Poiseuille number correlation for high speed micro-flows

Abstract

The Poiseuille number, the product of a friction factor and the Reynolds number (f · Re) for a quasi-fully developed high speed flow in a micro-channel of Re < 2300 and Mach number Ma < 0.7, was obtained numerically. The numerical methodology is based on the arbitrary-Lagrangian–Eulerian (ALE) method. Two-dimensional compressible momentum and energy equations with no-slip and slip boundary conditions were solved for constant wall temperatures that are lower or higher than the inlet temperature. The channel height ranges from 10 to 100 µm and the channel aspect ratio is 200. The stagnation pressure, pstg, is chosen such that the exit Mach number ranges from 0.1 to 0.7. The outlet pressure is fixed at atmospheric conditions. In the case of fast flow for both no-slip and slip boundary conditions, the value of f · Re is higher than 96 due to compressibility effects. However, in the case of slow flow for slip boundary conditions (Maout < 0.3), the value of f · Re is slightly lower than 96 due to rarefaction effects, even the flow is accelerated. The f · Re correlation for slip boundary conditions is obtained from that for no-slip boundary conditions and incompressible theory as a function of the Mach number and the Knudsen number. The f · Re correlation obtained for no-slip boundary conditions is also compared with that obtained for slip boundary conditions. The values of f · Re obtained for no-slip and slip boundary conditions almost coincide within 3% for the channel height in the range 10–100 µm.