Introduction
I.1. General points
The concept of a discrete body sets aside the notion of a continuum to establish the conservation laws of Mechanics. All of the theorems and mathematical properties are applied directly to objects of finite dimensions. Discrete Mechanics essentially takes elementary results from differential geometry to establish the laws of conservation along an edge. The Fundamental Law of Dynamics is the starting point for establishing the conservation of momentum equation.
A certain number of the concepts used in Continuum Mechanics will be abandoned; thus, the very notion of a continuum is not necessary in order to obtain discrete equations. Similarly, the hypothesis of Local Thermodynamic Equilibrium is set aside, because within the elementary volume, it is not necessary for a state equation to be satisfied. The concept of a tensor is replaced by the concepts of differential geometry and elementary operators, gradient, rotational and divergence, using which we are able to establish the link between single-point values, oriented vectors, oriented surfaces and volumes, and vice versa. Similarly, the projection onto an orthonormal axis system is not necessary to establish the conservation equations.
Although this formalism gives us a momentum balance equation which is different to the Navier–Stokes equation, their application to simple flows yields identical results. The Hodge–Helmholtz decomposition will play an essential role in showing how the momentum balance equation can be employed to separate any of the terms in this equation into a solenoidal part and an irrotational part.
In Fluid Mechanics, the pressure in the momentum balance equation cannot be reduced to the role of mechanical pressure. This equation represents an instantaneous equilibrium and the thermodynamic pressure, which plays the role of a stress accumulator, depends not only on properties such as the temperature or density, but also on the heat flux and the velocity. If we limit our discussion to mechanical and thermal effects, the role of the divergences of the flux and velocity in reading the thermodynamic pressure will become apparent.
The elements deriving from differential geometry, differential algebra, exterior calculus, and so on, will be omitted in the interests of a presentation which is as simple as possible, using the classic theorems such as those of Ostrogradski and Stokes, etc., and the properties of the standard differential operators.
I.2. Introduction
The laws of conservation in mechanics were established over two centuries ago, and have evolved very little since. The important contributions made by C. Truesdell and W. Noll [TRU 74, TRU 92] to integrate the laws of thermodynamics and the constitutive laws into the conservation of momentum- and energy laws, however, led to the establishment of the Navier–Stokes equations, which offer a very accurate representation of the physical reality of the phenomena being observed. Many more important contributions have helped construct the corpus of equations in Continuum Mechanics as it is taught today [LAN 59, BAT 67, SAL 02, GER 95, COI 07, GUY 91].
However, there are a certain number of difficulties inherent to the continuum theory which need to be taken into consideration:
– the concept of a continuum itself poses a significant problem: the reduction of the elementary volume to a single point, in order to define scalars, vectors and tensors, does away with any reference to the direction and orientation. In order to restore these concepts, it is necessary to place the domain in a frame of reference – e.g. to define a point velocity on the basis of its components;
– the introduction of the Cauchy tensor to express the local stress T = σ · n, for an isotropic fluid, brings into play two viscosity coefficients, μ and λ, which are interlinked by Stokes’ law 3λ + 2μ ≥ 0; we shall come back to this point in greater detail later on. The value of λ is very difficult to measure for fluids in general, and varies greatly depending on the authors and the measurement methods used. This law is not valid, in gneral [GAD 95]. It should also be noted, though, that in solids, the existence of this coefficient does not pose a problem;
– the concept of a tensor first appeared in the late 19th Century, and was further developed, in the context of Continuum Mechanics, before being used in other areas of physics. The absolute necessity of using tensors in the field of mechanics to describe the relations between the stresses and strains can, quite legitimately, be disputed. In fact, it was the simplistic interpretation of certain experiments in fluids and solids that guided this choice, which has remained the same ever since. The components of the Cauchy stress tensor have only been able to be reduced thanks to the principle of material frame indifference for an isotropic medium [TRU 74, SAL 02, GER 95, COI 07]. In spite of these reductions, the remaining coefficients are only linked by an inequality, which is confirmed by a thermodynamic approach;
– the formal link between the conservation equations and the Hodge–Helmholtz decomposition has not been established. Whilst Helmholtz’s theorem ensures that any vectorial field can be decomposed into an irrotational part and a solenoidal part in
for a decreasing field at infinity, its application is limited to the vectorial fields themselves, such as the velocity, for instance, which can be decomposed into two terms: the scalar and vectorial potentials;
