The dynamics of inextensible capsules in shear flow under the effect of the natural state

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.