In this article we present first a new distributed Lagrange multiplier/fictitious domain (DLM/FD) method for simulating fluid-particle interaction in Stokes flow. A conjugate gradient method driven by both pressure and distributed Lagrange multiplier, called one-shot method, has been developed to solve the discrete Stokes problem while enforcing the rigid body motion within the region occupied by the particle. The methodology is validated by comparing the numerical results of a neutrally buoyant particle of either a circular or elliptic shape with the associated Jeffery's solutions. We have successively combined the above methodology with an immersed boundary (IB) method and an elastic membrane modeled by a spring network to simulate the dynamics of a compound vesicle. In simple shear flow under creeping flow condition, the results are consistent with those obtained in literature. In Poiseuille flow, the compound vesicle motion is dominated by the motion of the vesicle membrane as expected and stays in the central region of the channel.

The effect of the natural state on the motion of an inextensible capsule in two-dimensional shear flow has been studied numerically. The energy barrier based on such natural state plays a role for having the transition between two well-known motions, tumbling and tank-treading (TT) with the long axis oscillating about a fixed inclination angle (a swinging mode), when varying the shear rate. Between tumbling and TT with a swinging mode, the intermittent region has been obtained for the capsule with a biconcave rest shape. The estimated critical value of the swelling ratio for having the intermittent region is <0.7, i.e., the capsule with the rest shape closer to a full disk has no intermittent behavior. The capsule intermittent behavior is a mixture of tumbling and TT. Just like the TT with a swinging mode, which can be viewed as TT with an incomplete tumbling, the membrane tank-treads backward and forward within a small range while tumbling. As the capillary number is very close to and below the threshold for pure TT with a swinging mode, the capsule tumbles once after several TT periods in each cycle. The number of TT periods in one cycle decreases when decreasing the capillary number, until the capsule has one tumble and one TT period alternatively and such alternating motion exists over a range of the capillary number; and then the capsule performs more tumbling between two consecutive TT periods when reducing the capillary number further, and finally has tumbling only.

The effect of the nature state on the motion of an inextensible capsule in simple shear flow has been studied in this paper. Besides the viscosity ratio of the internal fluid and external fluid of the capsule, the nature state effect also plays a role for having the transition between two well known motions, tumbling and tank-treading (TT) with the long axis oscillating about a fixed inclination angle (a swinging mode), when varying the shear rate. The intermittent region between tumbling and TT with a swinging mode of the capsule with a biconcave rest shape has been obtained in a narrow range of the capillary number. In such region, the dynamics of the capsule is a mixture of tumbling and TT with a swinging mode; when having the tumbling motion, the membrane tank-tread backward and forward within a small range. As the capillary number is very close to and below the threshold for the pure TT with a swinging mode, the capsule tumbles once after several TT periods in each cycle. The number of TT periods in one cycle decreases with respect to the decreasing of the capillary number, until the capsule has one tumble and one TT period alternatively and such alternating motion exists over a range of the capillary number; and then the capsule performs more tumbling between two consecutive TT periods when reducing the capillary number further, and finally shows pure tumbling. The critical value of the swelling ratio for having the intermittent region has been estimated.