Open Access. C. Escobar 1, F. Méndez*,1 and E. Luna 2

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1 The Ope Chemical Egieerig Joural Ope Access Determiatio of the Thickess of a Power-Law Fluid Drive by the Peetratio of a Log Gas Bubble i a Rectagular Chael Usig a Sigular Perturbatio Method C. Escobar F. Médez* ad E. Lua Facultad de Igeiería Uiversidad Nacioal Autóoma de México 45 Mexico Recuperació de Hidrocarburos Istituto Mexicao del Petróleo 77 Mexico Abstract: I the preset work a aalytical ad umerical study is preseted i order to determie the residual fluid film thickess of a power-law fluid o the walls of a rectagular horizotal chael whe it is displaced by aother immiscible fluid of egligible viscosity. The mathematical model describes the motio of the displaced fluid ad the iterface betwee both fluids. I order to obtai the residual film thickess m we used a sigular perturbatio techique: the matchig asymptotic method; i the limit of small capillary umber. The mai results idicated that the residual film thickess of the o-newtoia fluid decreases for decreasig values of the power-law idex which is i qualitative agreemet with experimetal results. Keywords: Gas-assisted power-law fluid displacemet gas bubble dyamics matched asymptotic lubricatio theory. INTRODUCTION The steady displacemet of Newtoia ad o- Newtoia fluids by log bubbles cofied i vertical ad horizotal cylidrical ducts or parallel plates has received cosiderable attetio durig the past decades due to the fudametal ad practical importace of this process i may idustrial applicatios. Typical examples appear i film coatig bubble colums gas-assisted ijectio moldig lubricatio theory oil recovery i aturally fractured reservoirs etc. The theoretical ad experimetal studies of this fluid dyamic problem are i geeral very complex because several physical difficulties associated with the formatio ad depositio of a liquid film i the wall of the cotaier are preseted. Usually i these problems the importat step is to determie the thickess of the film cosiderig two basic situatios: a) whe the plate is moved steadily out of a bath of the liquid with uiform velocity ad b) the slow displacemet of oe fluid by aother takig ito accout that the walls are fixed. For Newtoia fluids this problem has bee widely studied for differet cases ad partig from the pioeer aalytical works of Bretherto [] ad Cox [] ad the experimetal work of Taylor [] the basic mechaics of depositio films are ow well uderstood. I this directio we ca emphasize the followig works: Wilso [4] studied aalytically the film coatig i a plate whe it is draw steadily out of a bath of the liquid: the drag-out problem. Usig the method of matched asymptotic expasios to predict the thickess of the film for small values of the capillary umber this author showed that the aalysis of Ladau [5] for this same problem represets the leadig *Address correspodece to this author at the Facultad de Igeiería Uiversidad Nacioal Autóoma de México 45 Mexico; fmedez@servidor.uam.mx order of a asymptotic solutio where the higher order correctio terms were icluded to improve the Ladau's solutio. Park ad Homsy [6] studied i a Hele Shaw cell the two-phase displacemet i the gap betwee closely spaced plaes. These authors also used matched asymptotic expasios to estimate the thickess of the film for small values of the capillary umber cofirmig the previous results obtaied by Bretherto []. Schwartz et al. [7] i a effort to validate the theoretical predictio of Bretherto developed a experimetal procedure to measure the average thickess of the wettig film left behid durig the slow passage of a air bubble i a water-filled capillary tube. He cofirmed that for bubbles of legth less tha tube radii the agreemet is very good. The above works have a commo characteristic: the order of magitude of the experimetal ad theoretical predicted thickesses is sufficiet to avoid the presece of itermolecular forces. I this directio Teletzke et al. [8] distiguished the frotier betwee thick ad thi films recogizig that as liquid film thickess approaches molecular dimesios (thickesses m ) the itermolecular forces are of the same order of magitude of viscous ad capillary forces. A presetatio of these ew topics ca be foud i the excellet book of Middlema [9] where the presece of other effects as surface-active impurities slip at apparet cotact lies air etraimet i coatig fluids Maragoi shear stresses etc ca complicate eormously the treatmet of the problem. For istace Müch [] usig umerical ad asymptotic schemes derived the thickess of a film which forms at the tip of a capillary meiscus due to the presece of thermally iduced Maragoi shear stress effects. Therefore for Newtoia fluids the aalysis of thi ad ultrafilms is a active area of fudametal developmet that offers ew theoretical perspectives ad cotroversial aspects. 874-/8 8 Betham Ope

