Basics of Fluid Mechanics, 2014a

9.2. BUCKINGHAM–π–THEOREM 287
End Solution
An example of a ship 6 is be a typical example were more than one dimensionless
is to constructed. Also introduction of dimensional matrix is presented.
Example 9.5:
The modern ship today is equipped with a propeller as the main propulsion mechanism.
The thrust, T is known to be a function of the radius, r, the fluid density, ρ, relative
velocity of the ship to the water, U, rotation speed, rpm or N, and fluid viscosity,
μ. Assume that no other parameter affects the thrust, find the functionality of these
parameters and the thrust.
Solution
The general solution under these assumptions leads to solution of
T = Cr a ρ b U c N d μ e
(9.V.a)
It is convenient to arrange the dimensions and basic units in table. This table is referred
in the literature as the Dimensional matrix.
Table -9.4.
Dimensional matrix
T r ρ U N μ
M 1 0 1 0 0 1
L 1 1 -3 1 0 -1
t -2 0 0 -1 -1 -1
Using the matrix results in
MLt −2 = L a (Lt) b ( ML −3) c (
t
−t ) d (
ML −1 t −t) e
(9.V.b)
This matrix leads to three equations.
Mass,M 1= c + e
Length,L 1= a + b + −3c − e
time,t −2 = −c − d − e
(9.V.c)
6 This author who worked as ship engineer during his twenties likes to present material related to
ships.