Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2025-01-04T23:48:20.694Z Has data issue: false hasContentIssue false
  • English
  • Français

Lift and drag forces acting on a particle moving with zero slip in a linear shear flow near a wall

Published online by Cambridge University Press:  08 October 2020

Nilanka I. K. Ekanayake Open the ORCID record for Nilanka I. K. Ekanayake [Opens in a new window] ,
Joseph D. Berry Open the ORCID record for Joseph D. Berry [Opens in a new window] ,
Anthony D. Stickland ,
David E. Dunstan ,
Ineke L. Muir ,
Steven K. Dower  and
Dalton J. E. Harvie Open the ORCID record for Dalton J. E. Harvie [Opens in a new window]
Nilanka I. K. Ekanayake
Affiliation:
Department of Chemical Engineering, The University of Melbourne, Victoria3010, Australia
Joseph D. Berry
Affiliation:
Department of Chemical Engineering, The University of Melbourne, Victoria3010, Australia
Anthony D. Stickland
Affiliation:
Department of Chemical Engineering, The University of Melbourne, Victoria3010, Australia
David E. Dunstan
Affiliation:
Department of Chemical Engineering, The University of Melbourne, Victoria3010, Australia
Ineke L. Muir
Affiliation:
Bio21 Molecular Science and Biotechnology Institute, CSL, Victoria3052, Australia
Steven K. Dower
Affiliation:
Bio21 Molecular Science and Biotechnology Institute, CSL, Victoria3052, Australia
Dalton J. E. Harvie*
Affiliation:
Department of Chemical Engineering, The University of Melbourne, Victoria3010, Australia
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The lift and drag forces acting on a small spherical particle in a single wall-bounded linear shear flow are examined via numerical computation. The effects of shear rate are isolated from those of slip by setting the particle velocity equal to the local fluid velocity (zero slip), and examining the resulting hydrodynamic forces as a function of separation distance. In contrast to much of the previous numerical literature, low shear Reynolds numbers are considered ($10^{-3} \lesssim Re_{\gamma } \lesssim 10^{-1}$). This shear rate range is relevant when dealing with particulate flows within small channels, for example particle migration in microfluidic devices being used or developed for the biotech industry. We demonstrate a strong dependence of both the lift and drag forces on shear rate. Building on previous theoretical $Re_{\gamma } \ll 1$ studies, a wall-shear-based zero-slip lift correlation is proposed that is applicable when the wall lies both within the inner and outer regions of the disturbed flow. Similarly, we validate an improved wall-shear-based zero-slip drag correlation that more accurately captures the drag force when the particle is close to, but not touching, the wall. Application of the new correlations to predict the movement of a force-free particle shows that the examined shear-based lift force is as important as the previously examined slip-based lift force, highlighting the need to accurately account for shear when predicting the near-wall movement of force-free particles.

JFM classification

Multiphase and Particle-laden Flows: Particle/fluid flow Micro-/Nano-fluid dynamics: Microfluidics Complex Fluids: Suspensions
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 904 , 10 December 2020 , A6
Check for updates
Copyright
© The Author(s), 2020. Published by 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

