A generalization of the Lucas–Washburn–Rideal law to composite microchannels of arbitrary cross section

Abstract

Capillary microfluidics or capillarics has been lately gaining importance in the biotechnological and biological domains. In these domains where biological and chemical targets are transported by fluids, it has been shown that capillary actuation of fluids does not require bulky pumps or syringes and produces microsystems with a low cost of fabrication, which are user-friendly, portable and compatible with telemedicine. Capillary systems for biotechnology can be confined or open; i.e., the fluid moves inside a closed channel or in a channel with a boundary with air. In this work, we propose a general expression for the determination of the velocities of spontaneous capillary flows in composite, confined microchannels of arbitrary shapes. This expression generalizes the conventional Lucas–Washburn model which is valid for cylindrical channels. It is shown that the use of an equivalent hydraulic diameter in the Lucas–Washburn–Rideal model introduces a bias when the shape of the channel cross section differs notably from a circle. The approach also shows that relatively large velocities—at the scale of microsystems—can be reached by capillary microflows, depending on the shape of the channel, and that transport distances can be important.

Appendix: Numerical determination of the Poiseuille number

The numerical determination of the Poiseuille number—or equivalently of the hydraulic resistance—is made by considering a straight channel with the characteristic trapezoidal cross section. The liquid considered in the calculation is water; in fact, the choice of the liquid has no influence on the Poiseuille number since the quantity (1/μ) ∂P/∂z is independent of the nature of the fluid.

At inlet, a given velocity is imposed—for example, 1 mm/s. The Stokes model of the COMSOL numerical software is used to calculate the pressure at the nodes of the meshing in the channel. Let us recall that the Stokes equation results from a linearization of the Navier–Stokes equation valid in the case of a negligible inertia (Re < 1). More precisely, in this particular case, the flow velocity has only a z-component, denoted u, and the Stokes equation is

$$ \begin{aligned} {\nabla }P & = \frac{\partial P}{\partial z} = \mu \quad\Delta V = \mu \frac{{\partial^{2} u}}{{\partial z^{2} }} \\ \frac{\partial u}{\partial z} & = 0 \\ \end{aligned} $$

(31)

A linear pressure gradient ∂P/∂z is then derived from the contour plot of the pressure in the channel (Fig. 11). Finally, formula (23) is used to calculate the Poiseuille number after substitution of the value of the pressure gradient. A value of Po = 14.25—i.e., a value of fRe = 57—is found.

Fig. 11

a View of the channel and its meshing, b contour plot of the pressure for an imposed inlet velocity of 1 mm/s