This appendix outlines the role of several key dimensional and nondimensional parameters in micro- and nanoscale fluid mechanics that come from nondimensionalization of governing equations. A key advantage of nondimensionalization is that it leads to a compact description of flow parameters (i.e., Re) and thus leads to generalization. Nondimensionalization can be a powerful tool, but it is useful only if implemented with insight into the physics of the problems. Our stress here is the process of nondimensionalization, rather than a listing of nondimensional parameters, and we focus on only a few examples.

BUCKINGHAM Π THEOREM

The Buckingham Π theorem is a theorem in dimensional analysis that quantifies how many nondimensional parameters are required for specifying a problem. It also provides a process by which these nondimensional parameters can be determined. The Buckingham Π theorem states that a system with n independent physical variables that are a function of m fundamental physical quantities can be written as a function of n – m nondimensional quantities. As an example, the steady Navier–Stokes equations have four parameters: a characteristic length ℓ, a characteristic velocity U, the viscosity η, and the fluid density ρ. These are a function of three fundamental physical quantities: mass, length, and time. Thus the system can be described in terms of 4 − 3 = 1 nondimensional quantity, and it can be shown that the nondimensional quantity must be proportional to ρUℓ/η to some power.