Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-26T17:39:30.555Z Has data issue: false hasContentIssue false
  • English
  • Français

Analytical solutions for mass transport in hydrodynamic focusing by considering different diffusivities for sample and sheath flows

Published online by Cambridge University Press:  14 January 2019

Arman Sadeghi Open the ORCID record for Arman Sadeghi [Opens in a new window]
Arman Sadeghi*
Affiliation:
Department of Mechanical Engineering, University of Kurdistan, Sanandaj 66177-15175, Iran
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The fluid flow and mass transfer characteristics in two-dimensional hydrodynamic focusing are theoretically investigated by considering different physical properties for the sample and sheath flows. Adopting a single-domain formulation, which assigns the region variable physical properties, three-dimensional analytical solutions are obtained for species transport under hydrodynamically fully developed conditions. In addition, simplified analytical solutions are derived assuming a uniform velocity field appropriate to electrokinetic focusing. The results show that the normalized overall mean velocity is an increasing function of the height to width ratio and a decreasing function of the sheath to sample viscosity ratio. The dependence of this normalized mean velocity on the sheath to sample flow-rate ratio is, however, non-monotonic: it grows with the flow-rate ratio when the sample fluid is more viscous than the sheath fluid, whereas the opposite is true when the sheath fluid is more viscous. Moreover, although an increase in either the viscosity or flow-rate ratios results in creating a smaller value of the normalized focused width, varying the channel aspect ratio may lead to either thinner or thicker focused regions, depending on the viscosity ratio. The inspection of the mass transport characteristics reveals that only the viscosity ratio and the Péclet number can significantly alter the mixing length. Surprisingly, the minimum mixing length in the presence of significant axial diffusion is achieved for a single-phase flow. Finally, the dimensionless mixing length is reduced by increasing the Péclet number, although the changes are negligible when this parameter is above 10. This threshold is only true when the Péclet number is calculated based on the higher of the sample and sheath diffusion coefficients.

