Journal of Non-Newtonian Fluid Mechanics

Transcription

1 Joual of No-Newtoia Fluid Mechaics 93 (3) 9 Cotets lists available at SciVese ScieceDiect Joual of No-Newtoia Fluid Mechaics joual homepage: Combied effects of compessibility ad slip i flows of a Heschel Bulkley fluid Yiolada Damiaou a, Geogios C. Geogiou a,, Iee Moulitsas b a Depatmet of Mathematics ad Statistics, Uivesity of Cypus, P.O. Box 37, 67 Nicosia, Cypus b The Cypus Istitute, 7 Kypaoos Steet, 6 Nicosia, Cypus aticle ifo abstact Aticle histoy: Available olie 3 Septembe Keywods: Heschel Bulkley fluid Poiseuille flow Navie slip Pessue-depedet slip Compessibility I this wok, the combied effects of compessibility ad slip i Poiseuille flows of Heschel Bulkley fluids ae ivestigated. The desity is assumed to obey a liea equatio of state, ad wall slip is assumed to follow Navie s slip coditio with eo slip yield stess. The flow is cosideed to be weakly compessible so that the tasvese velocity compoet is eo ad the pessue is a fuctio of the axial coodiate. Appoximate semi-aalytical solutios of the steady, ceepig, plae ad axisymmetic Poiseuille flows ae deived ad the effects of compessibility, slip, ad the Bigham umbe ae discussed. I the case of icompessible flow, it is show that the velocity may become plug at a fiite citical value of the slip paamete which is ivesely popotioal to the yield stess. I compessible flow with slip, the velocity teds to become plug upsteam, which justifies the use of oe-dimesioal models fo viscoplastic flows i log tubes. The case of pessue-depedet slip is also ivestigated ad discussed. Ó Elsevie B.V. All ights eseved.. Itoductio Slip at the wall occus i may flows of complex fluids, such as suspesios, emulsios, polyme melts ad solutios, miscella solutios, ad foams, leadig to vey iteestig pheomea ad istabilities. The implicatios of slip have bee eviewed by vaious eseaches [,]. I ode to bette udestad ad simulate slip effects, it is ecessay to have ealistic slip velocity models. I a ecet eview, Hatikiiakos [] classified slip models ito static (weak slip) ad dyamic oes ad poited out that the fome ae ot valid i tasiet flows, sice slip elaxatio effects might become impotat, leadig to delayed slip ad othe pheomea. The expeimetal data show that the slip velocity is i geeal a fuctio of the wall shea stess, the wall omal stess (which icludes pessue), the tempeatue, the molecula weight ad its distibutio, ad the fluid/wall iteface, e.g. the iteactio betwee the fluid ad the solid suface ad suface oughess (see Ref. [] ad efeeces theei). Neto et al. [3] eviewed expeimetal studies of wall slip of Newtoia liquids ad discussed the effects of suface oughess, wettability, ad the pesece of gaseous layes. Moe ecetly, Sochi [4] eviewed slip at fluid solid itefaces fom diffeet pespectives, such as slip factos, mechaisms, ad measuemet, ad discussed, i paticula, slip with o-newtoia behavio, i.e. yield stess, viscoelasticity, ad time depedecy. I the peset wok we focus o the effects of wall shea stess ad pessue o the steady-state slip velocity. Theefoe, we discuss oly static slip models ad efe the eade to the eview of Coespodig autho. Tel.: ; fax: addess: geogios@ucy.ac.cy (G.C. Geogiou). Hatikiiakos [] fo dyamic slip models. Navie [] was the fist to popose a slip model elatig liealy the slip velocity u w, i.e. the fluid velocity elative to the adjacet wall, to the wall shea stess, s w : u w ¼ as w a beig the slip coefficiet. The slip coefficiet vaies i geeal with tempeatue, omal stess, ad pessue, molecula paametes, ad the chaacteistics of the fluid/wall iteface. Obviously, fo a =, we have o slip, while fo a? we get pefect slip. The slip coefficiet is also defied by a b g whee g is the viscosity ad b is the extapolatio legth, i.e. the chaacteistic legth equal to the distace that the velocity pofile at the wall must be extapolated to each eo. Moe complex, o-liea slip equatios have also bee poposed. A powe-law expessio, u w ¼ as m w whee m is the powe-law expoet, has bee widely employed by seveal ivestigatos (see, e.g., [6,7]). Expeimetal data o seveal fluid systems, such as liea polymes (maily polyethylees) [,9], highly etagled polymes [], pastes [], ad colloidal suspesios [], idicate that slip occus oly whe the stess exceeds a citical value s c, which is simila to a Coulomb fictio tem ad ca be viewed as a wall shea, o itefacial, o, simply, slip yield stess. Hatikiiakos ad Dealy [9] poited out that slip model (3) fails to descibe the slip velocity ðþ ðþ ð3þ 377-7/$ - see fot matte Ó Elsevie B.V. All ights eseved.

