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Stokes flow for a shrinking pore

Published online by Cambridge University Press:  23 December 2015

Christopher A. Aubin  and
Rolf J. Ryham
Christopher A. Aubin
Affiliation:
Department of Physics and Engineering Physics, Fordham University, Bronx, NY 10458, USA
Rolf J. Ryham*
Affiliation:
Department of Mathematics, Fordham University, Bronx, NY 10458, USA
*
†Email address for correspondence: [email protected]
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Abstract

We consider a sphere with a circular pore embedded in an unbounded viscous fluid, where the rim of the pore moves in such a way that the radius of the sphere is constant. Away from the pore, the surface area stretches or compresses uniformly. An exact form for the axisymmetric velocity field which describes the quasi-static motion of the bulk fluid is calculated. The resulting dissipation function yields an analytical value for the aqueous drag coefficient for the sphere with a shrinking pore. Additionally, we examine the small hole and small angle limits, which converge to the unsteady flow for the expansion of a hole in a plane wall, and for the contraction of a circular disk.

JFM classification

Biological Fluid Dynamics: Biological Fluid Dynamics Low-Reynolds-number flows: Low-Reynolds-number flows Biological Fluid Dynamics: Membranes
Type
Papers
Information
Journal of Fluid Mechanics , Volume 788 , 10 February 2016 , pp. 228 - 245
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Copyright
© 2016 Cambridge University Press 

