Up to this point little has been said about the physical
laws determing the flow of heat 
or the flux of
salt 
or the viscous stresses 
.
The laws governing these quantities represent additional
hypotheses, the justification of which lies in the satisfactory
comparison of deductions from these laws with experiment.
Such comparisons between theory and observation are
limited, particularly in respect to the more complex
situations which may arise where several different fluxes
exist simultaneously, and it is conceivable and even
probable that the classical ``laws'' of heat conduction,
diffusion, and momentum transfer must be altered to account
for phenomena which cannot be otherwise explained.
In short, the state of our knowledge of molecular transfer
of properties is far from complete. Nevertheless, there is
one guide in the selection of the hypothesis governing the
non-advective flux: it must be consistent with the second
law of thermodynamics as set forth in the inequality (VI-2).
In order to understand the nature of the restrictions imposed
by this law it is necessary to consider some examples. We will
start with the simplest case.
Fourier showed that the flow of heat in a solid could be explained if
where C, the coefficient of thermal conductivity, is
a property of the particular substance.
There is evidence that this law holds in a pure fluid
under certain conditions. The coefficient C may vary
with the temperature T, and certainly varies from one
substance to another, but the feature which it has in common
with all substances according to empirical evidence is that
it is always a positive constant. This of course makes
intuitive sense, at least if the molecular concept of
temperature is invoked; the heat (kinetic energy at the
microscopic scale in another sense) tends to flow from
regions of high concentration of internal energy to regions
of low internal energy, which is consistent with the
statement above that the heat flow is directed in the
direction of the gradient of the temperature
.
This is consistent with the second law of thermodynamics.
To show this we must make use of the first law in the
form (VI-1) which, when compared with (V-2),
indicates that the terms on the right hand side of (VI-1)
must at least be zero. Now, since the above law for
is independent of velocity gradients or
gradients of concentration such as would exist in a fluid
mixture, it is evidence that the last term
must be at least zero regardless of what value the other terms may have, for these can be zero independently (and would be in a solid under the conditions of Fourier's experiments). Therefor, making use of Eq. (5.01.1), it is required that
which in general requires that 
, and therefore
is consistent with experiment.