2 9 Y. Damiaou et al. / Joual of No-Newtoia Fluid Mechaics 93 (3) 9 i the eighbohood of s c, which is citical i udestadig polyme slip pheomea. They thus used the followig Bigham-type equatio: u w ¼ ; s w 6 s c as m w ; s w P s c The followig geeal pheomeological slip equatio u w ¼ ; s w 6 s c aðs w s c Þ m ; s w P s c has bee used by vaious eseaches i the aalysis of squeee flow of geealied Newtoia fluids with appaet wall slip [3,4]. As aleady metioed, the depedece of the slip velocity o the omal stess is weake tha that o the shea stess. I geeal, slip velocity deceases with pessue, i.e. slip occus ea the exit of a tube ad is educed upsteam. Expeimetal evidece fo this pheomeo was povided i the late sixties by Viogadov ad Ivaova [] who caied out capillay extusio expeimets showig that melt factue was suppessed at elevated pessues, a effect attibuted to the eductio of slip at high pessues. Hill et al. [6] poposed a famewok of adhesive failue betwee a highly-stessed polyme melt ad the wall based o the theoy of elastome adhesio, which leads to a slip elatio fo polyethylee melts that shows a powe-law depedece of the slip velocity o the wall shea stess ad a expoetial depedece o the isotopic pessue: u w ¼ a e a p s m w whee a ad a ae mateial costats ad p is the pessue. The same expessio was late used by Peso ad De [6] i a study of the flow of a powe-law fluid i a chael. Similaly, Hatikiiakos ad Dealy [7] fomulated a theoetical model based o a extesio of aalysis of Lau ad Schowalte [7], whose fom is simila to that of Eq. (), despite its diffeet theoetical basis. Howeve, they foud a stoge pessue depedece at modeate pessues ad satuatio at highe pessues [7]. Tag ad Kalyo [] developed a mathematical model descibig the time-depedet pessue-dive flow of compessible polymeic liquids subject to pessue-depedet slip ad epoted that udamped peiodic oscillatios i pessue ad mea velocity ae obseved whe the bouday coditio chages fom weak to stog slip. I ode to descibe the pessue depedece of the slip coefficiet, they used the followig expessio j p a ¼ a ð7þ p whee p is the atmospheic pessue. The positive expoet j is equal to fo Kudse flow (i.e. fo compessible gas flow) ad is detemied expeimetally fo polyme melts ad suspesios. With the exceptio of some ecet applicatios i mico- ad ao-fluidics, the issue of wall slip fo Newtoia fluids has bee of athe limited iteest []. The peset wok coces fluid systems with a yield stess, such as cocetated suspesios, gels, foams, dillig fluids, food poducts, ad aocomposites, fo which wall slip is commoplace [4] ad may lead to spectacula effects [3]. I the case of cocetated suspesios, slip is due to the displacemet of the dispese phase away fom solid walls [9]. Sochi [4] poits out that slip effects i o-newtoia systems become paticulaly impotat, sice they affect the shea ate ea the wall ad thus shea-ate-depedet paametes, such as the viscosity. I thei eview pape o squeee flow theoy, Egma et al. [] cosideed a wide class of mateials, icludig yieldstess ad viscoelastic fluids, ad poited out that eliable esults ca be obtaied oly if wall slip effects ae take ito accout. ð4þ ðþ ð6þ Thee ae umeous expeimetal woks demostatig slip with viscoplastic mateials. The hydaulic factuig gels studied by Jiag et al. [] obeyed a Heschel Bulkley costitutive equatio ad followed a powe-law slip equatio. Yilmae ad Kalyo [3] studied slip effects i capillay ad paallel disk tosioal flows of highly filled suspesios ad foud that the slip velocity iceased appoximately liealy with the shea stess (at high values of the wall shea stess). Moeove, they epoted that, due to wall slip, the velocity i capillay flows is almost plug above a citical shea stess. Piau [] studied the heology ad slip of cabopol gels i heometes. He used the Heschel Bulkley costitutive equatio to descibe thei heology ad slip Eq. () (with m =) to descibe slip at the wall. Foams ae also kow to exhibit both yield stess ad slip [3,4]. Ballesta et al. [] employed a liea slip equatio with a theshold shea stess (Eq. ()) fo had-sphee colloidal glasses obeyig the Heschel Bulkley costitutive equatio. Adakai et al. [] pefomed expeimets o a commecial toothpaste ad showed the existece of yield stess ad thixotopy. They also epoted that sevee slip occus i capillay flow with diffeet die desigs ad employed a liea elatioship fo slip, i ode to simulate toothpaste extusio. May impotat applicatios equie the study of weakly compessible flows of yield-stess fluids. A otable example is the time-depedet flow of waxy cude oils i a tube ivestigated umeically by Viay et al. [6], who employed the Bigham model ad a expoetial equatio of state ad used the augmeted Lagagia method. Belblidia et al. [7] solved the timedepedet, weakly compessible extudate-swell flow of a Heschel Bulkley fluid. Thei study showed that extudate swell is ualteed by compessibility ude o-slip wall coditios. They poited out that it is expected that this positio will be alteed ude slip-wall settigs. Taliadoou et al. [] deived appoximate semi-aalytical solutios of the axisymmetic ad plae Poiseuille flows of weakly compessible Heschel Bulkley fluid with o slip at the wall. The two-dimesioal axial velocity was assumed to be give by a expessio simila to that fo the icompessible flow, with the pessue-gadiet ad the yield stess poit assumed to be fuctios of the axial coodiate. The effects of compessibility have bee studied by usig a liea ad a expoetial equatio of state. These esults showed the pessue equied to dive the flow fo a give tube legth is educed with compessibility. Moeove, the two-dimesioal axial velocity was chaacteied by pluglike egios the sie of which iceases upsteam, i ageemet with moe sophisticated umeical simulatios [6]. The objective of the peset wok is to exted the wok of Taliadoou et al. [] allowig slip at the wall. I ode to ivestigate the combied effects of compessibility ad slip, we solve appoximately the plae ad axisymmetic Poiseuille flows of weakly compessible fluids with yield stess, i.e. fluids obeyig the Heschel Bulkley costitutive equatio, ude the lubicatio appoximatio assumptios used by Peso ad De [6] fo powe-law fluids. A liea equatio of state elatig the fluid desity to the pessue is employed [7]. To accout fo slip, Navie s slip coditio is assumed to hold at the wall ad the itefacial yield stess is take to be eo (s c = ). Moeove, the slip coefficiet is allowed to be pessue depedet. Both a liea ad a expoetial model ae employed to descibe the pessuedepedece of the slip coefficiet. The pape is ogaied as follows. I Sectio, we summaie the solutio of the oe-dimesioal icompessible axisymmetic Poiseuille flow of a Heschel Bulkley fluid with Navie (i.e. pessue-idepedet liea) slip at the wall ad discuss the udelyig assumptios. I Sectio 3, the esults ae exteded to the two-dimesioal weakly compessible Poiseuille flow. Aalytical ad semi-aalytical esults ae peseted fo both the icompessible ad compessible flows ad compaisos ae made with avail-

