A microscopic model able to describe simultaneously the dynamic viscosity and the self-diffusion coefficient of fluids is presented. This model is shown to emerge from the introduction of fractional calculus in a usual model of condensed matter physics based on an elastic energy functional. The main feature of the model is that all measurable quantities are predicted to depend in a non-trivial way on external parameters (e.g. the experimental set-up geometry, in particular the sample size). On the basis of an unprecedented comparative analysis of a collection of published experimental data, the modeling is applied to the case of water in all its fluid phases, in particular in the supercooled phase. It is shown that the discrepancies in the literature data are only apparent and can be quantitavely explained by the different experimental configurations. This approach makes it possible to reproduce the water viscosity with a better accuracy than the 2008 IAPWS formulation and also with a more physically satisfying modeling of the isochors. Moreover, it also allows the modeling within experimental accuracy of the translational self-diffusion data available in the literature in all water fluid phases. Finally, the formalism of the model makes it possible to understand the anomalies observed on the dynamic viscosity and self-diffusion coefficient and their possible links.
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