Stochastic Mode Reduction for the Immersed Boundary Method

We apply the formulation of a stochastic mode reduction method developed in a recent paper of Majda, Timofeyev, and Vanden-Eijnden [Comm. Pure Appl. Math., 54 (2001), pp. 891– 974] (MTV) to obtain simplified equations for the dynamics of structures immersed in a thermally fluctuating fluid at low Reynolds (or Kubo) number, as simulated by a recent extension of the immersed boundary (IB) method by Kramer and Peskin [Proceedings of the Second MIT Conference on Computational Fluid and Solid Mechanics, Elsevier Science, Oxford, UK, 2003, pp. 1755–1758]. The effective dynamics of the immersed structures are not obvious in the primitive equations, which involve both fluid and structure dynamics, but the procedure of MTV allows the rigorous derivation of a reduced stochastic system for the immersed structures alone. We find, in the limit of small Reynolds (or Kubo) number, that the Lagrangian particle constituents of the immersed structures undergo a drift-diffusive motion with several physically correct features, including the coupling between dynamics of different particles. The MTV procedure is also applied to the spatially discretized form of the IB equations with thermal fluctuations to assist in the design and assessment of numerical algorithms.