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Dynamics of strain-hardening and strain-softening capsules in strong planar extensional flows via an interfacial spectral boundary element algorithm for elastic membranes

Published online by Cambridge University Press:  25 November 2009

W. R. DODSON III  and
P. DIMITRAKOPOULOS
W. R. DODSON III
Affiliation:
Fischell Department of Bioengineering, University of Maryland, College Park, MD 20742, USA
P. DIMITRAKOPOULOS*
Affiliation:
Department of Chemical and Biomolecular Engineering, University of Maryland, College Park, MD 20742, USA
*
†Email address for correspondence: dimitrak@umd.edu
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Abstract

In the present study we investigate the dynamics of initially spherical capsules (made from elastic membranes obeying the strain-hardening Skalak or the strain-softening neo-Hookean law) in strong planar extensional flows via numerical computations. To achieve this, we develop a three-dimensional spectral boundary element algorithm for membranes with shearing and area-dilatation tensions in Stokes flow. The main attraction of this approach is that it exploits all the benefits of the spectral methods (i.e. high accuracy and numerical stability) but without creating denser systems. To achieve continuity of the interfacial geometry and its derivatives at the edges of the spectral elements during the interfacial deformation, a membrane-based interfacial smoothing is developed, via a Hermitian-like interpolation, for both the interfacial shape and the membrane elastic forces. Our numerical results show that no critical flow rate exists for both Skalak and neo-Hookean capsules in the moderate and strong planar extension flows considered in the present study. As the flow rate increases, both capsules reach elongated ellipsoidal steady-state configurations; the cross-section of the Skalak capsule preserves its elliptical shape, while the neo-Hookean capsule becomes more and more lamellar. The curvature at the pointed edges of these elongated steady-state shapes shows a very fast increase with the flow rate. The large interfacial deformations are accompanied with the development of strong membrane tensions especially for the strain-hardening Skalak capsule; the computed increase of the membrane tensions with the flow rate or the shape extension can be used to predict rupture of a specific membrane (with known lytic tension) due to excessive tensions. The type of the experiment imposed on the capsule as well as the applied flow rate affect dramatically the time evolution of the capsule edges owing to the interaction of the hydrodynamic forces with the membrane tensions; when a spherical Skalak capsule is let to deform in a strong flow, very large edge curvatures (with respect to the steady-state value) are developed during the transient evolution.

JFM classification

Biological Fluid Dynamics: Capsule/cell dynamics Interfacial Flows (free surface): Interfacial Flows (free surface) Low-Reynolds-number flows: Low-Reynolds-number flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 641 , 25 December 2009 , pp. 263 - 296
Copyright
Copyright © Cambridge University Press 2009

