Thus far our description of the equations of hydrodynamics has been exact. We will now derive an equation for the rate at which entropy is produced spontaneously in a nonequilibrium system. The second law of thermodynamics states that entropy is not a conserved quantity. In order to complete this derivation we must assume that we can apply the laws of equilibrium thermodynamics, at least on a local scale, in nonequilibrium systems. This assumption is called the local thermodynamic equilibrium postulate. We expect that this postulate should be valid for systems that are sufficiently close to equilibrium (de Groot and Mazur, 1962). This macroscopic theory provides no information on how small these deviations from equilibrium should be in order for local thermodynamic equilibrium to hold. It turns out however, that the local thermodynamic equilibrium postulate is satisfied for a wide variety of systems over a wide range of conditions. One obvious condition that must be met is that the characteristic distances over which inhomogeneities in the nonequilibrium system occur must be large in terms molecular dimensions. If this is not the case then the thermodynamic state variables will change so rapidly in space that a local thermodynamic state cannot be defined. Similarly the time scale for nonequilibrium change in the system must be large compared to the time scales required for the attainment of local equilibrium.

We let the entropy per unit mass be denoted as, s(r,t) and the entropy of an arbitrary volume V, be denoted by S. Clearly,

(2.27)

In contrast to the derivations of the conservation laws we do not expect that by taking account of convection and diffusion, we can totally account for the entropy of the system. The excess change of entropy is what we are seeking to calculate. We shall call the entropy produced per unit time per unit volume, the entropy source strength, σ(r,t).

(2.28)

In this equation J_ST(r,t) is the total entropy flux. As before we use the divergence theorem and the arbitrariness of V to calculate,

(2.29)

We can decompose J_ST(r,t) into a streaming or convective term ρ(r,t)s(r,t)u(r,t) in analogy with equation (2.8), and a diffusive term J_S(r,t). Using these terms (2.29) can be written as,

(2.30)

Using (2.5) to convert to total time derivatives we have,

(2.31)

At this stage we introduce the assumption of local thermodynamic equilibrium. We postulate a local version of the Gibbs relation TdS = dU + pdV. Converting this relation to a local version with extensive quantities replaced by the specific entropy energy and volume respectively and noting that the specific volume V∕M is simply ρ(r,t)^-1, we find that,

(2.32)

We can now use the mass continuity equation to eliminate the density derivative,

(2.33)

Multiplying (2.33) by ρ(r,t) and dividing by T(r,t) gives

(2.34)

We can substitute the energy continuity expression (2.26) for dU∕dt into (2.34) giving,

(2.35)

We now have two expressions for the streaming derivative of the specific entropy, ρ(r,t) ds(r,t)/dt, equation (2.31) and (2.35). The diffusive entropy flux J _S(r,t), using the time derivative of the local equilibrium postulate dQ = Tds, is equal to the heat flux divided by the absolute temperature and therefore,

(2.36)

Equating (2.31) and (2.35) using (2.36) gives,

(2.37)

We define the viscous pressure tensor Π⁽²⁾, as the nonequilibrium part of the pressure tensor.

(2.38)

Using this definition the entropy source strength can be written as,

(2.39)

A second postulate of nonlinear irreversible thermodynamics is that the entropy source strength always takes the canonical form (de Groot and Mazur, 1962),

(2.40)

This canonical form defines what are known as thermodynamic fluxes, J_i, and their conjugate thermodynamic forces, X_i. We can see immediately that our equation (2.39) takes this canonical form provided we make the identifications that: the thermodynamic fluxes are the various Cartesian elements of the heat flux vector, J _Q(r,t), and the viscous pressure tensor, Π(r,t). The thermodynamic forces conjugate to these fluxes are the corresponding Cartesian components of the temperature gradient divided by the square of the absolute temperature, T(r,t)^-2 ∇T(r,t), and the strain rate tensor divided by the absolute temperature, T(r,t)^-1 ∇u(r,t), respectively. We use the term corresponding quite deliberately; the α^th element of the heat flux is conjugate to the α^th element of the temperature gradient. There are no cross couplings. Similarly the α,β element of the pressure viscous pressure tensor is conjugate to the α,β element of the strain rate tensor.

There is clearly some ambiguity in defining the thermodynamic fluxes and forces. There is no fundamental thermodynamic reason why we included the temperature factors, T(r,t)^-2 and T(r,t)^-1, into the forces rather than into the fluxes. Either choice is possible. Ours is simply one of convention. More importantly there is no thermodynamic way of distinguishing between the fluxes and the forces. At a macroscopic level it is simply a convention to identify the temperature gradient as a thermodynamic force rather than a flux. The canonical form for the entropy source strength and the associated postulates of irreversible thermodynamics do not permit a distinction to be made between what we should identify as fluxes and what should be identified as a force. Microscopically it is clear that the heat flux is a flux. It is the diffusive energy flow across a comoving surface. At a macroscopic level however, no such distinction can be made.

Perhaps the simplest example of this macroscopic duality is the Norton constant current electrical circuit, and the Thevénin constant voltage equivalent circuit. We can talk of the resistance of a circuit element or of a conductance. At a macroscopic level the choice is simply one of practical convenience or convention.