Hostname: page-component-8448b6f56d-42gr6 Total loading time: 0 Render date: 2024-04-24T06:24:44.126Z Has data issue: false hasContentIssue false
  • English
  • Français

Inertial migration of an electrophoretic rigid sphere in a two-dimensional Poiseuille flow

Published online by Cambridge University Press:  12 July 2019

A. Choudhary ,
T. Renganathan  and
S. Pushpavanam Open the ORCID record for S. Pushpavanam [Opens in a new window]
A. Choudhary
Affiliation:
Department of Chemical Engineering, Indian Institute of Technology, Chennai, TN 600036, India
T. Renganathan
Affiliation:
Department of Chemical Engineering, Indian Institute of Technology, Chennai, TN 600036, India
S. Pushpavanam*
Affiliation:
Department of Chemical Engineering, Indian Institute of Technology, Chennai, TN 600036, India
*
†Email address for correspondence: spush@iitm.ac.in
Get access Rights & Permissions [Opens in a new window]

Abstract

There has been a recent interest in integrating external fields with inertial microfluidic devices to tune particle focusing. In this work, we analyse the inertial migration of an electrophoretic particle in a two-dimensional Poiseuille flow with an electric field applied parallel to the walls. For a thin electrical double layer, the particle exhibits a slip-driven electrokinetic motion along the direction of the applied electric field, which causes the particle to lead or lag the flow (depending on its surface charge). The fluid disturbance caused by this slip-driven motion is characterized by a rapidly decaying source-dipole field which alters the inertial lift on the particle. We determine this inertial lift using the reciprocal theorem. Assuming no wall effects, we derive an analytical expression for a ‘phoretic lift’ which captures the modification to the inertial lift due to electrophoresis. We also take wall effects into account, at the leading order, using the method of reflections. We find that for a leading particle, the phoretic lift acts towards the regions of high shear (i.e. walls), while the reverse is true for a lagging particle. Using an order-of-magnitude analysis, we obtain different components of the inertial force and classify them on the basis of the interactions from which they emerge. We show that the dominant contribution to the phoretic lift originates from the interaction of the source-dipole field (generated by the electrokinetic slip at the particle surface) with the stresslet field (generated due to particle’s resistance to strain in the background flow). Furthermore, to contrast the slip-driven phenomenon (electrophoresis) from the force-driven phenomenon (buoyancy) in terms of their influence on the inertial migration, we also study a non-neutrally buoyant particle. We show that the gravitational effects alter the inertial lift primarily through the interaction of the background shear with the buoyancy-induced Stokeslet field.

JFM classification

Micro-/Nano-fluid dynamics: Microfluidics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 874 , 10 September 2019 , pp. 856 - 890
Check for updates
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