JFM classification

Micro-/Nano-fluid dynamics: Microfluidics Multiphase and Particle-laden Flows: Multiphase flow
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 862 , 10 March 2019 , pp. 517 - 551
Copyright
© 2019 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ahmadian-Yazdi, A., Sadeghi, A. & Saidi, M. H. 2014 Rheology effects on cross-stream diffusion in a Y-shaped micromixer. Colloids Surf. A 456 (1), 296–306.Google Scholar
Barz, D. P. J., Zadeh, H. F. & Ehrhard, P. 2011 Measurements and simulations of time-dependent flow fields within an electrokinetic micromixer. J. Fluid Mech. 676, 265–293.Google Scholar
Beeby, S., Ensell, G., Kraft, M. & White, N. 2004 MEMS Mechanical Sensors. Artech House.Google Scholar
Bird, R. B., Stewart, W. E. & Lightfoot, E. N. 2002 Transport Phenomena, 2nd edn. Wiley.Google Scholar
Chang, C.-C., Huang, Z.-X. & Yang, R.-J. 2007 Three-dimensional hydrodynamic focusing in two-layer polydimethylsiloxane (PDMS) microchannels. J. Micromech. Microengng 17 (8), 1479–1486.Google Scholar
Cubaud, T. & Mason, T. G. 2006 Folding of viscous threads in diverging microchannels. Phys. Rev. Lett. 96, 114501.Google Scholar
Cubaud, T. & Mason, T. G. 2008 Formation of miscible fluid microstructures by hydrodynamic focusing in plane geometries. Phys. Rev. E 78 (5), 056308.Google Scholar
Cubaud, T. & Mason, T. G. 2009 High-viscosity fluid threads in weakly diffusive microfluidic systems. New J. Phys. 11 (7), 075029.Google Scholar
Cubaud, T. & Mason, T. G. 2012 Interacting viscous instabilities in microfluidic systems. Soft Matt. 8 (41), 10573–10582.Google Scholar
Datta, D., Gada, V. H. & Sharma, A. 2011 Analytical and level-set method-based numerical study for two-phase stratified flow in a plane channel and a square duct. Numer. Heat Transfer A 60 (4), 347–380.Google Scholar
Fu, L.-M., Yang, R.-J., Lin, C.-H., Pan, Y.-J. & Lee, G.-B. 2004 Electrokinetically driven micro flow cytometers with integrated fiber optics for on-line cell/particle detection. Anal. Chim. Acta 507 (1), 163–169.Google Scholar
Ghallab, Y. H. & Badawy, W. 2010 Lab-on-a-Chip; Techniques, Circuits, and Biomedical Applications. Artech House.Google Scholar
Haji-Sheikh, A. 2009 Determination of heat transfer in ducts with axial conduction by variational calculus. Trans. ASME J. Heat Transfer 131 (9), 091702.Google Scholar
Haji-Sheikh, A., Sparrow, E. M. & Minkowycz, W. J. 2005 Heat transfer to flow through porous passages using extended weighted residuals method – a Green’s function solution. Intl J. Heat Mass Transfer 48 (7), 1330–1349.Google Scholar
Ismagilov, R. F., Stroock, A. D., Kenis, P. J. A., Whitesides, G. & Stone, H. A. 2000 Experimental and theoretical scaling laws for transverse diffusive broadening in two-phase laminar flows in microchannels. Appl. Phys. Lett. 76 (17), 2376–2378.Google Scholar
Jacobson, S. C. & Ramsey, J. M. 1997 Electrokinetic focusing in microfabricated channel structures. Analyt. Chem. 69 (16), 3212–3217.Google Scholar
Jeffery, G. B. 1922 The motion of ellipsoidal particles immersed in a viscous fluid. Proc. R. Soc. Lond. A 102 (715), 161–179.Google Scholar
Jerry, M. C., Ming-Che, K. & Chien-Po, L. 2011 Control of droplet generation in flow-focusing microfluidic device with a converging-diverging nozzle-shaped section. Japan. J. Appl. Phys. 50 (10R), 107301.Google Scholar
Jiang, L., Wang, W., Chau, Y. & Yao, S. 2013 Controllable formation of aromatic nanoparticles in a three-dimensional hydrodynamic flow focusing microfluidic device. RSC Adv. 3 (39), 17762–17769.Google Scholar
Kamholz, A. E., Weigl, B. H., Finlayson, B. A. & Yager, P. 1999 Quantitative analysis of molecular interaction in a microfluidic channel: the T-sensor. Analyt. Chem. 71, 5340–5347.Google Scholar
Karnik, R., Gu, F., Basto, P., Cannizzaro, C., Dean, L., Kyei-Manu, W. & Farokhzad, O. C. 2008 Microfluidic platform for controlled synthesis of polymeric nanoparticles. Nano Lett. 8 (9), 2906–2912.Google Scholar
Knight, J. B., Vishwanath, A., Brody, J. P. & Austin, R. H. 1998 Hydrodynamic focusing on a silicon chip: mixing nanoliters in microseconds. Phys. Rev. Lett. 80 (17), 3863–3866.Google Scholar
Kunstmann-Olsen, C., Hoyland, J. D. & Rubahn, H.-G. 2012 Influence of geometry on hydrodynamic focusing and long-range fluid behavior in PDMS microfluidic chips. Microfluid. Nanofluid. 12 (5), 795–803.Google Scholar
Lee, G.-B., Chang, C.-C., Huang, S.-B. & Yang, R.-J. 2006 The hydrodynamic focusing effect inside rectangular microchannels. J. Micromech. Microengng 16 (5), 1024–1032.Google Scholar
Lee, G.-B., Hwei, B.-H. & Huang, G.-R. 2001 Micromachined pre-focused M × N flow switches for continuous multi-sample injection. J. Micromech. Microengng 11 (6), 654–661.Google Scholar
Li, D. 2005 Microfluidics in lab-on-a-chip: models, simulations and experiments. In Microscale Heat Transfer- Fundamental and Applications (ed. Kakac, S., Vasiliev, L. L., Bayazitoglu, Y. & Yener, Y.), Kluwer Academic.Google Scholar