3 Y. Damiaou et al. / Joual of No-Newtoia Fluid Mechaics 93 (3) 9 9 able petubatio solutios. I Sectio 4, the case of pessuedepedet slip is aalyed ad discussed. The equatios fo the plaa compessible Poiseuille flow ae give i Appedix A.. Icompessible Poiseuille flow with slip The tesoial fom of the costitutive equatio of a compessible Heschel Bulkley fluid with eo bulk viscosity (which implies that the viscosity foces ae oly due to shea ad ot to volume vaiatios [6]) is: < _c ¼ ; s 6 s : s ¼ s _c þ k _c _c 3 ui ; s P s whee s is the stess teso, u is the velocity vecto, I is the uit teso, ad _c is the ate-of-stai teso, i.e. _c u þðuþ T ð9þ whee u is the velocity-gadiet teso, ad the supescipt T deotes its taspose. Moeove, k is the cosistecy idex, is the powe law expoet, ad _c ad s ae espectively the magitudes of _c ad s, e.g. ffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi _c II _c ¼ _c : _c ðþ II beig the secod ivaiat of a teso. The powe-law fluid ad the Bigham plastic ae the special cases of the Heschel Bulkley model fo s = ad =, espectively. I the peset wok, we coside the steady, lamia axisymmetic Poiseuille flow of a Heschel Bulkley fluid i a tube of adius R, as show i Fig.. We also assume that slip occus alog the wall accodig to Navie s slip Eq. (). I Sectios ad 3, the slip coefficiet a is assumed to be costat. As aleady metioed, the limitig case a? coespods to full-slip, wheeas a = coespods to the o-slip bouday coditio. I Sectio 4, we ivestigate the moe geeal case whee a vaies with pessue. The solutio of the steady, icompessible Poiseuille flow of a Heschel Bulkley fluid with slip at the wall is staightfowad. This has bee povided ude diffeet foms by Kalyo ad co-wokes [9]. It is peseted hee i ode to show the aalogy with the weakly compessible solutio of Sectio 3 ad to itoduce the o-dimesioaliatio of the poblem. Ude the assumptios of uidiectioality ad eo gavity, the -mometum equatio becomes ðs Þ ¼ ðþ ðþ whee the pessue gadiet ( dp/d) is costat. The costitutive Eq. () is simplified as ¼ ; js j 6 s s ¼ s ðþ >: ; js j P The esultig axial velocity compoet is give by u ðþ ¼u w þ = = ð þ Þk = d < ðr Þ =þ ; 6 6 h i : ðr Þ =þ ð Þ =þ ; 6 6 R whee u w is the slip velocity, give by u w ¼ ar d ad s ¼ ð dp=dþ < R ð3þ ð4þ ðþ deotes the yield poit, i.e. the poit at which the mateial yields. Note that whe ( dp/d)<s /R the fluid moves with uifom velocity u w. The volumetic flow ate is give by Q ¼ pr p u w þ = R =þ3 =þ = ð3 þ Þk = d R þ þ R þ þ R ð6þ I what follows, it is pefeable to wok with dimesioless equatios. We thus scale legths by the tube adius, R, the velocity by the mea velocity, V, i the capillay, ad the pessue ad the stess compoets by kv. With these scaligs, the dimesioless R fom of the slip equatio is s w ¼ u w ð7þ A whee A kav R ðþ is the slip umbe. The o-slip ad full-slip limitig cases ae ecoveed whe A? ad, espectively. The dimesioless vesio of the costitutive equatio, i.e. of Eq. (), ¼ ; >: s ¼ B whee B s R kv js j 6 ; js j P ð9þ ðþ is the Bigham umbe. The dimesioless velocity pofile is witte as follows u ðþ ¼u w þ = = ð þ Þ d < ð Þ =þ ; 6 6 h i ðþ : ð Þ =þ ð Þ =þ ; 6 6 Fig.. Geomety of axisymmetic Poiseuille flow of a Heschel Bulkley fluid. whee u w ¼ A d ad ¼ B ð dp=dþ 6 ðþ ð3þ

4 9 Y. Damiaou et al. / Joual of No-Newtoia Fluid Mechaics 93 (3) 9 If ( dp/d) 6 B, the fluid is slidig with uifom velocity u w. Othewise, the dimesioless pessue-gadiet is a solutio of the followig equatio: = 3 þ ð u w Þ 3 ¼ =þ B d d " þ 4B # dp B þ ð4þ d þ d ð þ Þð þ Þ I the case of a Bigham plastic ( = ) flow with slip at the wall, Eq. (4) is educed to 3ð þ A Þ 4 ðb þ 3Þ 3 þ 6B 4 ¼ ðþ d d u u.... A = B=, =... A =... B=, = No slip B. A = Fig. 3. Vaiatio of the yield poit i icompessible flow of a Heschel Bulkley fluid fo vaious values of the powe-law expoet: o slip ad slip with A =. (the citical Bigham umbe is B = ). B.... A =.. A =..3. B=, = u w - =.. u. - - B. (c) Fig.. Velocity pofiles i icompessible flow with vaious values of the slip umbe: Newtoia fluid; Bigham fluid with B = (the citical slip umbe is A,cit =.); ad (c) Bigham fluid with B = (the citical slip umbe is A,cit =.). Fig. 4. Slip velocities fo icompessible flow vesus the Bigham umbe fo diffeet powe-law expoets ad A =.. Note that the citical Bigham umbe is. I the case of the flow of a powe-law fluid (B = ), oe gets: = 3 þ A ¼ = ð6þ d d

