Abstract
We study the flow of a Newtonian fluid through microfabricated hyperbolic contractions followed by a sudden expansion, with the aim of investigating the potential of this geometry to serve as an extensional microrheometer. A set of planar converging geometries, with total Hencky strains ranging from 1.0 to 3.7, were fabricated in order to produce a homogeneous extensional flow field within the contraction. The velocity field in various planes of the hyperbolic contraction was quantified by means of microparticle image velocimetry (μPIV) and the pressure drop across the converging geometry was also measured and found to vary approximately linearly with the flow rate. Additionally, an extensive range of numerical calculations were carried out using a finite-volume method to help assess the performance of this geometry as a microfluidic elongational rheometer. The measured velocity fields in the contraction and associated pressure drops compare very well (to within 10%) with the numerical predictions. For the typical dimensions used in the microfluidic devices, the steady viscous flow through the contraction is shown to be three-dimensional and it is demonstrated that regions with nearly constant strain rate can only be achieved using geometries with large total Hencky strains under Hele–Shaw (potential-like) flow conditions.
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References
Acheson DJ (1990) Elementary fluid dynamics. Clarendon Press, Oxford
Alves MA, Oliveira PJ, Pinho FT (2003) A convergent and universally bounded interpolation scheme for the treatment of advection. Int J Numer Meth Fl 41:47–75
Barnes HA, Hutton JF, Walters K (1989) An introduction to rheology. Elsevier, Amsterdam
Brown RA, McKinley GH (1994) Report on the VIIIth international workshop on numerical methods in viscoelastic flows. J Non-Newton Fluid Mech 52:407–413
Chiang TP, Sheu TWH, Wang SK (2000) Side wall effects on the structure of laminar flow over a plane-symmetric sudden expansion. Comput Fluids 29:467–492
Cogswell FN (1978) Converging flow and stretching flow: a compilation. J Non-Newton Fluid Mech 4:23–38
Einstein A (1905) On the movement of small particles suspended in a stationary liquid demanded by the molecular-kinetic theory of heat. In: Theory of Brownian Movement. Dover Publications Inc., New York, pp 1–18
Everage AE Jr, Ballman RL (1978) The extensional flow capillary as a new method for extensional viscosity measurement. Nature 273:213–215
Feigl K, Tanner FX, Edwards BJ, Collier JR (2003) A numerical study of the measurement of elongational viscosity of polymeric fluids in semi-hyperbolically converging die. J Non-Newton Fluid Mech 115:191–215
Groisman A, Quake SR (2004) A microfluidic rectifier: anisotropic flow resistance at low Reynolds numbers. Phys Rev Lett 92:1–4
Hassager O (1988) Working group on numerical techniques (Vth workshop on numerical methods in non-Newtonian flow). J Non-Newton Fluid Mech 29:2–5
Hermansky CG, Boger DV (1995) Opposing-jet viscometry of fluids with viscosity approaching that of water. J Non-Newton Fluid Mech 56:1–14
James DF (1991) Flow in a converging channel at moderate Reynolds number. AIChE J 37:59–64
James DF, Chandler GM, Armour SJ (1990) A converging channel rheometer for the measurement of extensional viscosity. J Non-Newton Fluid Mech 35:421–443
Karniadakis G, Beskok A, Aluru NR (2005) Microflows and nanoflows: fundamentals and simulation. Springer Verlag, New York, NY
Kang K, Koelling KW, Lee LJ (2006) Microdevice end pressure evaluations with Bagley correction. Microfluid Nanofluid 2:223–235
Kang K, Lee LJ, Koelling KW (2005) High shear microfluidics and its application in rheological measurement. Exp Fluids 38:222–232
Lauga E, Stroock AD, Stone HA (2004) Three-dimensional flows in slowly varying planar geometries. Phys Fluid 16:3051–3062
McDonald JC, Duffy DC, Anderson JR, Chiu DT, Wu HK, Schueller OJA, Whitesides GM (2000) Fabrication of microfluidic systems in poly(dimethylsiloxane). Electrophoresis 21:27–40
Meinhart CD, Wereley ST, Gray MHB (2000) Volume illumination for two-dimensional particle image velocimetry. Meas Sci Technol 11:809–814
