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Discrete & Continuous Dynamical Systems - S

2012 , Volume 5 , Issue 6

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Preface
Eduard Feireisl, Josef Málek and Mirko Rokyta
2012, 5(6): i-ii doi: 10.3934/dcdss.2012.5.6i +[Abstract](110) +[PDF](117.4KB)
Abstract:
This volume consists of five surveys that are focused on various aspects of theoretical fluid mechanics. In our opinion, mathematical analysis of fluid mechanics problems can lead to interesting and important results if the research shares at least two characteristics. First, it requires good understanding of how the models in consideration have been developed, what are their limitations, and in what situations can the models be applied. Second, the theoretical investigations must be driven by relevant and interesting problems in applications such as qualitative properties of the solution or the construction of numerical methods. Such a viewpoint has motivated the composition of the authors in this volume and resulted in five surveys that range from the constitutive theory of non-Newtonian fluids through the analysis of inhomogeneous incompressible fluid flows and the stability analysis of vortices up to the development of efficient computational methods for flows described by the incompressible Navier--Stokes equations and its generalizations.

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Motivation, analysis and control of the variable density Navier-Stokes equations
Enrique Fernández-Cara
2012, 5(6): 1021-1090 doi: 10.3934/dcdss.2012.5.1021 +[Abstract](193) +[PDF](831.8KB)
Abstract:
The main objective of these Notes is to provide an introduction to variable density NS: their motivation, some of the main mathematical problems connected with them, the main techniques used to solve these problems, the main results and open questions. First, we will describe the physical origin of the equations. Then, we will be concerned with existence, uniqueness, regularity and control of initial-boundary value problems in cylindrical domains $ Ω $ $\times (0,T)$; as usual, $ Ω $ is the spatial domain, an open set in $\mathbb{R}$2 or $\mathbb{R}$3 ``filled'' by the fluid particles and (0,T) is the time observation interval. Some open problems (not all them of the same difficulty) are also recalled.
Stability and interaction of vortices in two-dimensional viscous flows
Thierry Gallay
2012, 5(6): 1091-1131 doi: 10.3934/dcdss.2012.5.1091 +[Abstract](224) +[PDF](683.3KB)
Abstract:
The aim of these notes is to present in a comprehensive and relatively self-contained way some recent developments in the mathematical analysis of two-dimensional viscous flows. We consider the incompressible Navier-Stokes equations in the whole plane $\mathbb{R}$2, and assume that the initial vorticity is a finite measure. This general setting includes vortex patches, vortex sheets, and point vortices. We first prove the existence of a unique global solution, for any value of the viscosity parameter, and we investigate its long-time behavior. We next consider the particular case where the initial flow is a finite collection of point vortices. In that situation, we show that the solution behaves, in the vanishing viscosity limit, as a superposition of Oseen vortices whose centers evolve according to the Helmholtz-Kirchhoff point vortex system. The proof requires a careful stability analysis of the Oseen vortices in the large Reynolds number regime, as well as a precise computation of the deformations of the vortex cores due to mutual interactions.
The thermo-mechanics of rate-type fluids
K. R. Rajagopal
2012, 5(6): 1133-1145 doi: 10.3934/dcdss.2012.5.1133 +[Abstract](206) +[PDF](357.0KB)
Abstract:
In this short paper a brief review is provided concerning the modeling of the thermo-mechanical response of rate type fluid models. Recently, two different approaches have been used to develop thermodynamically compatible rate type fluid models, one that assumes the Helmholtz potential and the other the Gibbs potential for the fluids. These two perspectives are complimentary, not all models that can be modeled within the first procedure can be obtained from the second one, and vice-versa. The two approaches greatly enlarge the arsenal of a modeler and most models that are used can be derived within the purview of these two approaches. More importantly, the two methodologies lead to interesting and useful new models which can be used to describe the behavior of materials that have hitherto defied proper description.
A short course on numerical simulation of viscous flow: Discretization, optimization and stability analysis
Rolf Rannacher
2012, 5(6): 1147-1194 doi: 10.3934/dcdss.2012.5.1147 +[Abstract](207) +[PDF](1311.1KB)
Abstract:
This article contains part of the material of four introductory lectures given at the 12th school ``Mathematical Theory in Fluid Mechanics'', Spring 2011, at Kácov, Czech Republic, on ``Numerical simulation of viscous flow: discretization, optimization and stability analysis''. In the first lecture on ``Numerical computation of incompressible viscous flow'', we discuss the Galerkin finite element method for the discretization of the Navier-Stokes equations for modeling laminar flow. Particular emphasis is put on the aspects pressure stabilization and truncation to bounded domains. In the second lecture on ``Goal-oriented adaptivity'', we introduce the concept underlying the Dual Weighted Residual (DWR) method for goal-oriented residual-based adaptivity in solving the Navier-Stokes equations. This approach is presented for stationary as well as nonstationary situations. In the third lecture on ``Optimal flow control'', we discuss the use of the DWR method for adaptive discretization in flow control and model calibration. Finally, in the fourth lecture on ``Numerical stability analysis'', we consider the numerical stability analysis of stationary flows employing the concepts of linearized stability and pseudospectrum.
A framework for the development of implicit solvers for incompressible flow problems
David J. Silvester, Alex Bespalov and Catherine E. Powell
2012, 5(6): 1195-1221 doi: 10.3934/dcdss.2012.5.1195 +[Abstract](250) +[PDF](1691.2KB)
Abstract:
This survey paper reviews some recent developments in the design of robust solution methods for the Navier--Stokes equations modelling incompressible fluid flow. There are two building blocks in our solution strategy. First, an implicit time integrator that uses a stabilized trapezoid rule with an explicit Adams--Bashforth method for error control, and second, a robust Krylov subspace solver for the spatially discretized system. Numerical experiments are presented that illustrate the effectiveness of our generic approach. It is further shown that the basic solution strategy can be readily extended to more complicated models, including unsteady flow problems with coupled physics and steady flow problems that are nondeterministic in the sense that they have uncertain input data.

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