5 Y. Damiaou et al. / Joual of No-Newtoia Fluid Mechaics 93 (3) dp/d 3 = - A = The above equatio ca be solved aalytically oly fo cetai values of the powe-law expoet. Fo a Newtoia fluid ( = ) the stadad Poiseuille flow solutio with slip is ecoveed: ¼ ad p ¼ d þ A þ A ð7þ u w ¼ A þ A ðþ B Δp =/(B) Fig.. Pessue gadiet fo icompessible flow vesus the Bigham fo diffeet values of the slip umbe A ad =. The citical value of the pessue gadiet is /B. -dp/d -dp/d =.6.4. B= -4-4 =.6 A B=.4. B A Fig. 6. Pessue gadiet fo icompessible flow vesus the slip umbe A fo diffeet values of the powe-law expoet: B = (powe-law fluid) ad B = ; the citical value of A is.. ad u ðþ ¼ A þ ð Þ þ A ð9þ Similaly, fo = / ad /3 oe fids espectively ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ þ A d ð3þ ad qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi =3 ¼ 64A 3 d þ 9 A þ 4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi =3 ð3þ 64A 3 þ 9 þ 4 The coespodig velocity pofiles ae the give as special cases of Eq. () (with = ): u ðþ ¼A þ = ð =þ Þ ð3þ d = ð þ Þ d I the geeal case, fo ay values of B,, ad A, the oliea Eq. (4) is easily solved fo the pessue gadiet, ad the the velocity pofile ca be calculated by meas of Eq. (). I Fig., the velocity pofiles of a Newtoia (B = ) ad two Bigham fluids (B = ad ) ae show fo vaious values of the slip umbe. With the Newtoia fluid, the velocity teds to a plug pofile (u x = u w = ) i the limit of ifiite A (full slip). Iteestigly, with viscoplastic fluids, the plug velocity pofile is attaied at a fiite value of A at which the yield distace becomes. This pheomeo is illustated i Fig. 3, whee the yield distaces fo =.,, ad. ae plotted vesus the Bigham umbe. We obseve that the u u.. = - - B=, =, B=., No slip... = - - B=, =, B=., A =.. Fig. 7. Velocity pofiles acoss the tube i compessible Newtoia flow with B =.: o slip (A = ) ad pessue-idepedet slip (A =.).

6 94 Y. Damiaou et al. / Joual of No-Newtoia Fluid Mechaics 93 (3) 9 u u.. B=4, =, B=., No slip -... = - = - B=4, =, B=., A =. -. Fig.. Velocity pofiles acoss the tube i compessible Bigham flow with B = 4 ad B =.: o slip (A = ) ad pessue-idepedet slip (A =.). become (Fig. 4). Equivaletly, fo a give Bigham umbe, solutios beyod a citical value of A ae ot admissible. Note that fo values of A above a citical value the LHS of Eq. (4) becomes egative while the RHS is always positive, sice ( dp/d)>b. The citical value of A at which both sides of Eq. (4) vaish is A ;cit ¼ d cit ¼ B ð33þ Similaly, a citical Bigham umbe is defied whe the value of the slip umbe is specified. Theoetically, thee ca be plug flow i the limitig case ( dp/d)=b whe liea slip is allowed ad the slip umbe is give by Eq. (33). Fig. 4 also illustates the vaiatio of the slip velocity with the Bigham umbe. A plateau is obseved iitially coespodig to essetially Newtoia flow, but the u w iceases apidly eachig uity at the coespodig citical Bigham umbe. The combied effects of the Bigham ad slip umbes o the pessue gadiet equied to dive the flow ae illustated i Fig.. As the slip umbe is iceased, the depedece of the pessue gadiet o B ad the coespodig citical Bigham umbe ae educed. Fially, Fig. 6 shows plots of the pessue gadiet fo B = ad ad vaious values of the expoet. Obviously, the equied pessue gadiet iceases with. All cuves ae hoiotal iitially but as slip becomes stoge, ( dp/d) deceases apidly. I the case of a powe-law fluid, it goes asymptotically to eo, wheeas i the case of a yield-stess fluid it goes dow to the citical value B at a fiite value of the slip umbe. The above emaks fo the existece of a fiite citical slip umbe hold oly whe the itefacial yield stess is eo. If, fo exam- B=4, =, B=. 6 Newtoia, B=. -dp/d No slip A =. -dp/d 4 No slip A = Fig. 9. Vaiatio of the pessue gadiet withi the tube i the case of a compessible Newtoia fluid with B =. ad L =. No slip B=4, =, B=. A =. sie of the yielded egio is educed as the powe-law expoet is iceased. Whe o-slip is applied (Fig. 3a), teds to uity asymptotically as B goes to ifiity. Whe slip occus (Fig. 3b), becomes ad the velocity pofile is plug at a citical Bigham umbe, the value of which is idepedet of the powe-law expoet. I othe wods, the flow becomes plug at a citical wall shea stess, which agees, fo example, with expeimetal obsevatios o highly filled suspesios [3]. It is clea that whe Navie slip with eo slip yield stess is allowed, fo a give value of A thee is a uppe boud fo the Bigham umbe at which both the yield distace ad the slip velocity Fig.. Vaiatio of the pessue gadiet ad the yield poit withi the tube i the case of a compessible Bigham fluid with B=4, B =., ad L =.

