Three-dimensional numerical simulations using a front-tracking method are presented on the dynamics of oblate shape capsules in linear shear flow by considering a broad range of viscosity contrast (ratio of internal-to-external fluid viscosity), shear rate (or capillary number), and aspect ratio. We focus specifically on the coupling between the shape deformation and orientation dynamics of capsules, and show how this coupling influences the transition from the tank-treading to tumbling motion. At low capillary numbers, three distinct modes of motion are identified: a swinging or oscillatory (OS) mode at a low viscosity contrast in which the inclination angle theta(t) oscillates but always remains positive; a vacillating-breathing (VB) mode at a moderate viscosity contrast in which theta(t) periodically becomes positive and negative, but a full tumbling does not occur; and a pure tumbling mode (TU) at a higher viscosity contrast. At higher capillary numbers, three types of transient motions occur, in addition to the OS and TU modes, during which the capsule switches from one mode to the other as (i) VB to OS, (ii) TU to VB to OS, and (iii) TU to VB. Phase diagrams showing various regimes of capsule dynamics are presented. For all modes of motion (OS, VB, and TU), a large-amplitude oscillation in capsule shape and a strong coupling between the shape deformation and orientation dynamics are observed. It is shown that the coupling between the shape deformation and orientation is the strongest in the VB mode, and hence at a moderate viscosity contrast, for which the amplitude of shape deformation reaches its maximum. The numerical results are compared with the theories of Keller and Skalak, and Skotheim and Secomb. Significant departures from the two theories are discussed and related to the strong coupling between the shape deformation, inclination, and transition dynamics.
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