Advertisement

Hydrodynamics of Liquid Capsules Enclosed by Elastic Membranes

  • C. Pozrikidis
Conference paper
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 124)

Abstract

The flow-induced deformation of liquid capsules enclosed by elastic membranes is discussed with reference to the biofluid-dynamics of vesicles and biological cells. The deformation of a membrane from a reference configuration causes the development of non-isotropic in-plane elastic tensions, transverse shear tensions, and bending moments according to constitutive laws that reflect the membrane constitution. The type and degree of capsule deformation, the internal and external structure of the flow, and the macroscopic rheological properties of a suspension are determined by the magnitude of the developing elastic tensions and bending moments relative to the strength of the imposed flow. Integral representations of the flow past a collection of capsules are reviewed, and integral equations for the velocity distribution over the membranes are presented for two-dimensional, three-dimensional, and axisymmetric flow at vanishing Reynolds number. The velocity field due to the presence of the capsules is expressed in terms of distributions of Stokes flow singularities over the membranes, represented by the single-layer and double-layer hydrodynamic potential of Stokes flow. The strength of the distribution of the single-layer potential expressing interfacial distributions of point forces is identified with jump in the hydrodynamic traction across a membrane. Expressions for this jump are reviewed and derived in terms of the developing elastic tensions and bending moments under the formalism of the theory of thin shells.

