Editing Strouhal number (section)

===Scale Analysis===
In order to determine significance of the Strouhal number at varying scales, one may perform [[Scale analysis (mathematics)|scale analysis]]–a simplification method to analyze the impact of factors as they change with respect to some scale. When considered in the context of microrobotics and nanorobotics, size is the factor of interest when performing scale analysis.

Scale analysis of the Strouhal number allows for analysis of the relationship between mass and inertial forces as both change with respect to size. Taking its original underived form, <math>\tfrac{mU^2}{FL}</math>, we can then relate each term to size and see how the ratio changes as size changes.

Given <math>m=V\rho</math> where ''m'' is mass, ''V'' is volume, and <math>\rho</math> is density, we can see mass is directly related to size as volume scales with length (L). Taking the volume to be <math>L^3</math>, we can directly relate mass and size as

:<math>m \approx L^3</math>.

Characteristic speed (''U'') is in terms of <math>\tfrac{\text{distance}}{\text{time}}</math>, and relative distance scales with size, therefore

:<math>U^2 \approx L^2</math>.

The net external forces (''F'') scales in relation to mass and acceleration, given by <math>F=m\cdot a</math>. Acceleration is in terms of <math>\tfrac{\text{distance}}{\text{time}^2}</math>, therefore <math>a \approx L</math>. The mass-size relationship was established to be <math>m \approx L^3</math>, so considering all three relationships, we get

:<math>F \approx L^4</math>.

Length (''L'') already denotes size and remains ''L''.

Taking all of this together, we get

:<math>\dfrac{mU^2}{FL} \approx \dfrac{L^3L^2}{L^4L} \approx \dfrac{L^5}{L^5} \approx L^0 = 1</math>.

With the Strouhal number relating the mass to inertial forces, this can be expected as these two factors will scale proportionately with size and neither will increase nor decrease in significance with respect to their contribution to the body's behavior in the cyclic motion of the fluid.