2 9 The Ope Chemical Egieerig Joural 8 Volume Escobar et al. However the specialized literature to treat these problems for o-newtoia fluids is scarce i spite of that there are abudat evidece ad iterestig applicatios. The pioeerig works of Posliski et al. [] ad Posliski ad Coyle [] foud that the fractio of the o-newtoia fluid deposited o the walls is less tha that of a correspodig Newtoia fluid at low capillary umber. I particular their umerical results showed that fluids with shear thiig viscosity exhibit a smaller fractioal coverage tha Newtoia fluid at the same capillary umber. Fractioal coverage was also foud to be smaller for a fluid with a smaller shear thiig idex. Kamisli ad Rya [] studied the two dimesioal flow of a power-law fluid aalytically usig a sigular perturbatio method ad experimetally i order to determie the residual liquid film thickess for a circular tube ad for a rectagular chael. The aalytical results of this work idicated that the residual liquid film thickess of o- Newtoia fluids icreases with decreasig power-law idex. Later these authors Kamisli ad Rya [4] developed a mathematical aalysis to study the motio of log bubbles ito Newtoia ad o-newtoia fluid cofied i a horizotal circular tube rectagular chaels ad square crosssectioal chaels. The model results are i qualitative agreemet with previous experimetal observatios ad give also a good agreemet with a previous umerical solutio obtaied by Posliski et al. []. I order to idetify the effects of fluid elasticity o the fractioal coverage Huzyak ad Koellig [5] studied the peetratio of a log gas bubble through a tube filled with a viscoelastic fluid. The authors developed experimets with two Newtoia fluids ad two highly elastic costat shear viscosity fluids. The fractioal coverage was characterized i terms of the capillary ad Deborah umbers; ad foud that the fractioal coverage for viscoelastic fluids is a strog fuctio of the tube diameter. I the preset work we study the motio of a iviscid fluid ito a power-law fluid cofied i a horizotal rectagular chael. We develop a mathematical model usig a sigular perturbatio method i order to determie the residual fluid film thickess of pseudoplastic fluid (characterized by a power-law idex less tha uity < ) o the walls. METHODOLOGY Cosider that a fluid of egligible viscosity is ijected displacig with costat velocityu a power-law fluid cofied i a rectagular chael as show schematically i Fig. (). The walls of the chael are ifiite log i the directio perpedicular to the plae x y ad are separated a distace R small eough to eglect the gravitatioal effects so that the solutio is symmetric about the midplae. We assume that the displaced fluid totally wets the wall leavig a film o the wall as displacemet proceeds. The surface tesio betwee the fluids is kow ad uiform; therefore the tagetial force balace at the iterface is equal to zero. O the other had we cosider that the flow is very slow ad the capillary umber which will be defied later is small. Bretherto [] foud that for sufficietly small capillary umber the viscous stresses appreciably modify the static profile of the iterface for regios oly very ear to the wall; therefore the displacemet process ca be cosidered as a sigular perturbatio problem. Fig. (). Simplified physical model. I the preset work we propose that the fluid which is foud cofied is described by the power-law fluid model that is expressed by ij = K ij where ij is the stress tesor; ij is the strai-rate tesor K ad are material parameters; the first is the idex of cosistecy ad the later is the power-law idex. Usig the power-law model i the mass ad coservatio equatios i rtesia coordiates ad takig ito accout that the problem is statioary with uiform properties we obtai that the goverig equatios for the displaced fluid are u~ v~ + = () ~ x ~ y ~ ~ ~ ~ u ~ u P u + v = + ~ ~ ~ x y x ~ ~ ~ u u v K + ~ ~ ~ + ~ ~ x x y y x ~ ~ ~ ~ v ~ v P u + v = + ~ ~ ~ x y y ~ ~ u v K ~ + ~ ~ x y x ~ v + ~ y ~ y. I Eq. () () u ~ ad v ~ represet the axial ad radial velocities respectively; P is the pressure i the displaced fluid ad x ~ ad y ~ are the rtesia coordiates. These equatios must be solved with the appropriate boudary coditios. At the ier surface of the plates we impose the well kow o-slip coditio. At the iterface we have three coditios. The first is the kiematic boudary coditio which describes that the iterface is impermeable for both fluids ad is give by v ~ = u ~~ h ' (4) () ()

