Multiplicity of stable orbits for deformable prolate capsules in shear flow

Abstract: This work investigates the orbital dynamics of a fluid-filled deformable prolate capsule in unbounded shear flow. The motion of the capsule is simulated using a model that incorporates shear elasticity, area dilatation, and bending resistance. Here the deformability of the capsule is characterized by capillary number Ca, which represents the ratio of viscous stresses to elastic restoring stresses. For a capsule with small bending stiffness, at a given Ca, the orientation converges over time towards a unique stable orbit independent of the initial orientation. With increasing Ca, four dynamical modes are found for the stable orbit: rolling, wobbling, oscillating-swinging, and swinging. For a capsule with large bending stiffness, multiplicity in the orbit dynamics is observed. When the viscosity ratio $\lambda \lesssim 1$, the long-axis of the capsule always tends towards a stable orbit in the flow-gradient plane, either tumbling or swinging, depending on Ca. When $\lambda \gtrsim 1$, the stable orbit of the capsule is a tumbling motion at low Ca, irrespective of the initial orientation. Upon increasing Ca, there is a symmetry-breaking bifurcation away from the tumbling orbit, and the capsule is observed to adopt multiple stable orbital modes including nonsymmetric precessing and rolling, depending on the initial orientation. As Ca further increases, the nonsymmetric stable orbit loses existence at a saddle-node bifurcation, and rolling becomes the only attractor at high Ca, whereas the rolling state coexists with the nonsymmetric state at intermediate values of Ca. A symmetry-breaking bifurcation away from the rolling orbit is also found upon decreasing Ca. The regime with multiple attractors becomes broader as the aspect ratio of the capsule increases, while narrowing as viscosity ratio increases. We also report the particle contribution to the stress, which also displays multiplicity.