Formulation of the problem of the motion of a small body in a perturbed flow

Abstract

The classical problem of the hydrodynamic reactions on a body of arbitrary shape moving in a fluid at rest [1] was generalized by Sedov to the case of an accelerated translational flow [2]. In the expressions for the hydrodynamic reactions, the shape of the body is represented only by the coefficients λij of the added masses and the volume Ω of the body. In the general case of motion of a body in a nontranslational flow the shape of the body cannot be represented by a finite set of coefficients in the determination of the hydrodynamic reactions. An important simplification occurs in the small-body formulation, which again leads to expressions for the force and torque similar to the classical expressions. The problem of the motion of a small body in a perturbed nontranslational flow was posed by Grigoryan and Yakimov [3], and with allowance for deformation of the body by Yakimov [4]. Later studies containing this formulation have been reviewed by Vil'khovchenko and Yakimov [5]. The aim of the present paper is to formulate the small-body problem more precisely. The order of smallness of the terms in the earlier studies was estimated solely as a function of the power of a small parameter — the size of the body. In the present paper it is shown that if it is additionally required that the final expression for the reactions should contain only principal terms containing the components νi and ωi of the translational and angular velocities, and also terms describing the flow structure, then the expression found by Grigoryan and Yakimov [3] for the hydrodynamic reaction is valid. The terms are estimated on the basis of dimensional analysis. Such arguments have already been used by the author for special examples [6, 7].