Inspired by the physiological network of insects, which have dimensions on the order of micrometers to millimeters, we study the airflow within a single model insect tracheal tube. The tube undergoes localized rhythmic wall contractions. A theoretical analysis is given to model the airflow within the tracheal tube. Since flow motions at the microscale are dominated mainly by viscous effects, and the tube has radius, R, that is much smaller than its length, L, (i.e. δ = R/L ≪ 1), lubrication theory for axisymmetric, viscous, incompressible flows at low Reynolds number (Re ∼ δ) is used to model the problem mathematically. Expressions for the velocity field, pressure gradient, wall shear stress and net flow produced by the driving tube wall contractions are derived. The effect of the contraction amplitudes, time lag, and spacing between two sequences of contractions on the time-averaged net flow over a single cycle of wall motions is investigated. The study presents a new, insect-inspired mechanism for valveless pumping that can guide efforts to fabricate novel microfluidic devices that mimic these physiological systems. A x-ray image that shows the tracheal network of the respiratory system of an insect (Carabid beetle) and the associated locations of these rhythmic contractions are shown in figure (1) to promote this study.
Temporal stability analysis for multiple similarity solutions of viscous incompressible flows in porous channels with moving walls
