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1.
We introduce a concept of weak solution for a boundary value problem modelling the interactive motion of a coupled system consisting in a rigid body immersed in a viscous fluid. The fluid, and the solid are contained in a fixed open bounded set of R3. The motion of the fluid is governed by the incompresible Navier-Stokes equations and the standard conservation's laws of linear, and angular momentum rules the dynamics of the rigid body. The time variation of the fluid's domain (due to the motion of the rigid body) is not known apriori, so we deal with a free boundary value problem. Our main theorem asserts the existence of at least one weak solution for this problem. The result is global in time provided that the rigid body does not touch the boundary 相似文献
2.
V. N. Starovoitov B. N. Starovoitova 《Journal of Applied and Industrial Mathematics》2017,11(3):453-462
The existence of a generalized weak solution is proved for the nonstationary problem of motion of a rigid body in the flow of a viscous incompressible fluid filling a cylindrical pipe of arbitrary cross-section. The fluid flow conforms to the Navier–Stokes equations and tends to the Poiseuille flow at infinity. The body moves in accordance with the laws of classical mechanics under the influence of the surrounding fluid and the gravity force directed along the cylinder. Collisions of the body with the boundary of the flow domain are not admitted and, by this reason, the problem is considered until the body approaches the boundary. 相似文献
3.
V. N. Starovoitov 《Journal of Mathematical Sciences》2005,130(4):4893-4898
This paper is devoted to the problem on motion of a rigid body in a viscous incompressible fluid. It is proved that there exist at least two weak solutions of this problem if collisions of the body with the boundary of the flow domain are allowed. These solutions have different behavior of the body after the collision. Namely, for the first solution, the body goes away from the boundary after the collision. In the second solution, the body and the boundary remain in contact. Bibliography 15 titles.To Vsevolod Alekseevich Solonnikov on the occasion of his jubilee__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 306, 2003, pp. 199–209. 相似文献
4.
《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2019,36(5):1281-1309
We consider the motion of a rigid body in a viscoplastic material. This material is modeled by the 3D Bingham equations, and the Newton laws govern the displacement of the rigid body. Our main result is the existence of a weak solution for the corresponding system. The weak formulation is an inequality (due to the plasticity of the fluid), and it involves a free boundary (due to the motion of the rigid body). We approximate it by regularizing the convex terms in the Bingham fluid and by using a penalty method to take into account the presence of the rigid body. 相似文献
5.
E. S. Baranovskii 《Mathematical Methods in the Applied Sciences》2017,40(14):5035-5043
We consider the initial‐boundary value problem for a model of motion of aqueous polymer solutions in a bounded three‐dimensional domain subject to the Navier slip boundary condition. We construct a global (in time) weak solution to this problem. Moreover, we establish some uniqueness results, assuming additional regularity for weak solutions. Copyright © 2017 John Wiley & Sons, Ltd. 相似文献
6.
Borisov A. V. Kozlov V. V. Mamaev I. S. 《Proceedings of the Steklov Institute of Mathematics》2006,253(1):S24-S47
We consider a problem about the motion of a heavy rigid body in an unbounded volume of an ideal irrotational incompressible fluid. This problem generalizes a classical Kirchhoff problem describing the inertial motion of a rigid body in a fluid. We study different special statements of the problem: the plane motion and the motion of an axially symmetric body. In the general case of motion of a rigid body, we study the stability of partial solutions and point out limiting behaviors of the motion when the time increases infinitely. Using numerical computations on the plane of initial conditions, we construct domains corresponding to different types of the asymptotic behavior. We establish the fractal nature of the boundary separating these domains. 相似文献
7.
A boundary element method for potential flow problem coupled with the dynamics of rigid body was developed to determine numerically the resultant force and moment of force acting on an arbitrarily three-dimensional solid body and its motion in a current of an infinite fluid. An accurate integration method for singular integrands occurring in the boundary integral equations, a computational method for the tangential gradient of a velocity potential on a surface, and a method to properly treat the singularities appearing in the system of the dynamic equations of a rigid body, were proposed to complete the numerical solution of the problem. Several numerical examples were given to show the validity of the method. 相似文献
8.
