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The motion of long drops in rectangular microchannels at low capillary numbers

Published online by Cambridge University Press:  02 August 2018

Sai Sashankh Rao  and
Harris Wong Open the ORCID record for Harris Wong [Opens in a new window]
Sai Sashankh Rao
Affiliation:
Department of Mechanical and Industrial Engineering, Louisiana State University, Baton Rouge, LA 70803, USA
Harris Wong*
Affiliation:
Department of Mechanical and Industrial Engineering, Louisiana State University, Baton Rouge, LA 70803, USA
*
†Email address for correspondence: hwong@lsu.edu
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Abstract

Drop flow in rectangular microchannels has been utilized extensively in microfluidics. However, the pressure-gradient versus flow-rate relation is still not well understood. We study the motion of a long drop in a rectangular microchannel in the limit the capillary number $Ca\rightarrow 0$ ($Ca=\unicode[STIX]{x1D707}U/\unicode[STIX]{x1D70E}$, where $U$ is the constant drop velocity, $\unicode[STIX]{x1D707}$ is the viscosity of the carrier liquid and $\unicode[STIX]{x1D70E}$ is the interfacial tension). In this limit, the moving drop looks like the static drop and has two end caps connected by a long column, which is surrounded by thin films on the microchannel wall and by menisci along the microchannel corners. Integral axial force balances on the drop fluid and on the carrier liquid surrounding the drop relate the carrier-liquid pressure gradient to the drop-fluid pressure gradient and the contact-line drag. The contact-line drag is argued to be the same as that for a long bubble (which has been determined by Wong et al. (J. Fluid Mech., vol. 292, 1995b, pp. 95–110)) if the viscosity ratio $\unicode[STIX]{x1D706}\ll Ca^{-1/3}$ and $\unicode[STIX]{x1D706}\ll L$, where $\unicode[STIX]{x1D706}=\bar{\unicode[STIX]{x1D707}}/\unicode[STIX]{x1D707}$, $\bar{\unicode[STIX]{x1D707}}$ is the drop viscosity and $L~(\gg 1)$ is the dimensionless drop length. Thus, the force balances yield one equation relating the two pressure gradients. The two pressure gradients also drive unidirectional flows in the drop and in the corner channels along the long middle column. These coupled flows are solved by a finite-element method to yield another equation relating the two pressure gradients. From the two equations, we determine the pressure gradients and thus the unidirectional velocity fields inside and outside the drop for $\unicode[STIX]{x1D706}=0$–100 and various microchannel aspect ratios. We find that in the limit $LCa^{1/3}\rightarrow 0$, the contact-line drag dominates and the carrier liquid bypasses the drop through the corner channels alongside the drop. For $LCa^{1/3}\gg 1$, the contact-line drag is negligible and the corner fluid is stationary. Thus, the drop moves as a leaky piston. We extend our model to a train of long drops, and compare our model predictions with published experiments.

JFM classification

Drops and Bubbles: Drops Micro-/Nano-fluid dynamics: Microfluidics Multiphase and Particle-laden Flows: Multiphase flow
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 852 , 10 October 2018 , pp. 60 - 104
Copyright
© 2018 Cambridge University Press 

