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On the viscous flows of leak-out and spherical cap natation

Published online by Cambridge University Press:  12 December 2017

Rolf J. Ryham Open the ORCID record for Rolf J. Ryham [Opens in a new window]
Rolf J. Ryham*
Affiliation:
Department of Mathematics, Fordham University, Bronx, NY 10458, USA
*
†Email address for correspondence: rryham@fordham.edu
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Abstract

This paper deals with the hydrodynamics of a viscous liquid passing through the hole in a deflating hollow sphere. I employ the method of complementary integrals and calculate in closed form the pressure and streamfunction for the axisymmetric, creeping motion coming from changes in radius. The resulting flow fields describe the motion of a deformable spherical cap in a viscous environment, where the deformations include changes in the size of the spherical cap, the size of the hole and translation of the body along the axis of symmetry. The calculations yield explicit expressions for the jumps in pressure and resistance coefficients for the combined deformations. The equation for the translation force shows that a freely suspended spherical cap is able to propel as an active swimmer. The expression for pressure contains the classic Sampson flow rate equation as a limiting case, but simulations show that the pressure must also account for the velocity of hole widening to correctly predict outflow rates in physiology. Movies based on the closed-form solutions visualize the flow fields and pressures as part of physical processes.

JFM classification

Mathematical Foundations: General fluid mechanics Biological Fluid Dynamics: Propulsion
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 836 , 10 February 2018 , pp. 502 - 531
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Copyright
© 2017 Cambridge University Press 

