Stability of rotating equilibrium states of fluid deformable surfaces

Abstract

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 of 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, 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 also 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.

Derivation and simulation of a two-phase fluid deformable surface model

“…This equilibrium state coincides with the corresponding result for the reduced model without hydrodynamics, see Appendix D. However, without friction γ = 0 the dissipation potential in the reached axisymmetric state is invariant under surface rigid body motion. Such configurations have been explored for one-component fluid deformable surfaces (Reuther et al 2020;Krause & Voigt 2023;Nestler & Voigt 2023b;Olshanskii 2023). Indeed the shown configurations in figures 4 and 5 for κ = 0.5 undergo slight rigid body motions and as the resulting forces interact with the shape and the phase composition also slightly differ.…”