Question

what is the importance of mach number in fluid mechanics

Answer 1

In fluid mechanics, non-dimensionalization of the Navier–Stokes equations is the conversion of the Navier–Stokes equation to a nondimensional form. When skillfully performed, this eases the analysis of the problem which is at study, and reduces the number of free parameters. Small or large sizes of certain dimensionless parameters indicate the importance of certain terms in the equations for the studied flow. This may provide possibilities to neglect terms in certain (areas of the) considered flow. Further, non-dimensionalized Navier–Stokes equations can be beneficial if one is posed with similar physical situations – that is problems where the only changes are those of the basic dimensions of the system.

Scaling of Navier–Stokes equation refers to the process of selecting the proper spatial scales – for a certain type of flow – to be used in the non-dimensionalization of the equation. Since the resulting equations need to be dimensionless, a suitable combination of parameters and constants of the equations and flow (domain) characteristics have to be found. As a result of this combination, the number of parameters to be analyzed is reduced and the results may be obtained in terms of the scaled variables.