Evolution of Motions of a Rigid Body About its Center of Mass

Chernousko, F. L.; Акуленко, Л. Д.; Leshchenko, Dmytro

doi:10.1007/978-3-319-53928-7

Cited by 66 publications

(58 citation statements)

References 42 publications

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“…In [41], where homogeneous Dirichlet boundary conditions are adopted, the author shows that in absence of external forces, as time t → ∞ the velocity of the fluid tends to zero and density tends to a constant function. This cannot be true in our case, in general, since the magnitude of total angular momentum of the coupled system body-fluid must be conserved (see (3)). Consequently, the velocity of the whole system cannot be zero in the limit t → ∞ and also, contrary to the incompressible case, we cannot expect a constant density as a stationary solution.…”

Section: Introductionmentioning

confidence: 90%

On the Motion of a Body with a Cavity Filled with Compressible Fluid

Galdi

Mácha

Nečasová

2019

Arch Rational Mech Anal

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We study the motion of the system, S, constituted by a rigid body, B, containing in its interior a viscous compressible fluid, and moving in absence of external forces. Our main objective is to characterize the long time behavior of the coupled system body-fluid. Under suitable assumptions on the "mass distribution" of S, and for sufficiently "small" Mach number and initial data, we show that every corresponding motion (in a suitable regularity class) must tend to a steady state where the fluid is at rest with respect to B. Moreover, S, as a whole, performs a uniform rotation around an axis parallel to the (constant) angular momentum of S, and passing through its center of mass.

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Section: Introductionmentioning

confidence: 90%

On the Motion of a Body with a Cavity Filled with Compressible Fluid

Galdi

Mácha

Nečasová

2019

Arch Rational Mech Anal

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“…Successively, also due to the primary role played in several significant applications, the problem has been all along addressed, from different perspectives, by a large number of applied mathematicians, mostly belonging to the Russian school. The list of their names and corresponding contributions is too long to include here and, therefore, we refer the reader to the introductory chapter of the recent monograph [4].…”

Section: Introductionmentioning

confidence: 99%

“…However, it is only over the past few years, that a rigorous mathematical analysis has been initiated, with the objective of investigating a fundamental property of such coupled systems, namely, the characterization of their "ultimate dynamics" [24,11,5,12,15,9,14,21]. In fact, as shown by both experiment and qualitative analysis [27,3,4], the viscous liquid acts as a damper on the rigid body to the point, in some cases, of even bringing it to rest (see [13,15] for a rigorous mathematical explanation).…”

Section: Introductionmentioning

confidence: 99%

On weak solutions to the problem of a rigid body with a cavity filled with a compressible fluid, and their asymptotic behavior

Galdi

Mácha

Nečasová

2020

International Journal of Non-Linear Mechanics

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We prove the existence of a weak solution to the equations describing the inertial motions of a coupled system constituted by a rigid body containing a viscous compressible fluid. We then provide a weak-strong uniqueness result that allows us to completely characterize, under certain physical assumptions, the asymptotic behavior in time of the weak solution corresponding to smooth data of restricted "size," and show that it tends to a uniquely determined steady-state. 2"small" in size. As explained in the introductory section in [10] the analogous study in the larger class of weak solution corresponding to data possessing just finite energy appears, currently, out of reach. The main reason is ascribed to the fact that a necessarily detailed study of the Ω-limit requires that weak solutions become eventually smooth enough, a property that, while true in the incompressible case [5], need not be generally valid if the fluid is compressible [16].This drawback notwithstanding, one may wonder if, at least, a weak solution with smooth and sufficiently "small" initial data could satisfy the above property. The latter is certainly true, if one can show a weak-strong uniqueness result, which would, in particular, ensure that the weak solution is strong for all positive times.Objective of this article is to prove that this is, in fact, the case. More precisely, we show that, for generic initial data with finite energy, there corresponds at least one weak solution (see Theorem 3.2) suitably defined (see Definition 3). Furthermore, we show that if there is a strong solution (also suitably defined) possessing the same data, then necessarily it should coincide with the weak one (see Theorem 5.2). In view of this result, we may then conclude that, for smooth and "small" initial data, the weak solution must coincide with that constructed and used in [10], and this will ensure that, as established in [10], under the assumption of suitable "mass distribution" and "small" Mach numbers, the ultimate state reached by the weak solution must be a uniform rotation around a central axis of inertia.The plan of the paper is the following. In Section 2 we give a formulation of the problem and recall the relevant equations. Section 3 is dedicated to the definition of weak solutions and the proof of their existence (Theorem 3.2). In achieving the latter, we appropriately adjust to the case at hand an approximation procedure developed in [23,6]. Successively, in Section 4, we state results of local and global existence and uniqueness of strong solutions proved in [10]; see Theorems 4.1 and 4.2. The following Section 5 is devoted to the proof that the weak and the strong solution, dealt with before, corresponding to the same data must, in fact, coincide; see Theorem 5.2. In the proof of this result, we adapt the arguments of [7,18,2]; see also [18]. Once the weak-strong coincidence has been established, we dedicate the last two sections to recall and formulate the results proved in [10] concerning the asymptotic behavior of a strong solution for...

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“…Влияние диссипации на устойчивость вращения твердого тела рассматривалась во многих статьях. Достаточно хороший обзор этих статей и методов исследований можно найти, например, в работах [5][6][7][8][9]. В статье [8] рассмотрена задача о движении тяжелого динамически симметричного твёрдого тела вокруг неподвижной точки, находящегося под действием диссипативного и постоянного момента в инерциальной системе координат, а в работе [9] учтен еще и постоянный момент в системе отсчета, связанной с твердым телом.…”

Section: Introductionunclassified

Про Вплив Дисипативного І Двох Постійних Моментів На Стійкість Рівномірних Обертань Зв'язаних Гіроскопів Лагранжа.

Kononov¹,

Украины²,

Sviatenko³

et al. 2019

vestnik-a

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На основании иннорного подхода получены в виде системы трех неравенств условия асимптотической устойчивости равномерного вращения в сопротивляющейся среде двух тяжелых гироскопов Лагранжа, связанных сферическим шарниром. Нижний гироскоп имеет неподвижную точку. Вращение каждого гироскопа поддерживается постоянным моментом в инерциальной системе координат и в системе координат, связанной с твердым телом. В случае отсутствия одного из гироскопов вновь полученное условие устойчивости совпадают с известным. Рассмотрен случай вырождения сферического шарнира в универсальный шарнир (шарнир Гука). Показано, что невыполнение квадратного неравенства относительно угловой скорости собственного вращения ведет к неустойчивости.

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Evolution of Motions of a Rigid Body About its Center of Mass

Cited by 66 publications

References 42 publications

On the Motion of a Body with a Cavity Filled with Compressible Fluid

On the Motion of a Body with a Cavity Filled with Compressible Fluid

On weak solutions to the problem of a rigid body with a cavity filled with a compressible fluid, and their asymptotic behavior

Про Вплив Дисипативного І Двох Постійних Моментів На Стійкість Рівномірних Обертань Зв'язаних Гіроскопів Лагранжа.

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