7 Y. Damiaou et al. / Joual of No-Newtoia Fluid Mechaics 93 (3) 9 9 ple, s c = s ad slip is descibed by Eq. () with m =, the the dimesioless slip velocity i axisymmetic Poiseuille flow whe s w > s is give by u w ¼ A ad, thus, d B A ;cit ¼ B d cit ð34þ! ð3þ The aalysis of flow i the case of o-eo slip yield stess is out of the scope of the peset wok. 3. Weakly compessible Poiseuille flow with slip =. Fo a weakly compessible flow, it ca be assumed that the adial velocity compoet is eo. This assumptio is cosistet up to fist ode with Newtoia petubatio solutios i tems of compessibility [3]. Assumig futhe /@, the _c j@u =@j. I ceepig flow, the -mometum equatio is educed to Eq. () whee the pessue gadiet is ow a fuctio of. These assumptios ae valid i vey log tubes, i.e. whe R/ L [6]. With the temiology of Wachs et al. [3] the peset model is a.d model. Howeve, this ca simply be viewed as a lubicatio appoximatio model [6]. Beig a fuctio of the pessue, the desity also vaies acoss the tube, i.e. q = q(). Fo the mass to be coseved, thee must be qðþqðþ ¼Q ð36þ whee Q() is the volumetic flow ate ad Q = Q(). The dimesioless axial velocity (scaled by the mea velocity, V, at the exit of the capillay) is the give by u ð; Þ ¼A ðþþ = ðþ d = ð þ Þ d < ð ðþþ =þ ; 6 6 h i ð37þ : ð ðþþ =þ ð ðþþ =þ ; 6 6 whee is ow a fuctio of : B ðþ ¼ ð dp=dþðþ < ð3þ It is clea that at the capillay exit ( = ), Eqs. (37) ad (3) yield the icompessible flow solutio. As poited out by Viay et al. [6], () i steady compessible Poiseuille flow is just a pseudo-yield poit, i.e. a coveiet idealiatio. Sice the axial velocity vaies alog the /@ > ad thus _c is oeo, which implies that uyielded egios, simila to the classical plug egios, caot be obtaied. Viay et al. [6] have also calculated steady-state velocity pofiles at the ilet ad the outlet of the tube with the plug egio at the cete coespodig to half the pipe adius. The pessue depedece of the desity is take ito accout by meas of a liea themodyamic equatio of state. At costat 7 B=, =, B=, A =.... =... -dp/d 6.. = = = B=, =, B=, A =.. =... u w Fig.. Velocity cotous i icompessible Newtoia flow with pessuedepedet slip fo A =. ad vaious values of (expoetial model); L = Fig.. Pessue gadiet ad slip velocity i icompessible Newtoia flow with pessue-depedet slip fo A =. ad vaious values of (expoetial model); L =.

8 96 Y. Damiaou et al. / Joual of No-Newtoia Fluid Mechaics 93 (3) 9 = A =. A =. =. =. A = whee the efeece pessue, p, has bee set to eo, ad B is the compessibility umbe, B bkv R ð4þ The pessue gadiet ( dp/d)() acoss the capillay, i.e. fo 6, ca be calculated usig the cosevatio of mass, i.e. Eq. (36). It tus out that the pessue gadiet is a solutio of the followig equatio = 3 þ 3 d qðpþ u w ¼ =þ B d " þ 4B # dp B þ ð4þ d þ d ð þ Þð þ Þ which ivolves the pessue-depedet desity of the fluid. The pessue gadiet ca be viewed as a fuctio of p ad is expected to decease upsteam. Eq. (4) ca be itegated aalytically ad solved fo p oly i the case of a Newtoia fluid (B =, = ). It tus out that -dp/d Icompessible flow with = B=, =, B=., A = = B=, =, B=., A =. Icompessible flow with = Fig. 3. Velocity cotous i compessible Newtoia flow with pessue-depedet slip fo B =., A =. ad vaious values of (expoetial model); L =. tempeatue ad fo low pessues, the desity ca be epeseted by the liea appoximatio of the stadad expoetial expessio: q ¼ q ½ þ bðp p ÞŠ ð39þ whee b ð@t=@pþ p ;T =t is the isothemal compessibility assumed to be costat, t is the specific volume, q ad t ae, espectively, the desity ad the specific volume at the efeece pessue p, ad tempeatue, T. The equatio of state is odimesioalied scalig the desity q by q ad the pessue as i Sectio : q ¼ þ Bp ð4þ u w.3.. = Fig. 4. Pessue gadiet ad slip velocity i compessible Newtoia flow with pessue-depedet slip fo B =., A =. ad vaious values of (expoetial model); L =.

9 Y. Damiaou et al. / Joual of No-Newtoia Fluid Mechaics 93 (3) 9 97 = =... B = ; fo B = 4 a much lowe value was used (A =.), fo easos to be discussed below. Due to compessibility, the mea velocity is educed upsteam. Whe slip occus the adial depedece of the solutio is weake ad the velocity pofile teds to become flat. This effect becomes moe stikig i the case of Bigham fluids (Fig. ). The pessue gadiets acoss the tube fo the two flows of Figs. 7 ad ae plotted i Fig. 9 ad Fig. a. As demostated by Taliadoou et al. [] fo the o-slip case, i compessible flow the pseudo-yield poit moves towads the wall upsteam. This pheomeo is acceleated by slip, as illustated i Fig. b, whee the vaiatio of () acoss the tube is show. I the case of o slip, the pseudo-yield poit moves to the wall asymptotically. I the case of slip ad ude ou assumptios the B=, =, B=, A =. =. 9. =... -dp/d = B=, =, B=, A =. Fig.. Velocity cotous i icompessible Bigham flow with pessue-depedet slip fo B =,A =. ad vaious values of (expoetial model); L =. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! pðþ ¼ 6 B B þ A ad the coespodig velocity pofile is: A þ ð Þ u ð; Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð þ A Þ 6 þa B The slip velocity is obviously give by A u w ðþ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð þ A Þ 6 þa B ð43þ ð44þ ð4þ The esults i (43) (4) agee with the petubatio solutio of Poyiadji et al. [3] up to fist ode. I the geeal case, the pessue gadiet ad the pessue ae calculated umeically. Oce the pessue p() is kow at a poit (e.g., p() = ), the pessue gadiet ( dp/d)() ca be calculated fom Eq. (4). The the pessue ca be calculated by settig p = at the exit plae ( = ) ad itegatig the pessue gadiet movig upsteam []. The effect of compessibility o the shape of the velocity pofile acoss the tube i the case of slip is illustated i Figs. 7 ad fo B = (Newtoia) ad 4 (Bigham plastic), whee B was set to the athe high value of. i ode to exaggeate compessibility effects. As fo the slip coefficiet, the value A =. was used fo u w. = =... B=, =, B=, A = (c) Fig. 6. Pessue gadiet, yield poit positio, ad slip velocity (c) i icompessible Bigham flow with pessue-depedet slip fo B =,A =. ad vaious values of (expoetial model); L =.

