Consistent with our use of the local thermodynamic equilibrium postulate, which is assumed to be valid sufficiently close to equilibrium, a linear relation should hold between the conjugate thermodynamic fluxes and forces. We therefore postulate the existence of a set of linear phenomenological transport coefficients {L_ij} which relate the set forces {X_j} to the set of fluxes {J_i}. We use the term phenomenological to indicate that these transport coefficients are to be defined within the framework of linear irreversible thermodynamics and as we shall see there may be slight differences between the phenomenological transport coefficients L_ij and practical transport coefficients such as the viscosity coefficients or the usual thermal conductivity.

We postulate that all the thermodynamic forces appearing in the equation for the entropy source strength (2.40), are related to the various fluxes by a linear equation of the form

(2.41)

This equation could be thought of as arising from a Taylor series expansion of the fluxes in terms of the forces. Such a Taylor series will only exist if the flux is an analytic function of the force at X=0.

(2.42)

Clearly the first term is zero as the fluxes vanish when the thermodynamic forces are zero. The term which is linear in the forces is evidently derivable, at least formally, from the equilibrium properties of the system as the functional derivative of the fluxes with respect to the forces computed at equilibrium, X=0. The quadratic term is related to what are known as the nonlinear Burnett coefficients (see §9.5). They represent nonlinear contributions to the linear theory of irreversible thermodynamics.

If we substitute the linear phenomenological relations into the equation for the entropy source strength (2.40), we find that,

(2.43)

A postulate of linear irreversible thermodynamics is that the entropy source strength is always positive. There is always an increase in the entropy of a system so the transport coefficients are positive. Since this is also true for the mirror image of any system, we conclude that the entropy source strength is a positive polar scalar quantity. (A polar scalar is invariant under a mirror inversion of the coordinate axes. A pseudo scalar, on the other hand, changes its sign under a mirror inversion. The same distinction between polar and scalar quantities also applies to vectors and tensors.)

Suppose that we are studying the transport processes taking place in a fluid. In the absence of any external non-dissipative fields (such as gravitational or magnetic fields), the fluid is at equilibrium and assumed to be isotropic. Clearly since the linear transport coefficients can be formally calculated as a zero-field functional derivative they should have the symmetry characteristic of an isotropic system. Furthermore they should be invariant under a mirror reflection of the coordinate axes.

Suppose that all the fluxes and forces are scalars. The most general linear relation between the forces and fluxes is given by equation (2.41). Since the transport coefficients must be polar scalars there cannot be any coupling between a pseudo scalar flux and a polar force or between a polar flux and a pseudo scalar force. This is a simple application of the quotient rule in tensor analysis. Scalars of like parity only, can be coupled by the transport matrix L_ij.

If the forces and fluxes are vectors, the most general linear relation between the forces and fluxes which is consistent with isotropy is,

(2.44)

In this equation L _ij is a second rank polar tensor because the transport coefficients must be invariant under mirror inversion just like the equilibrium system itself. If the equilibrium system is isotropic then L _ij must be expressible as a scalar L_ij times the only isotropic second rank tensor I, (the Kronecker delta tensor I = δ_αβ ). The thermodynamic forces and fluxes which couple together must either all be pseudo vectors or polar vectors. Otherwise since the transport coefficients are polar quantities, the entropy source strength could be pseudo scalar. By comparing the trace of L _ij with the trace of L_ij I, we see that the polar scalar transport coefficients are given as,

(2.45)

If the thermodynamic forces and fluxes are all symmetric traceless second rank tensors J _i, X _i, where J _i = ¹/₂ (J _i + J _i ^T) - ¹/₃Tr (J _i) I, (we denote symmetric traceless tensors as outline sans serif characters), then

(2.46)

is the most linear general linear relation between the forces and fluxes. L _ij ⁽⁴⁾ is a symmetric fourth rank transport tensor. Unlike second rank tensors there are three linearly independent isotropic fourth rank polar tensors. (There are no isotropic pseudo tensors of the fourth rank.) These tensors can be related to the Kronecker delta tensor, and we depict these tensors by the forms,

(2.47a)

(2.47b)

(2.47c)

Since L _ij ⁽⁴⁾ is an isotropic tensor it must be representable as a linear combination of isotropic fourth rank tensors. It is convenient to write,

(2.48)

It is easy to show that for any second rank tensor A,

(2.49)

where A is the symmetric traceless part of A ⁽²⁾, A = ¹/₂(A - A ^T) is the antisymmetric part of A ⁽²⁾ (we denote antisymmetric tensors as shadowed sans serif characters), and A = ¹/₃Tr(A). This means that the three isotropic fourth rank tensors decouple the linear force flux relations into three separate sets of equations which relate respectively, the symmetric second rank forces and fluxes, the antisymmetric second rank forces and fluxes, and the traces of the forces and fluxes. These equations can be written as

(2.50a)

(2.50b)

