Dept. Engineering, Mechanical Engineering, University of Liverpool Liverpool L69 3GH, UK,

Transcription

1 R. J. Poole Dept. Engineering, Mechanical Engineering, University of Liverpool Liverpool L69 3GH, UK, M. A. Alves Departamento de Engenharia uímica, CEFT, Faculdade de Engenharia da Universidade do Porto, Portugal, A. Afonso Departamento de Engenharia uímica, CEFT, Faculdade de Engenharia da Universidade do Porto, Portugal, F. T. Pinho Escola de Engenharia, Universidade do Minho, Portugal, CEFT, Faculdade de Engenharia Universidade do Porto, Portugal, P. J. Oliveira Departamento de Eng. Electromecânica, Universidade da Beira Interior, Covilhã, Portugal, The Society of Rheology 79 th annual meeting, 7 th to th October 27 Salt Lake City, USA Outline Motivation and Previous work (UCM) Governing equations/numerical method Effect of solvent viscosity (Oldroyd-B) and inertia Effect of extensional viscosity (PTT) Conclusions

2 Motivation Arratia et al, Physical Review Letters 96, 4452 (26) 65 m Microfluidic flow in a cross channel geometry 5 m Newtonian: Re < -2 PAA Boger fluid: Re < -2 (De=4.5) Motivation and Previous work Poole, Alves and Oliveira, accepted in Physical Review Letters (27) Successfully used a numerical technique with a simple viscoelastic constitutive equation (UCM) to model this steady asymmetry under creeping flow conditions. Newtonian De=.4

3 Motivation and Previous work Poole, Alves and Oliveira, accepted in Physical Review Letters (27) Purely-elastic: inertia decreases the degree of asymmetry and stabilizes the flow.75 q q 2 q 2 q.5.25 D Flow asymmetry: q2 q q2 q D = = q + q 2 D= symmetric D=± completely asymmetric Re = Re = Re = 2 Re = 5 D = A (De - De CR ) De Motivation and Previous work Poole, Alves and Oliveira, accepted in Physical Review Letters (27) Purely-elastic: inertia decreases the degree of asymmetry and stabilizes the flow Re 4 3 Symmetric Re-De Map 2 Unsteady asymmetric Steady asymmetric De

4 Governing equations Incompressible Viscoelastic fluid " u = (Mass conservation) " s s # = o s + P T & o (" u + " u ) + ( # )" A Du ' = #" p + $& o" $ Dt % " DA T & $ (% u) A $ A( % u) # ' = f ( A) ( Dt ) (Momentum conservation) (Constitutive equation, based on the conformation tensor, A) Examples: " ( A I), # f ( A) = $ ( A I), # % Y (tr A)( A I), Upper Convected Maxwell - UCM (=) Oldroyd-B (<<) Phan-Thien and Tanner (<<), with ( A ) ( A " ) #$ + tr " 3 Y (tr A) = % $ exp &( tr 3 ') (linear) (exponential) Numerical method - brief description Finite Volume Method Oliveira, Pinho and Pinto (998) Oliveira and Pinho (999) Structured, collocated and non-orthogonal meshes. Discretization (formally 2nd order) Diffusive terms: central differences (CDS) Advective terms, high resolution scheme: CUBISTA Alves, Pinho and Oliveira (23) Dependent variables evaluated at cell centers; Special formulations for cell-face velocities and stresses;

5 Computational domain, boundary conditions =UD De=U/D 2D 2D 2D D q q 2 q 2 q 2D Inlet Boundary Conditions: Fully-developed u(y) and "(y) Outlet Boundary Conditions: " y = Effect of mesh refinement.5 y/d x/d NC DOF (x MIN )/D M M

6 Effect of mesh refinement (Oldroyd-B, =/9, De=.35 and Re=) u/u Txx x/d NC DOF (x MIN )/D M M Oldroyd-B Results effect Effect of increasing the solvent viscosity (creeping flow) q q 2 q 2 q.75.5 = =/9 Flow asymmetry: q2 q q2 q D = = q + q 2 D.25 =2/9 =3/9 D= symmetric D=± completely asymmetric Increases the critical De CR -.75 For ">3/9 flow became asymmetric unsteady; De

7 Oldroyd-B Results effect Streamlines (creeping flow and =/9) Oldroyd-B Results Inertia effect Effect of increasing the Reynolds number (=/9) Increases the critical De CR Decreases the degree of asymmetry;.75.5 Re=4 For Re>2, asymmetric unsteady flow..72 D.25 Re= Re= Re= D Re= De

