Surface Fluctuating Hydrodynamics Methods for the Drift-Diffusion Dynamics of Particles and Microstructures within Curved Fluid Interfaces

We introduce fluctuating hydrodynamics approaches on surfaces for capturing the drift-diffusion dynamics of particles and microstructures immersed within curved fluid interfaces of spherical shape. We take into account the interfacial hydrodynamic coupling, traction coupling with the surrounding bulk fluid, and thermal fluctuations. For fluid-structure interactions, we introduce Immersed Boundary Methods (IBM) and related Stochastic Eulerian-Lagrangian Methods (SELM) for curved surfaces. We use these approaches to investigate the statistics of surface fluctuating hydrodynamics and microstructures. For velocity autocorrelations, we find characteristic power-law scalings $\tau^{-1}$, $\tau^{-2}$, and plateaus can emerge depending on the physical regime associated with the geometry, surface viscosity, and bulk viscosity. This differs from the characteristic $\tau^{-3/2}$ scaling for bulk three dimensional fluids. We develop a theory explaining these observed power-laws that can be interpreted using time-scales associated with dissipation within the fluid interface and coupling to the bulk fluid. We then use our introduced methods to investigate a few example systems including the kinetics of passive particles and active microswimmers. We study how the drift-diffusion dynamics of microstructures compare with and without hydrodynamic coupling within the curved fluid interface.