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2 J. No-Newtoia Fluid Mech ) Cotets lists available at ScieceDirect Joural of No-Newtoia Fluid Mechaics joural homepage: Weakly compressible Poiseuille flows of a Herschel Bulkley fluid Elei Taliadorou a, Georgios C. Georgiou a,, Iree Moulitsas b a Departmet of Mathematics Statistics, Uiversity of Cyprus, P.O. Box 20537, 1678 Nicosia, Cyprus b The Cyprus Istitute, 17 Kypraoros Street, 1061 Nicosia, Cyprus article ifo abstract Article history: Received 28 September 2008 Received i revised form 14 November 2008 Accepted 17 November 2008 Keywords: Poiseuille flow Herschel Bulkley fluid Compressibility Equatio of state I this work, we derive approximate semi-aalytical solutios of the steady, creepig, weakly compressible plae axisymmetric Poiseuille flows of a Herschel Bulkley fluid. Sice the flow is weakly compressible, the radial velocity compoet is assumed to be zero the derivatives of the axial velocity with respect to the axial directio are assumed to be much smaller tha those with respect to the radial directio. The axial velocity is the give by a expressio similar to that holdig for the icompressible flow, the oly differece beig that the pressure-gradiet is a fuctio of the axial coordiate satisfies a o-liear equatio ivolvig the desity of the fluid. I the preset work, a liear as well as a expoetial equatio of state, relatig the desity of the fluid to the pressure, are cosidered. The pressure distributio alog the flow directio is calculated by meas of umerical itegratio the two-dimesioal axial velocity ca the be costructed. The effects of compressibility, the equatio of state, the Bigham umber the power-law expoet o the solutios are ivestigated Elsevier B.V. All rights reserved. 1. Itroductio Lamiar Poiseuille flows of weakly compressible materials have gaied iterest i the past two decades due to their applicatios i may processes ivolvig liquid flows i relatively log tubes, such as waxy crude oil trasport [1,2] polymer extrusio [3,4]. Numerical solutios of weakly compressible Poiseuille flows have bee reported for Newtoia fluids [3], geeralized Newtoia fluids, such as the Carreau fluid [4] the Bigham plastic [1], as well as for viscoelastic fluids [5]. The objective of the preset work is to solve approximately the plae axisymmetric Poiseuille flows of weakly compressible fluids with yield stress, i.e. fluids obeyig the Herschel Bulkley costitutive equatio, ivestigate the effects of compressibility by meas of two differet equatios of state, i.e. a liear a expoetial oe. A liear equatio of state has bee employed i previous umerical studies of the extrudate swell flow [6,7] by Hatzikiriakos Dealy [8] for HDPE, also for lamiar capillary flow by Veerus [9] for compressible Newtoia fluids, i our previous studies cocerig the simulatio of the stick-slip extrusio istability [3,4]. Expoetial equatios of state have bee employed, for example, by Ragaatha et al. [10] for a HDPE, more recetly, by Viay et al. [1] i simulatios of weakly compressible Bigham flows. Correspodig author. Tel.: ; fax: address: georgios@ucy.ac.cy G.C. Georgiou). The paper is orgaized as follows. I Sectio 2, the goverig equatios for the axisymmetric Poiseuille flow are preseted the assumptios uder which these are simplified are discussed. Aalytical semi-aalytical results are preseted for both the icompressible compressible flows of a Herschel Bulkley fluid the umerical method is briefly discussed. I Sectio 3, the umerical results for the compressible flows of Newtoia, powerlaw, Bigham, Herschel Bulkley fluids with both liear expoetial equatios of state are compared the effects of the compressibility the yield stress are ivestigated. Fially, Sectio 4 cotais the coclusios. 