Abramowitz, M. & Stegen, I. 1972 Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, vol. 55. Dover.Google Scholar
Anderson, J. L. 1989 Colloid transport by interfacial forces. Annu. Rev. Fluid Mech. 21 (1), 61–99.10.1146/annurev.fl.21.010189.000425Google Scholar
Becker, L. E., McKinley, G. H. & Stone, H. A. 1996 Sedimentation of a sphere near a plane wall: weak non-Newtonian and inertial effects. J. Non-Newtonian Fluid Mech. 63 (2–3), 201–233.10.1016/0377-0257(95)01424-1Google Scholar
Bike, S. G. & Prieve, D. C. 1995 Electrokinetic lift of a sphere moving in slow shear flow parallel to a wall: (ii). Theory. J. Colloid Interface Sci. 175 (2), 422–434.10.1006/jcis.1995.1472Google Scholar
Brenner, H. 1962 Effect of finite boundaries on the Stokes resistance of an arbitrary particle. J. Fluid Mech. 12 (1), 35–48.10.1017/S0022112062000026Google Scholar
Bretherton, F. P. 1962 The motion of rigid particles in a shear flow at low Reynolds number. J. Fluid Mech. 14 (2), 284–304.10.1017/S002211206200124XGoogle Scholar
Cevheri, N. & Yoda, M. 2014 Electrokinetically driven reversible banding of colloidal particles near the wall. Lab on a Chip 14 (8), 1391–1394.10.1039/c3lc51341fGoogle Scholar
Cox, R. G. & Brenner, H. 1968 The lateral migration of solid particles in Poiseuille flow. I. Theory. Chem. Engng Sci. 23 (2), 147–173.10.1016/0009-2509(68)87059-9Google Scholar
Faxén, H. 1922 The resistance to the movement of a rigid sphere in a zippered liquid enclosed between two parallel plane w. Ann. Phys. 373 (10), 89–119.10.1002/andp.19223731003Google Scholar
Feng, J., Hu, H. H. & Joseph, D. D. 1994 Direct simulation of initial value problems for the motion of solid bodies in a Newtonian fluid. Part 2. Couette and Poiseuille flows. J. Fluid Mech. 277, 271–301.10.1017/S0022112094002764Google Scholar
Guazzelli, E. & Morris, J. F. 2011 A Physical Introduction to Suspension Dynamics, vol. 45. Cambridge University Press.10.1017/CBO9780511894671Google Scholar
Happel, J. & Brenner, H. 2012 Low Reynolds Number Hydrodynamics: With Special Applications to Particulate Media, vol. 1. Springer Science & Business Media.Google Scholar
Ho, B. P. & Leal, L. G. 1974 Inertial migration of rigid spheres in two-dimensional unidirectional flows. J. Fluid Mech. 65 (2), 365–400.10.1017/S0022112074001431Google Scholar
Hood, K., Lee, S. & Roper, M. 2015 Inertial migration of a rigid sphere in three-dimensional Poiseuille flow. J. Fluid Mech. 765, 452–479.10.1017/jfm.2014.739Google Scholar
Keh, H.-J. & Anderson, J. L. 1985 Boundary effects on electrophoretic motion of colloidal spheres. J. Fluid Mech. 153, 417–439.10.1017/S002211208500132XGoogle Scholar
Keh, H. J. & Chen, S. B. 1988 Electrophoresis of a colloidal sphere parallel to a dielectric plane. J. Fluid Mech. 194, 377–390.10.1017/S0022112088003039Google Scholar
Kim, S. & Karrila, S. J. 2013 Microhydrodynamics: Principles and Selected Applications. Courier Corporation.Google Scholar
Kim, Y. W. & Yoo, J. Y. 2009a Axisymmetric flow focusing of particles in a single microchannel. Lab on a Chip 9 (8), 1043–1045.10.1039/b815286aGoogle Scholar
Kim, Y. W. & Yoo, J. Y. 2009b Three-dimensional focusing of red blood cells in microchannel flows for bio-sensing applications. Biosens. Bioelectr. 24 (12), 3677–3682.10.1016/j.bios.2009.05.037Google Scholar
Kirby, B. J. 2010 Micro- and Nanoscale Fluid Mechanics: Transport in Microfluidic Devices. Cambridge University Press.10.1017/CBO9780511760723Google Scholar
Lamb, H. 1975 Hydrodynamics, 6th edn. Cambridge University Press.Google Scholar
Lee, D., Nam, S. M., Kim, J.-a., Di Carlo, D. & Lee, W. 2018 Active control of inertial focusing positions and particle separations enabled by velocity profile tuning with coflow systems. Anal. Chem. 90 (4), 2902–2911.10.1021/acs.analchem.7b05143Google Scholar
Li, D. & Xuan, X. 2018 Electrophoretic slip-tuned particle migration in microchannel viscoelastic fluid flows. Phys. Rev. Fluids 3 (7), 074202.10.1103/PhysRevFluids.3.074202Google Scholar
Luke, J. H. 1989 Convergence of a multiple reflection method for calculating Stokes flow in a suspension. SIAM J. Appl. Maths 49 (6), 1635–1651.10.1137/0149099Google Scholar
