CO-ROTATIONAL MAXWELL FLUID ANALYSIS IN HELICAL SCREW RHEOMETER USING ADOMIAN DECOMPOSITION METHOD

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1 U.P.B. Sci. Bull. Seies A. Vol. 7 Iss. ISS - 77 CO-ROTATIOAL AXWELL FLUID AALYSIS I HELICAL SCREW RHEOETER USIG ADOIA DECOPOSITIO ETHOD. Zeb A.. Siddiqui T. Haoo This pape cosides a theoetical study o steady icompessible flow of cootatioal axwell fluid i helical scew heomete (HSR). The heological costitutie equatio fo co-otatioal axwell fluid model gies the secod ode oliea coupled diffeetial equatios which could ot be soled explicitly. A iteatie pocedue Adomia decompositio method (AD) is used to obtai the aalytical solutio. Expessios fo elocity compoets i θ ad z diectio ae obtaied. The olume flow ates ae calculated fo the azimuthal ad axial compoets of elocity field by itoducig the effect of flights. The esults hae bee discussed with the help of gaphs as well. We obsee that the elocity pofiles ae stogly deped o o-dimesioal paamete α with the icease i α pogessie icease see i the flow pofiles. We also oted that the paabolicity of flow pofiles icease with icease i the magitude of pessue gadiets. Thus the pofoud coclusio is that extusio pocess depeds o the ioled odimesioal paametes. Keywods: Helical Scew Rheomete; Adomia decompositio method; Secod ode oliea coupled diffeetial equatios. []Pimay 7-XX.. Itoductio The Helical Scew Rheomete (HSR) is used fo heological measuemets of fluid food suspesios. It cotais a helical scew i a tight fittig cylide with the ilet ad outlet pots closed the ie scew. Rotatio of scew ceates a pessue gadiet alog the axis of the scew. The geomety of a HSR matches to a sigle scew extude []. Extusio pocess is widely used i food pocessig i.e. cookie dough seai pastas beakfast ceeals fech fies baby food eady to eat sacks ad dy pet food. Extusio pocess also iclude fluids like multi-gade oils liquid detegets paits polyme solutios ad polyme melts the ijectio moldig pocess fo polymeic mateials the poductio of phamaceutical poducts ad pocessig of plastics [ ]. Duig pocessig physical ad chemical chages ca occu so it is desiable to moito the pocess to achiee excellet output ad quality cotol Assistat Pofesso Depatmet of athematics COSATS Istitute of Ifomatio Techology Attock Pakista zeb@ciit-attock.edu.pk Pofesso Depatmet of athematics Pesylaia State Uiesity Yok Campus Yok PA 7 USA Pofesso Depatmet of athematics COSATS Istitute of Ifomatio Techology Islamabad Pakista

2 . Zeb A.. Siddiqui T. Haoo []. Bid et al.[] peseted a asymptotic solutio ad abitay alues of the flow behaio idex fo the Powe-Law fluid i a ey thi aulus. Booy [7] studied the ifluece of chael cuatue o flow pessue distibutio ad powe equiemets of scew pumps ad melt extudes cosideig ewtoia fluid. Tamua et al.[] also iestigated the flow of ewtoia fluid i Helical Scew Rheomete. The classical aie-stokes equatios hae bee poed iadequate to descibe complete chaacteistics of o-ewtoia fluids. To study these fluids diffeet ew theoies hae bee deeloped [] ad diffeet models ae poposed [ ]. I the peset wok a attempt has bee made to study co-otatioal axwell fluid i Helical Scew Rheomete (HSR). We hae choose the cylidical coodiate system ( θ z) which seems to be a moe atual choice due to the geomety of HSR. The expessios fo the ad wcompoet of elocity pofiles ae obtaied fom the solutio of deeloped secod ode oliea coupled diffeetial equatios by usig Adomia decompositio method with ew modificatio suggested by Wazwaz [9]. Volume flow ates ae calculated by itoducig the effect of flights. The behaio of the elocity pofiles ae peseted though gaphs ad discussed. The pape is ogaized as follows. Sectio cotais the goeig equatios of the fluid model. I Sectio the poblem ude cosideatio is fomulated ad the goeig equatio of the poblem is soled usig Adomia decompositio method. I Sectio discussio about the behaio of the elocity pofiles is gie. Sectio cotais coclusio. Basic Equatios The basic equatios goeig the motio of a isothemal homogeeous ad icompessible co-otatioal axwell fluid ae: div () DV ρ ρf P. S () Dt whee ρ is the costat fluid desity V is the elocity ecto f is the body D foce pe uit mass P deotes the dyamic pessue ad the opeato deotes Dt D( ) the mateial time deiatie defied as ( ) ( V )( ) Dt t S is the exta stess teso which fo co-otatioal axwell fluid model is defied as S λs λ( S SA) ηaa () hee η ad λ ae zeo shea iscosity ad elaxatio time espectiely. The

