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Geometric scaling of a purely elastic flow instability in serpentine channels

Published online by Cambridge University Press:  01 October 2012

J. Zilz ,
R. J. Poole ,
M. A. Alves ,
D. Bartolo ,
B. Levaché  and
A. Lindner
J. Zilz
Affiliation:
PMMH UMR7636–ESPCI Paristech–CNRS–Paris 6–Paris 7, 10 rue Vauquelin F-75231 Paris CEDEX 05, France
R. J. Poole
Affiliation:
School of Engineering, University of Liverpool, Brownlow Hill, Liverpool L69 3GH, UK
M. A. Alves
Affiliation:
Departamento de Engenharia Química, Centro de Estudos de Fenómenos de Transporte, Faculdade de Engenharia da Universidade do Porto, Rua Doutor Roberto Frias, 4200-465 Porto, Portugal
D. Bartolo
Affiliation:
PMMH UMR7636–ESPCI Paristech–CNRS–Paris 6–Paris 7, 10 rue Vauquelin F-75231 Paris CEDEX 05, France
B. Levaché
Affiliation:
PMMH UMR7636–ESPCI Paristech–CNRS–Paris 6–Paris 7, 10 rue Vauquelin F-75231 Paris CEDEX 05, France
A. Lindner*
Affiliation:
PMMH UMR7636–ESPCI Paristech–CNRS–Paris 6–Paris 7, 10 rue Vauquelin F-75231 Paris CEDEX 05, France
*
†Email address for correspondence: [email protected]
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Abstract

A combined experimental, numerical and theoretical investigation of the geometric scaling of the onset of a purely elastic flow instability in serpentine channels is presented. Good qualitative agreement is obtained between experiments, using dilute solutions of flexible polymers in microfluidic devices, and three-dimensional numerical simulations using the upper-convected Maxwell model. The results are confirmed by a simple theoretical analysis, based on the dimensionless criterion proposed by Pakdel & McKinley (Phys. Rev. Lett., vol. 77, 1996, pp. 2459–2462) for onset of a purely elastic flow instability. Three-dimensional simulations show that the instability is primarily driven by the curvature of the streamlines induced by the flow geometry and not due to the weak secondary flow in the azimuthal direction. In addition, the simulations also reveal that the instability is time-dependent and that the flow oscillates with a well-defined period and amplitude close to the onset of the supercritical instability.

JFM classification

Low-Reynolds-number flows: Low-Reynolds-number flows Micro-/Nano-fluid dynamics: Microfluidics Non-Newtonian Flows: Viscoelasticity
Type
Papers
Information
Journal of Fluid Mechanics , Volume 712 , 10 December 2012 , pp. 203 - 218
Copyright
©2012 Cambridge University Press