Martel, J. M. & Toner, M. 2014 Inertial focusing in microfluidics. Annu. Rev. Biomed. Engng 16, 371–396.10.1146/annurev-bioeng-121813-120704Google Scholar
Matas, J. P., Morris, J. F. & Guazzelli, E. 2004 Lateral forces on a sphere. Oil Gas Sci. Technol. 59 (1), 59–70.10.2516/ogst:2004006Google Scholar
O’Brien, R. W. & Hunter, R. J. 1981 The electrophoretic mobility of large colloidal particles. Can. J. Chem. 59 (13), 1878–1887.10.1139/v81-280Google Scholar
O’Brien, R. W. & White, L. R. 1978 Electrophoretic mobility of a spherical colloidal particle. J. Chem. Soc. Faraday Trans. 74, 1607–1626.10.1039/f29787401607Google Scholar
Rossi, M., Marin, A., Cevheri, N., Kähler, C. J. & Yoda, M. 2019 Particle distribution and velocity in electrokinetically induced banding. Microfluid. Nanofluid. 23 (5), 67.10.1007/s10404-019-2227-9Google Scholar
Rubinow, S. I. & Keller, J. B. 1961 The transverse force on a spinning sphere moving in a viscous fluid. J. Fluid Mech. 11 (3), 447–459.10.1017/S0022112061000640Google Scholar
Saffman, P. G. T. 1965 The lift on a small sphere in a slow shear flow. J. Fluid Mech. 22 (2), 385–400.10.1017/S0022112065000824Google Scholar
Schnitzer, O., Frankel, I. & Yariv, E. 2012a Streaming-potential phenomena in the thin-debye-layer limit. Part 2. Moderate Péclet numbers. J. Fluid Mech. 704, 109–136.10.1017/jfm.2012.221Google Scholar
Schnitzer, O., Frankel, I. & Yariv, E. 2012b Streaming-potential phenomena in the thin-debye-layer limit. Part 2. Moderate Péclet numbers. J. Fluid Mech. 704, 109–136.10.1017/jfm.2012.221Google Scholar
Schnitzer, O. & Yariv, E. 2012 Macroscale description of electrokinetic flows at large zeta potentials: nonlinear surface conduction. Phys. Rev. E 86 (2), 021503.Google Scholar
Schnitzer, O. & Yariv, E. 2016 Streaming-potential phenomena in the thin-debye-layer limit. Part 3. Shear-induced electroviscous repulsion. J. Fluid Mech. 786, 84–109.10.1017/jfm.2015.647Google Scholar
Segre, G. & Silberberg, A. 1961 Radial particle displacements in Poiseuille flow of suspensions. Nature 189 (4760), 209–210.10.1038/189209a0Google Scholar
Segre, G. & Silberberg, A. J. 1962a Behaviour of macroscopic rigid spheres in Poiseuille flow. Part. 1. Determination of local concentration by statistical analysis of particle passages through crossed light beams. J. Fluid Mech. 14 (1), 115–135.10.1017/S002211206200110XGoogle Scholar
Segre, G. & Silberberg, A. J. 1962b Behaviour of macroscopic rigid spheres in Poiseuille flow. Part. 2. Experimental results and interpretation. J. Fluid Mech. 14 (1), 136–157.10.1017/S0022112062001111Google Scholar
Smoluchowski, M. 1903 Contribution à la théorie de l’endosmose électrique et de quelques phénomènes corrélatifs. Bull. Akad. Sci. Cracovie. 8, 182–200.Google Scholar
Strogatz, S. H. 2018 Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. CRC Press.10.1201/9780429492563Google Scholar
Vasseur, P. & Cox, R. G. 1976 The lateral migration of a spherical particle in two-dimensional shear flows. J. Fluid Mech. 78 (2), 385–413.10.1017/S0022112076002498Google Scholar
Yariv, E. 2006 ‘Force-free’ electrophoresis? Phys. Fluids 18 (3), 031702.10.1063/1.2185690Google Scholar
Yariv, E. 2016 Dielectrophoretic sphere–wall repulsion due to a uniform electric field. Soft Matter 12 (29), 6277–6284.10.1039/C6SM00462HGoogle Scholar
Yee, A. & Yoda, M. 2018 Experimental observations of bands of suspended colloidal particles subject to shear flow and steady electric field. Microfluid. Nanofluid. 22 (10), 113.10.1007/s10404-018-2136-3Google Scholar
Yuan, D., Pan, C., Zhang, J., Yan, S., Zhao, Q., Alici, G. & Li, W. 2016 Tunable particle focusing in a straight channel with symmetric semicircle obstacle arrays using electrophoresis-modified inertial effects. Micromachines 7 (11), 195.10.3390/mi7110195Google Scholar
Zhang, J., Yan, S., Yuan, D., Alici, G., Nguyen, N.-T., Warkiani, M. E. & Li, W. 2016 Fundamentals and applications of inertial microfluidics: a review. Lab on a Chip 16 (1), 10–34.10.1039/C5LC01159KGoogle Scholar
Supplementary material: File

Choudhary et al. supplementary material

Choudhary et al. supplementary material 1

Download Choudhary et al. supplementary material(File)
File 265.5 KB