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References

REFERENCES

Acrivos, A. & Lo, T. S. 1978 Deformation and breakup of a single slender drop in an extensional flow. J. Fluid Mech. 86, 641–672.Google Scholar
Barthès-Biesel, D. 1980 Motion of a spherical microcapsule freely suspended in a linear shear flow. J. Fluid Mech. 100, 831–853.Google Scholar
Barthès-Biesel, D. 1991 Role of interfacial properties on the motion and deformation of capsules in shear flow. Physica A 172, 103–124.Google Scholar
Barthès-Biesel, D., Diaz, A. & Dhenin, E. 2002 Effect of constitutive laws for two-dimensional membranes on flow-induced capsule deformation. J. Fluid Mech. 460, 211–222.Google Scholar
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.CrossRefGoogle Scholar
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, 119–136.CrossRefGoogle Scholar
Baskurt, O. K. & Meiselman, H. J. 2003 Blood rheology and hemodynamics. Sem. Thromb. Hem. 29, 435–450.Google ScholarPubMed
Bentley, B. J. & Leal, L. G. 1986 An experimental investigation of drop deformation and breakup in steady, two-dimensional linear flows. J. Fluid Mech. 167, 241–283.Google Scholar
Buckmaster, J. D. 1972 Pointed bubbles in slow viscous flow. J. Fluid Mech. 55, 385–400.Google Scholar
Canuto, C., Hussaini, M. Y., Quarteroni, A. & Zang, T. A. 1998 Spectral Methods in Fluid Dynamics. Springer.Google Scholar
Carin, M., Barthès-Biesel, D., Edwards-Lévy, F., Postel, C. & Andrei, D. C. 2003 Compression of biocompatible liquid-filled HSA–alginate capsules: determination of the membrane mechanical properties. Biotech. Bioengng 82, 207–212.CrossRefGoogle ScholarPubMed
Chang, K. S. & Olbricht, W. L. 1993 a Experimental studies of the deformation of a synthetic capsule in extensional flow. J. Fluid Mech. 250, 587–608.Google Scholar
Chang, K. S. & Olbricht, W. L. 1993 b Experimental studies of the deformation and breakup of a synthetic capsule in steady and unsteady simple shear flow. J. Fluid Mech. 250, 609–633.CrossRefGoogle Scholar
Diaz, A., Pelekasis, N. & Barthès-Biesel, D. 2000 Transient response of a capsule subjected to varying fluid conditions: effect of internal fluid viscosity and membrane elasticity. Phys. Fluids 12, 948–957.Google Scholar
Dimitrakopoulos, P. 2007 Interfacial dynamics in Stokes flow via a three-dimensional fully-implicit interfacial spectral boundary element algorithm. J. Comput. Phys. 225, 408–426.Google Scholar
Dimitrakopoulos, P. & Higdon, J. J. L. 1998 On the displacement of three-dimensional fluid droplets from solid surfaces in low-Reynolds-number shear flows. J. Fluid Mech. 377, 189–222.Google Scholar
Dimitrakopoulos, P. & Wang, J. 2007 A spectral boundary element algorithm for interfacial dynamics in two-dimensional Stokes flow based on Hermitian interfacial smoothing. Engng Anal. Bound. Elem. 31, 646–656.CrossRefGoogle Scholar
Doddi, S. K. & Bagchi, P. 2008 Effect of inertia on the hydrodynamic interaction between two liquid capsules in simple shear flow. Intl J. Multiphase Flow 34, 375–392.CrossRefGoogle Scholar
Dodson, W. R. III & Dimitrakopoulos, P. 2008 Spindles, cusps and bifurcation for capsules in Stokes flow. Phys. Rev. Lett. 101, 208102.Google Scholar
Eggleton, C. D. & Popel, A. S. 1998 Large deformation of red blood cell ghosts in simple shear flow. Phys. Fluids 10, 1834–1845.Google Scholar
Gould, P. L. 1999 Analysis of Shells and Plates. Prentice Hall.Google Scholar
Ha, J. W. & Leal, L. G. 2001 An experimental study of drop deformation and breakup in extensional flow at high capillary number. Phys. Fluids 13, 1568–1576.CrossRefGoogle Scholar
Hudson, S. D., Phelan, F. R. Jr., Handler, M. D., Cabral, J. T., Migler, K. B. & Amis, E. J., 2004 Microfluidic analog of the four-roll mill. Appl. Phys. Lett. 85 (2), 335–337.Google Scholar
Husmann, M., Rehage, H., Dhenin, E. & Barthès-Biesel, D. 2005 Deformation and bursting of nonspherical polysiloxane microcapsules in a spinning-drop apparatus. J. Colloid Interface Sci. 282, 109–119.Google Scholar
Kwak, S. & Pozrikidis, C. 2001 Effect of membrane bending stiffness on the axisymmetric deformation of capsules in uniaxial extensional flow. Phys. Fluids 13, 1234–1242.Google Scholar
Lac, E. & Barthès-Biesel, D., 2005 Deformation of a capsule in simple shear flow: effect of membrane prestress. Phys. Fluids 17, 072105.Google Scholar
Lac, E., Barthès-Biesel, D., Pelekasis, N. A., & Tsamopoulos, J. 2004 Spherical capsules in three-dimensional unbounded Stokes flows: effect of the membrane constitutive law and onset of buckling. J. Fluid Mech. 516, 303–334.CrossRefGoogle Scholar
Lubarda, V. A. 2002 Elastoplasticity Theory. CRC Press.Google Scholar
Mohandas, N. & Chasis, J. A. 1993 Red blood cell deformability, membrane material properties and shape: regulation by transmembrane, skeletal and cytosolic proteins and lipids. Sem. Hematol. 30, 171–192.Google Scholar
Muldowney, G. P. & Higdon, J. J. L. 1995 A spectral boundary element approach to three-dimensional Stokes flow. J. Fluid Mech. 298, 167–192.Google Scholar
Navot, Y. 1998 Elastic membranes in viscous shear flow. Phys. Fluids 10, 1819–1833.Google Scholar
Papastavridis, J. G. 1999 Tensor Calculus and Analytical Dynamics. CRC Press.Google Scholar
Pieper, G., Rehage, H. & Barthès-Biesel, D. 1998 Deformation of a capsule in a spinning drop apparatus. J. Colloid Interface Sci. 202, 293–300.Google Scholar
Popel, A. S. & Johnson, P. C. 2005 Microcirculation and hemorheology. Annu. Rev. Fluid Mech. 37, 43–69.Google Scholar
Pozrikidis, C. 1995 Finite deformation of liquid capsules enclosed by elastic membranes in simple shear flow. J. Fluid Mech. 297, 123–152.Google Scholar
Pozrikidis, C. 2001 Effect of membrane bending stiffness on the deformation of capsules in simple shear flow. J. Fluid Mech. 440, 269–291.CrossRefGoogle Scholar
Pozrikidis, C. 2001 Interfacial dynamics for Stokes flow. J. Comput. Phys. 169, 250–301.Google Scholar
Pozrikidis, C. (Ed.) 2003 Modelling and Simulation of Capsules and Biological Cells. Chapman and Hall.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Secomb, T. W., Hsu, R. & Pries, A. R. 1998 A model for red blood cell motion in glycocalyx-lined capillaries. Am. J. Physiol. Heart Circ. Physiol. 274, H1016–H1022.CrossRefGoogle Scholar
Secomb, T. W., Hsu, R. & Pries, A. R. 2001 Motion of red blood cells in a capillary with an endothelial surface layer: effect of flow velocity. Am. J. Physiol. Heart Circ. Physiol. 281, H629–H636.Google Scholar
Secomb, T. W., Hsu, R. & Pries, A. R. 2002 Blood flow and red blood cell deformation in non-uniform capillaries: effects of the endothelial surface layer. Microcirculation 9, 189–196.Google Scholar
Skalak, R., Tozeren, A., Zarda, R. P. & Chien, S. 1973 Strain energy function of red blood cell membranes. Biophys. J. 13, 245–264.Google Scholar
Steigmann, D. J. & Ogden, R. W. 1999 Elastic surface-substrate interactions. Proc. R. Soc. Lond. A 455, 437–474.Google Scholar
Taylor, G. I. 1934 The formation of emulsions in definable fields of flow. Proc. R. Soc. A 146, 501–523.Google Scholar
Wang, Y. & Dimitrakopoulos, P. 2006 A three-dimensional spectral boundary element algorithm for interfacial dynamics in Stokes flow. Phys. Fluids 18, 082106.Google Scholar
Waxman, A. M. 1984 Dynamics of a couple-stress fluid membrane. Stud. Appl. Math. 70, 63–86.Google Scholar