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References

Alam Shibly, S. U., Ghatak, C., Sayem Karal, M. A., Moniruzzaman, M. & Yamazaki, M. 2016 Experimental estimation of membrane tension induced by osmotic pressure. Biophys. J. 111, 2190–2201.CrossRefGoogle ScholarPubMed
Arroyo, M. & Desimone, A. 2009 Relaxation dynamics of fluid membranes. Phys. Rev. E 79, 031915.Google ScholarPubMed
Arroyo, M. & Trepat, X. 2017 Hydraulic fracturing in cells and tissues: fracking meets cell biology. Curr. Opin. Cell Biol. 44, 1–6.Google Scholar
Aubin, C. A. & Ryham, R. J. 2016 Stokes flow for a shrinking pore. J. Fluid Mech. 788, 228–245.Google Scholar
Becker, L. E., Koehler, S. A. & Stone, H. A. 2003 On self-propulsion of micro-machines at low Reynolds number: Purcell’s three-link swimmer. J. Fluid Mech. 490, 15–35.Google Scholar
Bocher, M. 1926 An Introduction to the Study of Integral Equations. Cambridge University Press.Google Scholar
Brenner, H. 1958 Dissipation of energy due to solid particles suspended in a viscous liquid. Phys. Fluids 1 (4), 338–346.Google Scholar
Brenner, M. P. & Gueyffier, D. 1999 On the bursting of viscous films. Phys. Fluids 11 (3), 737–739.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
Chabanon, M., Ho, J. C. S., Liedberg, B., Parikh, A. N. & Rangamani, P. 2017 Pulsatile lipid vesicles under osmotic stress. Biophys. J. 112 (8), 1682–1691.Google Scholar
Chang, C.-W., Chiang, C.-W. & Jackson, M. B. 2017 Fusion pores and their control of neurotransmitter and hormone release. J. Gen. Phys. 149, 1–22.CrossRefGoogle ScholarPubMed
Childress, S. 2012 A thermodynamic efficiency for Stokesian swimming. J. Fluid Mech. 705, 77–97.Google Scholar
Crowdy, D. 2016 Flipping and scooping of curved 2D rigid fibers in simple shear: the jeffery equations. Phys. Fluids 28 (5), 053105.Google Scholar
Dagan, Z., Weinbaum, S. & Pfeffer, R. 1982 An infinite-series solution for the creeping motion through an orifice of finite length. J. Fluid Mech. 115, 505–523.Google Scholar
Dancer, E. N., Daners, D. & Hauer, D. 2015 A Liouville theorem for p-harmonic functions on exterior domains. Positivity 19 (3), 577–586.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., 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
Dimova, R. & Riske, K. A. 2016 Electrodeformation, electroporation, and electrofusion of giant unilamellar vesicles. In Handbook of Electroporation (ed. Damijan, M.). Springer.Google Scholar
Dorrepaal, J. M. 1978 The stokes resistance of a spherical cap to translational and rotational motions in a linear shear flow. J. Fluid Mech. 84 (2), 265–279.Google Scholar
Dorrepaal, J. M. 1984 The resistance tensors for a spherical cap in Stokes flow. Q. J. Mech. Appl. Maths 37, 355–371.Google Scholar
Dorrepaal, J. M., O’Neill, M. E. & Ranger, K. B. 1976 Axisymmetric Stokes flow past a spherical cap. J. Fluid Mech. 75 (2), 273–286.Google Scholar
Eisenberg, R. S. 1999 From structure to function in open ionic channels. J. Mem. Bio. 171 (1), 1–24.Google Scholar
Evans, A. A., Spagnolie, S. E. & Lauga, E. 2010 Stokesian jellyfish: viscous locomotion of bilayer vesicles. Soft Matt. 6, 1737–1747.Google Scholar
Evans, E. & Rawicz, W. 1990 Entropy-driven tension and bending elasticity in condensed-fluid membranes. Phys. Rev. Lett. 64, 2094–2097.Google Scholar
Fournier, J.-B. 2007 On the stress and torque tensors in fluid membranes. Soft Matt. 3, 883–888.Google Scholar
Fournier, J.-B. 2015 On the hydrodynamics of bilayer membranes. Intl J. Nonlinear Mech. 75, 67–76.Google Scholar
Fournier, J.-B. & Barbetta, C. 2008 Direct calculation from the stress tensor of the lateral surface tension of fluctuating fluid membranes. Phys. Rev. Lett. 100, 078103.Google Scholar
Gordillo, L., Agbaglah, G., Duchemin, L. & Josserand, C. 2011 Asymptotic behavior of a retracting two-dimensional fluid sheet. Phys. Fluids 23 (12), 122101.Google Scholar
Happel, J. & Brenner, H. 1965 Low Reynolds Number Hydrodynamics with Special Applications to Particulate Media. Prentice Hall.Google Scholar
Helfield, B., Chen, X., Watkins, S. C. & Villanueva, F. S. 2016 Biophysical insight into mechanisms of sonoporation. Proc. Natl Acad. Sci. USA 113 (36), 9983–9988.Google Scholar
Hille, B. 2001 Ionic Channels of Excitable Membranes, 3rd edn. Sinauer Associates.Google Scholar
Ho, J. C. S., Rangamani, P., Liedberg, B. & Parikh, A. N. 2016 Mixing water, transducing energy, and shaping membranes: autonomously self-regulating giant vesicles. Langmuir 32 (9), 2151–2163.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
Ilton, M., Dimaria, C. & Dalnoki-Veress, K. 2016 Direct measurement of the critical pore size in a model membrane. Phys. Rev. Lett. 117, 257801.Google Scholar
Jackson, D. P. & Sleyman, S. 2010 Analysis of a deflating soap bubble. Am. J. Phys. 78, 990–994.Google Scholar
Jensen, K. H., Valente, A. X. C. N. & Stone, H. A. 2014 Flow rate through microfilters: influence of the pore size distribution, hydrodynamic interactions, wall slip, and inertia. Phys. Fluids 26 (5), 052004.CrossRefGoogle 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
Lauga, E. & Davis, A. M. J. 2012 Viscous Marangoni propulsion. J. Fluid Mech. 705, 120–133.CrossRefGoogle Scholar
Leal, L. G. 2006 Advanced Transport Phenomena: Fluid Mechanics and Convective Transport Processes. Cambridge University Press.Google Scholar
Lira, R. B., Steinkühler, J., Knorr, R. L., Dimova, R. & Riske, K. A. 2016 Posing for a picture: vesicle immobilization in agarose gel. Sci. Rep. 25254.Google Scholar
Manoussaki, D., Shin, W. D., Waterman, C. M. & Chadwick, R. S. 2015 Cytosolic pressure provides a propulsive force comparable to actin polymerization during lamellipod protrusion. Sci. Rep. 5, 12314.CrossRefGoogle ScholarPubMed
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.Google Scholar
Moffatt, H. K. 1964 Viscous and resistive eddies near a sharp corner. J. Fluid Mech. 18 (1), 1–18.Google Scholar
Onsager, L. 1931a Reciprocal relations in irreversible processes. Part I. Phys. Rev. 405–426.Google Scholar
Onsager, L. 1931b Reciprocal relations in irreversible processes. Part II. Phys. Rev. 2265–2279.Google Scholar
Peyret, A., Ibarboure, E., Tron, A., Beauté, L., Rust, R., Sandre, O., McClenaghan, N. D. & Lecommandoux, S. 2017 Polymersome popping by light-induced osmotic shock under temporal, spatial, and spectral control. Angew. Chem. Intl Ed. 56 (6), 1566–1570.CrossRefGoogle ScholarPubMed
Poignard, C., Silve, A. & Wegner, L. 2016 Different approaches used in modeling of cell membrane electroporation. In Handbook of Electroporation (ed. Damijan, M.). Springer.Google 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.CrossRefGoogle ScholarPubMed
Puech, P.-H. & Brochard-Wyart, F. 2004 Membrane tensiometer for heavy giant vesicles. Eur. Phys. J. E 15 (2), 127–132.Google ScholarPubMed
Purcell, E. M. 1977 Life at low Reynolds number. Am. J. Phys. 45, 3–11.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.CrossRefGoogle 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
Salipante, P. F. & Vlahovska, P. M. 2014 Vesicle deformation in DC electric pulses. Soft Matt. 10 (19), 3386–3393.CrossRefGoogle ScholarPubMed
Sampson, R. A. 1891 On Stokes’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
Sankhagowit, S., Wu, S.-H., Biswas, R., Riche, C. T., Povinelli, M. L. & Malmstadt, N. 2014 The dynamics of giant unilamellar vesicle oxidation probed by morphological transitions. B.B.A.-Biomembranes 1838 (10), 2615–2624.CrossRefGoogle ScholarPubMed
Savva, N. & Bush, J. W. M. 2009 Viscous sheet retraction. J. Fluid Mech. 626, 211–240.Google Scholar
Schroeder, A., Kost, J. & Barenholz, Y. 2009 Ultrasound, liposomes, and drug delivery: principles for using ultrasound to control the release of drugs from liposomes. Chem. Phys. Lipids 162 (1, 2), 1–16.CrossRefGoogle ScholarPubMed
Sengela, J. T. & Wallacea, M. I. 2016 Imaging the dynamics of individual electropores. P. Natl Acad. Sci. USA 113, 5281–5286.Google Scholar
Singer, A., Schuss, Z., Holcman, D. & Eisenberg, R. S. 2006 Narrow escape. Part I. J. Stat. Phys. 122 (3), 437–463.Google Scholar
Strutt, J. W. 1871 Some general theorems relating to vibrations. Proc. Lond. Math. Soc. s1‐4 (1), 357–368.Google Scholar
Weissberg, H. L. 1962 End correction for slow viscous flow through long tubes. Phys. Fluids 5 (9), 1033–1036.Google Scholar
Whittaker, E. T. & Watson, G. N. 2006 A Course of Modern Analysis. Cambridge University Press.Google Scholar
Zagier, D. 2007 The Dilogarithm Function. Springer.Google Scholar

