  We consider pressure-driven, steady state Poiseuille flow in straight
channels with various cross-sectional shapes: elliptic, rectangular,
triangular, and harmonic-perturbed circles. A given shape is characterized by
its perimeter P and area A which are combined into the dimensionless
compactness number C = P^2/A, while the hydraulic resistance is characterized
by the well-known dimensionless geometrical correction factor alpha. We find
that alpha depends linearly on C, which points out C as a single dimensionless
measure characterizing flow properties as well as the strength and
effectiveness of surface-related phenomena central to lab-on-a-chip
applications. This measure also provides a simple way to evaluate the hydraulic
resistance for the various shapes.
