Basics of Fluid Mechanics, 2014a

9.2. BUCKINGHAM–π–THEOREM 293
than the resistance is neglected and the fourth parameter is removed. In that case the
functional relationship can be written as
( )
ΔP Q
N 2 ,D 2 = f ND 3 (9.VIII.n)
End Solution
9.2.3.3 Mathematical Method: Constructing Dimensional Parameters
Advance material can be skipped
under construction please ignore for time being
In the progression of the development of the technique the new evolution is the
mathematical method. It can be noticed that in the previous technique the same matrix
was constructed with different vector solution (the right hand side of the equation). This
fact is the source to improve the previous method. However, it has to be cautioned
that this technique is overkill in most cases. Actually, this author is not aware for any
case this technique has any advantage over the “building block” technique.
In the following hypothetical example demonstrates the reason for the reduction
of variables. Assume that water is used to transport uniform grains of gold. The total
amount grains of gold is to be determined per unit length. For this analysis it is assumed
that grains of gold grains are uniformly distributed. The following parameters and their
dimensions are considered:
Table -9.5. Units and Parameters of gold grains
Parameters Units Dimension Remarks
grains amount q M/L total grains per unit length
cross section area A L 2 pipe cross section
grains per volume gr grains/L 3 count of grain per V
grain weight e M/grain count of grain per V
Notice that grains and grain are the same units for this discussion. Accordingly, the
dimensional matrix can be constructed as