2. Goverig equatios Let us cosider the steady, compressible axisymmetric Poiseuille flow of a geeralized Newtoia fluid. The geometry of the flow is give i Fig. 1. Assumig that the flow is creepig eglectig gravity, the mometum equatio is reduced to p + = 0 1) where p is the pressure is the stress tesor. Let us also deote the velocity vector by u the rate-of-strai tesor by, i.e. u + u) T, 2) where u is the velocity-gradiet tesor, the superscript T deotes its traspose. Uder the assumptio of zero bulk viscosity, which implies that the viscosity forces are oly due to shear ot to volume variatios [1], the viscous stress tesor for a geeralized Newtoia fluid is defied by a costitutive equatio of the /$ see frot matter 2008 Elsevier B.V. All rights reserved. doi: /j.jfm

3 E. Taliadorou et al. / J. No-Newtoia Fluid Mech ) Fig. 1. Geometry of compressible axisymmetric Poiseuille flow of a Herschel Bulkley fluid. followig geeral form: = ) 2 ui), 3) 3 where I is the idetity tesor, is the viscosity which depeds o the magitude of the rate-of-strai tesor: 1 = 2 II 1 = : 4) 2 II beig the secod ivariat of a tesor. The tesorial form of the Herschel Bulkley costitutive equatio is: = 0, 0 = 0 + k 1 ), 0 5) where 0 is the yield stress, k is the cosistecy idex, is the power law expoet, is the magitude of the stress tesor. The power-law fluid the Bigham plastic are the special cases of the Herschel Bulkley model for 0 =0 = 1, respectively. For a weakly compressible flow, we ca assume that the radial velocity compoet is zero. This assumptio is cosistet up to first order with Newtoia perturbatio solutios i terms of compressibility [11,9]. Whe u r = 0 the expressio for is simplified as follows: ) 2 ) 2 uz uz = 2 + 6) z r We further assume that u z / z 1 so that the secod term i the RHS of Eq. 3) is egligible u z r. 7) or z)qz) = Q 0 10) where Qz) is the volumetric flow rate Q 0 = Q0). I the followig subsectios we will first discuss the oe-dimesioal icompressible the the two-dimesioal compressible axisymmetric Poiseuille flow of a Herschel Bulkley fluid. The equatios for the plaar compressible Poiseuille flow are give i Appedix A Icompressible axisymmetric Poiseuille flow The solutio of the icompressible Poiseuille flow of a Herschel Bulkley fluid is straightforward well kow. However, it is preseted here i order to show the aalogy with the weakly compressible solutio to itroduce the odimesioalizatio of the problem. I icompressible flow, the pressure-gradiet the desity are costat the axial velocity compoet depeds oly o the radial coordiate [13]: u z r) = ) 1/ 2 1/ + 1)k 1/ R r 0 ) 1/+1, 0 r r 0 [R r 0 ) 1/+1 r r 0 ) 1/+1 ], r 0 r R 11) The from the r-mometum equatio it is deduced that p = pz) the z-mometum equatio is reduced to + 1 r r r rz) = 0, 8) where the pressure-gradiet is also a fuctio of z. It should be oted that the above assumptios are valid whe the radius of the tube is much smaller tha its legth [12]. Eq. 5) is simplified as follows: u z r = 0, rz 0 ) rz = 0 + k u 9) z, rz 0 r Beig a fuctio of the pressure, the desity also varies across the tube, i.e. = z). For the mass to be coserved, it must be R 2z) u z r, z)r dr = cost. 0 Fig. 2. Velocity profiles for the axisymmetric icompressible Poiseuille flow of a Herschel Bulkley fluid with = 0.5 various Bigham umbers.