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References

Al-Housseiny, T. T. & Stone, H. S. 2012 On boundary-layer flows induced by the motion of stretching surfaces. J. Fluid Mech. 706, 597–606.CrossRefGoogle Scholar
Ames, W. F. 1992 Numerical Methods for Partial Differential Equations. Academic.Google Scholar
Biria, A., Maleki, M. & Fried, E. 2013 Continuum Theory for the Edge of an Open Lipid Bilayer, Adv. in Appl. Mech., vol. 46. Elsevier.Google Scholar
Brenner, M. P. & Gueyffier, D. 1999 On the bursting of viscous films. Phys. Fluids 11 (3), 737–739.Google Scholar
Brochard, F., Gennes, P. G. & De Pfeuty, P. 1976 Surface tension and deformations of membrane structures: relation to two-dimensional phase transitions. J. Phys. France 37, 1099–1104.Google Scholar
Brochard-Wyart, F., de Gennes, P. G. & Sandre, O. 2000 Transient pores in stretched vesicles: role of leak-out. Physica A 278 (1–2), 32–51.CrossRefGoogle Scholar
Brown, F. L. H. 2011 Continuum simulations of biomembrane dynamics and the importance of hydrodynamic effects. Q. Rev. Biophys. 44 (04), 391–432.CrossRefGoogle ScholarPubMed
Cai, Z. & Kim, S. 2001 A finite element method using singular functions for the poisson equation: corner singularities. SIAM J. Numer. Anal. 39, 286–299.Google Scholar
Cohen, F. S. & Ryham, R. J. 2014 The aqueous viscous drag of a contracting open surface. Phys. Fluids 26 (2), 023101, 1–13.Google Scholar
Davis, A. M. J. & Stone, H. A. 1998 Slow translation, rotation or oscillation of a disk in a rotating fluid: effect of a plane wall or another disk. Q. J. Mech. Appl. Maths 51 (4), 495–513.Google Scholar
Debrégeas, G., Martin, P. & Brochard-Wyart, F. 1995 Viscous bursting of suspended films. Phys. Rev. Lett. 75 (21), 3886–3889.Google Scholar
Dimova, R., Aranda, S., Bezlyepkina, N., Nikolov, V., Riske, K. A. & Lipowsky, R. 2006 A practical guide to giant vesicles. Probing the membrane nanoregime via optical microscopy. J. Phys.: Condens. Matter 18 (28), S1151.Google Scholar
Dimova, R., Bezlyepkina, N., Jordö, M. D., Knorr, R. L., Riske, K. A., Staykova, M., Vlahovska, P. M., Yamamoto, T., Yang, P. & Lipowsky, R. 2009 Vesicles in electric fields: some novel aspects of membrane behavior. Soft Matt. 5, 3201–3212.Google Scholar
Dorrepaal, J. M., O’neill, M. E. & Ranger, K. B. 1976 Axisymmetric Stokes flow past a spherical cap. J. Fluid Mech. 75 (02), 273–286.Google Scholar
Eri, A. & Okumura, K. 2010 Bursting of a thin film in a confined geometry: rimless and constant-velocity dewetting. Phys. Rev. E 82 (3), 030601.Google Scholar
Happel, J. & Brenner, H. 1965 Low Reynolds Number Hydrodynamics With Special Applications to Particulate Media. Prentice-Hall.Google Scholar
Homma, S., Oka, H. & Koga, J. 2011 Retraction of the edge of an initially discotic liquid sheet. Can. J. Chem. Engng 89 (4), 691–695.Google Scholar
Honerkamp-Smith, A. R., Woodhouse, F. G., Kantsler, V. & Goldstein, R. E. 2013 Membrane viscosity determined from shear-driven flow in giant vesicles. Phys. Rev. Lett. 111 (3), 038103.Google Scholar
Hormel, T. T., Kurihara, S. Q., Brennan, M. K., Wozniak, M. C. & Parthasarathy, R. 2014 Measuring lipid membrane viscosity using rotational and translational probe diffusion. Phys. Rev. Lett. 112 (18), 188101.Google Scholar
Karatekin, E., Sandre, O., Guitouni, H., Borghi, N., Puech, P.-H. & Brochard-Wyart, F. 2003 Cascades of transient pores in giant vesicles: line tension and transport. Biophys. J. 84 (3), 1734–1749.Google Scholar
Martínez-Balbuena, L., Hernández-Zapata, E. & Santamaría-Holek, I. 2015 Onsager’s irreversible thermodynamics of the dynamics of transient pores in spherical lipid vesicles. Eur. Biophys. J. 44, 473–481.CrossRefGoogle ScholarPubMed
Moore, D. W., Saffman, P. G. & Maxworthy, T. 1969 The flow induced by the transverse motion of a thin disk in its own plane through a contained rapidly rotating viscous liquid. J. Fluid Mech. 39 (04), 831–847.Google Scholar
Onsager, L. 1931a Reciprocal relations in irreversible processes. I. Phys. Rev. 37, 405–426.Google Scholar
Onsager, L. 1931b Reciprocal relations in irreversible processes. II. Phys. Rev. 38, 2265–2279.Google Scholar
den Otter, W. K. & Shkulipa, S. A. 2007 Intermonolayer friction and surface shear viscosity of lipid bilayer membranes. Biophys. J. 93, 423–433.CrossRefGoogle ScholarPubMed
Payne, L. E. & Pell, W. H. 1960 The Stokes flow problem for a class of axially symmetric bodies. J. Fluid Mech. 7 (04), 529–549.CrossRefGoogle Scholar
Portet, R. & Dimova, R. 2010 A new method for measuring edge tensions and stability of lipid bilayers: effect of membrane composition. Biophys. J. 99 (10), 3264–3273.Google Scholar
Ranger, K. B. 1973 The Stokes drag for asymmetric flow past a spherical cap. Z. Angew. Math. Phys. 24 (6), 801–809.Google Scholar
Reyssat, É. & Quéré, D. 2006 Bursting of a fluid film in a viscous environment. Europhys. Lett. 76 (2), 236.Google Scholar
Riske, K. A. & Dimova, R. 2005 Electro-deformation and poration of giant vesicles viewed with high temporal resolution. Biophys. J. 88 (2), 1143–1155.Google Scholar
Ryham, R. J., Berezovik, I. & Cohen, F. S. 2011 Aqueous viscosity is the primary source of friction in lipidic pore dynamics. Biophys. J. 101 (12), 2929–2938.Google Scholar
Ryham, R. J., Cohen, F. S. & Eisenberg, R. 2012 A dynamic model of open vesicles in fluids. Commun. Math. Sci. 10 (4), 1273–1285.Google Scholar
Saffman, P. G. & Delbrück, M. 1975 Brownian motion in biological membranes. Proc. Natl Acad. Sci. USA 72, 3111–3113.Google Scholar
Salac, D. & Miksis, M. 2011 A level set projection model of lipid vesicles in general flows. J. Comput. Phys. 230 (22), 8192–8215.Google Scholar
Salipante, P. F. & Vlahovska, P. M. 2014 Vesicle deformation in dc electric pulses. Soft Matt. 10 (19), 3386–3393.Google Scholar
Sampson, R. A. 1891 On Stoke’s current function. Phil. Trans. R. Soc. Lond. A 182, 449–518.Google Scholar
Sandre, O., Moreaux, L. & Brochard-Wyart, F. 1999 Dynamics of transient pores in stretched vesicles. Proc. Natl Acad. Sci. USA 96, 10591–10596.Google Scholar
Strutt, J. W. 1871 Some general theorems relating to vibrations. Proc. Lond. Math. Soc. s1–4 (1), 357–368.Google Scholar
Tanzosh, J. P. & Stone, H. A. 1995 Transverse motion of a disk through a rotating viscous fluid. J. Fluid Mech. 301, 295–324.Google Scholar
Thoraval, M.-J. & Thoroddsen, S. T. 2013 Contraction of an air disk caught between two different liquids. Phys. Rev. E 88 (6), 061001.Google ScholarPubMed
Tien, H. T. 1974 Bilayer Lipid Membranes (BLM): Theory and Practice. Marcel Dekker.Google Scholar
Van Dyke, M. 1975 Perturbation Methods in Fluid Mechanics. The Parabolic Press.Google Scholar
Veerapaneni, S., Rahimian, A., Biros, G. & Zorin, D. 2011a A fast algorithm for simulating vesicle flows in three dimensions. J. Comput. Phys. 230 (14), 5610–5634.Google Scholar
Veerapaneni, S., Young, Y.-N., Vlahovska, P. & Bławzdziewicz, J. 2011b Dynamics of a compound vesicle in shear flow. Phys. Rev. Lett. 106 (15), 158103.Google Scholar
Wang, X. & Du, Q. 2008 Modelling and simulations of multi-component lipid membranes and open membranes via diffuse interface approaches. J. Math. Biol. 56 (3), 347–371.CrossRefGoogle ScholarPubMed
Weinstein, A. 1955 On a class of partial differential equations of even order. Ann. Mat. Pura Appl. 39, 245–254.Google Scholar