The time-dependent deformation of a capsule freely suspended in a linear shear flow

Abstract

An analysis is presented of the dynamics of a small deformable capsule freely suspended in a viscous fluid undergoing shear. The capsule consists of an elastic membrane which encloses another viscous fluid, and it deforms in response to the applied external stresses and the elastic forces generated within the membrane. Equations are derived which give its time-dependent deformation in the limit that the departure of the shape from sphericity is small. The form of the shear flow is arbitrary and a general (two-dimensional) elastic material is considered. Limiting forms are obtained for highly viscous capsules and for membranes which are area-preserving, and earlier results for surface tension droplets and incompressible isotropic membranes are derived as particular cases. Results for the viscosity of a dilute suspension of capsules are also given.The theoretical prediction for the relaxation rate of the shape is derived for an interface which has elastic properties appropriate for a red-blood-cell membrane, and is compared with experimental observations of erythrocytes.

Shear modulation of intercellular contact area between two deformable cells colliding under flow

“…Barthes-Biesel and colleagues (Barthes-Biesel and Rallison, 1981;Barthes-Biesel and Sgaier, 1985) studied the motion of an elastic capsule in a linear shear flow under the small deformation regime using perturbation analysis, and obtained the deformation and orientation of the capsule in the shear field. Deformation was found to increase with an increase in the capillary number.…”

Large deformation of red blood cell ghosts in a simple shear flow

“…Perturbation methods were used in the first models of fluid-capsule interactions. Assuming an initially unstressed spherical capsule with a quadratic strain energy function, linearized equations were solved to obtain the steady-state shape 16 and time evolution 17,18 of capsules in simple flows. Barthes-Biesel and Sgaier 19 used linear theory to study the effect of membrane viscosity and found that purely viscous membranes tumble while particles with viscoelastic membranes orient themselves to the streamlines when the shear rate and relaxation time are of the same order.…”

From Passive Motion of Capsules to Active Motion of Cells

“…1, No.1, 2006 where the shell deformations may be large and where the hydrodynamic forces are due to pressure and viscous shear stresses. When the capsule deformation is small, asymptotic solutions can be developed (12) . To first order in ε, the capsule takes an ellipsoidal shape, inclined at an angle Φ = 45 o with respect to the far field streamlines, while the membrane continuously rotates around the steady profile.…”

“…The validation of the numerical results is not easy. One way of doing it is to consider an intitially spherical capsule subjected to vanishingly small flow strength and to compare the numerical deformation to the asymptotic analytical deformation (12) developed for that case.…”

“…where V i and Ω i are the translational and rotational velocities of squirmer i. x i is the centre of squirmer i, and u s,i is the squirming velocity of squirmer i defined by (12). The swimming speed of a solitary squirmer is 2B 1 /3.…”