Journal of Fluid Mechanics

Effect of membrane bending stiffness on the deformation of capsules in simple shear flow

a1 Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, California 92093-0411, USA; e-mail:


The effect of interfacial bending stiffness on the deformation of liquid capsules enclosed by elastic membranes is discussed and investigated by numerical simulation. Flow-induced deformation causes the development of in-plane elastic tensions and bending moments accompanied by transverse shear tensions due to the non-infinitesimal membrane thickness or to a preferred configuration of an interfacial molecular network. To facilitate the implementation of the interfacial force and torque balance equations involving the hydrodynamic traction exerted on either side of the interface and the interfacial tensions and bending moments developing in the plane of the interface, a formulation in global Cartesian coordinates is developed. The balance equations involve the Cartesian curvature tensor defined in terms of the gradient of the normal vector extended off the plane of the interface in an appropriate fashion. The elastic tensions are related to the surface deformation gradient by constitutive equations derived by previous authors, and the bending moments for membranes whose unstressed shape has uniform curvature, including the sphere and a planar sheet, arise from a constitutive equation that involves the instantaneous Cartesian curvature tensor and the curvature of the resting configuration. A numerical procedure is developed for computing the capsule deformation in Stokes flow based on standard boundary-element methods. Results for spherical and biconcave resting shapes resembling red blood cells illustrate the effect of the bending modulus on the transient and asymptotic capsule deformation and on the membrane tank-treading motion.

(Received April 11 2000)
(Revised February 7 2001)

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