In this paper, we consider the interaction between a rigid body and an incompressible, homogeneous, viscous fluid. This fluid-solid
system is assumed to fill the whole space ℝ
d
, d = 2 or 3. The equations for the fluid are the classical Navier-Stokes equations whereas the motion of the rigid body is governed
by the standard conservation laws of linear and angular momentum. The time variation of the fluid domain (due to the motion
of the rigid body) is not known a priori, so we deal with a free boundary value problem.
We improve the known results by proving a complete wellposedness result: our main result yields a local in time existence
and uniqueness of strong solutions for d = 2 or 3. Moreover, we prove that the solution is global in time for d = 2 and also for d = 3 if the data are small enough.
Patricio Cumsille’s research was partially supported by CONICYT-FONDECYT grant (No. 3070040) and Takéo Takahashi’s research
was partially supported by Grant (JCJC06 137283) of the Agence Nationale de la Recherche. 相似文献
9.
Ana Leonor Silvestre 《Journal of Mathematical Analysis and Applications》2002,274(1):203-227
In this paper we study the Stokes approximation of the self-propelled motion of a rigid body in a viscous liquid that fills all the three-dimensional space exterior to the body. We prove the existence and uniqueness of strong solution to the coupled systems of equations describing the motion of the system body-liquid, for any time and any regular distribution of velocity on the boundary of the body. For the corresponding stationary problem we derive Lp-estimates for the solution in terms of the data. Finally, we prove that every steady solution is attainable as the limit, when t→∞, of an unsteady self-propelled solution which starts from rest. 相似文献
10.
F. L. Chernous'ko 《Journal of Applied Mathematics and Mechanics》1966,30(6):1167-1184
Many papers are concerned with the dynamics of a rigid body with a cavity filled with liquid (see the bibliography in [1]). The present paper deals with the motion of a rigid body having a cavity partly filled with a viscous incompressible liquid, and having a free surface. The shape of the cavity is arbitrary. The problem is considered in a linear formulation. The oscillations of the body with respect to its center of inertia and the motion of the liquid in the cavity are assumed small. The viscosity of the liquid is considered low. The solution of the problem of the oscillations of a body with a cavity partly filled with an ideal liquid is used as an initial approximation [1 to 6]. The viscosity is taken into consideration by the boundary layer method used before in similar problems [1 and 7 to 10). General equations are derived for the dynamics of a body filled with a liquid, for an arbitrary form of cavity. The coefficients of those integro-differential equations depend only on the solution of the problem of the oscillations of a body with a cavity of the given form filled with an ideal liquid. Since the corresponding problem has been solved for cavities of many forms [1 to 6, 11 and 12] in the case of an ideal liquid, the determination of the characteristic coefficients is reduced to the evaluation of quadratures. Several particular cases of motion are considered. 相似文献
11.
A.P. Markeyev 《Journal of Applied Mathematics and Mechanics》2011,75(2):132-139
The motion of a rigid body in a uniform gravity field is investigated. One of the points of the body (the suspension point) performs specified small-amplitude high-frequency periodic or conditionally periodic oscillations (vibrations). The geometry of the body mass is arbitrary. An approximate system of differential equations is obtained, which does not contain the time explicitly and describes the rotational motion of the rigid body with respect to a system of coordinates moving translationally together with the suspension point. The error with which the solutions of the approximate system approximate to the solution of the exact system of equations of motion is indicated. The problem of the stability with respect to the equilibrium of the rigid body, when the suspension point performs vibrations along the vertical, is considered as an application. 相似文献
12.
Evgeny V. Vetchanin Ivan S. Mamaev Valentin A. Tenenev 《Regular and Chaotic Dynamics》2013,18(1-2):100-117
An investigation of the characteristics of motion of a rigid body with variable internal mass distribution in a viscous fluid is carried out on the basis of a joint numerical solution of the Navier — Stokes equations and equations of motion for a rigid body. A nonstationary three-dimensional solution to the problem is found. The motion of a sphere and a drop-shaped body in a viscous fluid in a gravitational field, which is caused by the motion of internal material points, is explored. The possibility of self-propulsion of a body in an arbitrary given direction is shown. 相似文献
13.
