The dimensionless parameters for any particular problem can be determined in two ways

They can be deduced directly from the governing differential equations if these equations are known

Dimensionless parameters determined from the equations of motion are more readily interpreted and linked to the physical phenomena occurring in the flow
Knowledge of the relevant dimensionless parameters frequently aids the solution process, especially when assumptions and approximations are necessary to reach a solution

If the governing differential equations are unknown or the parameter of interest does not appear in the known equations, dimensionless parameters can be determined from dimensional analysis

Dimensional Analysis: The natural realm does not need mankind’s units of measurement to function. Natural laws are independent of any unit system imposed on them by human beings

Consider Newton’s second law, generically stated as force = (mass) × (acceleration); it is true whether a scientist or engineer uses cgs (centimeter, gram, second), MKS (meter, kilogram, second), or even US customary (inch or foot, pound, second) units in its application

All correct physical relationships can be stated in dimensionless form

the problem-simplification or scaling-law-development technique known as dimensional analysis

in any comparison, the units of the items being compared must be the same for the comparison to be valid

the principle of dimensional homogeneity

└─ Requires all terms in an equation to have the same dimension(s)

└─ If terms in an equation do not have the same dimension(s) then the equation is not correct

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