Keywords

Principal Curvature Thin Shell Strain Energy Function Meridional Plane Axisymmetric Flow 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Anderson, D.M., McFadden, G.B. & Wheeler, A.A., 1998, Diffuse-interface methods in fluid mechanics. Annu. Rev. Fluid Mech. 30, 139–165.MathSciNetCrossRefGoogle Scholar
  2. Aris, R., 1962, Vectors, Tensors, and the Basic Equations of Fluid Mechanics. Prentice-Hall.Google Scholar
  3. Barthès-Biesel, D., 1980, Motion of a spherical microcapsule freely suspended in a linear shear flow. J. Fluid Mech. 100, 831–853.MathSciNetCrossRefzbMATHGoogle Scholar
  4. Barthès-Biesel, D., 1991, Role of interfacial properties on the motion and deformation of capsules in shear flow. Physica A 172, 103–124.CrossRefGoogle Scholar
  5. Barthès-Biesel, D. & Rallison, J.M., 1981, The time-dependent deformation of a capsule freely suspended in a linear shear flow. J. Fluid Mech. 113, 251–267.CrossRefzbMATHGoogle Scholar
  6. Barthès-Biesel, D. & Sgaier, H., 1985 Role of membrane viscosity in the orientation and deformation of a spherical capsule suspended in shear flow. J. Fluid Mech. 160, 119–135.MathSciNetCrossRefzbMATHGoogle Scholar
  7. Beatty, M.F., 1987, Topics of finite elasticity: Hyperelasticity of rubber, elastomers, and biological tissues — with examples. Appl. Mech. Rev. 40, 1699–1734.CrossRefGoogle Scholar
  8. Boey, S.K., Boal, D.H. & Discher, D.E., 1998, Simulations of the erythrocyte cytoskeleton at large deformation. I. Microscopic models. Biophys. J. 75, 1573–1583.CrossRefGoogle Scholar
  9. Breyiannis, G. & Pozrikidis, C., 2000, Simple shear flow of suspensions of elastic capsules. Theor. Comp. Fluid Dyn. 13, 327–347.CrossRefzbMATHGoogle Scholar
  10. Budiansky, B. & Sanders, J.L. Jr., 1963, On the “Best” first-order linear shell theory. In Progress in Applied Mechanics, Prager Anniversary Volume, McMillan, pp. 129–140.Google Scholar
  11. Budiansky, B., 1968, Notes on nonlinear shell theory. J. Applied Mech. 35, 393–401.CrossRefGoogle Scholar
  12. Cai, W. & Lubensky, T.C., 1995, Hydrodynamics and dynamic fluctuations of fluid membranes. Phys. Rev. E 52, 4252–4266.CrossRefGoogle Scholar
  13. Charles, R. & Pozrikidis, C., 1998, Significance of the dispersed-phase viscosity on the simple shear flow of suspensions of two-dimensional liquid drops. J. Fluid Mech. 365, 205–234.MathSciNetCrossRefzbMATHGoogle Scholar
  14. Corneliussen, A.H. & Shield, R.T., 1961, Finite deformation of elastic membranes with application to the stability of an inflated and extended tube. Arch. Rational Mech. Anal. 7, 273–304.MathSciNetCrossRefzbMATHGoogle Scholar
  15. Diaz, A., Pelekasis, N. & Barthès-Biesel, D., 2000, Transient response of a capsule subjected to varying flow conditions: Effect of internal fluid viscosity and membrane elasticity. Phys. Fluids 12, 948–957.CrossRefzbMATHGoogle Scholar
  16. Discher, D.E., Boal, D.H. & Boey, S.K., 1998, Simulations of the erythrocyte cytoskeleton at large deformation. II. Micropipette aspiration. Biophys. J. 75, 1584–1597.CrossRefGoogle Scholar
  17. Edwards, D.A., Brenner, H. & Wasan, D.T., 1991, Interfacial Transport Processes and Rheology. Butterworth Heinemann, Massachusetts.Google Scholar
  18. Eggleton, C.D. & Popel, A.S., 1998, Large deformation of red blood cell ghosts in a simple shear flow. Phys. Fluids 10, 1834–1845.CrossRefGoogle Scholar
  19. Evans, E.A. & Skalak, R., 1980, Mechanics and Thermodynamics of Biomembranes. CRC Press.Google Scholar
  20. Flügge, W., 1973, Stresses in Shells. Springer-Verlag.Google Scholar
  21. Fung, Y.C., 1965, Foundations of Solid Mechanics. Prentice-Hall.Google Scholar
  22. Green, A.E. & Adkins, J.E., 1970, Large Elastic Deformations. Clarendon Press, Oxford.zbMATHGoogle Scholar
  23. Green, A.E. & Zerna, W., 1968, Theoretical Elasticity. Dover.Google Scholar
