Core Variation in the Entrance Region Flow of Herschel- Bulkley Fluid in an Annuli

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1 Poceedigs of the Wold ogess o Egieeig 4 Vol II, WE 4, July - 4, 4, Lodo, U.K. oe Vaiatio i the Etace Regio Flow of Heschel- Bulkley Fluid i a Auli Rekha G.Pai ad A.Kadasamy Abstact: The etace egio flow i chaels costitutes a poblem of fudametal iteest i egieeig applicatios such as uclea eactos, polyme pocessig idusties, haemodialyes ad capillay membae oxygeatos. I such istallatios, the behaio of the fluid i the etace egio may play a sigificat pat i the total legth of the chael ad the pessue dop may be makedly geate tha fo the case whee the flow is egaded as fully deeloped thoughout the chael. Recetly, thee has bee a iceasig iteest i poblems iolig mateials with aiable iscosity such as Bigham mateials, asso fluids ad Heschel-Bulkley fluids which ae chaacteied by a yield alue. The etace egio flow of a Heschel- Bulkley fluid i a aula cylide has bee iestigated umeically without makig pio assumptios o the fom of elocity pofile withi the bouday laye egio. This elocity distibutio is detemied as pat of the pocedue by coss sectioal itegatio of the mometum diffeetial equatio fo a gie distace fom the chael etace. Usig the macoscopic mass balace equatio the coe thickess has bee obtaied at each coss sectio of the auli fo specific alues of Heschel -Bulkley umbe, flow behaio idex ad aious alue of aspect atio. Idex Tems Bouday Laye Regio, oe Vaiatio, Etace Regio Flow, Heschel-Bulkley Fluid. T I. ITRODUTIO HE etace egio flow i chaels costitute a poblem of fudametal iteest i egieeig applicatios such as uclea eactos, polyme pocessig idusties, haemodialyes ad capillay membae oxygeatos. I such istallatios, the behaio of the fluid i the etace egio may play a sigificat pat i the total legth of the chael ad the pessue dop may be makedly geate tha fo the case whee the flow is egaded as fully deeloped thoughout the chael. Recetly, thee has bee a iceasig iteest i poblems iolig mateials with aiable iscosity such as Bigham mateials, asso fluids ad Heschel-Bulkley fluids which ae chaacteied by a yield alue. Bata ad Bigyai Jea [] iestigated stess-stai elatio i the etace egio of auli usig blood as asso fluid. Tado et al [] aalyed the blood flow i the ateies by assumig it as asso model. Dash et al [3] hae aalyed steady ad pulsatile flow cosideig pessue gadiet as a fuctio of time i a aow catheteied atey takig ito accout blood as asso fluid. Ramesh Gupta [4] has cosideed a costat popety of iscous fluid eteig a coduit ude appopiate itake coditios so that its elocity is pactically costat oe the ety coss-sectio. Maia ad Gaspaetto [5] applied fiite diffeece method fo the Powe-law fluids i the auli ad foud diffeece i etace geometies. Saka ad Hemalatha [6] hae studied the pulsatile flow of blood though catheteied atey by modelig blood as Heschel- Bulkley fluid ad cosideig atey as coaxial cicula cylides. Poole ad hhaba [7] epoted the esults of a systematic umeical iestigatio of deelopig lamia pipe flow of yield stess fluids. Recetly, Kadasamy ad Pai [8] hae iestigated the oe aiatio i the etace egio flow of asso Fluid i a Auli. The pupose of the peset wok is to aalye umeically the etace egio flow of a Heschel- Bulkley though aula cylide without makig assumptios i the fom of elocity pofile withi the bouday laye. The elocity distibutio is detemied as pat of the pocedue by cosssectioal itegatio of the mometum diffeetial equatio fo a gie distace fom the chael etace. Usig these elocity distibutio i the macoscopic mass balace equatio, the aiatio of the coe i the etace egio has bee obtaied fo aious alues flow chaacteistics ad geometical chaacteistics. II. AALYSIS We ae aalyig the etace egio flow of a Heschel- Bulkley fluid though a aula cylide. Fluid etes a hoiotal aula duct fom a lage chambe with a uifom elocity alog the axial diectio. The aalysis has bee caied out oe the wide age of aspect atios, that is, the atio of the adius of the ie cylide to that of the oute cylide. The deelopmet of bouday laye is isualied whe the fluid etes a aulus ad the fully deeloped elocity pofile is obseed i the egio statig fom the poit dow - steam whee the bouday layes meet asymptotically with the oute edge of the plug flow oe. Rekha G. Pai is a Assistat Pofesso i the Depatmet of Mathematics, Maipal Istitute of Techology, Maipal , Idia (pai.ekha@maipal.edu A.Kadasamy is a Pofesso, Depatmet of M.A..S,.I.T.K, Suathkal, Magaloe , Idia (phoe: ( , kady@itk.ac.i Geomety of the poblem is show i the Figue. ISB: ISS: (Pit; ISS: (Olie WE 4

