Dimensionless Numbers in Microfluidics - Microfluidics for Biotechnology - page 1

C h a p t e r 1
Dimensionless Numbers in Microluidics

1.1 Introduction

Scaling analysis and dimensionless numbers play a key role in physics. They indi-
cate the relative importance of forces, energies, or time scales in presence and lead
the way to simplification of complex problems. Besides, the use of dimensionless
parameters and variables in physical problems brings a universal character to the
system of equations governing the physical phenomena, transforming an individual
situation into a generic case.
The same remarks apply to microscale physics. Only forces, energies, and time
scales are different, and, even if some dimensionless numbers are the same as the
one used at the macroscale, many are specific to microscales. In this chapter, we
present the most widely used dimensionless numbers in microfluidics, after having
recalled the fundamental Buckingham's Pi theorem.

1.2 MicroluidicScales

Let us characterize the dimension of a system by the length scale L . Areas then
scale as L 2 and volume scales as L 3 . Surface forces are in general proportional to
the surface area and volume forces—like weight or inertia—are proportional to the
volume. The most typical change when switching from macroscopic to microscopic
scales is that the ratio between surfaces forces and volume forces increases as 1/ L . In
microsystems, surface forces tend to be dominant over volume forces. The scaling
laws of different physical quantities that frequently appear in the physics of micro-
systems as function of the length scale L are given in Table 1.1.
By looking at Table 1.1, it is deduced that when miniaturizing fluidic systems
(i.e., L ® 0), inertia and gravity become less important, whereas capillarity and
interface phenomena become dominant (Laplace pressure, capillary rise, and Ma-
rangoni force all scale as 1/ L ). Note the huge increase in hydraulic resistances (1/ L 4 )
and the importance of viscoelasticity at small scales, with the Deborah and elasto-
capillary numbers varying respectively as L -3/2 and L -2 .

1.3 Buckingham'sPiTheorem

Buckingham's pi theorem is a key theorem in dimensional physics [1]. The theorem
states that for a system of equations involving n physical variables, depending only on k
independent fundamental physical quantities (unities, for example), the system depends
only on p = n − k dimensionless variables constructed from the original variables.