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

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1 R. J. Poole pt. Engineering, Mechanical Engineering, University of Liverpool Liverpool L69 3GH, UK, M. A. Alves partamento de Engenharia uímica, CEFT, Faculdade de Engenharia da Universidade do Porto, Portugal, A. Afonso partamento 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 partamento 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 (=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 =.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 ( - CR ) 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- Map! 2 Unsteady asymmetric Steady asymmetric.3.4.5

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)! pendent variables evaluated at cell centers;! Special formulations for cell-face velocities and stresses;

5 Computational domain, boundary conditions =UD =!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, =.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 CR -.75 For ">3/9 flow became asymmetric unsteady;

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 CR creases the degree of asymmetry;.75.5 Re=4 For Re>2, asymmetric unsteady flow..72 D.25 Re= Re= Re= D Re=

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

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

10 Conclusions! Increasing the solvent viscosity (increasing!) increases the critical. 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 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.!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