Basics of Fluid Mechanics, 2014a

9.5. SUMMARY 319
It can be noticed that this can be achieved using Ohnesorge Number like this presentation.
End Solution
9.5 Summary
The two dimensional analysis methods or approaches were presented in this chapter.
Buckingham’s π technique is a quick “fix approach” which allow rough estimates and
relationship between model and prototype. Nusselt’s approach provides an heavy duties
approach to examine what dimensionless parameters effect the problem. It can be shown
that these two techniques in some situations provide almost similar solution. In other
cases, these technique proves different and even conflicting results. The dimensional
analysis technique provides a way to simplify models (solving the governing equation by
experimental means) and to predict effecting parameters.
9.6 Appendix summary of Dimensionless Form of Navier–
Stokes Equations
In a vector form Navier–Stokes equations can be written and later can be transformed
into dimensionless form which will yield dimensionless parameters. First, the typical
or characteristics values of scaling parameters has to be presented and appear in the
following table
Parameter Symbol Parameter Description Units
h characteristic length [L]
U 0
characteristic velocity
[ ] L
t
f
characteristic frequency
[ ] 1
t
ρ 0
P max − P ∞
characteristic density
maximum pressure drive
[ M
L 3 ]
[ M
Lt 2 ]
Basic non–dimensional form of the parameters
˜P =
˜t = ft
˜r = ⃗r
h
ŨU = ⃗ U
U 0
P − P ∞
P max − P ∞
˜∇ = h ∇ ˜ρ =
ρ
ρ 0
(9.50)