At the hydrodynamic level we are interested in the macroscopic evolution of densities of conserved extensive variables such as mass, energy and momentum. Because these quantities are conserved, their respective densities can only change by a process of redistribution. As we shall see, this means that the relaxation of these densities is slow, and therefore the relaxation plays a macroscopic role. If this relaxation were fast (i.e. if it occurred on a molecular time scale for instance) it would be unobservable at a macroscopic level. The macroscopic equations of motion for the densities of conserved quantities are called the Navier-Stokes equations. We will now give a brief description of how these equations are derived. It is important to understand this derivation because one of the objects of statistical mechanics is to provide a microscopic or molecular justification for the Navier-Stokes equations. In the process, statistical mechanics sheds light on the limits of applicability of these equations. Similar treatments can be found in de Groot and Mazur (1962) and Kreuzer (1981).
Let M(t) be the total mass contained in an arbitrary volume V, then
(2.1)
where ρ(r,t) is the mass density at position r and time t. Since mass is conserved, the only way that the mass in the volume V can change is by flowing through the enclosing surface, S (see Figure 2.1).
(2.2)
Here u(r,t) is the fluid streaming velocity at position r and time t. dS denotes an area element of the enclosing surface S, and ∇ is the spatial gradient vector operator, (∂/∂x,∂/∂y,∂/∂z). It is clear that the rate of change of the enclosed mass can also be written in terms of the change in mass density ρ(r,t), as
(2.3)
Figure 2.1. The change in the mass contained in an arbitrary closed volume V can be calculated by integrating the mass flux through the enclosing surface S.
If we equate these two expressions for the rate of change of the total mass we find that since the volume V was arbitrary,
(2.4)
This is called the mass continuity equation and is essentially a statement that mass is conserved. We can write the mass continuity equation in an alternative form if we use the relation between the total or streaming derivative, and the various partial derivatives. For an arbitrary function of position r and time t, for example a(r,t), we have
(2.5)
If we let a(r,t) ≡ ρ(r,t) in equation (2.5), and combine this with equation (2.4) then the mass continuity equation can be written as
(2.6)
In an entirely analogous fashion we can derive an equation of continuity for momentum. Let G(t) be the total momentum of the arbitrary volume V, then the rate of change of momentum is given by
(2.7)
The total momentum of volume V can change in two ways. Firstly it can change by convection. Momentum can flow through the enclosing surface. This convective term can be written as,
(2.8)
The second way that the momentum could change is by the pressure exerted on V by the surrounding fluid. We call this contribution the stress contribution. The force dF, exerted by the fluid across an elementary area dS, which is moving with the streaming velocity of the fluid, must be proportional to the magnitude of the area dS. The most general such linear relation is,
(2.9)
This is in fact the definition of the pressure tensor P. It is also the negative of the stress tensor. That the pressure tensor is a second rank tensor rather than a simple scalar, is a reflection of the fact that the force dF, and the area vector dS, need not be parallel. In fact for molecular fluids the pressure tensor is not symmetric in general.
As P is the first tensorial quantity that we have introduced it is appropriate to define the notational conventions that we will use. P is a second rank tensor and thus requires two subscripts to specify the element. In Einstein notation equation (2.9) reads dFα = -dSβPβα, where the repeated index β implies a summation. Notice that the contraction (or dot product) involves the first index of P and that the vector character of the force dF is determined by the second index of P. We will use bold san serif characters to denote tensors of rank two or greater. Figure 2.2 gives a diagrammatic representation of the tensorial relations in the definition of the pressure tensor.
Using this definition the stress contribution to the momentum change can be seen to be,
(2.10)
Combining (2.8, 2.10) and using the divergence theorem to convert surface integrals to volume integrals gives,
(2.11)
Since this equation is true for arbitrary V we conclude that,
(2.12)
This is one form of the momentum continuity equation. A simpler form can be obtained using streaming derivatives of the velocity rather than partial derivatives. Using the chain rule the left hand side of (2.12) can be expanded as,
(2.13)
Using the vector identity
and the mass continuity equation (2.4), equation (2.13) becomes
(2.14)
Now,
(2.15)
so that (2.14) can be written as,
(2.16)
The final conservation equation we will derive is the energy equation. If we denote the total energy per unit mass or the specific total energy as e(r,t), then the total energy density is ρ(r,t) e(r,t). If the fluid is convecting there is obviously a simple convective kinetic energy component in e(r,t). If this is removed from the energy density then what remains should be a thermodynamic internal energy density, ρ(r,t) U(r,t).
(2.17)
Here we have identified the first term on the right hand side as the convective kinetic energy. Using (2.16) we can show that,
(2.18)
The second equality is a consequence of the momentum conservation equation (2.16). In this equation we use the dyadic product of two first rank tensors (or ordinary vectors) u and ∇ to obtain a second rank tensor u∇. In Einstein notation (u∇)αβ ≡ u α ∇β . In the first form given in equation (2.18) ∇ is contracted into the first index of P and then u is contracted into the second remaining index. This defines the meaning of the double contraction notation after the second equals sign in equation (2.18) - inner indices are contracted first, then outer indices - that is u∇:P ≡ (u∇)αβ Pβα ≡ u α ∇ β P βα.
For any variable a, using equation (2.5) we have
(2.19)
Using the mass continuity equation (2.4)
(2.20)
If we let the total energy inside a volume V be E, then clearly,
(2.21)
Because the energy is conserved we can make a detailed account of the energy balance in the volume V. The energy can simply convect through the containing surface, it could diffuse through the surface and the surface stresses could do work on the volume V. In order these terms can be written,
(2.22)
In equation (2.22) J Q, is called the heat flux vector. It gives the energy flux across a surface which is moving with the local fluid streaming velocity. Using the divergence theorem, (2.22)can be written as,
(2.23)
Comparing equations (2.21) and (2.23) we derive the continuity equation for total energy,
(2.24)
We can use (2.20) to express this equation in terms of streaming derivatives of the total specific energy
(2.25)
Finally equations (2.17) and (2.18) can be used to derive a continuity equation for the specific internal energy
(2.26)
where the superscript T denotes transpose. The transpose of the pressure tensor appears as a result of our double contraction notation because in equation (2.25) ∇ is contracted into the first index of P.
The three continuity equations (2.6), (2.16) and (2.26) are continuum expressions of the fact that mass, momentum and energy are conserved. These equations are exact.