How valid is Taylor dispersion formula in slugs?

Abstract

In a landmark paper, Taylor predicted that shear flow increases the effective diffusivity of species [Taylor, Proc. Roy. Soc. A, 219:186-203,1953]. This paper focused on Poiseuille flow in a circular pipe and predicted the existence of an effective species diffusion much greater than molecular diffusion. The ratio between the effective and molecular diffusion was shown to scale with the square of the Peclet number (product of the pipe diameter with the mean flow velocity divided by the molecular diffusivity). Taylor's study assumed two infinite columns of miscible fluids initially juxtaposed in a pipe and transported by the flow. A question of high practical interest is how valid this prediction is when a finite-sized slug is considered instead of an infinite fluid column. This paper sheds light on the finite-size effects on the mixing of two miscible fluids in a slug and quantifies how accurate Taylor's prediction is for finite length liquid columns. Results show that Taylor's dispersion formula is most accurate for lower Peclet numbers and longer slugs. Results also show that mixing is quite insensitive to the Reynolds number. References

• Daniel A Beard. Taylor dispersion of a solute in a microfluidic channel. Journal of Applied Physics, 89(8):4667–4669, 2001. doi:10.1063/1.1357462.
• J Bico and D Quere. Liquid trains in a tube. EPL (Europhysics Letters), 51(5):546, 2000. doi:https://doi.org/10.1209/epl/i2000-00373-4.
• Jose Bico and David Quere. Falling slugs. Journal of colloid and interface science, 243(1):262–264, 2001. doi:https://doi.org/10.1006/jcis.2001.7891.
• Jose Bico and David Quere. Self-propelling slugs. Journal of Fluid Mechanics, 467:101–127, 2002. doi:10.1017/S002211200200126X.
• Wolfgang Buchegger, Christoph Wagner, Bernhard Lendl, Martin Kraft, and Michael J Vellekoop. A highly uniform lamination micromixer with wedge shaped inlet channels for time resolved infrared spectroscopy. Microfluidics and Nanofluidics, 10(4):889–897, 2011. doi:10.1007/s10404-010-0722-0.
• JR Burns and C Ramshaw. The intensification of rapid reactions in multiphase systems using slug flow in capillaries. Lab on a Chip, 1(1):10–15, 2001. doi:10.1039/B102818A.
• Brian Carroll and Carlos Hidrovo. Experimental investigation of inertial mixing in colliding droplets. Heat Transfer Engineering, 34(2-3):120–130, 2013. doi:10.1080/01457632.2013.703087.
• Rachid Chebbi. Dynamics of viscous slugs fall in dry capillaries. Journal of Adhesion Science and Technology, 28(16):1655–1660, 2014. doi:10.1080/01694243.2014.911645.
• GQ Chen and Zi Wu. Taylor dispersion in a two-zone packed tube. International Journal of Heat and Mass Transfer, 55(1):43–52, 2012. http://dx.doi.org/https://doi.org/10.1016/j.ijheatmasstransfer.2011.08.037 doi:https://doi.org/10.1016/j.ijheatmasstransfer.2011.08.037.
• Russell L Detwiler, Harihar Rajaram, and Robert J Glass. Solute transport in variable-aperture fractures: An investigation of the relative importance of taylor dispersion and macrodispersion. Water Resources Research, 36(7):1611–1625, 2000. doi:10.1029/2000WR900036.
• Roman O Grigoriev, Michael F Schatz, and Vivek Sharma. Chaotic mixing in microdroplets. Lab on a Chip, 6(10):1369–1372, 2006. doi:10.1039/B607003E.
• K Handique and Mark A Burns. Mathematical modeling of drop mixing in a slit-type microchannel. Journal of Micromechanics and Microengineering, 11(5):548, 2001. doi:https://doi.org/10.1088/0960-1317/11/5/316.