3 Co-otatioal axwell fluid aalysis [ ] heomete usig adomia decompositio method uppe cotaaiat coected deiatie desiged by oe S is defied as DS T S {( gad V) S S( gad V )} Dt ad ( ) ( ) T A gad V gad V is the fist Rili-Eickse teso. Poblem Fomulatio Steady lamia isothemal flow of a icompessible co-otatioal axwell fluid is cosideed i Helical scew heomete (HSR). The scew chael is assumed to be bouded by the bael ad scew oot sufaces ad by the two sides of a helical flight as show i Fig.. The geomety is appoximated as a shallow ifiite chael by assumig the width B of the chael lage compaed h with the depth h i.e. <<. Futhe moe side effects ca be igoed. Hee we B choose the cylidical coodiate system ( θ z) which is moe suitable choice fo the flow aalysis i HSR. A coguet elocity distibutio is assumed at paallel coss sectios though the chael. The iscosity of the fluid is assumed to be costat. The oute bael of adius is assumed to be statioay ad the scew oot of adius otates with agula elocity Ω [7 ]. Fig. Geomety of helical scew heomete The bouday coditios ae Ω w at () w at. The flow is assumed fully deeloped i the θ ad the z diectios so that V [ ( ) w ( )] S S ( ) () whee u ad w ae adial azimuthal ad axial elocity compoets espectiely. O substitutig these assumptios i equatio () we obtai ozeo compoets of exta stess teso S as

4 . Zeb A.. Siddiqui T. Haoo S S θ d dw η λ d d () d dw Sθ η Sz S z η d d (7) d dw Sθθ ηλ Szz ηλ () d d dw d Sθ z S zθ ηλ (9) d d whee. () d dw λ d d Fo highly iscous fluids the effect of acceleatio of fluid ad body foces ca be igoed []. Equatio () fo ceepig flow of a icompessible fluid ca be witte as gadp di S. () A umbe of simplificatios became possible by assumig ifiite aspect h atio << ad the coguet elocity distibutios at paallel coss sectios. B Velocities ae tagetial to cylidical sufaces i the limitig case of ifiite aspect atio. The pessue i the chael does ot chage with adius fo ey P shallow chaels theefoe []. I iew of ou assumptios equatio () is satisfied idetically ad substitutio of the calculated compoets of exta tess teso gie i equatios ()-(9) i the compoet fom of equatio () gie d dw d d d d d d d dw d dw λ λ d d d d () d d η d P () d θ d dw λ d d

5 Co-otatioal axwell fluid aalysis [ ] heomete usig adomia decompositio method dw d d P η. () d z d dw λ d d The equatios () ad () imply that P P( θ z) sice the left had sides of equatios () ad () ae fuctios of aloe ad P P() this implies P P costat ad costat. Ou cocetatio is o tosioal ad axial θ z flow so we will coside oly equatios () ad (). Itoducig o-dimesioal paametes z w P z w P Ω Ω ηω i equatios () ad () afte doppig * ad the itegatig with espect to `` " we get d K C d dw α () d d d dw C d dw K α. () d d d whee C ad C ae costat of itegatio ad α ( Wi) whee Wi λ Ω is the Weissebeg umbe aised i o-dimesiolizatio ad assume that P P K ad K. θ z The associated o-dimesioal bouday coditios ae w at (7) w at δ whee δ >. The esultat equatios () ad () ae coupled fist ode oliea odiay diffeetial equatios the exact solutio seems to be difficult. I the ext sectio we use Adomia decompositio method (AD) to obtai the appoximate solutio with the help of symbolic computatio softwae Wolfam athematica 7. Accodig to the AD the oliea odiay diffeetial equatio ca be witte i opeato fom as Lu Ru u g whee L is the highe ode liea diffeetial opeato which is assumed to be