  24. Gurtin, M.E. & Murdoch, A.I., 1975, A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57, 291–323.MathSciNetCrossRefzbMATHGoogle Scholar
  25. Hansen, J.C., Skalak, R., Chien, S. & Hoger, A., 1996, An elastic network model based on the structure of the red blood cell membrane skeleton. Biophys. J. 70, 146–166.CrossRefGoogle Scholar
  26. Hansen, J.C., Skalak, R., Chien, S. & Hoger, A., 1997a, Influence of network topology on the elasticity of the red blood cell membrane skeleton. Biophys. J. 72, 2369–2381.CrossRefGoogle Scholar
  27. Hansen, J.C., Skalak, R., Chien, S. & Hoger, A., 1997b, Spectrin properties and the elasticity of the red blood cell membrane skeleton. Biorheology 34, 327–348.CrossRefGoogle Scholar
  28. Kraus, M., Wintz, W., Seifert, U. & Lipowsky, R., 1996, Fluid vesicles in shear flow. Phys. Rev. Lett. 77, 3685–3688.CrossRefGoogle Scholar
  29. Kwak, S. & Pozrikidis, C., 2000. Effect of membrane bending stiffness on the deformation of capsules in uniaxial extensional flow. Submitted for publication.Google Scholar
  30. Ledret, H. & Raoult, A., 1995, The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity. J. Math. Pures Appl. 74, 549–578.MathSciNetGoogle Scholar
  31. Leyrat-Maurin, A. & Barthès-Biesel, D., 1994, Motion of a deformable capsule through a hyperbolic constriction. J. Fluid Mech. 279, 135–163.CrossRefzbMATHGoogle Scholar
  32. Li, X.Z., Barthès-Biesel, D. & Helmy, A., 1988, Large deformations and burst of a capsule freely suspended in an elongational flow. J. Fluid Mech. 187, 179–196.CrossRefzbMATHGoogle Scholar
  33. Li, X., Charles, R. & Pozrikidis, C., 1996, Simple shear flow of suspensions of liquid drops. J. Fluid Mech. 320, 395–416.CrossRefzbMATHGoogle Scholar
  34. Li, X. & Pozrikidis, C., 2000, Wall-bounded and channel flow of suspensions of liquid drops. Int. J. Multiphase Flow 26, 1247–1279.CrossRefzbMATHGoogle Scholar
  35. Libai, A. & Simmonds, J.G., 1998, The Nonlinear Theory of Elastic Shells, Cambridge University Press.Google Scholar
  36. Lipowsky, R., 1991, The conformation of membranes. Nature 349, 475–481.CrossRefGoogle Scholar
  37. Lipowsky, R. & Sackmann, E., 1995, Structure and Dynamics of Membranes, Vol. 1A, 1B. Elsevier.Google Scholar
  38. McDonald, P., 1996, Continuum Mechanics. PWS.Google Scholar
  39. Mohandas, N. & Evans, E., 1994, Mechanical properties of the red cell membrane in relation to molecular structure and genetic defects. Annu. Rev. Biophys. Biomol. Struct. 23, 787–818.CrossRefGoogle Scholar
  40. Møllmann, H., 1981, Introduction to the Theory of Thin Shells. Wiley.Google Scholar
  41. Naghdi, P.M., 1972, Theory of shells and plates. In Handbuch der Physik, vol VIa/2, C. Truesdell (Edt.), pp. 435–640.Google Scholar
  42. Navot, Y., 1998, Elastic membranes in viscous shear flow. Phys. Fluids 10, 1819–1833.MathSciNetCrossRefzbMATHGoogle Scholar
  43. Ogden, R.W., 1984, Non-linear Elastic Deformations. Dover.Google Scholar
  44. Pozrikidis, C., 1990, The axisymmetric deformation of a red blood cell in uniaxial straining flow. J. Fluid Mech. 216, 231–254.CrossRefzbMATHGoogle Scholar
  45. Pozrikidis, C., 1992, Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge University Press.Google Scholar
  46. Pozrikidis, C., 1994, Effects of surface viscosity on the finite deformation of a liquid drop and the rheology of dilute emulsions in simple shearing flow. J. Non-Newt. Fluid Mech. 51, 161–178.CrossRefGoogle Scholar
  47. Pozrikidis, C., 1995, Finite deformation of liquid capsules enclosed by elastic membranes in simple shear flow. J. Fluid Mech. 297, 123–152.CrossRefzbMATHGoogle Scholar
  48. Pozrikidis, C., 1997, Introduction to Theoretical and Computational Fluid Dynamics. Oxford University Press.Google Scholar
