We adopt a recent minimal mathematical model of a pumping mechanism in entomological respiratory systems and consider the model’s behavior in the slip flow regime, which occurs naturally in the distalmost portions of insect respiratory systems. In the model, a phase lag in the timing of two neighboring wall contractions in a rectangular microchannel produces a unidirectional flow. The current study investigates the results of incorporating slip effects into the model by introducing first-order accurate slip boundary conditions to investigate the method’s performance for slip flows at the microscale in the slip flow regime. The two-dimensional Navier–Stokes equations are solved with microscale and lubrication theory assumptions, and the tangential momentum accommodation coefficient is assumed to be one, so that the slip flow parameter \(\beta \) is identically equivalent to the Knudsen number, Kn. The variations of the axial velocity, pressure gradient, and total pressure along the channel are determined for three representative Knudsen numbers that span the continuum and slip flow regimes. It was observed that for the shear-driven flow investigated here, the overall effect of increasing the amount of slip is to decrease the volumetric flow rate and that the phase lag for producing maximum flow is in the range of \(63^\circ \)–\(67^\circ \), while in the no-slip case the optimum phase lag is approximately \(63^\circ \). The results suggest that shear-driven flows at the microscale in the slip flow regime may see a reduction in flow rate in contrast to pressure-driven microscale gas flows in the slip flow regime.
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The authors would like to thank the National Science Foundation (Grant No. 1437387) for providing the funding support for this study and Dr. Yasser Aboelkassem for providing valuable insights.
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Chatterjee, K., Staples, A. Slip flow in a microchannel driven by rhythmic wall contractions. Acta Mech 229, 4113–4129 (2018). https://doi.org/10.1007/s00707-018-2210-7