Microsystems with Only One Length Scale

Let the microsystem under consideration is characterized by only one length scale Lq and one velocity scale Vq. One example is the flow inside two infinite parallel plates. The expressions of coordinates and velocity in terms of dimensionless coordinates and velocity can be written as 

Note that a quantity often can be made dimensionless in more than one way. The pressure term can be nondimensionalized by using reference pressure Pq equal to VqILq or pVq. The 

we have introduced the dimensionless number Re (Reynolds number) as 

In the limit of small Reynolds number, the nonlinear N-S equation (2.9) reduces to linear 

We clearly observe from equation (2.9) that for Pe 1 the viscous term v u dominates, whereas in the steady state for 1, the inertia term (v V )v is the most important term. 

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