• Mranal Jain and K Nandakumar. Novel index for micromixing characterization and comparative analysis. Biomicrofluidics, 4(3):031101, 2010. doi:10.1063/1.3457121.
• Mark Johnson and Roger D Kamm. Numerical studies of steady flow dispersion at low dean number in a gently curving tube. Journal of Fluid Mechanics, 172:329–345, 1986. doi:10.1017/S0022112086001763.
• Madhvanand N Kashid and David W Agar. Hydrodynamics of liquid–liquid slug flow capillary microreactor: flow regimes, slug size and pressure drop. Chemical Engineering Journal, 131(1):1–13, 2007. doi:https://doi.org/10.1016/j.cej.2006.11.020.
• KP Mayock, JM Tarbell, and JL Duda. Numerical simulation of solute dispersion in laminar tube flow. Separation Science and Technology, 15(6):1285–1296, 1980. doi:10.1080/01496398008068505.
• Virginie Mengeaud, Jacques Josserand, and Hubert H Girault. Mixing processes in a zigzag microchannel: finite element simulations and optical study. Analytical chemistry, 74(16):4279–4286, 2002. doi:10.1021/ac025642e.
• Metin Muradoglu, Axel Gunther, and Howard A Stone. A computational study of axial dispersion in segmented gas-liquid flow. Physics of Fluids, 19(7):072109, 2007. doi:10.1063/1.2750295.
• Metin Muradoglu and Howard A Stone. Mixing in a drop moving through a serpentine channel: A computational study. Physics of Fluids, 17(7):073305, 2005. doi:10.1063/1.1992514.
• Peter E Neerincx, Roel PJ Denteneer, Sven Peelen, and Han EH Meijer. Compact mixing using multiple splitting, stretching, and recombining flows. Macromolecular Materials and Engineering, 296(3-4):349–361, 2011. doi:10.1002/mame.201000338.
• Mathieu Sellier, Claude Verdier, and Volker Nock. The spontaneous motion of a slug of miscible liquids in a capillary tube. International Journal of Nanotechnology, 14(1-6):530–539, 2017. doi:https://doi.org/10.1504/IJNT.2017.082475.
• Elliot J Smith, Wang Xi, Denys Makarov, Ingolf Monch, Stefan Harazim, Vladimir A Bolanos Quinones, Christine K Schmidt, Yongfeng Mei, Samuel Sanchez, and Oliver G Schmidt. Lab-in-a-tube: ultracompact components for on-chip capture and detection of individual micro-/nanoorganisms. Lab on a Chip, 12(11):1917–1931, 2012. doi:10.1039/C2LC21175K.
• Howard A Stone, Abraham D Stroock, and Armand Ajdari. Engineering flows in small devices: microfluidics toward a lab-on-a-chip. Annu. Rev. Fluid Mech., 36:381–411, 2004. doi:10.1146/annurev.fluid.36.050802.122124.
• Abraham D Stroock, Stephan KW Dertinger, Armand Ajdari, Igor Mezic, Howard A Stone, and George M Whitesides. Chaotic mixer for microchannels. Science, 295(5555):647–651, 2002. doi:10.1126/science.1066238.
• Wiroon Tanthapanichakoon, Nobuaki Aoki, Kazuo Matsuyama, and Kazuhiro Mae. Design of mixing in microfluidic liquid slugs based on a new dimensionless number for precise reaction and mixing operations. Chemical Engineering Science, 61(13):4220–4232, 2006. doi:https://doi.org/10.1016/j.ces.2006.01.047.
• Geoffrey Taylor. Dispersion of soluble matter in solvent flowing slowly through a tube. In Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, volume 219, pages 186–203. The Royal Society, 1953. doi:10.1098/rspa.1953.0139.
• Terje Tofteberg, Maciej Skolimowski, Erik Andreassen, and Oliver Geschke. A novel passive micromixer: lamination in a planar channel system. Microfluidics and Nanofluidics, 8(2):209–215, 2010. doi:10.1007/s10404-009-0456-z.