3 Determiatio of the Thickess of a Power-Law Fluid Drive The Ope Chemical Egieerig Joural 8 Volume 9 where h ~ is the iterface positio ad is oly fuctio of the ~ axial coordiate h = h ~ ( ~ x) ; ad the prime refers to differetiatio with respect to ~ ~ ~ dh x ; therefore h ' =. dx ~ The other coditios are the dyamic coditios: the tagetial ad ormal stress boudary coditios. The first oe expresses that the tagetial force balace at the iterface is zero because the surface tesio is uiform. The secod coditio establishes that the ormal stresses o the two sides of the iterface are balaced by surface tesio. Usig the power-law model for the displaced fluid the dyamic coditios are give by the followig expressios K u y + v x u K x ( h' ) v y h' = K u ( + h' ) y + v u x h' x h' K v + h' y + P = + R x R y ( ) Eq. (5) ad (6) represet the tagetial ad ormal stress boudary coditio respectively; where is the iterface surface tesio. I the ormal stress coditio Eq. (6) we have used the plaar approximatio for the curvature i the y directio equivalet to settig R y =. The pricipal radius of curvature R x ca be expressed i terms of the iterface positio as: R x = h'' + h' ( ) /. To obtai the fractio of the liquid deposited o the walls of the tube m we use the techique of matched asymptotic expasios. I order to apply the metioed techique we divided the domai ito three regios show i Fig. (): the costat film thickess regio (regio I) the frot regio (regio II) ad the static regio (regio III). I regio I the problem ca be solved usig the classical lubricatio theory if the film thickess is kow. I regio III the flow is parabolic i y directio ad accordig with the referece frame i the midplae this regio is static. A detailed aalysis of regio II is required to obtai the solutio. I the limit of small capillary umber the solutio i regio I does ot match smoothly with the solutio i the costat film thickess regio. Therefore i order to obtai the complete solutio the frot regio must be subdivided i two. The capillary static regio (regio B) i which the shape of the iterface is early circular ad pressure forces ad iterfacial tesio are importat. O the other had the trasitio regio (regio A) where the shape of the iterface is deformed by viscous tractio ad thus viscous forces also become importat. (5) (6) It is ecessary to examie regio II more carefully therefore i the followig we oly studied this regio. I order to obtai represetative dimesioless parameters ad to explore the most appropriate scalig for the goverig equatios preseted lies below; we apply a order of magitude aalysis usig the scales show i Fig. (). We cosider that the iterface tesio is costat ad that the fluid of egligible viscosity is ijected with a small costat velocity U. I the capillary static regio we use the followig scales to apply the order of magitude aalysis ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ x ~ R y R h R u U v U P ~. R Usig the characteristic scales i the goverig equatios ad i the boudary coditios we defie the Reyolds ad capillary umbers for a power-law fluid respectively as follow U R KU R Re = =. K Accordig with the characteristic values associated to the problem the Reyolds umber is very small Re << ; therefore i Eqs. () ad () the covective terms ca be eglected. From the order of magitude aalysis ad i the limit of small capillary umber we obtai that the pressure i the capillary static regio is costat ad cofirms that the iterfacial tesio ad the pressure forces are the domiat effects i the metioed regio. To apply the order of magitude aalysis to the trasitio regio we must cosider tha the thickess ad the legth L x of this regio are ukow. The scales for the trasitio regio are give by ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ x ~ Lx y h u U v V P ~. R Usig the scales give above i the goverig equatios ad the boudary coditios it is easily to obtai the ukow variables for this regio; which are give by the followig expressios: R L R V U. (7) x Usig the results of the order of magitude aalysis the equatios for the capillary static regio ca be odimesioalized usig the followig scales: x-ut y h u x= y= h= u= R R R U v P v= P=. U ó/r The dimesioless goverig equatios ca be writte for the capillary static regio as u v + = x y (8)

4 9 The Ope Chemical Egieerig Joural 8 Volume Escobar et al. u u P Reu + v = + x y x u u v + + x x y y x v v P Reu + v = + x y y u v v + + x y x y y (9) () Eq. (8) is the dimesioless mass coservatio equatio ad Eqs (9) ad () are the dimesioless motio equatios. The boudary coditios i dimesioless variables are give by the followig expressios: u = v = at y = () v = uh' at y = h () u y + v x at y = h u ( h' ) x v y h' = u ( + h' ) y + v u x h' x h' v h'' + h' y + P = + h' ( ) () ( ) / at y = h. (4) As we metioed previously the problem is suitable to apply the matched asymptotic expasio techique. For this problem the small parameter is the capillary umber; the we proposed the followig expasios for the capillary static regio: j ( + j= x) = ( x y) (5) j i which represet idistictly