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References

Anna, S. L. 2016 Droplets and bubbles in microfluidic devices. Annu. Rev. Fluid Mech. 48 (1), 285–309.Google Scholar
Baroud, C. N., Gallaire, G. & Dangla, R. 2010 Dynamics of microfluidic droplets. Lab on a Chip 10, 2032–2045.Google Scholar
Chaudhury, K., Acharya, P. V. & Chakraborty, S. 2014 Influence of disjoining pressure on the dynamics of steadily moving long bubbles inside narrow cylindrical capillaries. Phys. Rev. E 89 (5), 053002.Google Scholar
Chen, H., Li, Z. & Li, J. 2016 Thin-film profile around long bubbles in square microchannels measured by chromatic interference method. Appl. Phys. Lett. 109, 041604.Google Scholar
Cherlo, S. K. R., Kariveti, S. & Pushpavanam, S. 2010 Experimental and numerical investigations of two-phase (liquid–liquid) flow behavior in rectangular microchannels. Ind. Engng Chem. Res. 49 (2), 893–899.Google Scholar
Davis, S. H. 1980 Moving contact lines and rivulet instabilities. Part 1. The static rivulet. J. Fluid Mech. 98 (2), 225–242.Google Scholar
Fuerstman, M. J., Lai, A., Thurlow, M. E., Shevkoplyas, S. S., Stone, H. A. & Whitesides, G. M. 2007 The pressure drop along rectangular microchannels containing bubbles. Lab on a Chip 7 (11), 1479–1489.Google Scholar
Hammoud, N. H., Trinh, P. H., Howell, P. D. & Stone, H. A. 2017 Influence of van der Waals forces on a bubble moving in a tube. Phys. Rev. Fluids 2 (6), 063601.Google Scholar
Hoang, D. A., van Steijn, V., Portela, L. M., Kreutzer, M. T. & Kleijn, C. R. 2013 Benchmark numerical simulations of segmented two-phase flows in microchannels using the volume of fluid method. Comput. Fluids 86, 28–36.Google Scholar
Hodges, S. R., Jensen, O. E. & Rallison, J. M. 2004 The motion of a viscous drop through a cylindrical tube. J. Fluid Mech. 501, 279–301.Google Scholar
Huebner, A., Sharma, S., Srisa-Art, M., Hollfelder, F., Edel, J. B. & deMello, A. J. 2008 Microdroplets: a sea of applications? Lab on a Chip 8 (8), 1244–1254.Google Scholar
Jakiela, S., Korczyk, P. M., Makulska, S., Cybulski, O. & Garstecki, P. 2012 Discontinuous transition in a laminar fluid flow: a change of flow topology inside a droplet moving in a micron-size channel. Phys. Rev. Lett. 108 (13), 134501.Google Scholar
Jakiela, S., Makulska, S., Korczyk, P. M. & Garstecki, P. 2011 Speed of flow of individual droplets in microfluidic channels as a function of the capillary number, volume of droplets and contrast of viscosities. Lab on a Chip 11 (21), 3603–3608.Google Scholar
Jin, Y., Orth, A., Schonbrun, E. & Crozier, K. B. 2012 Measuring the pressures across microfluidic droplets with an optical tweezer. Opt. Exp. 20 (22), 24450–24464.Google Scholar
Jovanović, J., Rebrov, E. V., Nijhuis, T. A., Hessel, V. & Schouten, J. C. 2010 Phase-transfer catalysis in segmented flow in a microchannel: fluidic control of selectivity and productivity. Ind. Engng Chem. Res. 49 (6), 2681–2687.Google Scholar
Jovanović, J., Zhou, W., Rebrov, E. V., Nijhuis, T. A., Hessel, V. & Schouten, J. C. 2011 Liquid–liquid slug flow: hydrodynamics and pressure drop. Chem. Engng Sci. 66 (1), 42–54.Google Scholar
Kim, N., Murphy, M. C., Soper, S. A. & Nikitopoulos, D. E. 2014 Liquid–liquid segmented flows in polycarbonate microchannels with cross-sectional expansions. Intl J. Multiphase Flow 58 (0), 83–96.Google Scholar
Kinoshita, H., Kaneda, S., Fujii, T. & Oshima, M. 2007 Three-dimensional measurement and visualization of internal flow of a moving droplet using confocal micro-PIV. Lab on a Chip 7, 338–346.Google Scholar
Lac, E. & Sherwood, J. D. 2009 Motion of a drop along the centreline of a capillary in a pressure-driven flow. J. Fluid Mech. 640, 27–54.Google Scholar
Mathworks2017 Partial Differential Equation Toolbox: User’s Guide (R2013b). Retrieved December 17. http://www.mathworks.com/help/pdf_doc/pde/pde.pdf.Google Scholar