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References

Afonso, A., Oliveira, P. J., Pinho, F. T. & Alves, M. A. 2009 The log-conformation tensor approach in the finite-volume method framework. J. Non-Newtonian Fluid Mech. 157 (1–2), 55–65.CrossRefGoogle Scholar
Afonso, A., Oliveira, P. J., Pinho, F. T. & Alves, M. A. 2011 Dynamics of high-Deborah-number entry flows: a numerical study. J. Fluid Mech. 677, 272–304.Google Scholar
Alves, M. A., Oliveira, P. J. & Pinho, F. T. 2003 Benchmark solutions for the flow of Oldroyd-B and PTT fluids in planar contractions. J. Non-Newtonian Fluid Mech. 110, 45–75.Google Scholar
Alves, M. A. & Poole, R. J. 2007 Divergent flow in contractions. J. Non-Newtonian Fluid Mech. 144, 140–148.CrossRefGoogle Scholar
Berti, S. & Boffetta, G. 2010 Elastic waves and transition to elastic turbulence in a two-dimensional viscoelastic Kolmogorov flow. Phys. Rev. E 82, 036314.Google Scholar
Bird, R. B., Armstrong, R. C. & Hassager, O. 1987 Dynamics of Polymeric Liquids. Wiley.Google Scholar
Bistagnino, A., Boffetta, G., Celani, A., Mazzino, A., Puliafito, A. & Vergassola, M. 2007 Nonlinear dynamics of the viscoelastic Kolmogorov flow. J. Fluid Mech. 590, 61–80.Google Scholar
Boffetta, G., Celani, A., Mazzino, A., Puliafito, A. & Vergassola, M. 2005 The viscoelastic Kolmogorov flow: eddy-viscosity and linear stability. J. Fluid Mech. 523, 161–170.CrossRefGoogle Scholar
Bonhomme, O., Morozov, A., Leng, J. & Colin, A. 2011 Elastic instability in stratified core annular flow. Phys. Rev. E 83, 065301(R).Google Scholar
Burghelea, T., Segre, E., Bar-Joseph, I., Groisman, A. & Steinberg, V. 2004 Chaotic flow and efficient mixing in a micro-channel with a polymer solution. Phys. Rev. E 69, 066305.Google Scholar
Chen, K. P. 1991 Elastic instability of the interface in Couette flow of viscoelastic liquids. J. Fluid Mech. 40, 261–267.Google Scholar
Dean, W. R. 1927 Note on the motion in a curved pipe. Phil. Mag. 4, 208–223.Google Scholar
Fan, Y., Tanner, R. I. & Phan-Thien, N. 2001 Fully developed viscous and viscoelastic flows in curved pipes. J. Fluid Mech. 440, 327–357.CrossRefGoogle Scholar
Fardin, M. A., Ober, T. J., Gay, C., Gregoire, G., McKinley, G. H. & Lerouge, S. 2011 Criterion for purely elastic Taylor–Couette instability in the flows of shear-banding fluids. Eur. Phys. Lett. 96, 44004.CrossRefGoogle Scholar
Fattal, R. & Kupferman, R. 2004 Constitutive laws for the matrix-logarithm of the conformation tensor. J. Non-Newtonian Fluid Mech. 123, 281–285.Google Scholar
Grillet, A. M. & Shaqfeh, E. S. G. 1996 Viscoelastic instabilities in recirculation flows of Boger fluids. J. Non-Newtonian Fluid Mech. 64, 141–155.Google Scholar
Groisman, A. & Steinberg, V. 2000 Elastic turbulence in a polymer solution flow. Nature 405, 53–55.Google Scholar
Groisman, A. & Steinberg, V. 2001 Efficient mixing at low Reynolds numbers using polymer additives. Nature 410, 905–908.Google Scholar
Haward, S. J., Ober, T. J., Oliveira, M. S. N., Alves, M. A. & McKinley, G. H. 2012 Extensional rheology and elastic instabilities of a wormlike micellar solution in a microfluidic cross-slot device. Soft Matt. 8, 536–555.CrossRefGoogle Scholar
Joo, Y. L. & Shaqfeh, E. S. G. 1991 Viscoelastic Poiseuille flow through a curved channel: a new elastic instability. Phys. Fluids A 3, 1691–1694.Google Scholar
Joo, Y. L. & Shaqfeh, E. S. G. 1994 Observations of purely elastic instabilities in the Taylor–Dean flow of a Boger fluid. J. Fluid Mech. 262, 27–73.Google Scholar
Larson, R. G., Shaqfeh, E. S. G. & Muller, S. J. 1990 A purely elastic instability in Taylor–Couette flow. J. Fluid Mech. 218, 573–600.Google Scholar
Lee, J. S., Dylla-Spears, R., Teclemariam, N.-P. & Muller, S. J. 2007 Microfluidic four-roll mill for all flow types. Appl. Phys. Lett. 90, 074103.Google Scholar
Lindner, A., Vermant, J. & Bonn, D. 2003 How to obtain the elongational viscosity of dilute polymer solutions? Physica A 319, 125–133.CrossRefGoogle Scholar
McKinley, G. H., Byars, J. A., Brown, R. A. & Armstrong, R. C. 1991 Observations on the elastic instability in cone-and-plate and parallel-plate flows of a polyisobutylene Boger fluid. J. Non-Newtonian Fluid Mech. 10, 201–229.CrossRefGoogle Scholar
McKinley, G. H., Oztekin, A., Byars, J. A. & Brown, R. A. 1995 Self-similar spiral instabilities in elastic flows between a cone and a plate. J. Fluid Mech. 285, 123–164.Google Scholar
McKinley, G. H., Pakdel, P. & Oztekin, A. 1996 Rheological and geometric scaling of purely elastic flow instabilities. J. Non-Newtonian Fluid Mech. 67, 19–47.CrossRefGoogle Scholar
Meulenbroek, B., Storm, C., Morozov, A. N. & van Saarloos, W. 2004 Weakly non-linear subcritical instability of viscoelastic Poiseuille flow. J. Non-Newtonian Fluid Mech. 116, 235–268.Google Scholar
Morozov, A. N. & van Saarloos, W. 2007 An introductory essay on subcritical instabilities and the transition to turbulence in visco-elastic parallel shear flows. Phys. Rep. 447 (3–6), 112–143.Google Scholar
Oliveira, P. J., Pinho, F. T. & Pinto, G. A. 1998 Numerical simulation of non-linear elastic flows with a general collocated finite-volume method. J. Non-Newtonian Fluid Mech. 79, 1–43.Google Scholar
Pakdel, P. & McKinley, G. H. 1996 Elastic instability and curved streamlines. Phys. Rev. Lett. 77 (12), 2459–2462.Google Scholar
Pathak, J. A., Ross, D. & Migler, K. B. 2004 Elastic flow instability, curved streamlines, and mixing in microfluidic flows. Phys. Fluids 16, 4028–4034.Google Scholar
Poole, R. J., Alves, M. A. & Oliveira, P. J. 2007 Purely elastic flow asymmetries. Phys. Rev. Lett. 99 (16), 164503.CrossRefGoogle ScholarPubMed
Rodd, L. E., Cooper-White, J. J., Boger, D. V. & McKinley, G. H. 2007 Role of the elasticity number in the entry flow of dilute polymer solutions in micro-fabricated contraction geometries. J. Non-Newtonian Fluid Mech. 143 (2–3), 170–191.Google Scholar
Shaqfeh, E. S. G. 1996 Purely elastic instabilities in viscometric flows. Annu. Rev. Fluid Mech. 28, 129–185.Google Scholar
Su, Y.-Y. & Khomami, B. 1992 Purely elastic interfacial instabilities in superposed flow of polymeric fluids. Rheol. Acta 31, 413–420.CrossRefGoogle Scholar
Sudarsan, A. P. & Ugaz, V. M. 2006 Multivortex micromixing. Proc. Natl Acad. Sci. 103, 7228–7233.Google Scholar
Thomases, B. & Shelley, M. 2009 Transition to mixing and oscillations in a Stokesian viscoelastic flow. Phys. Rev. Lett. 103 (9), 094501.Google Scholar
White, F. M. 2005 Viscous Fluid Flow. McGraw-Hill.Google Scholar
Zell, A., Gier, S., Rafai, S. & Wagner, C. 2010 Is there a relation between the relaxation time measured in CaBER experiments and the first normal stress coefficient? J. Non-Newtonian Fluid Mech. 165 (19–20), 1265–1274.CrossRefGoogle Scholar