Dynamical similarity the Reynolds number

Nowadays wind-tuimel experiments are currently performed by airplane or car engineers to test the aerodynamicity oftheir machines. The physical principle of these 

For the present discussion it is sufficient to use the Navier-Stokes equation 1.32 in a reduced form by neglecting the contributions of the volume forces 

With the help of these new variables, the reduced Navier-Stokes equation is transformed to yield 

Re is a dimensionless parameter. Its definition constitutes a powerful tool for the transfer of information from experiments performed at the laboratory scale on various hydrodynamic phenomena to very large (e.g. airplanes) or very small (particles) scales. Via eq 1.36 Re also provides the possibilty of further simplification of the Navier-Stokes equation if the value of Re is very large or very small. 

The discussion of electrolyte solutions requires the estimation of the Reynolds number for the particular case where L is of the order of the mean diameter of the particles, i.e. 0.1 nm. All liquids commonly used as solvents show dynamic viscosities of the order of 1 cPoise and densities of the order of 1 g cm . Then the order of magnitude of Re can be evaluated if for U an estimate of the hydrodynamic velocity of the sphere in the liquid can be made. The order of magnitude of U can be derived from the linear transport theory, where the motion of a particle in a liquid is described at a local level by the action of a friction force F (eq 1.1). In the steady state of motion this force is supposed to equilibrate the thermodynamic force  

---