The Principle of Relativity for Slow Motions

The form of the above-written relations (8.21) can be specified more by applying some restrictions which follow from the assumption that the motion of structural elements of medium does not change very rapidly, so that the following relation is valid 

a is the characteristic size of the structural element, p is the density, which is approximately equal to 1 g/cm3, rj is the effective viscosity coefficient of the medium which is 10 2-10 Ps, and u is the characteristic velocity of motion of the particle, which is not more than the mean thermal velocity (T/to)1/2. It is easy to see that, at room temperature and with the above values of the parameters, condition (8.22) is valid if a 3 10 7 10 6 cm. 

As is well known, the equations of mechanics are covariant with the Galileo transform. This can be also said about relations (8.19) and (8.21). In the case, when the motions of the internal particles are slow (in the sense discussed above), we can state that a stronger principle is valid. It says that all the processes run in the same way and, consequently, should be described by similar equations in all the co-ordinate frames which are connected to each other by the transform 

Let us consider the restriction imposed on the form of the transfer equations by the discussed principle. It is easy to see that, when transformation (8.23) is applied to the co-ordinates, the tensor of velocity gradients transforms as 

The superscript point denotes differentiation with respect to time. 

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