Hostname: page-component-f554764f5-8cg97 Total loading time: 0 Render date: 2025-04-20T10:17:48.068Z Has data issue: false hasContentIssue false
  • English
  • Français

Mobility of particles embedded in a laterally bounded membrane

Published online by Cambridge University Press:  11 April 2025

Ehud Yariv Open the ORCID record for Ehud Yariv [Opens in a new window]  and
Ashok Shantilal Sangani Open the ORCID record for Ashok Shantilal Sangani [Opens in a new window]
Ehud Yariv*
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
Ashok Shantilal Sangani
Affiliation:
Department of Biomedical and Chemical Engineering, Syracuse University, Syracuse, NY 13244, USA
*
Corresponding author: Ehud Yariv, [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The hydrodynamic analysis of motion of small particles (e.g. proteins) within lipid bilayers appears to be naturally suitable for the framework of two-dimensional Stokes flow. Given the Stokes paradox, the problem in an unbounded domain is ill-posed. In his classical paper, Saffman (J. Fluid Mech., vol. 73, 1976, pp. 593–602) proposed several possible remedies, one of them based upon the finite extent of the membrane. Considering a circular boundary, that regularisation was briefly addressed by Saffman in the isotropic configuration, where the particle is concentrically positioned in the membrane. We investigate here the hydrodynamic problem in bounded membranes for the general case of eccentric particle position and a rectilinear motion in an arbitrary direction. Symmetry arguments provide a representation of the hydrodynamic drag in terms of ‘radial’ and ‘transverse’ coefficients, which depend upon two parameters: the ratio $\lambda$ of particle to membrane radii and the eccentricity $\beta$. Using matched asymptotic expansions we obtain closed-form approximations for these coefficients in the limit where $\lambda$ is small. In the isotropic case ($\beta = 0$) we find that the drag coefficient is $4\pi /(\ln ({1}/{\lambda })- {1})$, contradicting the value $4\pi /(\ln ({1}/{\lambda })- {1}/{2})$ obtained by Saffman. We explain the oversight in Saffman’s argument.

JFM classification

Biological Fluid Dynamics: Membranes Complex Fluids: Colloids
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 1008 , 10 April 2025 , A14
Check for updates
Copyright
© The Author(s), 2025. 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.)

Article purchase

Temporarily unavailable

References

Aponte-Rivera, C. & Zia, R.N. 2016 Simulation of hydrodynamically interacting particles confined by a spherical cavity. Phys. Rev. Fluids 1 (2), 023301.CrossRefGoogle Scholar
Barentin, C., Ybert, C., Di Meglio, J.-M. & Joanny, J.-F. 1999 Surface shear viscosity of Gibbs and Langmuir monolayers. J. Fluid Mech. 397, 331–349.CrossRefGoogle Scholar
Bussell, S.J., Koch, D.L. & Hammer, D.A. 1995 Effect of hydrodynamic interactions on the diffusion of integral membrane proteins: diffusion in plasma membranes. Biophys. J. 68 (5), 1836–1849.CrossRefGoogle ScholarPubMed
Charras, G.T., Coughlin, M., Mitchison, T.J. & Mahadevan, L. 2008 Life and times of a cellular bleb. Biophys. J. 94 (5), 1836–1853.CrossRefGoogle ScholarPubMed
Daripa, P. & Palaniappan, D. 2001 Singularity induced exterior and interior Stokes flows. Phys. Fluids 13 (11), 3134–3154.CrossRefGoogle Scholar
Dodd, T.L., Hammer, D.A., Sangani, A.S. & Koch, D.L. 1995 Numerical simulations of the effect of hydrodynamic interactions on diffusivities of integral membrane proteins. J. Fluid Mech. 293, 147–180.CrossRefGoogle Scholar
Einstein, A. 1956 Investigations on the Theory of the Brownian Movement. Courier Corporation.Google Scholar
Evans, A.A. & Levine, A.J. 2015 Membrane Rheology. Springer.CrossRefGoogle Scholar
Evans, E. & Sackmann, E. 1988 Translational and rotational drag coefficients for a disk moving in a liquid membrane associated with a rigid substrate. J. Fluid Mech. 194, 553–561.CrossRefGoogle Scholar
Fraenkel, L.E. 1969 On the methods of matched asymptotic expansions. Part I: a matching principle. Proc. Camb. Phil. Soc. 65 (1), 209–231.CrossRefGoogle Scholar
Fuller, G.G. & Vermant, J. 2012 Complex fluid-fluid interfaces: rheology and structure. Annu. Rev. Chem. Biomol. Engng 3 (1), 519–543.CrossRefGoogle ScholarPubMed
Happel, J. & Brenner, H. 1965 Low Reynolds Number Hydrodynamics. Prentice-Hall.Google Scholar
Henle, M.L. & Levine, A.J. 2010 Hydrodynamics in curved membranes: the effect of geometry on particulate mobility. Phys. Rev. E 81 (1), 011905.CrossRefGoogle ScholarPubMed
Hinch, E.J. 1991 Perturbation Methods. Cambridge University Press.CrossRefGoogle Scholar
Jeffrey, D.J. & Onishi, Y. 1981 The slow motion of a cylinder next to a plane wall. Q. J. Mech. Appl. Maths 34 (2), 129–137.CrossRefGoogle Scholar
Kim, S. & Karrila, S.J. 2005 Microhydrodynamics: Principles and Selected Applications. Butterworth-Heinemann.Google Scholar
Landau, L.D. & Lifshitz, E.M. 1987 Course of Theoretical Physics: Fluid Mechanics. Pergamon.Google Scholar
Leal, L.G. 2007 Advanced Transport Phenomena: Fluid Mechanics and Convective Transport Processes. Cambridge University Press.CrossRefGoogle Scholar
Liron, N. & Blake, J.R. 1981 Existence of viscous eddies near boundaries. J. Fluid Mech. 107, 109–129.CrossRefGoogle Scholar
Saffman, P.G. 1976 Brownian motion in thin sheets of viscous fluid. J. Fluid Mech. 73 (4), 593–602.CrossRefGoogle Scholar
Saffman, P.G. & Delbrück, M. 1975 Brownian motion in biological membranes. Proc. Natl Acad. Sci. USA 72 (8), 3111–3113.CrossRefGoogle ScholarPubMed
Scriven, L.E. 1960 Dynamics of a fluid interface: equation of motion for Newtonian surface fluids. Chem. Engng Sci. 12 (2), 98–108.CrossRefGoogle Scholar
Squires, T.M. & Mason, T.G. 2010 Fluid mechanics of microrheology. Annu. Rev. Fluid Mech. 42 (1), 413–438.CrossRefGoogle Scholar
Stone, H.A. & Ajdari, A. 1998 Hydrodynamics of particles embedded in a flat surfactant layer overlying a subphase of finite depth. J. Fluid Mech. 369, 151–173.CrossRefGoogle Scholar
Van Dyke, M. 1964 Perturbation Methods in Fluid Mechanics. Academic Press.Google Scholar
Yariv, E. & Peng, G.G. 2024 Long-range two-dimensional hydrodynamic interaction between a pair of mutually repellent disks. J. Fluid Mech. 988, A39.CrossRefGoogle Scholar