(2.50c)

where J _i is the antisymmetric part of J, and J = ¹/₃Tr(J). As J _i has only three independent elements it turns out that J _i can be related to a pseudo vector. This relationship is conveniently expressed in terms of the Levi-Civita isotropic third rank tensor ε ⁽³⁾. (Note: ε αβ γ = +1 if αβ γ is an even permutation, -1 if αβ γ is an odd permutation and is zero otherwise.) If we denote the pseudo vector dual of J _i as J _i ^ps then,

(2.51)

This means that the second equation in the set (2.50b) can be rewritten as,

(2.52)

Looking at (2.50) and (2.52) we see that we have decomposed the 81 elements of the (3-dimensional) fourth rank transport tensor L _ij ⁽⁴⁾, into three scalar quantities, L^s _ij, L^a _ij and L^tr _ij. Furthermore we have found that there are three irreducible sets of forces and fluxes. Couplings only exist within the sets. There are no couplings of forces of one set with fluxes of another set. The sets naturally represent the symmetric traceless parts, the antisymmetric part and the trace of the second rank tensors. The three irreducible components can be identified with irreducible second rank polar tensor component an irreducible pseudo vector and an irreducible polar scalar. Curie's principle states that linear transport couples can only occur between irreducible tensors of the same rank and parity.

If we return to our basic equation for the entropy source strength (2.40) we see that our irreducible decomposition of Cartesian tensors allows us to make the following decomposition for second rank fields and fluxes,

(2.53)

The conjugate forces and fluxes appearing in the entropy source equation separate into irreducible sets. This is easily seen when we realise that all cross couplings between irreducible tensors of different rank vanish; I : _i = I : = _i : = 0, etc. Conjugate thermodynamic forces and fluxes must have the same irreducible rank and parity.

We can now apply Curie's principle to the entropy source equation (2.39),

(2.54)

In writing this equation we have used the fact that the transpose of P is equal to P, and we have used equation (2.51) and the definition of the cross product ∇xu = - ε ⁽³⁾ : ∇u to transform the antisymmetric part of P ^T. Note that the transpose of P is equal to - P. There is no conjugacy between the vector J _Q(r,t) and the pseudo vector ∇xu(r,t) because they differ in parity. It can be easily shown that for atomic fluids the antisymmetric part of the pressure tensor is zero so that the terms in (2.54) involving the vorticity ∇xu(r,t) are identically zero. For molecular fluids, terms involving the vorticity do appear but we also have to consider another conservation equation - the conservation of angular momentum. In our description of the conservation equations we have ignored angular momentum conservation. The complete description of the hydrodynamics of molecular fluids must include this additional conservation law.

For single component atomic fluids we can now use Curie's principle to define the phenomenological transport coefficients.

(2.55a)

(2.55b)

(2.55c)

The positive sign of the entropy production implies that each of the phenomenological transport coefficients must be positive. As mentioned before these phenomenological definitions differ slightly from the usual definitions of the Navier-Stokes transport coefficients.

(2.56a)

(2.56b)

(2.56c)

These equations were postulated long before the development of linear irreversible thermodynamics. The first equation is known as Fourier's law of heat conduction. It gives the definition of the thermal conductivity λ. The second equation is known as Newton's law of viscosity (illustrated in Figure 2.3). It gives a definition of the shear viscosity coefficient η. The third equation is a more recent development. It defines the bulk viscosity coefficient η_V. These equations are known collectively as linear constitutive equations. When they are substituted into the conservation equations they yield the Navier-Stokes equations of hydrodynamics. The conservation equations relate thermodynamic fluxes and forces. They form a system of equations in two unknown fields - the force fields and the flux fields. The constitutive equations relate the forces and the fluxes. By combining the two systems of equations we can derive the Navier-Stokes equations which in their usual form give us a closed system of equations for the thermodynamic forces. Once the boundary conditions are supplied the Navier-Stokes equations can be solved to give a complete macroscopic description of the nonequilibrium flows expected in a fluid close to equilibrium in the sense required by linear irreversible thermodynamics. It is worth restating the expected conditions for the linearity to be observed:

After some tedious but quite straightforward algebra (de Groot and Mazur, 1962), the Navier-Stokes equations for a single component atomic fluid are obtained. The first of these is simply the mass conservation equation (2.4).

(2.57)

To obtain the second equation we combine equation (2.16) with the definition of the stress tensor from equation (2.12) which gives

(2.58)

We have assumed that the fluid is atomic and the pressure tensor contains no antisymmetric part. Substituting in the constitutive relations, equations (2.56b) and (2.56c) gives

(2.59)

Here we explicitly assume that the transport coefficients η_V and η are simple constants, independent of position r, time and flow rate u. The αβ component of the symmetric traceless tensor ∇u is given by

(2.60)

where as usual the repeated index γ implies a summation with respect to γ. It is then straightforward to see that

(2.61)

and it follows that the momentum flow Navier-Stokes equation is

(2.62)

The Navier-Stokes equation for energy flow can be obtained from equation (2.26) and the constitutive relations, equation (2.56). Again we assume that the pressure tensor is symmetric, and the second term on the right hand side of equation (2.26) becomes

(2.63)

It is then straightforward to see that

(2.64)