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 work is intended to supplement and complement standard texts on continuum mechanics by drawing attention to physical assumptions implicit in continuum modelling. Particular attention is paid to linking continuum concepts, fields, and relations with underlying molecular behaviour via local averaging in both space and time. The aim is to clarify physical interpretations of concepts and fields and in so doing provide a sound basis for future studies. The contents should be of interest to engineers, mathematicians, and physicists who study macroscopic material behaviour.

The contents are the result of a long-standing study of formal and axiomatic presentations of continuum mechanics. Some of the issues were first addressed in courses delivered under the auspices of CISM (Udine, 1986, 1987), University of Cairo (1994, 1996), and AMAS (Warsaw, 2002; Bydgoszcz, 2003), and other topics treated in published papers. Here the opportunity has been taken to elaborate upon and extend earlier works and to present a unified, more readily accessible treatment of the subject matter.

Given the differing backgrounds of the intended readership, two extensive appendices have been included which develop relevant mathematical concepts and results. In particular, the use of direct (i.e., co-ordinate-free) notation is explained and related to that of Cartesian tensors.

No work exists in isolation: the author is above all indebted to his teachers Mort Gurtin and Walter Noll who introduced him to the mathematical precision and clarity of exposition to be found in modern continuum mechanics.

Preamble

Here geometric and analytical pre-requisites for continuum modelling are developed and linked to the algebraic considerations of Appendix A. This material has been included in order to emphasise the direct (i.e., co-ordinate-free) approach employed, which may not be familiar to the reader. The aim has been to provide a reasonably self-contained basis for understanding the notation and methodology used in the main body of the work.

Continuum modelling of material behaviour requires, among other things, mathematical prescriptions of

1. (i) the location and distribution of matter for the physical system (or body) of interest at a given time,

2. (ii) changes in location of a body and any associated distortion,

3. (iii) spatial and temporal variation of local system descriptors (e.g., mass density or velocity), and

4. (iv) physical descriptors such as mass, momentum, and kinetic energy, which are additive over disjoint regions. (Such descriptors are termed extensive.)

Central to such prescriptions are the notions of point and space, here formalised in terms of Euclidean space ℰ. Distortion is described via one-to-one mappings of points into points (deformations), and local spatial variation of descriptors is treated in terms of generalisations of the derivative of a function of a real variable. Analysis of point (iv) involves relating values of extensive descriptors associated with finite regions to their local densities, and is accomplished via volume integration.

Preamble

Here we are concerned with fluid flow through a body which is accordingly ‘porous’ in some sense. In order that such flow be possible it is necessary that

1. (i) there is vacant space available ‘within’ the body to accommodate fluid, and

2. (ii) the space in which fluid can reside must be connected in order that the fluid can move through the body.

Vacant space within a body is termed pore space, a measure of which is porosity. Of course, not all pore space may be accessible to fluid: there may be isolated space inaccessible to fluid penetration. Such penetration, associated with connectivity, gives rise to the notion of permeability. Consider an insect attempting to crawl through a rectangular block of porous material from the centre of one face to exit through another particular face. This may or may not be possible. It could be that no connected route between the point of entry and the destination face exists, or that the insect is unable to squeeze through available ‘gaps’ en route. The former snag indicates that permeability is in some sense direction-dependent, while the latter draws attention to the scale dependence of both porosity and permeability.

Before addressing technicalities, it is worthwhile to note that the effects of porosity are crucial to our very existence: semipermeable membranes help to govern vital processes throughout our bodies, and the porous nature of bone provides structural strength without undue mass. Further, plant life and soil properties depend in part on relevant porosities, while the presence of subterranean water sources (aquifers) and oil-bearing shale derives from porosity within the surface of the Earth. More mundanely, we utilise sponges for cleaning and filtration systems in our water supplies but can be inconvenienced by dampness in the fabric of buildings and swelling of kitchen worktops due to water ingress. The foregoing serves to illustrate the diversity of porous system effects and the range of associated length scales.