REFERENCES

Ambari, A., Gauthier, M. B. & Guyon, E. A. 1984 Wall effects on a sphere translating at constant velocity. J. Fluid Mech. 149 (1), 235–253.CrossRefGoogle Scholar
Asmolov, E. S. 1989 Lift force exerted on a spherical particle in a laminar boundary layer. Fluid Dyn. 24 (5), 710–714.CrossRefGoogle Scholar
Asmolov, E. S. 1990 Dynamics of a spherical particle in a laminar boundary layer. Fluid Dyn. 25 (6), 886–890.CrossRefGoogle Scholar
Asmolov, E. S. 1999 The inertial lift on a spherical particle in a plane poiseuille flow at large channel Reynolds number. J. Fluid Mech. 381, 63.CrossRefGoogle Scholar
Asmolov, E. S., Dubov, A. L., Nizkaya, T. V., Harting, J. & Vinogradova, O. I. 2018 Inertial focusing of finite-size particles in microchannels. J. Fluid Mech. 840, 613–630.CrossRefGoogle Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Bird, R. B., Stewart, W. E. & Lightfoot, E. N. 2002 Transport Phenomena, 2nd edn.John Wiley & Sons.Google 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.CrossRefGoogle Scholar
Cherukat, P. & McLaughlin, J. B. 1994 The inertial lift on a rigid sphere in a linear shear flow field near a flat wall. J. Fluid Mech. 263 (1), 1–18.CrossRefGoogle Scholar
Cherukat, P. & McLaughlin, J. B. 1995 The inertial lift on a rigid sphere in a linear shear flow field near a flat wall. J. Fluid Mech. 285, 407.CrossRefGoogle 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.CrossRefGoogle Scholar
Cox, R. G. & Hsu, S. K. 1977 The lateral migration of solid particles in a laminar flow near a plane. Intl J. Multiphase Flow 3 (3), 201–222.CrossRefGoogle Scholar
Dandy, D. S. & Dwyer, H. A. 1990 A sphere in shear flow at finite Reynolds number: effect of shear on particle lift, drag, and heat transfer. J. Fluid Mech. 216, 381–410.CrossRefGoogle Scholar
Di Carlo, D., Edd, J. F., Humphry, K. J., Stone, H. A. & Toner, M. 2009 Particle segregation and dynamics in confined flows. Phys. Rev. Lett. 102 (9), 094503.CrossRefGoogle ScholarPubMed
Drew, D. A. 1988 The lift force on a small sphere in the presence of a wall. Chem. Engng Sci. 43 (4), 769–773.CrossRefGoogle Scholar
Ekanayake, N. I., Berry, J. D., Stickland, A. D., Muir, I. L., Dower, S. K. & Harvie, D. J. 2018 Lift and drag forces on a particle near a wall at low Reynolds numbers. In 21st Australasian Fluid Mechanics Conference, Adelaide, Australia.Google Scholar
Faxen, H. 1922 Der widerstand gegen die bewegung einer starren kugel in einer zähen flüssigkeit, die zwischen zwei parallelen ebenen wänden eingeschlossen ist. Ann. Phys. 373 (10), 89–119.CrossRefGoogle Scholar
Fischer, T. M. & Rosenberger, R. 1987 A boundary integral method for the numerical computation of the forces exerted on a sphere in viscous incompressible flows near a plane wall. Z. Angew. Math. Phys. 38 (3), 339–365.CrossRefGoogle Scholar
Geuzaine, C. & Remacle, J. 2009 Gmsh: A 3-D finite element mesh generator with built-in pre- nd post-processing facilities. Intl J. Numer. Meth. Engng 79 (11), 1309–1331.CrossRefGoogle Scholar
Goldman, A. J., Cox, R. G. & Brenner, H. 1967 a Slow viscous motion of a sphere parallel to a plane wall—I motion through a quiescent fluid. Chem. Engng Sci. 22 (4), 637–651.CrossRefGoogle Scholar
Goldman, A. J., Cox, R. G. & Brenner, H. 1967 b Slow viscous motion of a sphere parallel to a plane wall—II. Couette flow. Chem. Engng Sci. 22 (4), 653–660.CrossRefGoogle Scholar
Halow, J. S. & Wills, G. B. 1970 Experimental observations of sphere migration in Couette systems. Ind. Engng Chem. Fundam. 9 (4), 603–607.CrossRefGoogle Scholar
Happel, J. & Brenner, H. 1981 Low Reynolds Number Hydrodynamics. [Electronic Resource]: With Special Applications to Particulate Media. Springer.CrossRefGoogle Scholar
