Pressure driven, steady state flow of an incompressible fluid through a straight microchannel is typically laminar in nature, where the viscous effects dominate and inertial effects can be neglected as characterized by very small Reynolds's number. Viscous effects leads to internal friction in the fluid and creates resistance to the flow of the fluid. Just like electrical resistance leads to Joule heating, hydraulic resistance leads to dissipation of kinetic energy into heat.
Understanding hydraulic resistance is vital to the design of a microchannel. Mathematically it is represented by the Hagen-Poiseuille law that relates constant pressure drop dP resulting in a constant fluid flow Q
dP = Rhyd Q
Hagen-Poiseuille law is analogous to the Ohm's law dV = RI that relates the current I through a wire of resistance R and an electrical potential drop of dV. Similar to electrical resistors connected in series, two channels with different hydraulic resistance can be modeled by summing the two hydraulic resistance together as shown in the picture above. On similar lines, two hydraulic resistors connected in parallel follows the law of additivity of inverse of their hydraulic resistances as shown in the figure.
Use this design form to calculate the hydraulic resistance of a parabola shaped microfluidic channel with a constant cross-sectional area. If the channel has varying cross sectional area, the hydraulic resistance for individual channels with constant cross section can be found using this design form and then added together using the additivity law explained above. The flow rate can then be estimated using the Hagen-Poiseuille law. The flow velocity can then be calculated. To find the time the fluid takes to travel the full length of the channel, divide the length of the channel with the flow velocity.
The plot shows the dependence of hydraulic resistance of a parabolic channel to the width of the channel.