Application to Hydrodynamics
Hydrodynamics is fully defined by these factors:
Mass density \rho, speed \nu, pressure p, viscosity \mu, and the acceleration due to gravity g.
Take for example the capillary effect:
| Symbol | Description | Base Dimensions |
|---|---|---|
| h | Distance water is drawn into the tube | L |
| d | Diameter of the tube | L |
| \sigma | Surface tension of the water | MT^{-2} |
| \rho | Mass density of water | ML^{-3} |
| g | Acceleration due to gravity | LT^{-2} |
h is some function of the other three quantities:
h = f\left( d, \sigma, \rho, g \right)
Then
\textbf{A} = \begin{bmatrix} 1 & 0 & -3 & 1 \\ 0 & 1 & 1 & 0 \\ 0 & -2 & 0 & -2 \\ \end{bmatrix}
The null space of \textbf{A} is linear combinations of the vector \left(-2, 1, -1, 1\right)
Therefore
h = d \cdot g\left(\frac{\sigma g}{d^{2}p}\right)