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

## TL;DR

This paper develops fluctuating hydrodynamics methods for curved fluid interfaces to study particle dynamics, revealing unique power-law behaviors and providing theoretical explanations for surface-specific dissipation phenomena.

## Contribution

It introduces novel surface fluctuating hydrodynamics approaches, including immersed boundary and stochastic Eulerian-Lagrangian methods, for curved interfaces with thermal fluctuations.

## Key findings

- Power-law velocity autocorrelation scalings depend on geometry and viscosities.
- Surface hydrodynamics differ from bulk fluid behavior, showing unique dissipation time-scales.
- Methods successfully model passive particles and microswimmers on curved interfaces.

## Abstract

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. This depends 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 theory explaining these observed power-laws associated with time-scales for dissipation within the fluid interface and coupling to the surrounding fluid. We then use our introduced methods to investigate a few example systems and roles of hydrodynamic coupling and thermal fluctuations including for the kinetics of passive particles and active microswimmers in curved fluid interfaces.