Relationship between mean and thermodynamic pressure

By definition, the mean pressure p = —1/3 5j rkk only includes normal stresses. In order to create a link between mean and thermodynamic pressure we will consider a cubic fluid element at a temperature T and of specific volume v, Fig. Al. We will now assume that the cube is at rest at time t = 0, so that the thermodynamic pressure p prevails inside the element. Now let us assume the mean pressure p is being exerted on the element from outside. When p p the cube is compressed, should p p then it expands. So, work — p dy is carried out by the external pressure p. This is equal to the work done during the volume change in the gas — p dy and the dissipated work. It therefore holds that dW = —p dV = —p dy + dWdi88 with the the dissipated work as 

It is clearly reasonable that the speed dy /dt of the volume change or dwi/dxi is a monotonically decaying function of p — p, Fig. A2, as the larger the overpressure p — p, the faster the volume of the cube reduces. It is therefore suggested, that 

The factor defined by this, 0, is the volume viscosity (SI units kg/sm). It has to be determined either experimentally or using methods of statistical thermodynamics, which is only possible for substances with simple molecules. It can be seen that the mean and thermodynamic pressures only strictly agree when C = 0 or the fluid is incompressible, dwi/dxi = 0. 

---