dimesioless pressure velocity or iterface positio ad the subscript j refers to the order of the variable. Substitutig the above expasio ito Eq. (8) (4) ad collectig terms of the same power of we obtai a set of equatios i the capillary static regio. The leadig order of Eqs. (9) ad () are give by P x P = y = (6) where the subscript deotes the zeroth order approximatio. Eq. (6) implies that the leadig order to the pressure is costat as it was metioed i the order of magitude aalysis. I order to determie the value of the costat pressure the iterfacial boudary coditios have to be examied. The zeroth order iterface positio ca be obtaied from the ormal stress boudary coditio subjected to the coditios at the frot tip of the iterface: h' whe x L ad h = at x = L. We obtai the zero order solutio as: ( ) / h = P ( x L) P (7) where P is the pressure jump across the iterface ad L is the distace from the tip of the frot to the origi of the referece frame. I the trasitio regio the problem has to be rescaled. I this regio the pressure viscous force ad iterfacial tesio are all importat. Usig the order of magitude results Eq. (7) we proposed the followig chage of variables to obtai the dimesioless goverig equatios. x + l = v V = + + y + = + P = P. h = + u = u I the chage of variables l is a shift of coordiates that is determied with the matchig coditio. Usig a equivalet expasio as i the capillary static regio the zeroth order equatios for the trasitio regio are give by u V + = P u + P =. = (8) (9) () u = V = at () = V = u at = () u = at = () P = at =. (4) From the above Eq. (8) (4) it is possible to obtai a ordiary differetial equatio for the iterface give by ( ) + ( ) + d d =. (5) Eq. (5) is a o-liear ad third-order differetial equatio i which meas the leadig order for the costat film thickess i the trasitio regio that is determied by a matchig coditio. For the umerical itegratio Eq. (5) is trasformed by usig a chage of variable origially proposed by Park ad Homsy [6]. The trasformatio is give by

5 Determiatio of the Thickess of a Power-Law Fluid Drive The Ope Chemical Egieerig Joural 8 Volume 9 H = X = + s + ( )/ ( ) + t /. (6) I the above equatio s is a arbitrary costat that shifts the coordiates ad is determied by the matchig coditio. Usig Eq. (6) ito Eq. (5) we obtai the caoical form for a power-law fluid expressed as: d H dx ( ) H = H ( ) +. (7) Eq. (7) must be solved with appropriate boudary coditios which are provided by the matchig with the uiform film thickess regio (I) ad with the ear regio (III) to the frot tip. Therefore we proposed for the first regio that for X ; H. (8) For simplicity we propose a ew chage of variable for Eq. (7) of the followig form: = H. Therefore the matchig coditio is X ;. Usig the chage of variable i Eq. (8) ad liearizig it about H = accordig with the matchig coditio we obtai the followig equatio: d =. (9) dx Eq. (9) presets two differet cases. The first oe is the Newtoia limit with =. Uder this coditio Eq. (9) is a ordiary liear equatio ad equi-dimesioal i ; therefore the solutio for the Newtoia case valid for the matchig coditio ca be give by = F e () X where F is a costat ad may take ay value due to the arbitrary shift of coordiates s. For < (pseudoplastic fluid) Eq. (7) is a o-liear equatio therefore it does ot accept a expoetial solutio [6]. The secod boudary coditios for Eq. (7) is give whe the iterface is closed to the tip. This matchig coditio with regio III is H >> whe X. Therefore Eq. (7) i this limit ca be rewritte as follow H d dx. The solutio for Eq. () is () H AX +BX+C. () I the above equatio A B ad C are costats that will be determie by umerical itegratio of Eq. (7) usig a fourth order Ruge-Kutta method. The results for differet values of the power-law idex are show i Table ad i Fig. (). The iitial values to itegrate Eq. (7) are give by Eq. () for ay value of the power idex. For o- Newtoia fluids we use a iterative form. Table. Values for Costats of Itegratio A B ad C for Differet Values of the Power-Law Idex A B C Fig. (). Results of the umerical itegratio of Eq. (7) for differet power-law idexes. Oce the costats of itegratio are kow the Eq. () ca be expressed i dimesioless variables for the trasitio regio is ( )= A As ( ) / + + ( ) / ( + As )/ + + ( )/ + Bs ( ) / + ( ) / + + ( )/ + B ( + C )/. ( ) / + ( )/ + () The solutio is icomplete util the thickess t is determied. This variable ad all other quatities are determied by matchig with the solutio i the capillary static regio. The matchig coditio is give by the followig equatio: lim h( x)= lim ( xl )/+ ( ). (4) By expadig h(x) about x = l usig Taylor series expasio rewritig the expasio i trasitio regio variables ad comparig it with the left-had side term by term matchig coditios for each order ca easily be determied.