4 164 E. Taliadorou et al. / J. No-Newtoia Fluid Mech ) Fig. 3. Velocity profiles for the axisymmetric icompressible Poiseuille flow of a Herschel Bulkley fluid with B = 10 various values of the power-law expoet. Fig. 4. Positio of the yield poit i axisymmetric icompressible Poiseuille flow of Herschel Bulkley fluids. Fig. 5. Pressure distributios for four differet fluids obtaied with the liear solid) the expoetial dashed) equatios of state i axisymmetric Poiseuille flow with B = 0 icompressible flow),

5 E. Taliadorou et al. / J. No-Newtoia Fluid Mech ) where R is the capillary radius, dp/) is the costat pressuregradiet, 2 0 r 0 = <R 12) dp/) deotes the yield poit, i.e. the poit at which the material yields. Note that flow occurs oly if dp/) > 2 0 /R. The volumetric flow rate is give by Q = ) 1/ R 1/+3 2 1/ 3 + 1)k 1/ [ r r R + 1 R 1 r 0 R ) 1/+1 ]}. 13) I the cases of a Bigham plastic = 1) a power-law fluid 0 = 0, r 0 = 0), Eq. 13) is reduced to ) [ ) 4 ] R 4 Q = 8k r 0 R r0 R 14) Q = ) 1/ R 1/+3 15) 2 1/ 3 + 1)k 1/ respectively. I what follows, it is preferable to work with dimesioless equatios. Legths are scaled by the tube radius, R, the velocity by the mea velocity, V 0, i the capillary, the pressure by kv 0 /R. With these scaligs, the dimesioless velocity profile is writte as follows: u z r) = 2 1/ + 1) ) 1/ 1 r 0 ) 1/+1, 0 r r 0 [1 r 0 ) 1/+1 r r 0 ) 1/+1 ], r 0 r 1 16) where all quatities are ow dimesioless, 2B r 0 = dp/) < 1 17) B = 0R kv 18) 0 is the Bigham umber. The dimesioless versio of the costitutive equatio, i.e. of Eq. 9), is: u z r = 0, rz B ) rz = B + u 19) z, rz B r It should be oted that flow occurs oly if dp/) > 2B. For give values of B, Eq. 20) is easily solved for the pressure-gradiet usig the Newto Raphso method, the the velocity profile ca be costructed usig Eq. 16). I Fig. 2, the velocity profiles calculated for = 0.5 various Bigham umbers are show. I Fig. 3, the velocity profiles obtaied with B = 10 = 0.5, are compared. With fixed volumetric flow rate, the size of the yielded regio is reduced as the power-law expoet is icreased. This is also show i Fig. 4, where the yield poit r 0 is plotted as a fuctio of the Bigham umber for various values of Compressible axisymmetric Poiseuille flow I the case of compressible flow, the pressure-gradiet the desity are fuctios of z so are r 0 the volumetric flow rate. It is easily deduced the that the dimesioless axial velocity scaled by the mea velocity, V 0, at the exit of the capillary) is give by u z r, z) = ) 1/ z) 2 1/ + 1) where [1 r 0 z)] 1/+1, 0 r r 0 [1 r 0 z)] 1/+1 [r r 0 z)] 1/+1 }, r 0 r 1 23) 2B r 0 z) = dp/)z). 24) It is clear that at the capillary exit z = 0), Eqs. 23) 24) give the icompressible flow solutio. It should be poited out that i steady compressible Poiseuille flow r 0 z) is just a coveiet idealizatio ot a real yield poit. Sice the axial velocity varies alog the tube, u z / z > 0 thus is ozero, which implies that uyielded regios caot exist. Moreover, the dimesioless pressure-gradiet is a solutio of the followig equatio: 1/ ) 3 [ = ) ] 1/+1 [ 2B ) 2 + 4B ) B 2 + 1)2 + 1) ]. 20) I the case of a power-law fluid B = 0), the solutio of Eq. 20) is simply ) ) = 2. 21) I the case of a Bigham-plastic = 1), Eq. 20) is reduced to 3 ) 4 8B + 3) ) B 4 = 0 22) Fig. 6. Velocity cotours for four differet fluids obtaied for the axisymmetric Poiseuille flow with the liear equatio of state with B = 0 icompressible flow),