Jaime Ortega Lionel Rosier Takéo Takahashi 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2007
We consider the motion of a rigid body immersed in a bidimensional incompressible perfect fluid. The motion of the fluid is governed by the Euler equations and the conservation laws of linear and angular momentum rule the dynamics of the rigid body. We prove the existence and uniqueness of a global classical solution for this fluid–structure interaction problem. The proof relies mainly on weighted estimates for the vorticity associated with the strong solution of a fluid–structure interaction problem obtained by incorporating some viscosity. 相似文献
14.
S. Chen 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2003,289(1):387-409
15.
A hybrid vibrational system containing a solid (a cylinder) with an elastic connection to a coaxial cylindrical cavity, completely filled with a heavy ideal stably stratified two-layer fluid, is considered. The combined self-consistent vibrations of the body and the fluid (of the internal waves) are studied. An explicit solution of the internal boundary value problem of an inhomogeneous liquid in an annular domain for a specified motion of the body is obtained. An integrodifferential equation of the Newton type is constructed on the basis of this. This equation describes the self-consistent oscillations of the cylinder. In the case of weak coupling of the interaction between the solid and the medium, an approximate solution is obtained using asymptotic methods and an analysis is carried out. Qualitative effects of the mutual effect of the motions of the cylinder and the fluid are found. 相似文献
16.
Ye. M. Rudoi 《Journal of Applied Mathematics and Mechanics》2011,75(6):731-738
The plane problem in the linear theory of elasticity for a body with a rigid inclusion located within it is investigated. It is assumed that there is a crack on part of the boundary joining the inclusion and the matrix and complete bonding on the remaining part of the boundary. Zero displacements are specified on the outer boundary of the body. The crack surface is free from forces and the stress state in the body is determined by the bulk forces acting on it. The variation in the energy functional in the case of a variation in the rigid inclusion and the crack is investigated. The deviation of the solution of the perturbed problem from the solution of the initial problem is estimated. An expression is obtained for the derivative of the energy functional with respect to a zone perturbation parameter that depends on the solution of the initial problem and the form of the vector function defining the perturbation. Examples of the application of the results obtained are studied. 相似文献
17.
The existence of a weak solution of a boundary value problem for a fractional viscoelasticity model that is a fractional analogue of the anti-Zener model with memory along trajectories of motion is proved. The rheological equation of the given model involves fractional-order derivatives. The proof relies on an approximation of the original problem by a sequence of regularized ones and on the theory of regular Lagrangian flows. 相似文献
18.
A.A. Gurchenkov V.V. Korneyev M.V. Nosov 《Journal of Applied Mathematics and Mechanics》2008,72(6):653-659
The Cauchy problem for the motion of a dynamically symmetrical rigid body with a cavity, filled with an ideal liquid, which is perturbed from uniform rotation, is considered in a linear formulation. The problem of the simultanious solution of the equations of hydrodynamics and the mechanics of a rigid body is reduced to the solution of an eigenvalue problem which depends solely on the geometry of the cavity and the subsequent integration of a system of differential equations. 相似文献
19.
We consider two problems from the rigid body dynamics and use new methods of stability and asymptotic behavior analysis for
their solution. The first problem deals with motion of a rigid body in an unbounded volume of ideal fluid with zero vorticity.
The second problem, having similar asymptotic behavior, is concerned with motion of a sleigh on an inclined plane. The equations
of motion for the second problem are non-holonomic and exhibit some new features not typical for Hamiltonian systems. A comprehensive
survey of references is given and new problems connected with falling motion of heavy bodies in fluid are proposed.
相似文献
20.
The existence of a weak solution of a boundary value problem for a viscoelasticity model with memory on an infinite time interval is proved. The proof relies on an approximation of the original boundary value problem by regularized ones on finite time intervals and makes use of recent results concerning the solvability of Cauchy problems for systems of ordinary differential equations in the class of regular Lagrangian flows. 相似文献