8 Oldroyd-B Results Inertia effect Re.vs.De map (=/9) 4 Symmetric 3 Re 2 Unsteady asymmetric Steady asymmetric De PTT Results Effect of varying # parameter in PTT model (creeping flow and =/9) Increases the critical De CR Decreases the degree of asymmetry (#<.4);.75.5 #= #=.2 #=.4 #=.6 Increases the degree of asymmetry and in De extension (#>.4); D #=.8 For #>.8, asymmetric stable flow desappears De

9 PTT Results #.vs.de stability map (Creeping flow and =/9). Symmetric Steady asymmetric.5 Unsteady asymmetric De PTT Results Streamlines (creeping flow, #=.2 and =/9)

10 Conclusions Increasing the solvent viscosity (increasing ) increases the critical De. For >2/9 the first steady instability disappears and the flow becomes unsteady (but still asymmetric); Increasing the level of inertia (increasing Re) shifts the onset of the instability to higher De and decreases the degree of asymmetry. Essentially inertia appears to stabilize the flow. For the PTT model, decreasing the extensional viscosity (increasing #) increases the critical De. We propose that this flow would make a good benchmark case for purely-elastic instabilities Acknowledgments Fundação para a Ciência e Tecnologia: PROJECT: BD/28288/27 PROJECT POCI/EME/59338 Fundação Luso-Americana: PROJECT: 544/27 The Society of Rheology: Student Travel Grant

11 THE SOCIETY OF RHEOLOGY 79 TH ANNUAL MEETING PROGRAM AND ABSTRACTS Hilton Salt Lake City Center Salt Lake City, Utah October 7 -, 27 Program Committee: Mataz Alcoutlabi University of Utah Patrick Anderson Eindhoven University of Technology Ralph Colby Pennsylvania State University John Dorgan Colorado School of Mines Pat Doyle Massachusetts Institute of Technology Eric Furst University of Delaware Michael Graham"#$%#&'() University of Wisconsin Erik Hobbie National Institute of Standards & Technology Ravi P. Jagadeeshan Monash University Rajesh Khare Texas Tech University Daniel Klingenberg"#$%#&'() University of Wisconsin Local Arrangements: Jules Magda University of Utah Abstract Book Editor and Webmaster: Matt Liberatore Colorado School of Mines James Oberhauser University of Virginia Jai Pathak US Naval Research Lab Bob Prud homme Princeton University Jonathan Rothstein University of Massachusetts Jay Schieber Illinois Institute of Technology Nina Shapley Columbia University Phil Sullivan Schlumberger Technology Corp Sachin Velankar University of Pittsburgh Henning Winter University of Massachusetts Andy Kraynik Sandia National Laboratories Albert Co, University of Maine

12 Meeting Schedule Monday, October 8, 27 Tuesday, October 9, 27 Wednesday, October, 27 Thursday, October, 27 +,-. /'23443"567 8,9. #$4433 8,:; <#7 /=7 =>7 5<7 7.,7. <#9 /=9 =>9 5<9 7.,-; <#- /=- =>- 5<- 77,.. <#: /=: =>: 5<: 77,9; <#; /=; =>; 5<; 77,;. 7,-. <#B /=B =>B 5<B 7,;; <#C /=C =>C 5<C 9,9. <#+ /=+ =>+ 5<+ 9,:; <#8 /=8 =>8 5<8 -,7. #$4433 -,-; <#7. /=7. =>7. 5<7. :,.. <#77 /=77 =>77 5<77 :,9; <#79 /=79 =>79 5<79 :,;. <#7- /=7- =>7-5<7- ;,7; +,-. F/G)'EH"569 8,9. #$4433 8,:; /=7: =>7: 5<7: 7.,7. <#7; /=7; =>7; 5<7; 7.,-; <#7B /=7B =>7B 5<7B 77,.. <#7C /=7C =>7C 5<7C 77,9; <#7+ /=7+ =>7+ 5<7+ 77,;. 7,-. <#78 /=78 =>78 5<78 7,;; <#9. /=9. =>9. 5<9. 9,9. <#97 /=97 =>97 5<97 9,:; <#99 /=99 =>99 -,7. #$4433 -,-; <#9- /=9- =>9- G<7 :,.. <#9: GD7 =>9: G<9 :,9; <#9; GD9 <=7 G<- :,;. <#9B GD- <=9 G<: ;,7; ;,-. C,.. +,.. +,-. FL63M(I"56-8,9. #$4433 8,:; <#9C GD: <=- G<; 7.,7. <#9+ GD; <=: G<B 7.,-; <#98 GDB <=; G<C 77,.. <#-. GDC <=B G<+ 77,9; <#-7 GD+ <=C G<8 77,;. 7,-. <#-9 GD8 <=+ G<7. 7,;; <#-- GD7. <=8 G<77 9,9. <#-: GD77 <=7. G<79 9,:; <#-; GD79 <=77 G<7- -,7. #$4433 -,-; <#-B GD7- <=79 G<7: :,.. <#-C GD7: <=7- G<7; :,9; <#-+ GD7; <=7: G<7B :,;. <#-8 GD7B <=7; G<7C ;,7; <#:. <=7B ;,:. B,.. +,.; <#:7 GD7C <=7C <7 +,-. <#:9 GD7+ <=7+ <9 +,;; D57 GD78 <=78 <- 8,9. D59 <=9. <: 8,:; #$ ,7. D5- GD97 <=97 <; 7.,-; D5: GD99 <=99 <B 77,.. D5; GD9- <=9- <C 77,9; D5B GD9: <=9: <+ 77,;. D5C GD9; <=9; <8 79,7; Session Codes <HIJ3TI

