In this paper, a computational model for the natural state of an inextensible capsule has been successfully combined with a spring model of the capsule membrane to simulate the motion of the capsule in two-dimensional shear flow. Besides the viscosity ratio of the internal fluid and external fluid of the capsule, the natural state also plays a role for having the transition between two well known motions, tumbling and tank-treading (TT) with the long axis oscillates about a fixed inclination angle (a swinging mode), when varying the shear rate. Between tumbling and tank-treading, the intermittent behavior has been obtained for the capsule with a biconcave rest shape. The estimated critical value of the swelling ratio for having the intermittent transition behavior is less than 0.7, i.e., the capsules with rest shape closer to a full disk do not have the intermittent behavior in shear flow. The intermittent dynamics of the capsule in the transition region is a mixture of tumbling and TT with a swinging mode. Just like the motion of TT with a swing mode, which can be viewed as a tank-treading with an incomplete tumbling, the membrane tank-treads backward and forward within a small range during the tumbling motion.

Amid the recent interest in the role of membrane viscosity in the deformation of a fluid-filled capsule, we consider the role of various capsule properties (shear elasticity, membrane bending stiffness and viscosity) in determining the response and recovery times of a spherical capsule in shear flow. These times are determined by fitting exponential functions to results for the Taylor deformation parameter Dxy. We focus on the relationship between the membrane and fluid viscosity ratios, as suggested by Diaz et al, and whether adjustments to the fluid viscosity ratio may be used to approximate the effects of membrane viscosity. Based on its ability to reproduce response and recovery times, our results suggest that such an approach holds promise.

The lattice Boltzmann method (LBM) with an elastic model is applied to the simulation of two-phase flows containing a deformable body with a viscoelastic membrane. The numerical method is based on the LBM for incompressible two-phase fluid flows with the same density. The body has an internal fluid covered by a viscoelastic membrane of a finite thickness. An elastic model is introduced to the LBM in order to determine the elastic forces acting on the viscoelastic membrane of the body. In the present method, we take account of changes in surface area of the membrane and in total volume of the body as well as shear deformation of the membrane. By using this method, we calculate two problems, the behavior of an initially spherical body under shear flow and the motion of a body with initially spherical or biconcave discoidal shape in square pipe flow. Calculated deformations of the body (the Taylor shape parameter) for various shear rates are in good agreement with other numerical results. Moreover, tank-treading motion, which is a characteristic motion of viscoelastic bodies in shear flows, is simulated by the present method.

We consider a dense granular shear flow in a two-dimensional system. Granular systems(composed of a large number of macroscopic particles) are far from equilibrium due toinelastic collisions between particles: an external driving is needed to maintain themotion of particles. Theoretical description of driven granular media is especiallychallenging for dense granular flows. This paper focuses on a gravity-driven densegranular Poiseuille flow in a channel. A special focus here is on the intriguingphenomenon of fluid-solid coexistence: a solid plug in the center of the system,surrounded by fluid layers. To find and analyze various flow regimes, a multi-scaleapproach is taken. On macro scale, granular hydrodynamics is employed. On micro scale,event-driven molecular dynamics simulations are performed. The entire phase diagram ofparameters is explored, in order to determine which flow regime occurs in various regionsin the parameter space.

The disturbance flow due to the motion of a small sphere parallel to the streamlines of an unbounded linear shear flow is evaluated at small Reynolds number using the method of matched expansions. Decaying laws are obtained for all velocity components in a far inviscid region and viscous wakes. The z component (in the direction of the shear-rate gradient) of the disturbance velocity is cylindrically symmetric in the inviscid region. It decays with the distance r from the sphere like r−5/3, while the y component (in the direction of vorticity) decays like r−4/3. The widths of two viscous wakes, located upstream and downstream of the sphere, grow with the longitudinal coordinate x as yw ~ zw ~ |x|1/3. The maximum x and z components of the velocity are located in the wake cores; they scale like |x|−2/3 and |x|−1 respectively. For two particles interacting through their outer regions, the migration velocity of each particle is the sum of the velocity of an isolated particle and of a disturbance velocity induced by the other one. Particles placed in the normal or transversal directions repel each other. When each particle is located in a wake of the other one, they may either attract or repel each other.