Ryham supplementary movie 1

A contracting spherical cap (thick black arc) expels fluid by decreasing in size. The spherical radius R is initially 20 μm and decreases monotonically to the final value 12 μm. Because the cap is freely suspended, the distance traveled by the centre of the sphere is identical to the initial minus the final radius (see (4.21)). The top of the spherical cap is a stagnation point and does not move. In this simulation, the angle α = 150° is constant. Tracer particles (small black dots) show the velocity field in the aqueous phase. The simulation uses the creeping motion (2.3, 2.4) with U1 = d R/dt, U2= 0 and U1 + U3= 0 to define the particle trajectories. The particles pass through the hole opening and do not cross the spherical cap surface.

Download Ryham supplementary movie 1(Video)
Video 7.3 MB

Ryham supplementary movie 2

Complementary to the flow pattern in Movie 1, hole widening draws the spherical cap (thick black arc) and its contents backwards. In this simulation, spherical radius R = 15 μm is constant and the hole radius grows from 3 μm to 15 μm (the angle α decreases monotonically from 168° to the final angle 90° over time). The creeping motion (2.3, 2.4) driven by changes in the hole angle uses U1 = 0 and U2 = Rdα/dt. The translation velocity U3 derives from momentum conservation (3.16) with F3 = 0.

Download Ryham supplementary movie 2(Video)
Video 6.9 MB

Ryham supplementary movie 3

The stretched liposome is under tension. At first, leak-out cannot relieve the tension because the initial pore (radius 0.05 μm) is too small to allow passage of the viscous fluid. Instead, the pore widens, decreasing surface area and releasing energy stored in stretching. Leak-out initiates once the pore is sufficiently wide, and continues to expel fluid until the pore has collapsed. The liposome and its contents shift backwards in hole widening and shift forwards during hole closure. In this simulation, the parameters R, α and z0 are the time course for kinetic equations (7.2-7.4). The creeping motion (2.3, 2.4) defines the tracer particle trajectories in the aqueous phase. In this slow motion movie, one second of realtime corresponds to 3.3 milliseconds. But to make the movements more apparent, the slow, linear closure stage plays ten times faster. Once the pore has collapsed, the final liposome volume is about 80 % that of the initial volume.

Download Ryham supplementary movie 3(Video)
Video 10.6 MB

Ryham supplementary movie 4

The stretched liposome exerts a pressure on the fluid inside the sphere. Initially, intracellular pressure (red, about 5 Pa) behaves as the Young-Laplace pressure for an intact sphere, and dissipates to about 0.1 Pa with the decreasing surface area and decreasing intracellular volume. The ambient reference pressure is 0 Pa (white). The blue colouration shows the inverse square root singularity just outside the edge of the hole. There, the pressure is negative because the edge is retracting. Around 15 s realtime, the edge singularities are positive (red spots) as the surface extends into the fluid. Hole closure increases surface area at a rate greater than that of leak-out, restoring some stretching. The pressure is about 0.5 Pa (pink) at the time of hole collapse. This movie uses the same velocities as in movie 3, and the analytical expression (5.9) to plot the pressure fields.

Download Ryham supplementary movie 4(Video)
Video 2.8 MB