Abstract

We consider the problem of the existence and finite dimensionality of attractors for some classes of two-dimensional turbulent boundarydriven flows that naturally appear in lubrication theory. The flows admit mixed, non-standard boundary conditions and time-dependent driving forces. We are interested in the dependence of the dimension of the attractors on the geometry of the flow domain and on the boundary conditions.

Introduction

This work gives a survey of the results obtained in a series of papers by Boukrouche & Łukaszewicz (2004, 2005a,b, 2007) and Boukrouche, Łukaszewicz, & Real (2006) in which we consider the problem of the existence and finite dimensionality of attractors for some classes of twodimensional turbulent boundary-driven flows (Problems I–IV below). The flows admit mixed, non-standard boundary conditions and also time-dependent driving forces (Problems III and IV). We are interested in the dependence of the dimension of the attractors on the geometry of the flow domain and on the boundary conditions. This research is motivated by problems from lubrication theory. Our results generalize some earlier ones devoted to the existence of attractors and estimates of their dimensions for a variety of Navier–Stokes flows. We would like to mention a few results that are particularly relevant to the problems we consider.

Most earlier results on shear flows treated the autonomous Navier–Stokes equations. In Doering & Wang (1998), the domain of the flow is an elongated rectangle ω = (0, L) × (0, h), L ≫ h.

Abstract

Since the 1970s the use of statistical solutions of the Navier–Stokes equations has led to a number of rigorous results for turbulent flows. This paper reviews the concept of a statistical solution, its role in the mathematical foundation of the theory of turbulence, some of its successes, and the theoretical and applied challenges that still remain. The theory is illustrated in detail for the particular case of a two-dimensional flow driven by a uniform pressure gradient.

Introduction

It is believed that turbulent fluid motions are well modelled by the Navier–Stokes equations. However, due to the complicated nature of these equations, most of our understanding of turbulence relies to a great extent on laboratory experiments and on heuristic and phenomenological arguments. Nevertheless, a number of rigorous mathematical results have been obtained directly from the Navier–Stokes equations, particularly in the last two decades.

Of great interest in turbulence theory are mean quantities, which are in general well behaved, in contrast to the corresponding instantaneous values, which tend to vary quite dramatically in time. The treatment of mean values, however, is a delicate problem, as remarked by Monin & Yaglom (1975). In practice time and space averages are the most generally used, while in theory averages with respect to a large ensemble of flows avoid some analytical difficulties and have a more universal character.

This volume is the result of a workshop, “Partial Differential Equations and Fluid Mechanics”, which took place in the Mathematics Institute at the University of Warwick, May 21st–23rd, 2007.

Several of the speakers agreed to write review papers related to their contributions to the workshop, while others have written more traditional research papers. All the papers have been carefully edited in the interests of clarity and consistency, and the research papers have been externally refereed. We are very grateful to the referees for their work. We believe that this volume therefore provides an accessible summary of a wide range of active research topics, along with some exciting new results, and we hope that it will prove a useful resource for both graduate students new to the area and to more established researchers.

We would like to express their gratitude to the following sponsors of the workshop: the London Mathematical Society, the Royal Society, via a University Research Fellowship awarded to James Robinson, the North American Fund and Research Development Fund schemes of Warwick University, and the Warwick Mathematics Department (via MIR@W). JCR is currently supported by the EPSRC, grant EP/G007470/1.

Finally it is a pleasure to thank Yvonne Collins and Hazel Higgens from the Warwick Mathematics Research Centre for their assistance during the organization of the workshop.

Abstract

This paper gives a brief summary of some of the main results concerning the regularity of solutions of the three-dimensional Navier–Stokes equations. We then outline the basis of a numerical algorithm that, at least in theory, can verify regularity for all initial conditions in any bounded subset of H1.

Introduction

The aim of this paper is to present some partial results concerning the problem of regularity of global solutions of the three-dimensional Navier–Stokes equations. Since these equations form the fundamental model of hydrodynamics it is a matter of great importance whether or not they can be uniquely solved. However, one hundred and fifty years after the Navier–Stokes model was presented for the first time, we still lack an existence and uniqueness theorem, and the most significant contributions to the subject remain those of Leray (1934) and Hopf (1951).

Nevertheless, there have been many advances since their work, and it would not be possible to give an exhaustive presentation of these in a short article. We give a brief overview of some of the main results, and then concentrate on one specific and in some ways non-standard approach to the problem, with a discussion of the feasibility of testing for regularity via numerical computations following Chernysehnko et al. (2007), Dashti & Robinson (2008), and Robinson & Sadowski (2008). In some ways this contribution can be viewed as a companion to the introductory review by Robinson (2006).

This book provides a self-contained mathematical exposition of the theory of monochromatic wave propagation in layered viscoelastic media. It provides analytic solutions and numerical results for fundamental wave-propagation problems in arbitrary linear viscoelastic media not published previously in a book. As a text book with numerical examples and problem sets, it provides the opportunity to teach the theory of monochromatic wave propagation as usually taught for elastic media in the broader context of wave propagation in any media with a linear response without undue complications in the mathematics. Formulations of the expressions for the waves and the constitutive relation for the media afford considerable generality and simplification in the mathematics required to derive analytic solutions valid for any viscoelastic solid including an elastic medium. The book is intended for the beginning student of wave propagation with prerequisites being knowledge of differential equations and complex variables.

As a reference text, this book provides the theory of monochromatic wave propagation in more than one dimension developed in the last three to four decades. As such, it provides a compendium of recent advances that show that physical characteristics of two- and three-dimensional anelastic body and surface waves are not predictable from the theory for one-dimensional waves. It provides the basis for the derivation of results beyond the scope of the present text book. The theory is of interest in the broad field of solid mechanics and of special interest in seismology, engineering, exploration geophysics, and acoustics for consideration of wave propagation in layered media with arbitrary amounts of intrinsic absorption, ranging from low-loss models of the deep Earth to moderate-loss models for soils and weathered rock.

A theoretical closed-form solution for the problem of a general P, SI, or SII wave incident on a plane welded boundary between HILV media, V and V′, is one for which the characteristics of the reflected and refracted waves are expressed in terms of the assumed characteristics of the incident wave. Application of the boundary conditions at the boundary allows the amplitude and phase for the reflected and refracted waves to be expressed in terms of the properties of the media and those given for the incident wave. The directions of the propagation and attenuation vectors for the reflected and refracted waves are determined in terms of those of the incident wave by showing that the complex wave number for each solution must be the same. For problems involving incident P and SI waves, the boundary conditions are most readily applied using the solutions involving displacement potentials, namely, (4.2.1), (4.2.2), (4.2.16), and (4.2.17). For problems involving incident SII waves, the boundary conditions can be applied most easily using solutions involving only one component of the displacement field, namely (4.2.26) and (4.2.27).

Boundary-Condition Equations for General Waves

The welded boundary between media V and V′ is specified mathematically by requiring that the stress and displacement are continuous across the boundary. For purposes of brevity, application of these boundary conditions to the general solutions specifying each type of wave as incident, reflected, or refracted allows a general set of equations to be derived from which a particular problem of interest can be solved by choosing the incident wave of interest.