Theoretical and numerical analysis of the flow through a diffuser/nozzle element in pulsatile laminar conditions

Diffuser/nozzle pipes produce a directional flow resistance that is often exploited in microcirculation to generate a pumping action. This work presents an approximate time-dependent theoretical solution based on the mechanical energy conservation equation to predict the laminar flow rate through an ideal diffuser/nozzle pump. The theoretical solution is then used to characterize the dimensionless parameters that control the dynamics of the valveless pump in the pulsatile flow regime. A suitable numerical model is also implemented to solve the flow in a parametrized two-dimensional axial-symmetric domain subjected to an oscillating pressure, and its results are used to assess the theoretical solution. The pump dynamics and the main model parameters, such as the energy-loss coefficients, result in the following dependence on the ratios between the viscous force, the advective inertia, and the temporal inertia, i.e., the Reynolds (Re-d), Womersley (Wo(d)), and Strouhal (St) numbers referred to throat diameter. In particular, The Womersley number plays an essential role in controlling the global energy loss when Re-d < 100. The flow transition is also investigated and found when Red exceeds a critical value, which increases with Wo(d). Finally, the pump efficiency is found to reach its maximum when the convective and temporal inertia become comparable, i.e., St = O ( 1 ), consistent with the observed range of St in real-world diffuser/nozzle pumps. This optimum range of functioning of the pump is also observed for cerebrospinal pulsatile flow in the Sylvius aqueduct, suggesting that the modeled mechanism is used to promote or enhance cerebrospinal fluid circulation.