discussion

n of the parameter plane

xial CouettePoiseuille ow of Bingham uids throughnnuli, all eight types of owhave been solved fully ana-e derive eight critical axial Couette numbers Co1 Co8ponding value ranges for B, which are

+ 1 )B +[ln + 1

2(1 2)

]B 0

B) ln

1 B +12[1 ( + B)2] 0 B 1

B) ln ( + B) + 12[(1 B)2 2] 0 B 1

n + 1 )B +[2 ln + 1

2(1 2)

]0 B 2

) ln (B ) + 12[(1 B)2 2] 2 B 1 +

ln B(1 ) + 12(1 2) 2 B 2

+ 1 )B +[ln + 1

2(1 2)

]B 2

1) lnB 1

+ 1

2[1 (B )2] 1 + B 2

(72)

g to (72), distributions of various ow cases areith Co1 Co8 drawn on the parameter plane CoB.inctions between [0, 1/3] and [1/3, 1] in casetrate the distributions by setting =0.25 and =0.5,2.5], as shown in Figs. 9 and 10. Red lines innd 10(a) refer to case VIII, while red circles stand for

demarspondiand B=

Sintion po increin Figs[2,2]minesvalue rexists,long aspheno

Bygram wVI arehowev

4.2. Co

FillRobertfor Binrally, abe per


Fig. 10

In [4], Fider V to beis equal todependingthemovingdimensionl

T0 =20PR

=PR2K

(Apparen

between {Co =

(

Co = (

First we, their ow

(1) P > 0(2) P > 0(3) P > 0(4) P < 0(5) P < 0(6) P < 0. Phase diagram with =0.5: (a) full map and (b) upper half.

lip and David set the axial velocity of the inner cylin-positive all along, while the pressure gradient P, whichdp/dz in this paper, can be either positive or negative,on assisting or opposing the drag on the uid caused byinner cylinder. Besides the radius ratio , the other twoess parameters are dened as:

(73)

R

V

)n(74)

tly, and T0 are equal to our and B. The connectionand Co isn=1)1

P > 0n=1)1

P < 0(75)

perform some qualitative comparisons. Basing on P andcases consist of the following six ones

> cr1,2cr1,2 cr2,3 < cr2,3 > cr4,5cr4,5 cr5,6 < cr5,6

(76)

Fig.

which corrI(Co 0(2) P > 0(3) P > 0(4) P < 0(5) P < 0(6) P < 011. Phase diagram with =0.9: (a) full map; (b) upper half.

espond consecutively to our cases I (Co>0), III, V,and IV, and are essentially the same as those in plane,7]. None of them involve cases VI and VII in this paper.come to quantitative comparisons. The critical num-are

( + T0 1) T1/n0 + 1

+T0

[ ( + T0)

]1/nd

}n

( 1) T1/n0 + 1

[ ( T0)

]1/nd

}n

+ T0 1) T1/n0 + 1T0

(1 T0

)1/n

d

]n

1) T1/n0 + 1

(1 + T0

)1/n

d

]n(77)

ting (77) into (76), with n=1 and (75), we have

0 < Co < Co3Co3 Co Co4Co > Co4Co2 < Co < 0Co1 Co Co2Co < Co1

(78)


where

Co3 =(

cr

Co4 =(

cr

Co2 = (

Co1 = (

ExpressiAs seen

to those foCo1 Co4 eFig. 10, (78correct in thstart to app

4.3. Analys

It is neceVI and VII illustrated i

It iswellby stress pcan be writonic decrethe stress pemergencestress protionally, altof the stresFig. 12. Illustration of analyses in Sectio

1,2

n=1)1

2,3

n=1)1

cr4,5

n=1)1

cr5,6

n=1)1

(79)

ons of Co1 Co4 are in (72).from above, when results of Filip and David [4] reducer Bingham uids, only cases IV and critical numbersmerge, with no restrictions for B at all. From Fig. 9 and) match the lower half of the parameter plane, but aree upper half only if B2. Once B>2, cases VI and VIIear, which their results cannot describe.

