Surface Fluctuating Hydrodynamics Methods for the Drift-Diffusion Dynamics of Particles and Microstructures within Curved Fluid Interfaces (1906.01146v3)

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.

• The paper introduces a novel framework that integrates fluctuating hydrodynamics with stochastic simulations to model particle drift-diffusion on curved interfaces.
• It employs Immersed Boundary and Stochastic Eulerian-Lagrangian Methods to capture interfacial coupling and curvature effects.
• Numerical results reveal unique velocity autocorrelation scalings and enhanced particle mixing distinct from flat interface dynamics.

Analyzing Surface Fluctuating Hydrodynamics for Drift-Diffusion Dynamics in Curved Fluid Interfaces

This academic paper introduces a comprehensive framework for simulating and analyzing the hydrodynamics of particles and microstructures embedded in curved fluid interfaces, particularly those with spherical geometry. The authors present a series of methods that leverage fluctuating hydrodynamics approaches to capture the complex dynamics that arise in such systems. These methods integrate various factors, including interfacial hydrodynamic coupling, thermal fluctuations, and interactions with bulk surrounding fluids.

The paper builds upon established models in the field, such as the Saffman-Delbrück hydrodynamics, but extends them to account for significant geometric and topological effects engendered by the curvature of fluid interfaces. It challenges previous assumptions that were primarily based on flat interfaces, addressing how surface curvature can significantly alter hydrodynamic responses.

Methodologies

To tackle the complex fluid-structure interactions within curved interfaces, the paper employs Immersed Boundary Methods (IBM) and Stochastic Eulerian-Lagrangian Methods (SELM). These methods are tailored to handle the unique challenges posed by curved geometries and address both the inertial and overdamped quasi-steady regimes. By utilizing generalized vector potentials through the Hodge decomposition for manifolds, the authors effectively manage the incompressibility constraints typical of hydrodynamic fields on curved surfaces.

Importantly, the authors introduce stochastic numerical methods designed for simulating the drift-diffusion dynamics of microstructures, which are essential for understanding the detailed statistical mechanics of surface fluctuating hydrodynamics. This includes the development of mobility tensors that encapsulate the collective hydrodynamic coupling for both passive and active microstructures.

Results and Implications

The numerical investigations reveal several interesting phenomena. Key observations include the presence of velocity autocorrelation power-law scalings of τ^−1 and τ^−2, alongside plateau behaviors, which emerge under different physical regimes linked to the interface geometry, surface viscosity, and bulk viscosity. These scalings diverge from standard three-dimensional bulk dynamics, highlighting the unique effects of curvature-mediated hydrodynamic interactions.

The paper's examples elucidate the roles of hydrodynamic coupling and thermal fluctuations in determining the kinetics of passive particles and active microswimmers. For particles and microswimmers on spherical interfaces, the simulated behaviors demonstrate enhanced mixing and correlated diffusion phenomena that significantly differ from those observed in flat systems.

These findings bear both theoretical and practical implications. Theoretically, this work advances the understanding of fluid dynamics at curved interfaces, providing tools to predict behaviors not previously accounted for in existing models. Practically, the insights gained from such simulations could be pivotal in designing and controlling material properties in soft condensed matter systems or biological membranes, where the mechanics of embedded entities are crucial.

Future Directions

The methods and findings presented in this paper lay the groundwork for further research into more complex geometries and interaction regimes. Future research could extend beyond simple spherical interfaces to explore interfaces with varying curvatures and topologies. Additionally, the incorporation of different constitutive materials and the transitory dynamics of bulk surrounding fluids offer promising avenues for expanding this work, potentially impacting a broader spectrum of applications in materials science and biology.

In conclusion, the paper presents a substantial contribution to the field of fluctuating hydrodynamics and fluid-structure interactions at curved interfaces. By integrating sophisticated mathematical tools with computational techniques, it sets a new standard for modeling and understanding the intricate dynamics that emerge in these systems.