Stability of rotating equilibrium states of fluid deformable surfaces

We consider rotating equilibrium states of fluid deformable surfaces. These states are characterized by a force balance between centrifugal and bending forces, involve surface Killing vector fields and are independent on the surface viscosity. Considering a continuum description based on the incompressible surface Navier Stokes equations with bending forces and conserved enclosed volume we numerically demonstrate that these rotating equilibrium states can be reached, but also that these states are not stable. Any perturbation in shape or rotating flow field leads to dissipation and destroys the rotating equilibrium states. After breaking symmetry the evolution reaches other rotating states with a lower energy for which the symmetry axis and the rotation axis are not aligned. Such flow fields could be characterized by three-dimensional Killing vector fields. However, also these states are not stable. Based on these numerical results we postulate a cascading mechanism of disturbance - force balance reconfiguration - dissipation that contains various rotating equilibrium states as transient configurations but eventually leads to the classical equilibrium shapes of the Helfrich energy.