On the correct use of mathematical constructions in physical models

The physical limitations of the mathematical constructions used in developing or modifying mathematical models are discussed. All reasonings are illustrated by examples from fluid mechanics. The following topics are considered: means of description; correct approach to model modification and the physical meaning of model development stages. In the first case, the method of describing physical objects using numbers as well as corresponding restrictions are investigated, followed by developing general recommendations on procedures for modifying mathematical models of fluid dynamics. The well-known procedure of averaging the viscous fluid model equations to obtain the turbulent fluid model is used as an illustration. Since we are considering the models of physical phenomena, it is natural to provide physical interpretation for each stage of model development. Unfortunately, some of the transformations used are often treated as purely technical tricks, therefore denoting the lack of the physical meaning in such cases, which does not make a mathematical procedure unacceptable, but does mark out the model's place which requires reasonable interpretation. In this paper, we are considering two variants of this kind of interpretation, namely the case of using imaginary quantities, and the case of applying integral transformations. Meanwhile, all the above-mentioned restrictions are not always given due attention. Sometimes this leads to various undesirable consequences, including excessive task complication, implicit substitution of a declared problem with another one, or, finally, lack of solution to the formulated problem.