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1.
We perform a mathematical analysis of the steady flow of a viscous liquid, L{\mathcal{L}} , past a three-dimensional elastic body, B{\mathcal{B}} . We assume that L{\mathcal{L}} fills the whole space exterior to B{\mathcal{B}} , and that its motion is governed by the Navier–Stokes equations corresponding to non-zero velocity at infinity, v
∞. As for B{\mathcal{B}} , we suppose that it is a St. Venant–Kirchhoff material, held in equilibrium either by keeping an interior portion of it
attached to a rigid body or by means of appropriate control body force and surface traction. We treat the problem as a coupled
steady state fluid-structure problem with the surface of B{\mathcal{B}} as a free boundary. Our main goal is to show existence and uniqueness for the coupled system liquid-body, for sufficiently
small |v
∞|. This goal is reached by a fixed point approach based upon a suitable reformulation of the Navier–Stokes equation in the
reference configuration, along with appropriate a priori estimates of solutions to the corresponding Oseen linearization and
to the elasticity equations. 相似文献
2.
Thieu Huy Nguyen 《Archive for Rational Mechanics and Analysis》2014,213(2):689-703
We prove the existence and uniqueness of periodic motions to Stokes and Navier–Stokes flows around a rotating obstacle \({D \subset \mathbb{R}^3}\) with the complement \({\Omega = \mathbb{R}^3 \backslash D}\) being an exterior domain. In our strategy, we show the C b -regularity in time for the mild solutions to linearized equations in the Lorentz space \({L^{3,\infty}(\Omega)}\) (known as weak-L 3 spaces) and prove a Massera-typed Theorem on the existence and uniqueness of periodic mild solutions to the linearized equations in weak-L 3 spaces. We then use the obtained results for such equations and the fixed point argument to prove such results for Navier–Stokes equations around a rotating obstacle. We also show the stability of such periodic solutions. 相似文献
3.
We consider the two-dimensional motion of the coupled system of a viscous incompressible fluid and a rigid disc moving with
the fluid, in the whole plane. The fluid motion is described by the Navier–Stokes equations and the motion of the rigid body
by conservation laws of linear and angular momentum. We show that, assuming that the rigid disc is not allowed to rotate,
as the radius of the disc goes to zero, the solution of this system converges, in an appropriate sense, to the solution of
the Navier–Stokes equations describing the motion of only fluid in the whole plane. We also prove that the trajectory of the
centre of the disc, at the zero limit of its radius, coincides with a fluid particle trajectory. 相似文献
4.
The initial boundary value problem for the compressible Navier–Stokes equation is considered in an infinite layer of . It is proved that if the Reynolds and Mach numbers are sufficiently small, then strong solutions to the compressible Navier–Stokes equation around parallel flows exist globally in time for sufficiently small initial perturbations. The large time behavior of the solution is described by a solution of a one-dimensional viscous Burgers equation. The proof is given by a combination of spectral analysis of the linearized operator and a variant of the Matsumura–Nishida energy method. 相似文献
5.
We propose a new outflow boundary condition, a unilateral condition of Signorini’s type, for the incompressible Navier–Stokes equations. The condition is a generalization of the standard free-traction condition. Its variational formulation is given by a variational inequality. We also consider a penalty approximation, a kind of the Robin condition, to deduce a suitable formulation for numerical computations. Under those conditions, we can obtain energy inequalities that are key features for numerical computations. The main contribution of this paper is to establish the well-posedness of the Navier–Stokes equations under those boundary conditions. Particularly, we prove the unique existence of strong solutions of Ladyzhenskaya’s class using the standard Galerkin’s method. For the proof of the existence of pressures, however, we offer a new method of analysis. 相似文献
6.
We consider the asymptotic limit for the complete Navier–Stokes–Fourier system as both Mach and Froude numbers tend to zero.
The limit is investigated in the context of weak variational solutions on an arbitrary large time interval and for the ill-prepared
initial data. The convergence to the Oberbeck–Boussinesq system is shown.
相似文献
7.
8.
We study the resolvent equation associated with a linear operator L{\mathcal{L}} arising from the linearized equation for perturbations of a steady Navier–Stokes flow U*{\mathbf{U^*}}. We derive estimates which, together with a stability criterion from [33], show that the stability of U*{\mathbf{U^*}} (in the L2-norm) depends only on the position of the eigenvalues of L{\mathcal{L}}, regardless the presence of the essential spectrum. 相似文献
9.
10.
Mikhail V. Korobkov Konstantin Pileckas Remigio Russo 《Archive for Rational Mechanics and Analysis》2013,207(1):185-213
We study the nonhomogeneous boundary value problem for Navier–Stokes equations of steady motion of a viscous incompressible fluid in a two-dimensional, bounded, multiply connected domain ${\Omega = \Omega_1 \backslash \overline{\Omega}_2, \overline\Omega_2\subset \Omega_1}$ . We prove that this problem has a solution if the flux ${\mathcal{F}}$ of the boundary value through ?Ω 2 is nonnegative (inflow condition). The proof of the main result uses the Bernoulli law for a weak solution to the Euler equations and the one-sided maximum principle for the total head pressure corresponding to this solution. 相似文献
11.