6 . Zeb A.. Siddiqui T. Haoo ietible ad R is a liea liea diffeetial opeato of ode less tha L is d oliea opeato ad g is a souce tem. I ou poblem L L () d d R C d dw u α K d d C d dw u α K d ad d C C g K g K. AD suggest that the ukow fuctios ad wbe expessed as w w ad the oliea tem u ca be exploe i the fom of AD polyomials A s as u A. Adomia decompositio method simplifies the deeloped equatios () ad () i compoets fom alog with the bouday coditios i the followig way.. Zeoth compoet Solutio Zeoth compoet poblem K C C L () C w C L K (9) alog with the bouday coditios w at () w at δ gie K l( ) () w ( ) l( ) () Equatios () ad () ae the solutios fo liealy iscous fluid [].

7 Co-otatioal axwell fluid aalysis [ ] heomete usig adomia decompositio method 7. Fist compoet Solutio Fist compoet poblem C L α L ( A ) () C w L α L ( B ) () togethe with bouday coditios w at () w at δ whee A ad B ae Adomia polyomials i ad w gie as K C d dw A () d d C d dw B K. (7) d d Equatios () ad () esult i the solutio 7 α ( ) l( ) 9 () 7 w α ( )l( ) 9 (9). Secod Ode Solutio The secod compoet poblem C L α L ( A ) () C w L α L ( B ) () with the bouday coditios w at () w at δ. cotaiig Adomia polyomials A ad B i w ad w gie as K C d d dw dw A d d d d

8 . Zeb A.. Siddiqui T. Haoo d dw d d C () d dw d dw d d d d C K B d dw d d C () esults i the solutio l 7 9 α 9 7 () l w α l 9 7 () whee 9 9 i j i j ae costat coefficiets. Sice the AD solutio ca be accumulated as. w w ad (7) Thus the AD solutio fo the azimuthal ad axial elocity compoets up to secod iteatio afte substitutig i (7) become ) l( ) ( ) l( K α α l 9 7 () ) )l( ( ) l( ) ( w α l 9 7 α

9 Co-otatioal axwell fluid aalysis [ ] heomete usig adomia decompositio method 9 7 l 9. (9). Volume flow ate i θ -diectio The olume flow ate i θ diectio as gie i [] is Volume flow ate () i o-dimesioal fom is Qθ π ( taφ) d. () Q θ δ πδ taφ d () Qθ whee Qθ. Doppig * we get Ω K α α Q ( l ) ( θ πδ taφ δ δ δ α α ) ( α α α l )( ) δ α9 α 9 δ δ α α α7 α ( ) δ δ α δ δ 7 ( δ ) ( δ lδ ). (). Volume flow ate i z -diectio The olume flow ate i z -diectio ca be calculated as Q π wd. () z The dimesioless fom of equatio() ca be witte as Qz whee Qz. ow doppig * we Ω δ Q z π wd () Qz π δ δ α α ( ) δ δ lδ ( ) l ( ) ( )( ) α α δ α α α α δ δ ( ) ( ( ) )( )

10 . Zeb A.. Siddiqui T. Haoo ( α α )( δ ) ( α α )( δ ) α δ δ δ ( ). Results ad Discussio I this wok we hae cosideed steady flow of a icompessible cootatioal axwell fluid though HSR. We obtaied coupled secod ode oliea ODEs. Its exact solutios seems to be difficult appoximate aalytical solutios ae obtaied with the help of AD as this method gies coeget solutio i fiite domai. Expessios fo azimuthal () ad axial elocity w() ae calculated. The olume flow ates i θ ad z diectios ae also deied. Hee we discussed the effect of ioled flow paametes o the elocity pofiles with the help of gaphs. Figue (a) is plotted fo the elocity fo diffeet alues of fluid paamete α steadily icease obseed i the elocity fom scew towad bael ad the elocity attais maximum alues i betwee the chael which show shea thiig due to iceases i the alue of α. Figue (b) is sketched fo the elocity pofile w fo diffeet alues of α the elocity pofile is seem to be paabolic i atue ad also attais maximum alues at some poits i the chael. The behaio of elocity w shows that it is esposible to take the fluid towad the exit. Figues (a) ad (b) ae show fo the elocity fo diffeet alues of pessue gadiets P θ ad P z espectiely it ca be see that elocity iceases with the icease i pessue gadiets. It is oticed that P z esist the elocity as gaphs show the smalle magitude of fo P z. Similaly figues (a) ad (b) ae plotted fo the elocity w fo diffeet alues of P θ ad P z. With the icease i the alue of P θ ad P z icease i the w is obseed howee the effect of P θ is obseed less o w which show P θ ty to esist the flow i axial diectio. ()