10 9 Y. Damiaou et al. / Joual of No-Newtoia Fluid Mechaics 93 (3) 9 flow becomes plug at a fiite distace upsteam the exit. Similaly, the pessue gadiet is educed dow to the citical value B. qðpþ½ þ AðpÞŠ ¼ d ð46þ 4. Poiseuille flows with pessue-depedet slip which esults i the followig expessio fo the velocity pofile: I the case that the slip coefficiet is pessue depedet, the above aalysis still holds ude the assumptios u y = with A i Eq. () ow eplaced by a fuctio of pessue, A(p). As aleady metioed, the aalysis of Peso ad De [6] fo the flow of a powe-law fluid i a slit with pessue-depedet wall slip is based o the same assumptios (lubicatio appoximatio). Fo a Newtoia fluid (B = ad = ), oe gets: 9 B=, =, B=., A =... =. -dp/d = = B=, =, B=., A =. =. A =. =. A = u w A= =.. B=, =, B=., A =. (c) Fig. 7. Velocity cotous i compessible Bigham flow with pessue-depedet slip fo B =,B =., A =. ad vaious values of (expoetial model); L =. Fig.. Pessue gadiet, yield poit positio, ad slip velocity (c) i compessible Bigham flow with pessue-depedet slip fo B =, B =., A =. ad vaious values of (expoetial model); L =.

11 u ð; Þ ¼ AðpÞþ qðpþ½ þ AðpÞŠ ð47þ I the peset wok we cosideed two models descibig the pessue depedece of A. The fist is liea AðpÞ ¼A p; 6 p 6 A = ð4þ while the secod oe is expoetial [6]: AðpÞ ¼A e p whee is a dimesioless costat: B= B=. Y. Damiaou et al. / Joual of No-Newtoia Fluid Mechaics 93 (3) ð49þ ka V R ðþ Obviously whe =, the pessue-idepedet Navie slip coditio is ecoveed i both cases. The two models ae equivalet fo small values of ad small pessues. Substitutig Eqs. (4) ad (4) ito Eq. (46) we get ð þ BpÞ½ þ A pš ¼ ðþ d Fom the above equatio it is diectly deduced that the effect of pessue-depedet slip is opposite to that of compessibility. Itegatig Eq. () esults i a cubic equatio that ca be solved aalytically fo the pessue p. I the case of icompessible flow (B =) oe gets: s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! pðþ ¼ þ A ad þ ð þ A Þ u ð; Þ ¼ ða pþþð Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð þ A Þ þ ðþa Þ 6 B=. B=, =, A =., =. ðþ ð3þ B=.. -dp/d B= B=, =, A =., =. B=..4.. u w.3. B=... B= Fig. 9. Velocity cotous i Newtoia flow with pessue-depedet slip fo vaious compessibility umbes; A =., =. (expoetial model), ad L =. Fig.. Pessue gadiet ad slip velocity i Newtoia flow with pessuedepedet slip fo vaious compessibility umbes; A =., =. (expoetial model), ad L =.

12 Y. Damiaou et al. / Joual of No-Newtoia Fluid Mechaics 93 (3) 9 The umeical esults peseted below have bee obtaied usig the expoetial model i Eq. (49), with which o aalytical solutio ca be obtaied. The velocity cotous i icompessible Newtoia flow fo A =. ad vaious values of ae give i Fig.. Fo eo o small values of the cotou lies ae hoiotal idicatig that the flow is essetially oe-dimesioal. At highe values of the depedece of u o becomes stoge ad thus the cotous ae bedig towads the axis of symmety. Fig. shows the pofiles of the pessue gadiet ad the slip velocity acoss the tube fo A =. ad vaious values of. Obviously, fo = (o pessue-depedece) both the pessue gadiet ad the slip velocity ae costat, sice the flow is oedimesioal. Fo small values of (weak pessue-depedece), the pessue gadiet deceases ad the slip velocity iceases liealy acoss the tube. Fo high values of (stog pessue depedece), slip is esticted oly ea the exit of the tube. Upsteam, the slip velocity is essetially eo ad the pessue gadiet is costat. The combied effects of compessibility ad pessue-depedet slip fo the Newtoia case ae illustated i Fig. 3, whee the velocity cotous fo B =., A =. ad vaious values of ae show. As the pessue depedece is iceased, the velocity cotous ted to become hoiotal ad a velocity peak appeas close to the tube exit. The vaiatio of the pessue gadiet ad the slip velocity with is illustated i Fig. 4. Note that due to compessibility both the pessue gadiet ad the slip velocity ae iceasig with the axial distace whe =. As expected, the slip velocity is educed ad the pessue gadiet iceases with. At high values of, howeve, thee appeas a maximum of the pessue gadiet which moves towads the exit as is iceased ad slip becomes moe localied. The effects of the slip decay paamete i the case of Bigham flow ae illustated i Figs., whee velocity cotous ad esults fo the pessue gadiet, the yield poit, ad the slip velocity ae peseted fo both the icompessible ad compessible cases. I icompessible flow (B =.), the sie of the pseudo-uyielded egio is educed upsteam as the slip velocity is educed (Figs. ad 6 fo B = ad B = ). I the compessible case, the esults ae moe iteestig, sice at high values of the adius of the pseudo-uyielded egio appeas to pass though a miimum (Fig. ) leadig to the appeaace of uyielded islads (Fig. 7). This pheomeo becomes moe poouced as the Bigham umbe is iceased. To ivestigate futhe the combied effects of slip pessuedepedecy ad compessibility, we fixed the slip paametes to A =. ad =. (always fo the expoetial model) ad vaied the compessibility umbe. It should be oted that the velocity pofile at the exit plae ( = ) is idepedet of the compessibility umbe, sice this is by assumptio the pofile fo icompessible flow with costat slip. I Fig. 9, a evesal of the velocity cotou patte is obseved as the compessibility umbe is iceased. Iitially, the effects of the slip pessuedepedecy pevail but as B is iceased these ae coutebalaced ad the suppessed by compessibility effects. Fo B =. the competig effects of ad B ae equivalet so that the velocity cotous ae almost eveywhee hoiotal. The coespodig pessue-gadiet ad slip velocity distibutios ae give i Fig.. We obseve i Fig. a that fo small compessibility um- B=, =, B=, A =., =. B=, =, B=., A =., =.. - =. = u u (c) B=, =, B=., A =., =. B=, =, B=., A =., =.. - =. = u u (d) Fig.. Velocity pofiles at the ilet ad the outlet i Newtoia flow with pessue-depedet slip fo A =., =. (expoetial model) ad vaious compessibility umbes; L =

