Reference-invariant Representation of a Material Function

At this point the remarks given in the introduction to this chapter should be recalled Water is of no use to investigate the performance of a petrochemical plant in a cold model . 

In general, standard representation depends upon the choice of the reference point. The question is posed Do mathematical functions exist whose standard representations do not depend on the choice of the reference point and therefore could be named reference-invariant functions In case of an affirmative answer on the one hand the reference point p0 - here T0 - could be omitted (constriction of the pi-space by one pi-number) and on the other hand the dimensionless representation of the material function would stretch over the entire recorded range. 

It is mathematically proven that only one class of reference-invariant standard representations exists and that this can be represented by one-parametric x (u, tp) functions  

The regions of existence and appearance of reference-invariant functions % (u, i j) are represented in Fig. 10. Curves with maxima and minima cannot be described in a reference-invariant manner. In this case, both the dimensional-analytical representation and the model material system are confined to the region close to the standardization range . 

The reference-invariant representations of the temperature dependence of viscosity in Fig. 8 a and b were obtained by ip = -0.179 and -0.106 respectively. 

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