Hi all! I am having problems with understanding the scaling process of the N-S equations in fluid dynamics. From textbooks, I see that each quantity say velocity, time, length...etc are all divided some some reference values in order to obtain some dimensionaless quantity V*, t*, p*, g* etc.. And the N-S equations are then rewrite into a dimensionless form, the coefficients beceome the Reynolds number, Froude number etc... And the writer says after having this dimensionless equation, we can know the importance of the terms by just looking at the coefficients. That's what the textbook said, and I don't really understand. I can't catch the reason for making it in a dimensionless form. Can't I still judge the importance of the terms by looking at the coefficients of the terms when the equation have dimensions? Why must we transform it to be dimensionless? Can anyone help me out?
Certainly, you don't have to use dimensionless variables. However, using dimensionless variables can simplify the analysis of a problem. First, it cuts down the number of symbols in the equation. (Less writing!) This often highlights the "[standard] form" of an equation. It may be easier to recognize the mathematics, and possibly draw analogies between different physical systems. (For example, it may help you recognize that the mass-spring system is analogous to an inductor-capacitor system.) Additionally, one can't compare A with B (i.e. one can't say A>B) if they carry different dimensions. However, if [tex]\alpha[/tex] and [tex]\beta[/tex] are dimensionless, then one can compare them. In particular, it may useful to know that [tex]\alpha\gg\beta[/tex] so that the [tex]\beta[/tex]-term in an expression like [tex]\alpha \blacksquare + \beta \blacksquare +\ldots [/tex] may be neglected. Implicit in the above is the idea of "scaling". For example, if you know how a problem scales, you can experimentally model it less expensively. (For example, wind tunnel tests for airplanes. Another example: special effects using miniatures and slow-motion.)