es of cases VI and VII

ssary to analyzemore detailly the appearances of casesrstly presented in this paper. The following analyses aren Fig. 12.known that velocity proles are essentially determinedroles. For ow in plane channels, the stress proletten as xy = y+C, which exhibits a linearly mono-asing distribution. By altering the integral constant C,role is shifted parallelly throughout, leading to theof cases IV. For that in annular channels, however, thele is rz = r+C/r, which is not linear any more. Addi-erations of C give rise to more complicated variationss prole. For C>0, the stress prole remains monotonic

decreasing,although rthen monodecreasingVI appears.decreasingnot speciallsolving pro

Physicalof the plugrz |r=r1 = means theygradient torz |r=r2 = tangent stras well as thfound in plastresses exiOn the otheleading to dit enables tcases VII an

5. Conclus

We deCouettePoannuli, whdimensionln 4.3.

leading again to the emergence of cases IV. For C


B and the axial Couette number Co. Domains belonging to all formsare validated in the parameter plane CoB for various radiusratios with fully analytical boundaries.

Compared to earlier researches, we emphasize that two brand-new ow cases are discovered, and restrictions concerning thedimensionless parameters for each of the eight forms are preciselyveried. After careful comparisonwith results from Filip and David[4], we show that they missed these two ow cases, and gave outincorrect restrictions for several cases.

Finally, we explain why the two brand-new ow cases,which are impossible to appear in plane channels, emerge inannular channels both mathematically and physically. Resultsand analyses presented in this paper might be instructive forsolving ows of viscoplastic uids in annuli or other geome-tries.

Acknowledgements

This research is supportedby theNationalNatural Science Foun-dation of China under Grant No. 10772097. The nal manuscript

benets from comments and suggestions provided by two refer-ees. In particular, we are indebted to a referee whose questionshave enriched the contents of the paper.

References

[1] R.B. Bird, G.C. Dai, B.J. Yarusso, The rheology and ow of viscoplastic materials,Rev. Chem. Eng. 1 (1983) 170.

[2] E.J. Fordham, S.H. Bittleston, M.A. Tehrani, Viscoplastic ow in centered annuli,pipes and slots, Ind. Eng. Chem. Res. 30 (1991) 517524.

[3] S.H. Bittleston, O. Hassager, Flow of viscoplastic uids in a rotating concentricannulus, J. Non-Newtonian Fluid Mech. 42 (1992) 1936.

[4] P. Filip, J. David, Axial CouettePoiseuille ow of power-law viscoplastic uidsin concentric annuli, J. Petrol. Sci. Eng. 40 (2003) 111119.

[5] J. Peng, K.Q. Zhu, Linear stability of Bingham uids in spiral Couette ow, J. FluidMech. 512 (2004) 2145.

[6] S. Tsangaris, C. Nikas, G. Tsangaris, P. Neofytou, Couette owof a Binghamplasticin a channel with equally porous parallel walls, J. Non-Newtonian Fluid Mech.144 (2007) 4248.

[7] Y.L. Chen, K.Q. Zhu, CouettePoiseuille ow of Bingham uids between twoporous parallel plates with slip conditions, J. Non-Newtonian Fluid Mech. 153(2008) 111.

[8] I.A. Frigaard, S.D. Howison, I.J. Sobey, On the stability of Poiseuille ow of aBingham uid, J. Fluid Mech. 263 (1994) 133150.

Axial CouettePoiseuille flow of Bingham fluids through concentric annuliIntroductionGeometry, governing equations and boundary conditionsAnalytical solutionsCases corresponding to those in Filip and David [4]Case ICase IICase IIICase IVCase V

New cases discovered by usCase VICase VIICase VIII

Results and discussionDivision of the parameter planeComparison to results of Filip and David [4]Analyses of cases VI and VII

ConclusionAcknowledgementsReferences

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158970944 Axial Couette Flow Bingham Fluid