6 166 E. Taliadorou et al. / J. No-Newtoia Fluid Mech ) Hece, r 0 z) will be referred to as the pseudo-yield poit. The fact that the classical plug regio flow caot be obtaied i a compressible case was first emphasized by Viay et al. [1]. However, these authors also calculated steady-state velocity profiles at the ilet the outlet of the tube with the plug regio at the ceter correspodig to half the pipe radius. The pressure-gradiet dp/)z) across the capillary, i.e. for z 0, ca be calculated usig the coservatio of mass, i.e. Eq. 10). It turs out that the pressure-gradiet is a solutio of the followig equatio: 1/ ) 3 = p) [ ) ] 1/+1 [ 2B ) 2 + 4B ) B 2 + 1)2 + 1) ] 25) which ivolves the pressure-depedet desity of the fluid. The pressure-gradiet is obviously a fuctio of p is expected to decrease upstream. The pressure depedece of the desity is take ito accout by meas of a thermodyamic equatio of state. At costat temperature low pressures, the desity ca be represeted by the liear approximatio = 0 [1 + ˇp p 0 )], 26) where ˇ / p) p0,t / 0 is the isothermal compressibility assumed to be costat, is the specific volume, 0 0 are, respectively, the desity the specific volume at a referece pressure p 0, T is the temperature. For compariso purposes, the followig expoetial equatio is also used: = 0 eˇp p 0 ). 27) This is equivalet to the liear equatio of state for sufficietly small values of ˇ low pressures. A disadvatage of this equatio is the fast growth of the desity for high values of ˇ). O the other h, the liear model may lead to egative values of the desity. Obviously more sophisticated equatios of state should be used for highly compressible flows. The equatios of state are odimesioalized scalig the desity by 0 the pressure as above. We thus get = 1 + Bp 28) = e Bp, 29) where the referece pressure, p 0, has bee set to zero, B is the compressibility umber, B ˇkV 0 R. 30) Fig. 7. Velocity profiles at z = 0, for four differet fluids obtaied with the liear equatio of state i axisymmetric Poiseuille flow with B = 0.1.

7 E. Taliadorou et al. / J. No-Newtoia Fluid Mech ) Oce the pressure pz) is kow at a poit e.g. p0) = 0), the pressure-gradiet dp/)z) ca be calculated from Eq. 25), usig the Newto Raphso method, as before. Hece, we ca write = f p), 35) where the fuctio f is implicitly kow. If the pressure p i at a poit z i is give, the the poit z i+1 at which the pressure becomes p i+1 = p i + p ca be foud by itegratig the above equatio: z i+1 = z i pi + p p i dp f p). 36) Fig. 8. Positio of the pseudo-yield poit i axisymmetric Poiseuille flow of a Bigham fluid for B = 10 various compressibility umbers. The solid lies correspod to the liear equatio of state the dashed oes to the expoetial oe. The Mach umber is defied by Ma = V 0 /c, where c is the speed of soud i the fluid. I the preset work, we cosider subsoic flows such that Ma 1. Eq. 25) ca be itegrated aalytically i the case of a power-law fluid B = 0). With the liear equatio of state oe fids: [ pz) = 1 1 ) ] 1/+1) ) B + 3 Bz 1} 31) The itegral i the RHS of the above equatio was calculated usig the composite Simpso s rule with 101 poits p = 0.1. At each itegratio poit, the pressure is kow the correspodig pressure-gradiet is calculated solvig Eq. 25). It is also clear that we start at the chael exit z 0 = 0) march to the left, up to ay desired distace upstream. The umerical code has bee tested agaist the aalytical expressios for the pressure distributio i the case of a power-law fluid. u z r, z) = 3 + 1)1 r 1/+1 ) + 1)[ )1/ + 3) Bz] 1/+1) 32) Similarly, with the expoetial equatio of state oe gets: [ pz) = 1 1 ) ] B l Bz u z r, z) = 33) 3 + 1)1 r 1/+1 ) + 1)[1 21/ + 3) Bz] 1/ 34) Nevertheless, i the geeral case the pressure-gradiet the pressure are calculated umerically. Fig. 9. Velocity cotours i axisymmetric Poiseuille flow of a Bigham fluid with B = 10 differet compressibility umbers usig the liear the expoetial equatios of state. Fig. 10. Velocity profiles at differet distaces from the capillary exit i axisymmetric Poiseuille flow of a Bigham fluid with B = 10 B = 0.01: a) liear equatio of state b) expoetial equatio of state.

