# On the Thermodynamic Theory of the Second Viscosity

J. Meixner

## Abstract

It seems possible to reconcile the Navier-Stokes equations and the thermodynamic theory of relaxation, but a full insight into the problem of the two viscosities may be obtained by considering the liquid as a limiting case of an elastic body with 'after-effects'. The after-effect theory, when applied to a generalized stress-strain relation that contains temperature and entropy besides the components of the strain and stress tensors, leads to interesting results; for instance, it can be proved that the compression and shear moduli, divided by p, are positive functions in p (=i$\omega$, where $\omega$ is the frequency), in the sense of the electric network terminology. In such materials one can define an apparent viscosity matrix; its essential components in the special case of isotropic visco-elastic substances are the first and the second viscosities, which are positive functions too. It is not possible to represent an arbitrary after-effect behaviour of a material by a thermodynamic theory of relaxation in the usual sense, except if the relaxation times are real and positive and some other conditions are fulfilled, but it seems that an appropriate extension of the thermodynamic theory can be made. In those cases that are suitable for a thermodynamic treatment, the internal variables representing the internal mechanisms divide, for isotropic substances, into two groups; the one containing invariant variables, active only in compression, the other containing sets of five variables with tensor character, and active only in shear deformation. It is essential for a thermodynamic treatment of shear relaxation to represent each molecular mechanism by such a set of five internal variables.