13 Contents Monday Morning... Monday Afternoon... 9 <?IN3@I($@IS#$22$(EI'@E)'@?2')=3E('8 V$@%V3MJ$@('@/2?(E=3A&'@(AI77 =(A)$)&3$2$KHS=(A)$42?(E(AI'@E#$@4(@3E<HIJ3TI7-5$2HT3)<$2?J($@I7; Tuesday Morning @')H63AJ?)3I78 <?IN3@I($@IS#$22$(EI'@E)'@?2')=3E('78 V$@%V3MJ$@('@/2?(E=3A&'@(AI9. =(A)$)&3$2$KHS=(A)$42?(E(AI'@E#$@4(@3E<HIJ3TI99 5$2HT3)<$2?J($@I9- Tuesday Afternoon <?IN3@I($@IS#$22$(EI'@E)'@?2')=3E('9C V$@%V3MJ$@('@/2?(E=3A&'@(AI98 G23@EISDT?2I($@I'@E=?2J(N&'I3/2?(EI-7 =(A)$)&3$2$KHS=(A)$42?(E(AI'@E#$@4(@3E<HIJ3TI-9 D@J'@K23E<$2?J($@I'@E=32JI-: 5$2HT3)<$2?J($@I-: G($2$K(A'2'@E<324%'II3TU23E<HIJ3TI-; Wednesday Morning @')H63AJ?)3I-C <?IN3@I($@IS#$22$(EI'@E)'@?2')=3E('-C G23@EISDT?2I($@I'@E=?2J(N&'I3/2?(EI-8 D@J'@K23E<$2?J($@I'@E=32JI:. G($2$K(A'2'@E<324%'II3TU23E<HIJ3TI:9 Wednesday Afternoon <?IN3@I($@IS#$22$(EI'@E)'@?2')=3E(':; G23@EISDT?2I($@I'@E=?2J(N&'I3/2?(EI:C D@J'@K23E<$2?J($@I'@E=32JI;. G($2$K(A'2'@E<324%'II3TU23E<HIJ3TI;9 W&3<$A(3JH$4>&3$2$KHC8J&L@@?'2=33J(@KSXAJ$U3)9..C (