Park, C.-W. & Homsy, G. 1984 Two-phase displacement in Hele Shaw cells: theory. J. Fluid Mech. 139 (1), 291–308.Google Scholar
Patzek, T. W. & Kristensen, J. G. 2001 Shape factor correlations of hydraulic conductance in noncircular capillaries: II. Two-phase creeping flow. J. Colloid Interface Sci. 236 (2), 305–317.Google Scholar
Raj, R., Mathur, N. & Buwa, V. V. 2010 Numerical simulations of liquid–liquid flows in microchannels. Ind. Engng Chem. Res. 49 (21), 10606–10614.Google Scholar
Ransohoff, T. C. & Radke, C. J. 1988 Laminar flow of a wetting liquid along the corners of a predominantly gas-occupied noncircular pore. J. Colloid Interface Sci. 121 (2), 392–401.Google Scholar
Rao, S. S.2015 Modeling two-phase flow and heat transfer in polygonal microchannels. Doctoral dissertation, Louisiana State University, LA. https://digitalcommons.lsu.edu/gradschool_dissertations/1087.Google Scholar
Rao, S. S. & Wong, H. 2015 Heat and mass transfer in polygonal micro heat pipes under small imposed temperature differences. Intl J. Heat Mass Transfer 89, 1369–1385.Google Scholar
Sarrazin, F., Bonometti, T., Prat, L., Gourdon, C. & Magnaudet, J. 2008 Hydrodynamic structures of droplets engineered in rectangular micro-channels. Microfluid. Nanofluid. 5 (1), 131–137.Google Scholar
Sarrazin, F., Loubière, K., Prat, L., Gourdon, C., Bonometti, T. & Magnaudet, J. 2006 Experimental and numerical study of droplets hydrodynamics in microchannels. AIChE J. 52 (12), 4061–4070.Google Scholar
Shikhmurzaev, Y. D. 1997 Moving contact lines in liquid/liquid/solid systems. J Fluid Mech. 334, 211–249.Google Scholar
Snoeijer, J. H. & Andreotti, B. 2013 Moving contact lines: scales, regimes, and dynamical transitions. Annu. Rev. Fluid Mech. 45, 269–292.Google Scholar
Soares, E. J. & Thompson, R. L. 2009 Flow regimes for the immiscible liquid–liquid displacement in capillary tubes with complete wetting of the displaced liquid. J. Fluid Mech. 641, 63–84.Google Scholar
Song, H., Chen, D. L. & Ismagilov, R. F. 2006 Reactions in droplets in microfluidic channels. Angew. Chem. Intl Ed. Engl. 45 (44), 7336–7356.Google Scholar
Stone, H. A., Stroock, A. D. & Ajdari, A. 2004 Engineering flows in small devices. Annu. Rev. Fluid Mech. 36 (1), 381–411.Google Scholar
Tan, J., Xu, J. H., Li, S. W. & Luo, G. S. 2008 Drop dispenser in a cross-junction microfluidic device: scaling and mechanism of break-up. Chem. Engng J. 136 (2), 306–311.Google Scholar
Teh, S.-Y., Lin, R., Hung, L.-H. & Lee, A. P. 2008 Droplet microfluidics. Lab on a Chip 8 (2), 198–220.Google Scholar
Thompson, P. A. & Troian, S. M. 1997 A general boundary condition for liquid flow at solid surfaces. Nature 389, 360–362.Google Scholar
Vanapalli, S. A., Banpurkar, A. G., van den Ende, D., Duits, M. H. G. & Mugele, F. 2009 Hydrodynamic resistance of single confined moving drops in rectangular microchannels. Lab on a Chip 9 (7), 982–990.Google Scholar
Vladimir, S. A. & Homsy, G. M. 2006 Modeling shapes and dynamics of confined bubbles. Annu. Rev. Fluid Mech. 38 (1), 277–307.Google Scholar
White, F. M. 1991 Viscous Fluid Flow, 2nd edn., p. 120. McGraw-Hill.Google Scholar
Wong, H., Fatt, I. & Radke, C. J. 1996 Deposition and thinning of the human tear film. J. Colloid Interface Sci. 184 (1), 44–51.Google Scholar
Wong, H., Morris, S. & Radke, C. J. 1992 Three-dimensional menisci in polygonal capillaries. J. Colloid Interface Sci. 148 (2), 317–336.Google Scholar
Wong, H., Radke, C. J. & Morris, S. 1995a The motion of long bubbles in polygonal capillaries. Part 1. Thin films. J. Fluid Mech. 292, 71–94.Google Scholar
Wong, H., Radke, C. & Morris, S. 1995b The motion of long bubbles in polygonal capillaries. II: drag, fluid pressure and fluid flow. J. Fluid Mech. 292, 95–110.Google Scholar
Yong, Y., Yang, C., Jiang, Yi., Joshi, A., Shi, Y. & Yin, X. 2011 Numerical simulation of immiscible liquid–liquid flow in microchannels using lattice Boltzmann method. Sci. China Chem. 54 (1), 244–256.Google Scholar