  49. Pozrikidis, C., 2000a, Effect of membrane bending stiffness on the deformation of capsules in simple shear flow. Submitted for publication.Google Scholar
  50. Pozrikidis, C., 2000b. Interfacial dynamics for Stokes flow. J. Comp. Phys. To appear.Google Scholar
  51. Ramanujan, S. & Pozrikidis, C., 1998, Deformation of liquid capsules enclosed by elastic membranes in simple shear flow: large deformations and the effect of fluid viscosities. J. Fluid Mech. 361, 117–143.MathSciNetCrossRefzbMATHGoogle Scholar
  52. Reissner, E., 1949, On the theory of thin elastic shells. In Contributions to Applied Mechanics, H. Reissner Anniversary Volume, J.W. Edwards (Edt.), Ann Arbor, pp. 231–247.Google Scholar
  53. Reissner, E., 1950, On axisymmetrical deformations of thin shells of revolution. Proceedings, Third Symposium in Applied Mathematics, pp. 27–52.Google Scholar
  54. Reissner, E., 1963, On the equations for finite symmetrical deflections of thin shells of revolution. In Progress in Applied Mechanics, Prager Anniversary Volume, McMillan, pp. 171–178.Google Scholar
  55. Reissner, E., 1969, On finite symmetrical deflections of thin shells of revolution. J. Appl. Mech. 36, Trans. ASME 91, Series E, 267–270.CrossRefzbMATHGoogle Scholar
  56. Sanders, J.L., 1963, Nonlinear theories of thin shells. Quart. Appl. Math. 21, 21–36.MathSciNetGoogle Scholar
  57. Scriven, L.E., 1960, Dynamics of a fluid interface. Chem. Eng. Sci. 12, 98–108.CrossRefGoogle Scholar
  58. Secomb, T.W. & Skalak, R., 1982, Surface flow of viscoelastic membranes in viscous fluids. Q. J. Mech. Appl. Maths 35, 233–247.CrossRefzbMATHGoogle Scholar
  59. Secomb, T.W., Skalak, R., Özkaya, N. & Gross, J.F., 1986, Flow of axisymmetric red blood cells in narrow capillaries. J. Fluid Mech. 163, 405–423.CrossRefGoogle Scholar
  60. Seifert, U., 1997, Configurations of fluid membranes and vesicles. Adv. Phys. 46, 13–137.CrossRefGoogle Scholar
  61. Seifert, U., 1998, Modelling nonlinear red cell elasticity. Biophys J. 75 1141-1142.Google Scholar
  62. Simmonds, J.G. & Danielson, D.A., 1972, Nonlinear shell theory with finite rotation and stress-function vectors. J. Applied Mech. 39, 1098–1090.CrossRefGoogle Scholar
  63. Skalak, R., Özkaya, N. & Skalak, T.C., 1989, Biofluid mechanics. Annu. Rev. Fluid Mech. 21, 167–204.CrossRefGoogle Scholar
  64. Skalak, R. Tözeren, A., Zarda, P.R. & Chien, S., 1973, Strain energy function of red blood cell membranes. Biophys. J. 13, 245–264.CrossRefGoogle Scholar
  65. Slattery, J.C., 1990, Interfacial Transport Phenomena. Springer-Verlag.Google Scholar
  66. Steigmann, D.J. & Ogden, R.W., 1997, Plane deformations of elastic solids with intrinsic boundary elasticity. Proc. R. Soc. London A 453, 853–877.MathSciNetCrossRefzbMATHGoogle Scholar
  67. Steigmann, D.J. & Ogden, R.W., 1999, Elastic surface-substrate interactions. Proc. R. Soc. London A 455, 437–474.MathSciNetCrossRefzbMATHGoogle Scholar
  68. Valid, R., 1995, The Nonlinear Theory of Shells through Variational Principles. Wiley.Google Scholar
  69. Waxman, A.M., 1984, Dynamics of a couple-stress fluid membrane. Stud. Appl. Math. 70, 63–86.MathSciNetzbMATHGoogle Scholar
  70. Zarda, P.R., Chien, S. & Skalak, S., 1977, Elastic deformations of red blood cells. J. Biomechanics. 10, 211–221.CrossRefGoogle Scholar
  71. Zinemanas, D. & Nir, A., 1988, On the viscous deformation of biological cells under anisotropic surface tension. J. Fluid Mech. 193, 217–241.MathSciNetCrossRefzbMATHGoogle Scholar
  72. Zhou, H. & Pozrikidis, C., 1995, Deformation of liquid capsules with incompressible interfaces in simple shear flow. J. Fluid Mech. 283, 175–200.CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • C. Pozrikidis
    • 1
  1. 1.Department of Mechanical and Aerospace EngineeringUniversity of CaliforniaSan Diego, La JollaUSA

Personalised recommendations