6 94 The Ope Chemical Egieerig Joural 8 Volume Escobar et al. We obtai the followig set of equatios up to terms of secod order. The zero order are ( l) h = (5) ( l) h ' = (6) / + '' ( ) h l = A ( ). + / The first order equatios are give by ( l) (7) h = (8) h' + / + ( l) = As ( + ) B / ( ) ( l). / / (9) h '' = (4) h Fially the secod order equatios are ( l) = ( + ) + Bs ( ) As / / ( l) + C / / (4) h ' = (4) ( l). h '' = (4) Eqs. (5) ad (6) give the zeroth order for pressure ad to defie the positio of the referece frame for the trasitio regio. The pressure jump across the iterface ad the shift of coordiates ca be determied from the previous equatios combied with Eq. (7). The the leadig order for the pressure jump ad the shift of coordiates are give by the followig expressios P = (44) l = L. (45) Kowig P ad l Eq. (7) ca be solved ad the residual film thickess is determied. The the film thickess expressed i dimesioless variables of the capillary static regio is give by + / + + t = A. (46) The fractio of liquid deposited o the walls is defied AT AB as: m = ; where A T is the trasverse area ad A B AT is trasverse area of the ijected fluid. For this particular case we have a rectagular trasverse area; therefore m = t. The the liquid fractio is + + / + m = A (47) The higher order correctios for both regios ca be determied i a similar way. However the details are ot preseted i this work. RESULTS I order to validate the model developed i this work we compare our theoretical predictios with the results obtaied by Park ad Homsy [6]. These authors studied the two-phase displacemet i three dimesios i a Hele Shaw cell ad preseted the followig expressios for the thickess of the film ad the pressure jump across the iterface. The results are give by t =.7 / + f ''+ f ' 4 O 4/ ( ) P = +.8 / 4 f '' + O ( / ). I the above equatios [ ] L T + (48) (49) R = where L T is the width of the Hele Shaw cell; f represets the tip projectio of the iterface oto the x z plae ad is fuctio of the z coordiate. I the case that we are studyig L = ad f = L. Therefore the results from Eqs. (44) ad (46) takig ito accout higher order correctios are / t =.75 (5) / P = (5) For Newtoia fluid case Eqs. (5) ad (5) are i good agreemet with Eqs. (48) ad (49). Eq. () expresses the fractio m of fluid remaiig after passage a fluid of egligible viscosity for ay power-law Table. Fluid Fractio Expressios for Differet Values of the Power-Law Idex Power-law idex T Fluid Fractio m /.675 /.8.7 / / / / 7..8

7 Determiatio of the Thickess of a Power-Law Fluid Drive The Ope Chemical Egieerig Joural 8 Volume 95 idex. From this equatio is clearly show that the fractio m depeds o two parameters: the capillary umber ad the fluid power idex. The ifluece of both parameters o the fluid fractio is clearly show i Table. I Figs. () ad (4) we show the fluid fractio for differet values of the power idex. It is clear that for smallest values of the capillary umber we obtai the miimum fluid fractio idepedetly of the fluid properties. O the other had for ay give capillary umber with decreasig the power-law idex the fluid fractio decreases. This result is i qualitative agreemet with the experimetal results obtaied by Kamisli ad Rya []. Fig. (). Fluid fractio versus capillary umber. capillary umber ad is idepedet of the power-law idex. For ay power-law idex the dimesioless variables for the trasitio regio suggested by Kamisli ad Rya [] are give by x + l y + h = = = / / v V = P = P. / / u = u We ca appreciate that the expasios ad the scalig of the dimesioless variables proposed by Kamisli ad Rya [] produces a divergece betwee experimetal ad umerical results. However our scalig is i accordace with i the determiatio of the residual film thickess. These ca be see i Eq. (5) ad i the dimesioless variables for the trasitio regio. For the Newtoia fluid = we recover the expasios ad dimesioless variables proposed by the authors metioed above. Nevertheless for the expasios proposed i [] are ot valid because the residual film thickess depeds o the displaced fluid properties. I order to aalyze this behavior we preset Fig. (5) i which we have plotted the ratio of the o-newtoia fluid fractio m with the Newtoia