Complete expressions are derived for the inertia forces and moments acting on a small body in a six-degree-of-freedom motion in a three-dimensional unsteady flow in an unbounded ideal fluid. The far-field approximation of the body motion is represented by a series of multipoles located at the origin of the body. Unsteady terms are expanded in a dual series to the multipole series. Lagally integrals are expressed in terms of multipoles as well, by using Legendre polynomial expansions. New inertia force expressions are derived by truncating the multipole series after the quadrupoles. Corresponding terms for moments are also developed. The derived formulas are still compact enough for engineering applications. Many practical problems involving fixed and oscillating cylinders, piles, and risers are studied numerically. Comparisons to the Morison equation formulation prove that the nonlinear convective terms are not negligible in multidimensional relative flows.

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