2 Poceedigs of the Wold ogess o Egieeig 4 Vol II, WE 4, July - 4, 4, Lodo, U.K. Fig.. Geomety of a auli Fig.. Fluid flow i aious egios i a auli We coside a hoiotal aula duct cosistig of ie cylide of adius ad oute cylide of adius. A icompessible, lamia ad isothemal Heschel-Bulkley fluid etes fom a lage chambe to this duct with steady elocity. We use cylidical pola coodiates system (, θ, with axial symmety ad oigi at the cete of the cylides at the ilet with axis coicidig with the axis of the cylides. is the elocity compoet i diectio, is the elocity i the diectio. Sice Heschel-Bulkley fluid possesses a yield alue, thee is a plug coe fomatio away fom the walls. O each wall thee is bouday laye fomatio sepaated by the coe. As the two egios about the axis ae symmetical, the solutio of the poblem will be cosideed i the uppe half oly. We hae ie bouday laye egio with thickess ( ad oute bouday laye egio with thickess ( ad the plug flow egio o Plug oe Regio Flow Regio Flow Regio sepaatig the two bouday layes, whee the coe is moig with costat elocity. I the plug coe egio, the shea stess τ is less tha o equal to the yield stess τ ad the elocity at each coss sectio is costat. Ude the assumptios of bouday laye theoy ad eglectig ietia tems, the equatio of motio of a isotopic, icompessible fluid ca be witte as τ dp d ( whee p is the pessue of the fluid. The costitutie equatio of a Heschel - Bulkley fluid is gie by η, τ τ τ τ ( Whee epesets coefficiet of iscosity fo Heschel Bulkley fluid ad is the flow behaio idex. Equatios ( ad ( will be soled ude the followig bouday coditios: i The axial elocity compoets ae eo at the wall, i.e., (,, (, (3a ii At the edge of the bouday layes, is equal to the plug coe elocity, i.e., (,, (, c c (3b iii The shea stess at the edge of the lowe ad uppe bouday of the plug coe egio ae equal to - ad τ espectiely. Hece Eq. ( educes to, (3c Usig equatios (,( ad coditios (3a, (3b, (3c we obtai the elocity pofiles of aious egios i the odimesioal fom as follows: ( ( c (R ε (c ( (3 (3 c (R ε (c (4 8 V ( ( c (R ε (c ( (- (4 (4 c (R ε ( c (3 8, ε R V (- ( c ( ε ( c ( (3 (3 c ( ε ( c (4 8 (- ( c ( ε ( c ( (- (4- (4 c ( ε ( c (3 8, R τ (4 (5 ISB: ISS: (Pit; ISS: (Olie WE 4

3 Poceedigs of the Wold ogess o Egieeig 4 Vol II, WE 4, July - 4, 4, Lodo, U.K. ( (R ( ( (3 (R ( (4 8 V ( ( (R ε ( ( (- (4 (4 (R ε ( (3 8, R V (- ( ( ( (3- ( ( (4 8 ( - ( ( ( (- (4 - ( ( (3 8, R Whee the o-dimesioal quatities ae defied as follows Z Z V, V, V, c η c, Z, - R, 4 ρ V p, ε, ρ P, ε Aspect Ratio, τ Heschel - Bulkley umbe (8 η Hee,, ad ukows fist we equate (6 (7 ae ukows. To fid these V V whee both epesets coe elocities. The, we get followig algebaic equatio i tems of pessue gadiet. ( ( ( (4 8 ( ( (- ( (3 8 ( ( ( (4 8 (- (- ( ( ε (3 (- (3- (- ε ( ( (- ( (3 8 ε (4- ( ( ( ( (4- (3 ( ε ( (4 ( ( The macoscopic mass balace equatio is gie by V R dr (9 ( Usig the elocity equatios (4, (6 ad (7 i (, we get a equatio i the fom A + B + = ( Whee ( ( (3 - ( (4 8 ( A ( ε (3 - (5 - (4 - (3 ε (5 - ε (4 - ( - ( (3 8 ( - (3 8 (6 - (5 (4 ε ε ε (6 - ε ε 3 ε ε 4 ε ε ( (6 ( ( ( ( ISB: ISS: (Pit; ISS: (Olie WE 4