fluid fractio m for three values of the capillary umber. Fig. (5) shows that the ratio of fluid fractios decreases for decreasig values of the power-law idex; ad cofirms that the miimum fluid fractio is obtaied with smaller capillary umbers. Fig. (5). Fluid fractio ratio versus power-law idex. Fig. (4). Fluid fractio versus capillary umber (log-log). The results show i the preset study cotrast with the theoretical results obtaied by Kamisli ad Rya [] usig perturbatio methods to determie the residual film thickess i gas-assisted power-law displacemet. These authors cocluded that a perturbatio aalysis usig a power-law costitutive equatio does ot correctly predict the variatio of the residual liquid film thickess as a fuctio of the power law idex. I the cited study above the expasios for the capillary static ad the trasitio regios are developed i / terms of a small parameter where is the modified DISCUSSION AND CONCLUSION The residual liquid film thickess is determied applyig a perturbatio aalysis usig a power-law costitutive equatio. The theory predicted that the residual fractio icreases with icreasig values of the power-law idex which is i qualitative agreemet with the experimetal observatios of previous ivestigators. Furthermore it is possible to recover the results obtaied previously by may authors for the Newtoia case. It is importat to ote the relevace of the order of magitude aalysis i this kid of problems because the appropriate scales of the trasitio regio ad the series expasios for the perturbatio techique are fudametal to

8 96 The Ope Chemical Egieerig Joural 8 Volume Escobar et al. obtai the residual film thickess ad the pressure jump across the iterface. Accordig to the results for ay power-law idex the miimum residual liquid film thickess is obtaied for the smaller capillary umbers. This result is importat to determie the best coditios for oil recovery i aturally fractured reservoirs. ACKNOWLEDGEMENTS This work has bee supported by the research grats IN6 of the DGAPA UNAM ad 4-Y of the Cosejo Nacioal de Ciecia y Tecología at Mexico. REFERENCES [] F.P. Bretherto The motio of log bubbles i tubes J. Fluid Mech. vol. pp [] B.G. Cox O drivig a viscous fluid out of a tube J. Fluid Mech. vol. 4 pp [] S.G.I. Taylor Depositio of a viscous fluid o the wall of a tube J. Fluid Mech. vol. pp [4] S.D.R. Wilso The drag-out problem i film coatig theory J. Eg. Math. vol. 6 pp [5] L. Ladau ad B. Levich Draggig of a liquid by a movig plate Acta Physicochim. URSS. vol. 7 pp [6] C.W. Park ad G.M. Homsy Two-phase displacemet i Hele Shaw cells: theory J. Fluid Mech. vol. 9 pp [7] L.W. Schwartz H.M. Price ad A.D. Kiss O the motio of bubbles i capillary tubes J. Fluid Mech. vol. 7 pp [8] G.F. Teletzke H.T. Davis ad L.E. Scrive Wettig hydrodyamics Revue Phys. Appl. vol. pp [9] S. Middlema Modelig Axisymmetric Flows: Dyamics of Films Jets ad Drops. Ed. Academic Press Limited 995. [] A. Müch The thickess of a Maragoi-drive thi liquid film emergig from a meiscus Siam J. Appl. Math. vol. 6 pp [] A.J. Posliski P.R. Oehler ad V.K. Stokes Isothermal gasassisted displacemet of viscoplastic liquids i tubes Polym. Eg. Sci. vol. 5 pp [] A.J. Posliski ad D.J. Coyle Steady gas peetratio through o-newtoia liquids i tube ad slit geometries: isothermal shear-thiig effects i Proc. Polymer Processig Society th Aual Meetig 994 pp. 9. [] F. Kamisli ad M.E. Rya Perturbatio method i gas-assisted power-law fluid displacemet i a circular tube ad a rectagular chael Chem. Eg. J. vol. 75 pp [4] F. Kamisli ad M.E. Rya Gas-assisted o-newtoia fluid displacemet i circular tubes ad ocircular chaels Chem. Eg. Sci. vol. 56 pp [5] P.C. Huzyak ad K.W. Koellig The peetratio of a log bubble through a viscoelastic fluid i a tube J. No-Newtoia Fluid Mech. vol. 7 pp [6] C.M. Beder ad S.A. Orszag Advaced Mathematical Methods for Scietists ad Efieers Ed. McGraw-Hill 978. Received: April 8 Revised: Jue 6 8 Accepted: July 8 Escobar et al.; Licesee Betham Ope. This is a ope access article distributed uder the terms of the Creative Commos Attributio Licese ( which permits urestrictive use distributio ad reproductio i ay medium provided the origial work is properly cited.