13 Y. Damiaou et al. / Joual of No-Newtoia Fluid Mechaics 93 (3) 9 bes the pessue gadiet is deceasig acoss the tube. Howeve, this ted chages as B iceases. Fo a citical value betwee. ad. the pessue gadiet is oughly costat ad the velocity pofile is weakly depedet o. This effect is illustated i Fig. whee the velocity pofiles at the ilet ad the outlet ae compaed fo B =,.,., ad.. As expected, the mea velocity is educed upsteam due to the cosevatio of mass. At some low value of B (Fig. b) the velocity pofiles at the ilet ad the outlet ae quite simila, but at highe values of B the velocity at the ilet is educed damatically ad teds to become flat. This pheomeo B= becomes moe stikig i the case of Bigham flow i which the velocity i the uyielded egio is flat ayway. I Fig., we show esults obtaied with B =,A =. ad =.. As i the Newtoia case, the velocity cotou patte chages damatically as compessibility is iceased. Fo small compessibility umbes, the pseudo-yield distace is slightly educed upsteam, but at highe compessibility umbes this ted is evesed. As i the case of pessue-idepedet slip, the flow upsteam teds to become plug whe the compessibility of the fluid is take ito accout.. Coclusios B=. B=.... Appoximate semi-aalytical solutios of the steady, ceepig, weakly compessible plae ad axisymmetic Poiseuille flows of a Heschel Bulkley fluid with slip at the wall have bee deived, ude lubicatio appoximatio assumptios, employig a liea equatio of state ad Navie s slip coditio with eo slip yield stess. The combied effects of compessibility, slip, ad yield stess ad the case of pessue-depedet slip have bee ivestigated. I ageemet with pevious woks [6,], it was show that whe slip is peset ad the yield stess fluid is compessible, the velocity upsteam, teds to become plug, which justifies the use of aveaged models i solvig viscoplastic flows i log tubes. This effect is ehaced with the pesece of slip. It has also bee demostated that the effect of slip pessue-depedecy is opposite to that of compessibility. As fo the futue plas, we ited to exted the peset aalysis to.d time-depedet models. Appedix A. Compessible plae Poiseuille flow with slip B=. B=. B=. Fig.. Velocity cotous i Bigham flow with pessue-depedet slip fo B = ad vaious compessibility umbes; A =., =. (expoetial model), ad L =... I plae Poiseuille flow, legths ae scaled by the chael-halfwidth, H, the velocity by the mea velocity, V, at the exit of the chael, ad the pessue by kv /H. Ude the same assumptios used fo the axisymmetic flow, the dimesioless velocity pofile i the case of compessible plae flow is witte as follows: u x ðx; yþ ¼u w ðxþþ = dp ðxþ þ dx ( ð4þ ð y ðxþþ =þ ; 6 y 6 y ½ð y ðxþþ =þ ðy y ðxþþ =þ Š; y 6 y 6 whee the slip velocity ad the yield poit ae give by u w ¼ A dx ad B y ¼ ð dp=dxþðxþ A ad The slip ad Bigham umbes ae espectively defied by akv H B s H kv ðþ ð6þ ð7þ ðþ It tus out that the dimesioless pessue-gadiet is a solutio of the followig equatio: þ dx qðpþ u w ¼ =þ B dx þ dx þ B : ð9þ

