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

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.