Fluctuating hydrodynamics for curved fluid interfaces: An Extended Saffman-Delbruck approach for drift-diffusion dynamics of particle inclusions

We develop fluctuating hydrodynamics approaches that extend Saffman-Delbruck theory to capture the collective drift-diffusion dynamics of inclusion particles within curved fluid interfaces. Our extended SD theory and computational methods take into account the two dimensional hydrodynamics of the curved interface coupled with the three dimensional hydrodynamics of the surrounding bulk fluid. Using analytic and computational approaches, we show how Gaussian curvature can significantly impact dissipation within the curved two dimensional membrane fluid to augment the collective drift-diffusion dynamics of particle inclusions. We also present general results on the collective drift-diffusion dynamics when heterogeneous curved structures are present in the membrane geometry showing how these local Gaussian curvature effects influence hydrodynamic coupling in some interesting ways. Lastly we present approaches for computing these hydrodynamic equations on arbitrary compact manifolds, as well as methods for parallelizing our implementation for high-performance computing. We demonstrate our results for applications to dynamics of proteins within curved lipid bilayer membranes and colloidal self-assembly in curved fluid sheets.