Katrin Schumacher 《Journal of Mathematical Fluid Mechanics》2009,11(4):552-571
We investigate the solvability of the instationary Navier–Stokes equations with fully inhomogeneous data in a bounded domain
W ì \mathbbRn \Omega \subset {{\mathbb{R}}^{n}} . The class of solutions is contained in Lr(0, T; Hb, qw (W))L^{r}(0, T; H^{\beta, q}_{w} (\Omega)), where Hb, qw (W){H^{\beta, q}_{w}} (\Omega) is a Bessel-Potential space with a Muckenhoupt weight w. In this context we derive solvability for small data, where this smallness can be realized by the restriction to a short
time interval. Depending on the order of this Bessel-Potential space we are dealing with strong solutions, weak solutions,
or with very weak solutions. 相似文献
12.
13.
Jürgen Saal 《Journal of Mathematical Fluid Mechanics》2006,8(2):211-241
We study the initial-boundary value problem for the Stokes equations with Robin boundary conditions in the half-space
It is proved that the associated Stokes operator is sectorial and admits a bounded H∞-calculus on
As an application we prove also a local existence result for the nonlinear initial value problem of the Navier–Stokes equations
with Robin boundary conditions. 相似文献
14.
Walter Rusin 《Journal of Mathematical Fluid Mechanics》2012,14(2):383-405
In this paper, we consider the Cauchy problem for a nonlinear parabolic system ${u^\epsilon_t - \Delta u^\epsilon + u^\epsilon \cdot \nabla u^\epsilon + \frac{1}{2}u^\epsilon\, {\rm div}\, u^\epsilon - \frac{1}{\epsilon}\nabla\, {\rm div}\, u^\epsilon = 0}$ in ${\mathbb {R}^3 \times (0,\infty)}$ with initial data in Lebesgue spaces ${L^2(\mathbb {R}^3)}$ or ${L^3(\mathbb {R}^3)}$ . We analyze the convergence of its solutions to a solution of the incompressible Navier?CStokes system as ${\epsilon \to 0}$ . 相似文献
15.
《Comptes Rendus Mecanique》2019,347(10):677-684
Some implications of the simplest accounting of defects of compatibility in the velocity field on the structure of the classical Navier–Stokes equations are explored, leading to connections between classical elasticity, the elastic theory of defects, plasticity theory, and classical fluid mechanics. 相似文献
16.
17.
Alexander A. Shapiro 《Transport in Porous Media》2018,122(3):713-744
A new three-dimensional hydrodynamic model for unsteady two-phase flows in a porous medium, accounting for the motion of the interface between the flowing liquids, is developed. In a minimum number of interpretable geometrical assumptions, a complete system of macroscale flow equations is derived by averaging the microscale equations for viscous flow. The macroscale flow velocities of the phases may be non-parallel, while the interface between them is, on average, inclined to the directions of the phase velocities, as well as to the direction of the saturation gradient. The last gradient plays a specific role in the determination of the flow geometry. The resulting system of flow equations is a far generalization of the classical Buckley–Leverett model, explicitly describing the motion of the interface and velocity of the liquid close to it. Apart from propagation of the two liquid volumes, their expansion or contraction is also described, while rotation has been proven negligible. A detailed comparison with the previous studies for the two-phase flows accounting for propagation of the interface on micro- and macroscale has been carried out. A numerical algorithm has been developed allowing for solution of the system of flow equations in multiple dimensions. Sample computations demonstrate that the new model results in sharpening the displacement front and a more piston-like character of displacement. It is also demonstrated that the velocities of the flowing phases may indeed be non-collinear, especially at the zone of intersection of the displacement front and a zone of sharp permeability variation. 相似文献
18.
We consider asymptotic behavior of Leray’s solution which expresses axis-symmetric incompressible Navier–Stokes flow past
an axis-symmetric body. When the velocity at infinity is prescribed to be nonzero constant, Leray’s solution is known to have
optimum decay rate, which is in the class of physically reasonable solution. When the velocity at infinity is prescribed to
be zero, the decay rate at infinity has been shown under certain restrictions such as smallness on the data. Here we find
an explicit decay rate when the flow is axis-symmetric by decoupling the axial velocity and the horizontal velocities.
The first author was supported by KRF-2006-312-C00466. The second author was supported by KRF-2006-531-C00009. 相似文献
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20.
Lorenzo Brandolese 《Archive for Rational Mechanics and Analysis》2009,192(3):375-401
We study the solutions of the nonstationary incompressible Navier–Stokes equations in , of self-similar form , obtained from small and homogeneous initial data a(x). We construct an explicit asymptotic formula relating the self-similar profile U(x) of the velocity field to its corresponding initial datum a(x). 相似文献