Vorticity fields that are not overly damped develop extremely complex spatial structures exhibiting a wide range of scales. These structures wax and wane in coherence; some are intense and most of them weak; and they interact nonlinearly. Their evolution is strongly influenced by the presence of boundaries, shear, rotation, stratification and magnetic fields. We label the multitude of phenomena associated with these fields as turbulence and the challenge of predicting the statistical behaviour of such flows has engaged some of the finest minds in twentieth century science.

The progress has been famously slow. This slowness is in part because of the bewildering variety of turbulent flows, from the ideal laboratory creations on a small scale to heterogeneous flows on the dazzling scale of cosmos. Philip Saffman (Structure and Mechanisms of Turbulence II, Lecture Notes in Physics 76, Springer, 1978, p. 273) commented: “… we should not altogether neglect the possibility that there is no such thing as ‘turbulence’. That is to say, it is not meaningful to talk about the properties of a turbulent flow independently of the physical situation in which it arises. In searching for a theory of turbulence, perhaps we are looking for a chimera … Perhaps there is no ‘real turbulence problem’, but a large number of turbulent flows and our problem is the self-imposed and possibly impossible task of fitting many phenomena into the Procrustean bed of a universal turbulence theory.”

Introduction

Superfluids can flow without friction and display two-fluid phenomena. These two properties, which have quantum mechanical origins, lie outside common experience with classical fluids. The subject of superfluids has thus generally been relegated to the backwaters of mainstream fluid dynamics. The focus of low-temperature physicists has been the microscopic structure of superfluids, which does not naturally invite the attention of experts on classical fluids. However, perhaps amazingly, there exists a state of superfluid flow that is similar to classical turbulence, qualitatively and quantitatively, in which superfluids are endowed with quasiclassical properties such as effective friction and finite heat conductivity. This state is called superfluid or quantum turbulence (QT) [Feynman (1955); Vinen & Niemela (2002); Skrbek (2004); Skrbek & Sreenivasan (2012)]. Although QT differs from classical turbulence in several important respects, many of its properties can often be understood in terms of the existing phenomenology of its classical counterpart. We can also learn new physics about classical turbulence by studying QT. Our goal in this article is to explore this interrelation. Instead of expanding the scope of the article broadly and compromising on details, we will focus on one important aspect: the physics that is common between decaying vortex line density in QT and the decay of three-dimensional (3D) turbulence that is nearly homogeneous and isotropic turbulence (HIT), which has been a cornerstone of many theoretical and modeling advances in hydrodynamic turbulence. A more comprehensive discussion can be found in a recent review by Skrbek & Sreenivasan (2012).

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, June 15th–19th, 2010.

Several of the speakers agreed to write review papers related to their contributions to the workshop, while others have written more traditional research papers. 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, EPSRC (via a conference grant EP/I001050/1), and the Warwick Mathematics Department. JCR is currently supported by an EPSRC Leadership Fellowship (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 The aim of this contribution is to make a connection between two recent results concerning the dynamics of vortices in incompressible planar flows. The first one is an asymptotic expansion, in the vanishing viscosity limit, of the solution of the two dimensional Navier-Stokes equation with point vortices as initial data. In such a situation it is known, see Gallay (Arch. Ration. Mech. Anal.200 (2011) 445–490), that the solution behaves to leading order like a linear superposition of Oseen vortices whose centres evolve according to the point vortex system. However, higher order corrections can also be computed and these describe the deformation of the vortex cores due to mutual interactions. The second result is the construction by Smets & Van Schaftingen (Arch. Ration. Mech. Anal.198 (2010) 869–925) of “desingularized” solutions of the two-dimensional Euler equation. These solutions are stationary in a uniformly rotating or translating frame, and converge either to a single vortex or to a vortex pair as the size parameter ∈ goes to zero. We consider here the particular case of a pair of identical vortices, and we show that the solution of the weakly viscous Navier-Stokes equation is accurately described at time t by an approximate steady state of the rotating Euler equation which is a desingularized solution in the sense of Smets & Van Schaftingen (2010) with Gaussian profile and size

Introduction

Numerical simulations of freely decaying turbulence show that vortex interactions play a crucial role in the dynamics of two-dimensional viscous flows, see McWilliams (1984, 1990).

