Hydrodynamic moment acting on a moving deformable body in a linear potential flow of an ideal incompressible liquid

Abstract

The solution of the problem of a solid body motion in a parallel flow is discussed in [1]. To investigate the motion of a body with a deformable surface in a potential flow, one can use the following approximate method. By the external flow outside the body we shall understand the difference between the resulting flow, and the flow with a potential regular outside the body due to the presence of the body. For an infinite problem the notions of external and unperturbed flow (the flow in the absence of the body) coincide. We investigate the class of external flows for which the Taylor series of the velocity field converges in some sphere containing the body. The flows described by the partial sums S1, S2,... of this series approximate with increasing accuracy the external flow in the sphere. The exact solution Sn of the body motion in a flow without boundaries is an approximate solution of the original problem (not necessarily without boundaries). The flow S1 is parallel. We shall call the flow S2 linear because it is a linear function of the radius vector. Evidently, S2 is the simplest nonparallel flow Sn for n ≥ 2. Apparently, [2] is the first work that investigated the linear flow in three dimensions. Yakimov [3] obtained an expression for the force acting on a deformable body in the linear flow. Votsnov et al. [4] also solved the problem of hydrodynamic action on the solid body, in the linear-flow approximation, but in the expression for the moment he did not obtain all terms that appear in the exact solution. In the present work we shall obtain an expression for the moment acting on an arbitrary (deformable) body in a linear potential flow. The result is expressed in terms of the local characteristics of the external flow, and in terms of the shape of the body. We investigate bodies whose surface has planes of symmetry, and revolution bodies. We compare our results with the results for parallel flow and with the results of [4].