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# Dimensional Analysis

The treatment of this chapter goes far beyond the field of fluid mechanics. Although the concepts of dimensional analysis apply in all the fields of science, it finds extensive use in fluid mechanics as a tool for studying various phenomena. It, therefore, occupies center stage in this book. This chapter elaborates on several topics:

– It sets out the consequences of a system’s behavior being independent from the system of units used to describe it. This idea is linked to the notion of homogeneity that should be verified by any mathematical expression describing a physical system.

– Dimensional analysis helps to establish the general form of a relation that exists between the various parameters involved in a problem. Let us consider the example of a fluid flow in a pipe with a diameter D and a length L, which we discuss later. This flow is generated by the pressure change between the inlet and outlet, shifting from P1 to P2. The streamwise velocity, U, is therefore expressed by a relation in the form F(P1, P2, U, D, L, v, ρ, …) = 0 between the different physical parameters that define the system. Dimensional analysis shows that this relation should associate the parameters in a way respecting consistency in regards to units attached to the parameters. It also enables us to identify a reduction in the number of variables in the relation describing the system.

– An offshoot of dimensional analysis is the theory of similarities, which sets out a way ...

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