Abstract We review recent global existence and uniqueness results of solutions for models of complex fluids in ℝd. We describe results concerning the Oldroyd-B and related models.

Introduction

Complex fluids are ubiquitous in nature and manifest a rather large number of different behaviours. There is no single accepted general model for all these, and the presence of a large array of complicated models is an indication of the difficulties encountered at a fundamental level. In this article I will describe some of the mathematical issues. A complex fluid is a mixture between a solvent, which is treated as a normal fluid, and particulate matter in it. The particles are sufficiently many and sufficiently small compared to the characteristic scales of the motions of the solvent, so that one may hope for a description that does not have to resolve the fluid mechanical problem of flow past the particles. The particles themselves are treated in a simplified manner as objects m ∈ M, where M is a finite-degrees-of-freedom configuration space accounting for the salient features of the particles. For instance M can be a subset of ℝN or a more complicated metric space. Models have been devised to deal with microscopic elastic thread-like objects such as polymers (Doi & Edwards, 1998, Öttinger, 1996). The complicated hydrodynamic interactions are simplified using the separation of scales, replacing the many degrees of freedom due to them by a few representative ones.

Abstract The motion of the position of the maximum of vorticity ∥ζ∥∞ and growth of enstrophy are explored numerically using a recent calculation at moderately high resolution. We provide an exact analytic formula for the motion of ∥ζ∥∞ in the symmetry plane and use it to validate the numerical data. This motion drifts with respect to the local Lagrangian frame of reference due to the local gradient of vortex stretching, and can thus be associated with depletion of the circulation in the symmetry plane from the vicinity of ∥ζ∥∞. Despite this depletion, the numerical data is consistent with singular growth of the 3D enstrophy and symmetry-plane enstrophy that is bounded by inequalities analogous to the type known for the three-dimensional Navier-Stokes equations, using their respective enstrophies and with the viscosity being replaced by the circulation invariant.

Introduction

After the proof that the time integral of maximum vorticity controlled any possible singularities of the three-dimensional incompressible Euler equations (Beale, Kato, & Majda, 1984), it was realized that this was a quantity of low enough order to be accessible by numerical simulations, and the quest to determine numerically whether or not the Euler equations develop singularities ensued.

However, apparent singular growth in one quantity is insufficient for claiming consistency with singular growth as it is difficult to distinguish singular power-law growth from strong, but non-singular growth such as the exponential of an exponential. At a minimum, the growth in stretching near the position of maximum vorticity needs to be tracked independently and should grow in a manner consistent with the mathematical bounds, as first suggested by Pumir & Siggia (1990).

Abstract This paper considers the feedback stabilization problem for the Navier-Stokes equations defined in a bounded domain. Control via a forcing term (both distributed and impulsive) supported in a subdomain, via the initial condition, and via boundary conditions are studied, and relationships between these different kinds of control are presented. The precise meaning of ‘feedback control’ is discussed, and a feedback map providing control via the state variable is constructed for initial and distributed control (in the latter case for the linear Oseen equations only). Numerical algorithms for the calculation of stable invariant manifolds and projection operators on these sets are discussed. Finally the results of a numerical stabilization of a particular fluid flow are presented.

Introduction

The aim of this paper is to give a relatively short presentation of mathematical and numerical results concerning stabilization of the Navier-Stokes equations by feedback control. A description of the mathematical stabilization construction will be accompanied by a discussion of how mathematical notions of stabilization theory can be adapted to calculations.

Control theory for partial differential equations has been developed very intensively over the last few decades, and is now a very wide and farreaching field, even if one excludes extremal theory for partial differential equations. For some idea of this field more broadly, see the recent books by Coron (2007) and Tucsnak & Weiss (2009), as well as the earlier survey in Fursikov & Imanuilov (1999), together with the references in these publications.

The stabilization problem for the 2D Navier-Stokes system by a feedback distributed control supported in the whole fluid domain was studied by Barbu & Sritharan (1998).