Harvie, D. J. E. 2010 An implicit finite volume method for arbitrary transport equations. ANZIAM J. 52, C1126.CrossRefGoogle Scholar
Ho, B. P. & Leal, L. G. 1974 Inertial migration of rigid spheres in two-dimensional unidirectional flows. J. Fluid Mech. 65 (02), 365.CrossRefGoogle 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.CrossRefGoogle Scholar
Krishnan, G. & Leighton, D. 1995 Inertial lift on a moving sphere in contact with a plane wall in a shear flow. Phys. Fluids 7 (11), 2538–2545.CrossRefGoogle Scholar
Kurose, R. & Komori, S. 1999 Drag and lift forces on a rotating sphere in a linear shear flow. J. Fluid Mech. 384, 183–206.CrossRefGoogle Scholar
Lee, H. & Balachandar, S. 2010 Drag and lift forces on a spherical particle moving on a wall in a shear flow at finite Re. J. Fluid Mech. 657, 89–125.CrossRefGoogle Scholar
Leiderman, K. & Fogelson, A. L. 2011 Grow with the flow: a spatial-temporal model of platelet deposition and blood coagulation under flow. Math. Med. Biol. 28 (1), 47–84.CrossRefGoogle ScholarPubMed
Leighton, D. & Acrivos, A. 1985 The lift on a small sphere touching a plane in the presence of a simple shear flow. Z. Angew. Math. Phys. 36 (1), 174–178.CrossRefGoogle Scholar
Magnaudet, J., Takagi, S. H. U. & Legendre, D. 2003 Drag, deformation and lateral migration of a buoyant drop moving near a wall. J. Fluid Mech. 476, 115–154.CrossRefGoogle Scholar
McLaughlin, J. B. 1991 Inertial migration of a small sphere in linear shear flows. J. Fluid Mech. 224, 261–274.CrossRefGoogle Scholar
McLaughlin, J. B. 1993 The lift on a small sphere in wall-bounded linear shear flows. J. Fluid Mech. 246, 249–265.CrossRefGoogle 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.CrossRefGoogle Scholar
Saffman, P. G. T. 1965 The lift on a small sphere in a slow shear flow. J. Fluid Mech. 22 (02), 385–400.CrossRefGoogle Scholar
Schonberg, J. A. & Hinch, E. J. 1989 Inertial migration of a sphere in poiseuille flow. J. Fluid Mech. 203, 517–524.CrossRefGoogle Scholar
Segre, G. & Silberberg, A. 1962 Behaviour of macroscopic rigid spheres in poiseuille flow. Part 2. Experimental results and interpretation. J. Fluid Mech. 14 (1), 136–157.CrossRefGoogle Scholar
Shi, P. & Rzehak, R. 2020 Lift forces on solid spherical particles in wall-bounded flows. Chem. Engng Sci. 211, 115264.CrossRefGoogle Scholar
Takemura, F. 2004 Migration velocities of spherical solid particles near a vertical wall for Reynolds number from 0.1 to 5. Phys. Fluids 16 (1), 204.CrossRefGoogle Scholar
Takemura, F. & Magnaudet, J. 2003 The transverse force on clean and contaminated bubbles rising near a vertical wall at moderate Reynolds number. J. Fluid Mech. 495, 235–253.CrossRefGoogle Scholar
Takemura, F. & Magnaudet, J. 2009 Lateral migration of a small spherical buoyant particle in a wall-bounded linear shear flow. Phys. Fluids 21 (8), 083303.CrossRefGoogle Scholar
Takemura, F., Magnaudet, J. & Dimitrakopoulos, P. 2009 Migration and deformation of bubbles rising in a wall-bounded shear flow at finite Reynolds number. J. Fluid Mech. 634, 463–486.CrossRefGoogle Scholar
Vasseur, P. & Cox, R. G. 1976 The lateral migration of a spherical particle in two-dimensional shear flows. J. Fluid Mech. 78 (02), 385.CrossRefGoogle Scholar
Vasseur, P. & Cox, R. G. 1977 The lateral migration of spherical particles sedimenting in a stagnant bounded fluid. J. Fluid Mech. 80 (3), 561–591.CrossRefGoogle Scholar
Zeng, L., Balachandar, S. & Fischer, P. 2005 Wall-induced forces on a rigid sphere at finite Reynolds number. J. Fluid Mech. 536 (1), 1.CrossRefGoogle Scholar
Zeng, L., Fady, N., Balachandar, S. & Fischer, P. 2009 Forces on a finite-sized particle located close to a wall in a linear shear flow. Phys. Fluids 21 (3), 033302.CrossRefGoogle Scholar