14 Thursday Morning Poster Session Author Index Paper Index &JJN,[[MMM)&3$2$KH$)K[I$).C'[ ((

15 /=8 Purely elastic instabilities in a cross-slot flow >$U3)JF5$$23 7 S='@?32LL2Y3I 9 SL23Z'@E)3L4$@I$ 9 S/3)@'@E$W5(@&$ - S'@E5'?2$FX2(Y3()' : Department of Engineering, University of Liverpool, Liverpool L69 3GH, United Kingdom; 2 Departamento de Engenharia uímica, CEFT, Faculdade de Engenharia da Universidade do Porto, Porto , Portugal; 3 Centro de Estudos de Fenómenos de Transporte, Faculdade de Engenharia da Universidade do Porto, Porto , Portugal; 4 Electromechanical Engineering Department, University of Beira Interior, Covilhã, Castelo Branco 62-, Portugal \@ ' )3A3@J N'N3) L))'J(' 3J '2 ]5&HI >3Y 63JJ ^$2 8B"7:"9..B_ E3T$@IJ)'J3E 3ZN3)(T3@J'22H J&'J J&3 2$M 42$M $4' Y(IA$32'IJ(AN$2HT3)I$2?J($@(@'T(A)$42?(E(AA)$II%I2$JK3$T3J)HA'@N)$E?A3JM$JHN3I$4(@IJ'U(2(J(3I\@J&34()IJ(@IJ'U(2(JH'U$Y3'A)(J(A'2 `3U$)'&@?TU3)J&342$MU3A$T3I'IHTT3J)(ASU?J)3T'(@IIJ3'EHaN$@(@A)3'I(@KJ&3`3U$)'&@?TU3)IJ(224?)J&3)'I3A$@E(@IJ'U(2(JHI3JI(@ '@EJ&342$MU3A$T3IIJ)$@K2HJ(T3E3N3@E3@J \@J&3A?))3@JN'N3)M3E3T$@IJ)'J3@?T3)(A'22HS?I(@K'4(@(J3%Y$2?T3T3J&$ESJ&'JU$J&J&3I3(@IJ'U(2(J(3IA'@U3N)3E(AJ3E?I(@KJ&3?NN3)% A$@Y3AJ3E='ZM322T$E32?@E3)A)33N(@K42$MA$@E(J($@IS(@I$E$(@KE3T$@IJ)'J(@KJ&'JU$J&(@IJ'U(2(J(3I')3N?)32H32'IJ(A(@@'J?)33'2I$ I&$MJ&3IJ'U(2(b(@K3443AJ$4(@3)J('(@)3E?A(@KJ&342$M'IHTT3J)HS'@EM3&(K&2(K&JJ&3I33443AJI(@'De%ReT'NS(E3@J(4H(@KJ&3)3K($@I$4 IJ3'EHIHTT3J)(ASIJ3'EH'IHTT3J)(A'@E?@IJ3'EH"'IHTT3J)(A42$M)3K(T3IW&33443AJI$4)$?@E(@KJ&3A$)@3)I'@EJ&3?I3$4T$)3)3'2(IJ(A Y(IA$32'IJ(AA$@IJ(J?J(Y33O?'J($@IM(22'2I$U3E(IA?II3E =$@E'H-,-;L2N(@33IJ /=7. A mechanism for oscillatory instability in viscoelastic cross-slot flow 6(c('@E=(A&'32`)'&'T Department of Chemical and Biological Engineering, University of Wisconsin-Madison, Madison, WI , United States \@J3)($) IJ'K@'J($@ N$(@J 42$MI $4 Y(IA$32'IJ(A 2(O?(EI ')(I3 (@ ' M(E3 Y')(3JH $4 'NN2(A'J($@I (@A2?E(@K 3ZJ3@I($@'2 Y(IA$T3J)HS N$2HT3) N)$A3II(@K'@ET(A)$42?(E(AIDZN3)(T3@J'22HSJ&3I342$MI&'Y32$@KU33@d@$M@J$3Z&(U(J(@IJ'U(2(J(3ISU?JJ&3T3A&'@(ITI?@E3)2H(@KJ&3T &(K&2HIJ)3JA&3EN$2HT3)A&'(@I'II$A('J3EM(J&J&3$?J42$M4)$TJ&3IJ'K@'J($@N$(@J'J&(K&3(II3@U3)K@?TU3)LEE(J($@'22HSM3E3IA)(U3 J&3T3A&'@(IT$4(@IJ'U(2(JHSM&(A&')(I3I4)$TJ&3A$?N2(@K$442$MM(J&3ZJ3@I($@'2IJ)3II3I'@EJ&3()IJ33NK)'E(3@JI(@J&3IJ'K@'J($@N$(@J )3K($@ =$@E'H:,..L2N(@33IJ /=77 Low inertia mixing of viscous fluids by a chemically triggered shear flow instability W3$E$)\G?)K&323' 7 Sf3)IJ(@(32'K3%G?)A&')E 7 S\'@L/)(K'')E 9 S'@E=')d`=')J(@3b - Department of Mathematics, University of British Columbia, Vancouver, British Columbia V6T Z2, Canada; 2 University of British Columbia, Vancouver, Canada; 3 Department of Chemical and Biological Engineering, University of British Columbia, Vancouver, British Columbia V6T Z4, Canada 3IJ?EH3ZN3)(T3@J'22HJ&3T(Z(@K$4JM$Y(IA$?I42?