4 Poceedigs of the Wold ogess o Egieeig 4 Vol II, WE 4, July - 4, 4, Lodo, U.K. B (- ( ( ( (3- ( ( (4 8 (- ( ( ( (- (4- ( ( (3 8 (3- (- ( ( (3 - (5- (- ( (4 8 (5 - (4- (- ( ( (4 - (6- (- (- ( (3 8 (6 - Equatios (9 ad ( ae umeically soled usig multi aiable ewto-raphso method to obtai ad pessue gadiet fo diffeet alues of Heschel-Bulkley umbe, aspect atio ad flow behaio idex. III RESULTS AD DISUSSIO The solutio of the etace egio flow of a Heschel- Bulkley fluid though a aula cylide has bee iestigated umeically without pio assumptios o the fom of the elocity pofile i the deelopig bouday laye. Iitially, a oliea algebaic equatio fo detemiig the pessue gadiet as a fuctio of coe thickess has bee deied usig the mass balace equatio. This equatio is soled usig iteatie method to obtai. Usig the alues of pessue gadiet so obtaied, the coe thickess is obtaied fo aious alues of Heschel- Bulkley umbe, aspect atio ad flow behaio idex. Figues 3-6 shows gaphically the aiatio of coe thickess alog the axial distace fo aious alues of aspect atio at a paticula alue of Heschel-Bulkley umbe ad flow behaio idex. It is obseed that as aspect atio iceases, fo paticula Heschel-Bulkley umbe ad flow behaio idex, the coe thickess deceases. Also, fom figues 7 ad 8, it is obseed that at ay coss sectio Z, fo fixed auli aspect atio ad flow behaio idex, the coe thickess iceases with the icease i the alue of Heschel-Bulkley umbe. This idicates that plug coe will be ey high fo mateials haig thick iscosity, like mieal oils ad polyme thickeed fluid. Simila esults ae also obseed fo othe aspect atios with aious flow behaio idices as show i figues 4, 9, ad. Futhe, fom the esults it is eidet that as the alue of flow behaio idex iceases, the coe thickess deceases fo a fixed alue of Heschel- Bulkley umbe ad auli aspect atio. REFEREES [] R.L. Bata ad BigyaiJea, Etace Regio Flow of Blood i ocetic Aulus, Iteatioal Joual of Egieeig Sciece, Vol. 8, pp.47-49, 99. [] P..Tado, L.M.Siastaa ad K.Kushwaha, Deelopig Blood Flow i the Etace Regio of a Atey, Iteatioal Joual of Bio-Medical omputig, Vol. 36, pp , 994. [3] R.K.Dash, G.Jayaama ad K..Mehta, Estimatio of iceased flow esistace i a aow catheteied atey A Theoetical Model, J. Biomechaics, Vol. 9, pp , 996. [4] Ramesh.Gupta, O Deelopig Lamia o-ewtoia Flow i Pipes ad haels, oliea Aalysis: Real Wold Applicatios, Vol., pp. 7-93, Jue. [5] M..A.Maia ad.a.gaspaetto, A umeical Solutio fo the Etace Regio of o- ewtoia Flow i Auli, Bailia Joual of hemical Egieeig, Vol., pp.-8, 3. [6] D.S.Saka ad K.Hemalatha, Pulsatile flow of Heschel-Bulkley fluid though catheied ateies- A mathematical model, Applied Mathematical Modelig, pp , 7. [7] R.J.Poole ad R.P.hhaba, Deelopmet legth equiemets fo fully deeloped Lamia pipe flow of yield stess fluids, J. Fluids Eg, Vol.3, pp ,. [8] A.Kadasamy ad Rekha.G.Pai, oe Vaiatio i the Etace Regio Flow of asso fluid i auli, Applied Mechaics ad Mateials, Vol.39,pp , 3. Fig.3. The aiatio of coe thickess alog axial distace fo ε =.3, =., =.55. ISB: ISS: (Pit; ISS: (Olie WE 4

5 Poceedigs of the Wold ogess o Egieeig 4 Vol II, WE 4, July - 4, 4, Lodo, U.K. Fig.4. The aiatio of coe thickess alog axial distace fo ε =.4, =., =.55. Fig.5. The aiatio of coe thickess alog axial distace fo ε =.5, =., =.55. Fig.6. The aiatio of coe thickess alog axial distace fo ε =.6, =., =.55. Fig.7. The aiatio of coe thickess alog axial distace fo ε =.5, =.5, =.55. Fig.8. The aiatio of coe thickess alog axial distace fo ε =.5, =.8, =.55. ISB: ISS: (Pit; ISS: (Olie WE 4

6 Poceedigs of the Wold ogess o Egieeig 4 Vol II, WE 4, July - 4, 4, Lodo, U.K. Fig.9.The aiatio of coe thickess alog axial distace fo ε =.5, =.5, =.6. Fig.. The aiatio of coe thickess alog axial distace fo ε =.5, =., =.6. Fig.. The aiatio of coe thickess alog axial distace fo ε =.4, =.5, =.55. ISB: ISS: (Pit; ISS: (Olie WE 4

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