Asymptotic Analysis of Steady and Nonsteady Navier-Stokes Equations for Barotropic Compressible Flow
- PDF / 512,756 Bytes
- 19 Pages / 439.37 x 666.142 pts Page_size
- 91 Downloads / 177 Views
Asymptotic Analysis of Steady and Nonsteady Navier-Stokes Equations for Barotropic Compressible Flow Rostislav Vodák
Received: 4 September 2007 / Accepted: 23 February 2009 / Published online: 6 March 2009 © Springer Science+Business Media B.V. 2009
Abstract We study the asymptotic behavior of solutions to steady and nonsteady NavierStokes equations for barotropic compressible fluids with slip boundary conditions in small channels whose diameters converge to zero. We also derive the corresponding asymptotic one-dimensional equations and we analyze the sets, where L1 -weak convergence of the pressure terms fails. Keywords Navier-Stokes equations · Asymptotic analysis · Channels Mathematics Subject Classification (2000) 35Q30 · 35B99 · 76N25 1 Introduction There is a vast literature concerning asymptotic analysis of solutions to compressible NavierStokes equations. Let us mention [15], where the stabilization of solutions was studied for time tending to infinity, and [3, 7, 11], where incompressible limits of compressible NavierStokes equations were derived. In [12], the authors studied what happens with the solutions if p(ρ) = ρ γ and γ → ∞. For the dependence of the solutions on domains, we refer to [4] and [6]. In this paper, we analyze the behavior of solutions to steady and nonsteady barotropic compressible Navier-Stokes equations with slip boundary conditions in small channels with diameters converging to zero. The problem is solved by a transformation to a reference domain and performing a limit. Up to now, the limit process was used only for elastic materials described by a linear model (see for instance [1, 2, 9, 13, 14]). As a result we obtain the limit triplet (ρ, u, π(ρ)) (see [12] for the same concept of a limit solution) solving asymptotic one-dimensional equations, where π(ρ) is a Radon measure representing a limit
Supported by the Grant No. 201/07/P165 of Grant Agency of Czech Republic. R. Vodák () Department of Mathematical Analysis and Applications of Mathematics, Faculty of Science, Palacky University, Tomkova 40, 779 00 Olomouc-Hejˇcín, Czech Republic e-mail: [email protected]
992
R. Vodák
of the averages of pressure terms over the cross-sections of the channels. We show how to treat the convective term and how the Radon measure is connected with the components of limit stress tensor. We also mention what happens in the case of steady flow. At the end, we discuss why the contemporary theory developed for passing the nonlinearity in pressure for compressible flows fails here. The paper is organized as follows: In Sect. 2, we introduce the basic notation and the nonsteady equations. Section 3 is devoted to the basic convergences derived by means of the energy inequality and to the construction of one-dimensional approximations of the first component of the velocity fields. We further derive the asymptotic one-dimensional equations here. We state the main result in Theorem 3.3. Section 4 deals with properties of the sets, where L1 -weak compactness of pressure terms fails. In Sect. 5, we show t
Data Loading...