8 168 E. Taliadorou et al. / J. No-Newtoia Fluid Mech ) Numerical results Numerical results have bee obtaied usig both the liear expoetial equatios of state i order to ivestigate the effects of compressibility i Poiseuille flow of fluids with a yield stress. The effects of the three dimesioless parameters cotrollig the flow, i.e. the Bigham umber, the compressibility umber, the power-law expoet have also bee studied. The pressure distributios for a Newtoia, a power-law, a Bigham a Herschel Bulkley fluid obtaied usig both the equatios of state for B = 0, are show i Fig. 5. Note that the latter value of B is very high correspods to a highly compressible flow; it is used here oly for illustratio purposes. It is clear that the pressure-gradiet the pressure required to drive the flow are reduced as compressibility is icreased icrease with the Bigham umber the power-law expoet. The two equatios of state give essetially the same results oly for sufficietly low compressibility umbers /or ear the die exit. Therefore, a careful selectio of the equatio of state is ecessary whe oe studies compressible Poiseuille flow i very log chaels. Oce the pressure-gradiet is kow as a fuctio of z, the twodimesioal axial velocity ca be costructed by meas of Eq. 23). The velocity cotours correspodig to the flows of Fig. 5 are show Fig. 12. Effect of the Bigham umber o the velocity cotours i axisymmetric compressible Poiseuille flow of a Bigham fluid; liear equatio of state, B = i Fig. 6. Sice the desity becomes higher, the flow decelerates upstream forces the higher cotours to bed towards the symmetry axis. I the case of fluids with a yield stress, this pheomeo is more abrupt, sice just before the disappearace of a cotour lie, this is vertical to the symmetry plae exteds up to the correspodig pseudo-yield poit. The results for the Bigham plastic = 1) the Herschel Bulkley fluid = 0.5) are quite similar. We ca clearly observe that the pseudo-yield poit moves towards the wall as we move upstream. The velocity profiles for the four fluids at z = 0, obtaied usig the liear equatio of state with B = 0.1 are give i Fig. 7. As already metioed, the presece of uyielded regios i steady compressible viscoplastic flow is oly a idealizatio. However, regios of plug-like flow may still exist as idicated by the steady-state umerical results of Viay et al. [1]. I Fig. 8, the positios of the pseudo-yield poit calculated usig both equatios of state for three compressibility umbers are show. I the icompressible flow, the yield poit is, of course, idepedet of the axial distace. I the compressible flow, r 0 moves towards the wall as we move upstream, which implies that the size of the plug-like regio icreases. This pheomeo is better observed i the expoetial case due to the faster icrease of the desity. I Fig. 9, we plot the velocity cotours of a Bigham fluid with B = 10 B = 0, usig both equatios of state. Upstream, the velocity reduces rapidly i the case of the expoetial equatio of state, which is expected because of the faster icrease of the desity. As a result, the velocity cotours are crowded towards the exit plae. I Fig. 10, we plot the velocity profiles at differet distaces from the capillary exit of a Bigham fluid with B = 10 B = 0.01 usig agai both equatios of state. Fig. 11 shows the effects of the Bigham umber o the pressure distributio the positio of the pseudo-yield poit i the case of Bigham flow = 1) usig the liear equatio of state with B = We observe that the pressure icreases upstream the pseudo-yield poit moves faster towards the wall as the Bigham umber icreases. This is more clearly show i Fig. 12, where the velocity cotours for differet Bigham umbers are show. As B is icreased the uyielded regio moves towards the exit of the die. 4. Coclusios Fig. 11. Effect of the Bigham umber i axisymmetric Poiseuille flow of a Bigham fluid with the liear equatio of state B = 0.01: a) pressure distributio b) positio of the pseudo-yield poit. We have derived approximate semi-aalytical solutios of the axisymmetric plae Poiseuille flows of weakly compressible Herschel Bulkley fluid. The two-dimesioal axial velocity is give by a expressio similar to that for the icompressible flow, with the pressure-gradiet the yield stress poit assumed to be fuctios of the axial coordiate. The pressure-gradiet is calculated by meas of umerical itegratio startig at the exit of the tube marchig upstream. The effects of compressibility have bee studied by usig a liear a expoetial equatio of state.