14 Y. Damiaou et al. / Joual of No-Newtoia Fluid Mechaics 93 (3) 9 I the case of a powe-law fluid, Eq. (9) is simplified as follows: þ qðpþ A dx ¼ = ð6þ dx I the case of a Bigham-plastic, Eq. (6) is educed to ð þ 3A Þ 3 3 B þ þ B 3 ¼ ; ð6þ dx qðpþ dx which has the followig solutio: B ¼ dx þ 3A þ qðpþ " " ( )## þ cos 3 cos þ ð 3A Þ B 3 ðb þð=qðpþþþ 3 Refeeces ð6þ [] M.M. De, Extusio istabilities ad wall slip, A. Rev. Fluid Mech. 33 () 6 7. [] S.G. Hatikiiakos, Wall slip of molte polymes, Pog. Polym. Sci. 37 () [3] C. Neto, D.R. Evas, E. Boaccuso, H.J. Butt, V.S.J. Caig, Bouday slip i Newtoia liquids: a eview of expeimetal studies, Rep. Pog. Phys. 6 () [4] T. Sochi, Slip at fluid solid iteface, Polym. Rev. () [] C.L.M.H. Navie, Su les lois du mouvemet des fluides, Mem. Acad. Roy. Sci. Ist. F. 6 (7) [6] D.A. Hill, T. Hasegawa, M.M. De, O the appaet elatio betwee the adhesive failue ad melt factue, J. Rheol. 34 (99) 9 9. [7] S.G. Hatikiiakos, J.M. Dealy, Wall slip of molte high desity polyethylees. II. Capillay heomete studies, J. Rheol. 36 (99) [] D.S. Kalika, M.M. De, Wall slip ad extudate distotio i liea low-desity polyethylee, J. Rheol. 3 (97) 34. [9] S.G. Hatikiiakos, J.M. Dealy, Wall slip of molte high desity polyethylee. I. Slidig plate heomete studies, J. Rheol. 3 (99) [] J.M. Piau, N. El Kissi, Measuemet ad modellig of fictio i polyme melts duig macoscopic slip at the wall, J. No-Newtoia Fluid Mech. 4 (994) 4. [] M.J. Adams, I. Aydi, B.J. Biscoe, S.K. Siha, A fiite elemet aalysis of the squeee flow of a elasto-viscoplastic paste mateial, J. No-Newtoia Fluid Mech. 7 (997) 4 7. [] P. Ballesta, G. Petekidis, L. Isa, W.C.K. Poo, R. Besselig, Wall slip ad flow of cocetated had-sphee colloidal suspesios, J. Rheol. 6 () -37. [3] U. Yilmae, D.M. Kalyo, Slip effects i capillay ad paallel disk tosioal flows of highly filled suspesios, J. Rheol. 33 (99) 97. [4] P. Estellé, C. Laos, Squeee flow of Bigham fluids ude slip with fictio bouday coditio, Rheol. Acta 46 (7) [] G.V. Viogadov, L.I. Ivaova, Wall slippage ad elastic tubulece of polymes i the ubbey state, Rheol. Acta 7 (96) [6] T.J. Peso, M.M. De, The effect of die mateials ad pessue-depedet slip o the extusio of liea low-desity polyethylee, J. Rheol. 4 (997) [7] H.C. Lau, W.R. Schowalte, A model of adhesive failue of viscoelastic fluids duig flow, J. Rheol. 3 (96) [] H.S. Tag, D.M. Kalyo, Usteady cicula tube flow of compessible polymeic liquids subject to pessue-depedet wall slip, J. Rheol. () 7. [9] H.A. Baes, A eview of the slip (wall depletio) of polyme solutios, emulsios ad paticle suspesios i viscometes: its cause, chaacte, ad cue, J. No-Newtoia Fluid Mech. 6 (99). [] J. Egma, C. Seais, A.S. Bubidge, Squeee flow theoy ad applicatios to heomety: a eview, J. No-Newtoia Fluid Mech. 3 () 7. [] T.Q. Jiag, A.C. Youg, A.B. Mete, The heological chaacteiatio of HPG gels: measuemet of slip velocities i capillay tubes, Rheol. Acta (96) [] J.M. Piau, Cabopol gels: elastoviscoplastic ad slippey glasses made of idividual swolle spoges: meso-ad macoscopic popeties, costitutive equatios ad scalig laws, J. No-Newtoia Fluid Mech. 44 (7) 9. [3] A. Lide, P. Coussot, D. Bo, Viscous figeig i a yield stess fluid, Phys. Rev. Let. () [4] S. Mae, D. Lagevi, A. Sait-Jalmes, Aqueous foam slip ad othe egimes detemied by heomety ad multiple light scatteig, J. Rheol. () 9. [] H.A. Adakai, E. Mitsoulis, S.G. Hatikiiakos, Thixotopic flow of toothpaste though extusio dies, J. No-Newtoia Fluid Mech. 66 () [6] G. Viay, A. Wachs, J.-F. Agassat, Numeical simulatio of weakly compessible Bigham flows: the estat of pipelie flows of waxy cude oils, J. No-Newtoia Fluid Mech. 36 (6) 93. [7] F. Belblidia, T. Haoo, M.F. Webste, The dyamics of compessible Heschel Bulkley fluids i die-swell flows, i: T. Boukhaouda, et al., (Eds.), Damage ad Factue Mechaics: Failue Aalysis of Egieeig Mateials ad Stuctues, 9, pp [] E. Taliadoou, G.C. Geogiou, I. Moulitsas, Weakly compessible Poiseuille flows of a Heschel Bulkley fluid, J. No-Newtoia Fluid Mech. (9) [9] D.M. Kalyo, P. Yaas, B. Aal, U. Yilmae, Rheological behavio of a cocetated suspesio: a solid ocket fuel stimulat, J. Rheol. 37 (993) 3 3. [3] E.G. Taliadoou, M. Neophytou, G.C. Geogiou, Petubatio solutios of Poiseuille flows of weakly compessible Newtoia liquids, J. No- Newtoia Fluid Mech. (9) [3] A. Wachs, G. Viay, I. Figaad, A.D umeical model fo the stat up of weakly compessible flow of a viscoplastic ad thixotopic fluid i pipelies, J. No-Newtoia Fluid Mech. 9 (9) 94. [3] S. Poyiadji, G.C. Geogiou, K. Kaoui, K.D. Housiadas, Petubatio solutios of weakly compessible Newtoia Poiseuille flows with Navie slip at the wall, Rheol. Acta () 497.