## Non-modal Floquet analysis for stabilizing soft particles in large-amplitude oscillatory extension

We analyzed the stability of a capsule in large-amplitude oscillatory extensional (LAOE) flow, as often used to study the rheology and dynamics of suspensions. Such a flow is typically established in a cross-slot configuration, with the particle (or particles) of interest observed in the stagnation region. However, controlling this configuration is challenging because the flow is unstable. We quantify such an instability for spherical elastic capsules suspended near the stagnation point using a non-modal global Floquet analysis, which is formulated to include full coupling of the capsule-viscous-flow dynamics. The flow is shown to be transiently, though not asymptotically, unstable. For each case considered, two predominant transient instabilities are identified: intra-period growth for translational capsule perturbations and period-to-period growth for certain capsule distortions. The amplitude of the intra-period instability depends linearly on the flow strength and oscillation period, and the period-to-period growth saturates over several periods, commensurate with the asymptotic stability of the flow.

## A Floquet stability analysis for capsules in shear flow

Observations in experiments and simulations show that the kinematic behaviour of an elastic capsule, suspended and rotating in shear flow, depends upon the flow strength, the capsule membrane material properties and its at-rest shape. We developed a linear stability description of the periodically rotating base state of this coupled system, as represented by a boundary integral flow formulation with spherical harmonic basis functions describing the elastic capsule geometry. This gave Floquet multipliers that classified the stability of the capsule motion for varying elastic capillary numbers. Viscous dissipation rapidly damps most perturbations. However, for all cases, a single component grows or decays slowly, depending upon the flow strength, over many periods of the rotation. The transitions in this stability behavior correspond to the different classes of rotating motion observed in previous studies.

**S. H. Bryngelson** and J. B. Freund, “Floquet stability analysis of capsules in viscous shear ﬂow,” *J. Fluid Mech.* **852**, 663–677 (2018)