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This paper provides a new insight to the classical Björknes’s problem. We examine a mechanical system “solid+fluid” consisted of a solid and a point source (singlet) of fluid, whose intensity is a given function of time. First we show that this system is governed by the least action (Hamilton’s) principle and derive an explicit expression for the Lagrangian in terms of the Green function of the solid. The Lagrangian contains a linear in velocity term. We prove that it does not produce a gyroscopic force only in the case of a spherical solid. Then we consider the periodical high-frequency pulsations (vibrations) of the singlet. In order to construct the high-frequency asymptotic solution we employ a version of the multiple scale method that allows us to obtain the “slow” Lagrangian for the averaged motions directly from Hamilton’s principle. We derive such a “slow” Lagrangian for a general solid. In details, we study the “slow” dynamics of a spherical solid, which can be either homogeneous or inhomogeneous in density. Finally, we discuss the “Björknes’s dynamic buoyancy” for a solid of general form.
|Title of host publication||Regular and Chaotic Dynamics|
|Subtitle of host publication||Hamiltonian Dynamics, Vortex Structures, Turbulence: Proceedings of the IUTAM Symposium held in Moscow, 25–30 August, 2006|
|Editors||Alexey Borisov, Valery Kozlov, Ivan Mamaev, Mikhail Sokolovskiy|
|Number of pages||15|
|Publication status||Published - 2008|
- Fluid Dynamics