Ng JMK, Gitlin I, Stroock AD, Whitesides GM (2002) Components for integrated poly(dimethylsiloxane) microfluidic systems. Electrophoresis 23:3461–3473
Oliveira PJ (2003) Asymmetric flows of viscoelastic fluids in symmetric planar expansion geometries. J Non-Newton Fluid Mech 114:33–63
Oliveira PJ, Pinho FT (1999) Numerical procedure for the computation of fluid flow with arbitrary stress-strain relationships. Numer Heat Tr B-Fund 35:295–315
Oliveira PJ, Pinho FT, Pinto GA (1998) Numerical simulation of non-linear elastic flows with a general collocated finite-volume method. J Non-Newton Fluid Mech 79:1–43
Olsen MG, Adrian RJ (2000) Out-of-focus effects on particle image visibility and correlation in microscopic particle image velocimetry. Exp Fluids 29:S166–S174
Prasad AK, Adrian RJ, Landreth CC, Offutt PW (1992) Effect of resolution on the speed and accuracy of particle image velocimetry interrogation. Exp Fluids 13:105–116
Rodd LE (2006) Planar entry flow of low viscosity elastic fluids in micro-fabricated contraction geometries. Ph.D. Thesis, Department of Chemical and Biomolecular Engineering, University of Melbourne, Melbourne, VIC, Australia
Rodd LE, Scott TP, Boger DV, Cooper-White JJ, McKinley GH (2005) The inertio-elastic planar entry flow of low-viscosity elastic fluids in micro-fabricated geometries. J Non-Newton Fluid Mech 129:1–22
Rodd LE, Cooper-White JJ, Boger DV, McKinley GH (2007) Role of the elasticity number in the entry flow of dilute polymer solutions in micro-fabricated contraction geometries. J Non-Newton Fluid Mech (in press)
Santiago JG, Wereley ST, Meinhart CD, Beebe DJ, Adrian RJ (1998) Particle image velocimetry systems for microfludics. Exp Fluids 25:316–319
Scott TP (2004) Contraction/expansion flow of dilute elastic solutions in microchannels. M.S. Thesis, Mechanical Engineering Department, MIT, Cambridge, MA, USA
Sharp KV, Adrian RJ (2004) Transition from laminar to turbulent flow in liquid filled microtubes. Exp Fluids 36:741–747
Shaw MT (1975) Flow of polymer melts through a well-lubricated, conical die. J Appl Polym Sci 19:2811–2816
Squires TM, Quake SR (2005) Microfluidics: fluid physics at the nanoliter scale. Rev Mod Phys 77:977–1026
Stone HA, Stroock AD, Ajdari A (2004) Engineering flows in small devices: microfluidics toward a Lab-on-a-Chip. Annu Review Fluid Mech 36:381–411
Tsai C-H, Chen H-T, Wang Y-N, Lin C-H, Fu L-M (2006) Capabilities and limitations of 2-dimensional and 3-dimensional numerical methods in modeling the fluid flow in sudden expansion microchannels. Microfluid and Nanofluid. doi:10.1007/s10404-006-0099-2
Townsend P, Walters K (1994) Expansion flows of non-Newtonian liquids. Chem Eng Sci 49:749–763
Wereley ST, Meinhart CD (2004) Micron-resolution particle image velocimetry. In: Breuer KS (ed) Microscale diagnostic techniques. Springer, Berlin
Wereley ST, Gui L, Meinhart CD (2002) Advanced algorithms for microscale particle image velocimetry. AIAA J 40:1047–1055
White FM (1991) Viscous fluid flow. McGraw-Hill, New York
Whitesides GM, Stroock AD (2001) Flexible methods for microfluidics. Phys Today 54:42–48
Wille R, Fernholz H (1965) Report on first European mechanics colloquium on coanda effect. J Fluid Mech 23:801–819
Acknowledgments
M. S. N. Oliveira would like to thank Fundação para a Ciência e a Tecnologia (FCT), Portugal for the financial support (SFRH/BPD/15005/2004). M. S. N. Oliveira, M. A. Alves and F. T. Pinho acknowledge the financial support provided under program POCI2010 by FCT and FEDER: project POCI/EME/59338/2004 (M. S. N. Oliveira, M. A. Alves, and F. T. Pinho) and project POCI/EQU/59256/2004 (M. A. Alves). The experimental portion of this work was carried out in the Hatsopoulos Microfluids Laboratory at MIT using equipment provided by the National Science Foundation under grant CTS-0116486.
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Oliveira, M.S.N., Alves, M.A., Pinho, F.T. et al. Viscous flow through microfabricated hyperbolic contractions. Exp Fluids 43, 437–451 (2007). https://doi.org/10.1007/s00348-007-0306-2
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DOI: https://doi.org/10.1007/s00348-007-0306-2