11 Co-otatioal axwell fluid aalysis [ ] heomete usig adomia decompositio method Fig. : (a) () fo diffeet alues of α keepig P. P. ad δ θ θ z z w () fo diffeet alues of α keepig P. P. ad δ.. (b) Fig. : (a) () fo diffeet alues of P θ keepig α. P z. ad δ. (b) () fo diffeet alues of P z keepig α. P θ. ad δ. Fig. : (a) w () fo diffeet alues of P θ keepig α. P z. ad δ. (b) w () fo diffeet alues of P z keepig α. P θ. ad δ. It is mesioed hee the ioled coeffeiciets ae ot fuctio of ay paametes like α these coefficiets ae just costats depedig o costat atio

12 . Zeb A.. Siddiqui T. Haoo P P δ > ad costat pessue gadiets i.e. K ad K. It is θ z also mesioed hee the AD esults i the coeget solutio i the fiite domai. The poblem gie i () (7) has fiite domai fom to so the solutios with espect to AD ae coeget. Coclusio This pape cosides a theoetical study o steady icompessible flow of co-otatioal axwell fluid i helical scew heomete (HSR). The heological costitutie equatio fo co-otatioal axwell fluid model gies the secod ode oliea coupled diffeetial equatios which could ot be soled explicitly. A iteatie pocedue Adomia decompositio method (AD) is used to obtai the aalytical solutio. It is to be metioed hee that the AD gies coeget solutio i fiite domai. Expessios fo elocity compoets i θ ad z diectio ae obtaied. The olume flow ates ae calculated fo the azimuthal ad axial compoets of elocity field by itoducig the effect of flights. The esults hae bee discussed with the help of gaphs as well. We obsee that the elocity pofiles ae stogly deped o o-dimesioal paamete ad pessue gadiets. Thus the pofoud coclusio is that extusio pocess depeds o the ioled o-dimesioal paametes alog with the othe factos ioled i food pocessig. R E F E R E C E S [].S. Tamua J.. Hedeso R.L. Powell C.F. Shoemake Aalysis of the helical scew heomete fo fluid food J. Food Pocess Eg. (99) o []. Zeb S. Islam A.. Siddiqui T. Haoo Aalysis of Thid-gade fluid i Helical Scew Rheomete J. Appl. ath. Aticle ID -. [] A.. Siqqiqui T. Haoo. Zeb Aalysis of Eyig-Powell Fluid i Helical Scew Rheomete Sci. Wold J. Aticle ID 9 -. [] A.. Siddiqui T. Haoo S. Ium Tosioal flow of thid gade fluid usig modified homotopy petubatio method Comp. ad ath. with Appl. (9) o [].A. Rao Rheology of liquid foods: A eiew J. Textue Studies (977) o. -. [] R. B. Bid R. C. Amstog ad O. Hassage Ehacemet of Axial Aula Flow by Rottaig Ie Cylide Dyamics of Ploymeic Liquids Fluid echaics Wiley ew Yok 97. [7]. L. Booy Ifluece of Chael Cuatue o Flow Pessue Distibutio ad Powe Requiemets of Scew Pumps ad elt Extudes Polym. Eg. Sci. (9) o [] A.. Siddiqui. Hammed B.. Siddiqui Q. K. Ghoi Use of Adomia decompositiomethod i the study of paallel plate flow of a thid gade fluid Commu oliea Sci. ume. Simulat. () o [9] A.. Wazwaz ad S.. El-Sayed A ew modificatio of the Adomia decompositio method fo liea ad oliea opeatos Appl. ath. Comput. () o. pp. 9-. []. Zeb T. Haoo A.. Siddiqui Homotopy Petubatio Solutio fo Flow of a Thid gade Fluid i Helical Scew Rheomete U.P.B. Sci. Bull. Seies A 7() o