(EIJ)3'TIUH'2$M>3H@$2EI@?TU3)I&3')42$M(@IJ'U(2(JHJ)(KK3)3EUH'4'IJA&3T(A'2 )3'AJ($@W&3Y(IA$?I42?(EI')33Y3@2H(@g3AJ3EI(E3UHI(E3(@'W%I&'N3EIJ)'(K&JA&'@@32L@'A(E%U'I3)3'AJ($@J'd(@KN2'A3'JJ&3(@J3)4'A3 U3JM33@'V3MJ$@('@42?(E'@E'#')U$N$2%8:.I$2?J($@23'EIJ$'IJ)$@KY(IA$I(JHIJ)'J(4(A'J($@SM&(A&2$A'22HE3IJ'U(2(b3IJ&342$MLI$@3 'EY'@A3IE$M@IJ)3'TSA$TN23ZI3A$@E')H42$MN'JJ3)@IE3Y32$N)3I?2J(@K(@344(A(3@JT(Z(@K3'2I$N)3I3@JN)32(T(@')H'@'2HI(I$4J&342$M '@E(@IJ'U(2(JHSY(''I(TN2(4(3ET$E32U'I3E$@'Y3)'K(@KJ&)$?K&J&3A&'@@32&3(K&J =$@E'H:,9;L2N(@33IJ /=79 The effects of poly(ethylene oxide) on the stability boundaries of flow regimes in co- and counter-rotating Taylor-Couette flow #')(<`?JA&3)'@E<?I'@F=?223) Department of Chemical Engineering, University of California, Berkeley, CA 9472, United States W&3N)3I3@A3$4g?IJ'43MN')JIN3)T(22($@$4&(K&T$23A?2')M3(K&J2(@3')N$2HT3)I(Id@$M@J$&'Y3'I(K@(4(A'@J(TN'AJ$@J?)U?23@J43'J?)3I (@42$MS@'T32HN$2HT3)%(@E?A3EJ?)U?23@JE)'K)3E?AJ($@W$A$@J)(U?J3 J$ J&3O?'@J(J'J(Y3?@E3)IJ'@E(@K $4 J&(I E)'T'J(A N&3@$T3@$@S J&3 '?J&$)I&'Y3 IJ?E(3E J&3(@42?3@A3 $4 E(2?J3 N$2H3J&H23@3 $Z(E3"5DX I$2?J($@I $@(I$2'J3E I3A$@E')H 42$M 43'J?)3I UH T'NN(@KJ&3 IJ'U(2(JH U$?@E')(3I $4 T?2J(N23 42$M )3K(T3I (@ ' W'H2$)%#$?3JJ3 "W# K3$T3J)HW&(I M$)d 3ZN'@EI $@N)3Y($?I M$)d (@Y$2Y(@K 5DX I$2?J($@I UH T'NN(@K J&3 U$?@E')(3I 4$) U$J& A$%'@EA$?@J3)% )$J'J($@'2 42$MIS M(J& ' 4$A?I $@ T$E(4(A'J($@ $4 &(K&3) $)E3) J)'@I(J($@I "?NJ$>3 (@@3) h O"7. - W&3 32'IJ(A(JHS E34(@3E 'I J&3 )'J($ $4 J&3 N$2HT3) )32'Z'J($@ J(T3 J$ J&3 (@3)J('2 J(T3 IA'23S 4$) J&3 5DX I$2?J($@I (@ ' Y(IA$I(4(3E 'O?3$?I I$2Y3@JS )'@K3I 4)$T O"7. %: J$7. %7 W&3 N$2HT3)(A I$2?J($@I ')3 A&')'AJ3)(b3E UH $4 (@E3N3@E3@J J3A&@(O?3IS (@A2?E(@K EH@'T(A'@EIJ3'EHI&3')42$MIS3ZJ3@I($@'242$MI"#'GD>SI3II(23E)$N3ZN3)(T3@JI'@EEH@'T(A2(K&JIA'JJ3)(@K#&'@K3I(@IJ'U(2(JH(@W# 3ZN3)(T3@JIM3)34$?@EE?)(@K'E('U'J(A(@A)3'I3I$4J&3(@@3)AH2(@E3)>3H@$2EI@?TU3)?I(@KIN3AJ)'2'@'2HI(I'@E42$MY(I?'2(b'J($@(@9` N2'@3I$4)'E('2S'Z('2SN)$g3AJ3E'b(T?J&'2'@EJ(T3E(T3@I($@IW&3)3I?2J'@J42$MIJ'J3J)'@I(J($@I')3A$TN')3EJ$N)3Y($?IIJ'U(2(JHT'NI4$) V3MJ$@('@ 42?(EI $UJ'(@3E(@ $?) 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