– the level of modeling of the effects of pressure and those of viscosity is not the same in the Navier–Stokes equation [SAL 02]. Whilst particular attention has been paid to viscous effects, enabling us to describe the transfers of momentum within a fluid, the effects of compression or decompression are only taken into account by means of a scalar – the pressure – without any first-order link being established between the pressure and the velocity. In order to make this connection, we have to use other conservation laws: those relating to the conservation of mass and energy;
– for a long time, thermodynamics has had an important role to play, which Trusdell [TRU 74] integrated into the equations of mechanics during the last century. It conferred the status of a law on the relation between the different measurable values, such as the density, pressure and temperature, for example. The structural coupling links between the momentum equation, the conservation of mass equation and the state law lead to confusion as to the role played by each of these relations. For example, the conservation of mass must serve to set the density as a function of the external actions, but not to calculate the pressure. In addition, there is no condition sine qua non which means that the state law has to be satisfied at all points and at all times;
– on its own, the conservation of energy carries the notion of flux and of energy, and the conservation of flux is completely absent from the classic formalism used in Continuum Mechanics. Although the conservation of momentum equation is associated with the conservation of mass, the heat flux can easily be introduced into this relation by a simplistic law forming the link between the flux and the temperature gradient: Fourier’s law. This is considered an experimental, phenomenological law, serving to bring closure to the system of equations;
– the boundary conditions between two immiscible fluids or on the edge of a domain are written on the basis of the stresses defined by the Cauchy tensor, and are difficult to apply in practical terms; they need to be supplemented by compatibility relations – for instance in the case of shockwaves. Also, they are strongly imposed – for example, for a fluid flow entering into a domain, we impose the normal velocity, thereby violating the equilibrium conditions described by the various terms in the conservation equation.
The continuum formalism is essentially linked to the relations between stresses and strains (also known as deformations), which are represented by a stress tensor, of varying degrees of complexity. The most complete tensors, such as the Green–Lagrange tensor, enable us to take account of significant deformations, whilst certain tensors found by linearization, such as the Cauchy tensor, are limited to small transformations. The displacement field gradient thus introduces the notion of a tensor which can be decomposed into symmetric and antisymmetric parts. The purpose of this operation is to filter out the rigid translational motion which does not give rise to any force within the material.
The issue that we tackle in the area of Discrete Mechanics is based on the laying aside of the idea of a continuum, where all the scalar and vectorial variables are defined at a single point. In a discrete medium, the scalar values are associated with a point, whereas the vectorial values are defined on an edge which can be as short as it needs to be, provided its direction is preserved. To begin with, we shall work in the context of small displacements, and in order to counter the disadvantages that come with that approach, we shall introduce the principle of accumulation, whereby each equilbrium state is conserved so as to represent the evolution of the physical system.
There are a certain number of principles which seem indispensable in order to model all the mechanical effects:
– the medium is at equilibrium in space if it is not subject to any force, body or contact;
– the principle of action and reaction must be borne in mind, although the surface stress does not have the same meaning here as it does in Continuum Mechanics;
– the rigidifying overall translational and rotational motions may lead to the inadequacy of the formulation where rotation plays an important role. We shall suppose that the rotation rate is zero at infinity, which is one of the use conditions for the Stokes theorem. The only essential condition is that any rigid or rigidifying translational or rotational motion must not affect the acceleration;
– the dissipation of the mechanical effects, wave propagation and viscous effects must be positive.
The theory taking shape in this book is founded upon the fundamental law of dynamics and on some elementary experiments, using these as bases upon which to construct coherent and balanced models of the observed effects – in particular the diffusion of momentum and the propagation of waves. It also draws on certain elements which were established in a previous publication by this author on the subject [CAL 01] to specify and supplement the derivation of the scalar and vectorial conservation equations without using tensors whose order is equal to or higher than two.
Next, we go on to discuss the general properties of these equations. In particular, the differences between the Navier–Stokes equation and the momentum balance equation stemming from this theory will be illustrated; also, the two forms of the dissipation term will be compared.