A Stochastic Immersed Boundary Method for Fluid-Structure Dynamics at Microscopic Length Scales

TL;DR

This paper introduces a stochastic immersed boundary method that incorporates thermal fluctuations for fluid-structure interactions at microscopic scales, allowing for efficient long-time simulations with accurate statistical and physical properties.

Contribution

It extends the immersed boundary method to include thermal fluctuations and develops a numerical scheme that handles system stiffness and various resolution regimes while maintaining accuracy.

Findings

Correct Boltzmann equilibrium statistics for particles

Proper scaling of diffusion in three dimensions

Reproduction of long-time hydrodynamic effects like velocity autocorrelation decay

Abstract

In this work it is shown how the immersed boundary method of (Peskin2002) for modeling flexible structures immersed in a fluid can be extended to include thermal fluctuations. A stochastic numerical method is proposed which deals with stiffness in the system of equations by handling systematically the statistical contributions of the fastest dynamics of the fluid and immersed structures over long time steps. An important feature of the numerical method is that time steps can be taken in which the degrees of freedom of the fluid are completely underresolved, partially resolved, or fully resolved while retaining a good level of accuracy. Error estimates in each of these regimes are given for the method. A number of theoretical and numerical checks are furthermore performed to assess its physical fidelity. For a conservative force, the method is found to simulate particles with the correct Boltzmann equilibrium statistics. It is shown in three dimensions that the diffusion of immersed particles simulated with the method has the correct scaling in the physical parameters. The method is also shown to reproduce a well-known hydrodynamic effect of a Brownian particle in which the velocity autocorrelation function exhibits an algebraic tau^(-3/2) decay for long times. A few preliminary results are presented for more complex systems which demonstrate some potential application areas of the method.