Reexamination of Hagen-Poiseuille flow: Shape dependence of the hydraulic resistance in microchannels

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 $\mathcal{P}$ and area $\mathcal{A}$ which are combined into the dimensionless compactness number $\mathcal{C}={\mathcal{P}}^2∕\mathcal{A}$, while the hydraulic resistance is characterized by the well-known dimensionless geometrical correction factor $\alpha$. We find that $\alpha$ depends linearly on $\mathcal{C}$, which points out $\mathcal{C}$ 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.

Figure 1

An arbitrary cross-sectional shape $\Omega$ with boundary $\partial \Omega$ of a straight fluid channel with pressure-driven steady-state flow. The contours show the velocity $v(x,y)$ obtained numerically from Eq. (3) by a finite-element method. The velocity is zero at the boundary and maximal near the center-of-mass.

Figure 2

The correction factor versus compactness for the elliptical, rectangular, and triangular classes. The solid lines are the exact results, and the dashed lines indicate Eqs. (9, 14, 15). Numerical results from a finite-element simulation are also included (엯, ◻, and ▵). Note that in the case of triangles all classes (right, isosceles, and acute/obtuse scalene triangles—marked by different grayscale triangles) fall on the same straight line.

Figure 3

(a) The geometry of the unperturbed and analytically solvable cross section, the unit circle, described by coordinates $(\tilde{x},\tilde{y})$ or $(\rho ,\theta )$. (b) The geometry of the perturbed cross section described by coordinates $(x,y)$ or $(r,\varphi )$ and the perturbation parameter $ϵ$. Here $a=1$, $k=5$, and $ϵ=0.2$.