Li, H., Wong, T. N. & Nguyen, N.-T. 2009 Electroosmotic control of width and position of liquid streams in hydrodynamic focusing. Microfluid. Nanofluid. 7 (4), 489–497.Google Scholar
Li, Y., Ward, K. R. & Burns, M. A. 2017 Viscosity measurements using microfluidic droplet length. Analyt. Chem. 89 (7), 3996–4006.Google Scholar
Lu, M., Ozcelik, A., Grigsby, C. L., Zhao, Y., Guo, F., Leong, K. W. & Huang, T. J. 2016 Microfluidic hydrodynamic focusing for synthesis of nanomaterials. Nano Today 11 (6), 778–792.Google Scholar
Mouheb, N. A., Malsch, D., Montillet, A., Solliec, C. & Henkel, T. 2012 Numerical and experimental investigations of mixing in T-shaped and cross-shaped micromixers. Chem. Engng Sci. 68, 278–289.Google Scholar
Myint-U, T. & Debnath, L. 2007 Linear Partial Differential Equations for Scientists and Engineers, 4th edn. Birkhäuser.Google Scholar
Nguyen, N.-T. & Huang, X. 2005 Mixing in microchannels based on hydrodynamic focusing and time-interleaved segmentation: modelling and experiment. Lab on a Chip 5 (11), 1320–1326.Google Scholar
Nguyen, N.-T., Yap, Y.-F. & Sumargo, A. 2008 Microfluidic rheometer based on hydrodynamic focusing. Meas. Sci. Technol. 19 (8), 085405.Google Scholar
Rhee, M., Valencia, P. M., Rodriguez, M. I., Langer, R., Farokhzad, O. C. & Karnik, R. 2011 Synthesis of size-tunable polymeric nanoparticles enabled by 3d hydrodynamic flow focusing in single-layer microchannels. Adv. Mater. 23 (12), H79–H83.Google Scholar
Riseman, J. & Ullman, R. 1951 The concentration dependence of the viscosity of solutions of macromolecules. J. Chem. Phys. 19 (5), 578–584.Google Scholar
Ruey-Jen, Y., Chih-Chang, C., Sung-Bin, H. & Gwo-Bin, L. 2005 A new focusing model and switching approach for electrokinetic flow inside microchannels. J. Micromech. Microengng 15 (11), 2141–2148.Google Scholar
Sadeghi, A. 2016 Analytical solutions for species transport in a T-sensor at low Peclet numbers. AIChE J. 62 (11), 4119–4130.Google Scholar
Sadeghi, A., Saidi, M. H., Waezi, Z. & Chakraborty, S. 2013 Variational formulation on Joule heating in combined electroosmotic and pressure driven microflows. Intl J. Heat Mass Transfer 61 (1), 254–265.Google Scholar
Schrum, D. P., Culbertson, C. T., Jacobson, S. C. & Ramsey, J. M. 1999 Microchip flow cytometry using electrokinetic focusing. Analyt. Chem. 71 (19), 4173–4177.Google Scholar
Shivhare, P. K., Bhadra, A., Sajeesh, P., Prabhakar, A. & Sen, A. K. 2016 Hydrodynamic focusing and interdistance control of particle-laden flow for microflow cytometry. Microfluid. Nanofluid. 20 (6), 86.Google Scholar
Simonnet, C. & Groisman, A. 2005 Two-dimensional hydrodynamic focusing in a simple microfluidic device. Appl. Phys. Lett. 87 (11), 114104.Google Scholar
Stiles, P. J. & Fletcher, D. F. 2004 Hydrodynamic control of the interface between two liquids flowing through a horizontal or vertical microchannel. Lab on a Chip 4 (2), 121–124.Google Scholar
Takayama, S., McDonald, J. C., Ostuni, E., Liang, M. N., Kenis, P. J. A., Ismagilov, R. F. & Whitesides, G. M. 1999 Patterning cells and their environments using multiple laminar fluid flows in capillary networks. Proc. Natl Acad. Sci. 96 (10), 5545–5548.Google Scholar
Taylor, G. 1953 Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. R. Soc. Lond. A 219 (1137), 186–203.Google Scholar
Tripathi, S., Chakravarty, P. & Agrawal, A. 2014 On non-monotonic variation of hydrodynamically focused width in a rectangular microchannel. Curr. Sci. 107 (8), 1260–1274.Google Scholar
Valencia, P. M., Basto, P. A., Zhang, L., Rhee, M., Langer, R., Farokhzad, O. C. & Karnik, R. 2010 Single-step assembly of homogenous lipid-polymeric and lipid-quantum dot nanoparticles enabled by microfluidic rapid mixing. ACS Nano 4 (3), 1671–1679.Google Scholar
Walker, G. M., Ozers, M. S. & Beebe, D. J. 2004 Cell infection within a microfluidic device using virus gradients. Sensors Actuators B 98 (2), 347–355.Google Scholar
Weber, H. J. & Arfken, G. B. 2003 Essential Mathematical Methods for Physicists. Academic Press.Google Scholar
Williams, M. C. 1975 Migration of two liquid phases in capillary extrusion: an energy interpretation. AIChE J. 21 (6), 1204–1207.Google Scholar
Wong, P. K., Lee, Y.-K. & Ho, C.-M. 2003 Deformation of DNA molecules by hydrodynamic focusing. J. Fluid Mech. 497, 55–65.Google Scholar
Wu, T., Luo, Z., Ding, W., Cheng, Z. & He, L. 2017 Monodisperse droplets by impinging flow-focusing. Microfluid. Nanofluid. 21 (8), 129.Google Scholar
Wu, Z. & Nguyen, N.-T. 2005 Hydrodynamic focusing in microchannels under consideration of diffusive dispersion: theories and experiments. Sensors Actuators B 107 (2), 965–974.Google Scholar
Yoshiko, Y., Takeshi, H., Maria Portia, B., Kenichi, Y., Masaya, M., Hiroyuki, N. & Hideaki, M. 2006 Influence of gravity on a laminar flow in a microbioanalysis system. Meas. Sci. Technol. 17 (12), 3162–3166.Google Scholar
Zhang, M., Lian, Y., Harnett, C. & Brehob, E. 2012 Investigation of hydrodynamic focusing in a microfluidic coulter counter device. Trans. ASME J. Biomech. Engng 134 (8), 081001.Google Scholar