## Appendix 1

## Review of Mathematical Notations and Relations

The purpose of this appendix is to facilitate the understanding of theoretical developments by recalling some mathematical notions and relations used in fluid mechanics. These reminders are restricted to the mathematical tools used in this treatise.

## A.1. Scalars, vectors, and tensors

The behavior of a fluid is described by calling upon mathematical concepts with proper properties and submitted to well-defined operation rules. Thus, we introduce:

– One-dimensional scalar quantities represented by a number: this is the case of the pressure *p*, density *ρ*, temperature *T*, etc.

– Vectors which in a three-dimensional space are represented by three numbers, namely, their components in some coordinate system: this is the case of the velocity , acceleration , force , tension , etc.

In a Cartesian system of coordinates, with unit vectors along directions *Ox*, *Oy*, *Oz*, a vector will be noted:

where *u*, *v*, *w* are the vector components in the chosen ...