9 E. Taliadorou et al. / J. No-Newtoia Fluid Mech ) The effects of the yield stress the power-law expoet o the pressure-gradiet the velocity have also bee ivestigated. Our calculatios lead to the followig coclusios: a) The pressure required to drive the flow for a give tube legth is reduced with compressibility. b) The liear the expoetial equatios of state give similar predictios oly for sufficietly low compressibility umbers /or for short tubes. Hece, the equatio of state should be chose very carefully i umerical simulatios of compressible flow i log tubes. c) The two-dimesioal axial velocity is characterized by pluglike regios the size of which icreases upstream, i agreemet with the more sophisticated umerical simulatios of Viay et al. [1]. d) With the expoetial equatio of state, the upstream growth of the pseudo-uyielded regio is much faster tha with the liear equatio of state. Appedix A. Compressible plae Poiseuille flow I plae Poiseuille flow, legths are scaled by the chaelhalfwidth, H, the velocity by the mea velocity, V 0, at the exit of the chael, the pressure by kv 0 /H. Uder the same assumptios used for the axisymmetric flow, the dimesioless velocity profile i the case of compressible plae flow is writte as follows: u x x, y) = ) 1/ x) + 1 dx [1 y 0 x)] 1/+1, 0 y y 0 [1 y 0 x)] 1/+1 [y y 0 x)] 1/+1 }, y 0 y 1 37) where B y 0 x) = 38) dp/dx)x) B = 0H kv 39) 0 is the Bigham umber. The dimesioless pressure-gradiet is a solutio of the followig equatio: ) 2 [ = ) ] 1/+1 [ B p) dx dx + 1 B + )]. dx 40) It is clear that at the chael exit x = 0), Eqs. 37) 38) yield the solutio for icompressible flow. I the case of a power-law fluid, the solutio of Eq. 40) is simply ) ) =. 41) dx p) I the case of a Bigham-plastic, Eq. 40) is reduced to 2 ) 3 3 B + 2 ) ) 2 + B 3 = 0, 42) dx p) dx which has the followig solutio: ) B = dx ) p) [ [ }]] 1 2B cos 3 cos 1 1 B + 2/p))) 3. 43) Detailed results for the compressible plae Poiseuille flow ca be foud i Ref. [14]. Refereces [1] G. Viay, A. Wachs, J.-F. Agassat, Numerical simulatio of weakly compressible flows: the restart of pipelie flows of waxy crude oils, J. No-Newto. Fluid Mech ) [2] M.R. Davidso, Q.D. Nguye, H.P. Røigse, Restart model for a multi-plug gelled waxed oil pipelie, J. Petrol. Sci. Eg ) [3] G.C. Georgiou, M.J. Crochet, Compressible viscous flow i slits with slip at the wall, J. Rheol ) [4] G.C. Georgiou, The time-depedet compressible Poiseuille extrudateswell flows of a Carreau fluid with slip at the wall, J. No-Newto. Fluid Mech ) [5] F. Belblidia, I.J. Keshtiba, M.F. Webster, Stabilised computatios for viscoelastic flows uder compressible cosideratios, J. No-Newto. Fluid Mech ) [6] C.R. Beverly, R.I. Taer, Compressible extrudate swell, Rheol. Acta ) [7] G.C. Georgiou, The compressible Newtoia extrudate swell problem, It. J. Numer. Methods Fluids ) [8] S.G. Hatzikiriakos, J.M. Dealy, Role of slip fracture i the oscillatig flow of a HDPE i a capillary, J. Rheol ) [9] D.C. Veerus, Lamiar capillary flow of compressible viscous fluids, J. Fluid Mech ) [10] M. Ragaatha, M.R. Mackley, P.H.J. Spitteler, The applicatio of the multipass rheometer to time-depedet capillary flow measuremets of a polyethylee melt, J. Rheol ) [11] L.W. Schwartz, A perturbatio solutio for compressible viscous chael flows, J. Eg. Math ) [12] G. Viay, A. Wachs, I. Frigaard, Start-up trasiets efficiet computatio of isothermal waxy crude oil flows, J. No-Newto. Fluid Mech ) [13] R.R. Huilgol, Z. You, Applicatio of the augmeted Lagragia method to steady pipe flows of Bigham, Casso Herschel Bulkley fluids, J. No-Newto. Fluid Mech ) [14] E. Taliadorou, Numerical simulatios of weakly compressible geeralized Newtoia flows, Ph.D. Thesis, Departmet of Mathematics Statistics, Uiversity of Cyprus, 2008.