(170e) Solution of the Boltzmann Equation for Fluid Flows In Microchannels at Finite Knudsen Numbers with a Third-Order Quadrature-Based Moment Method
Flows in microchannels have a characteristic length scale comparable to the mean molecular free path, causing the microscopic fluid-wall interactions and the kinetic phenomena to show their effect at the macroscale. Under these conditions the Navier-Stokes equation is not valid, and the solution of the Boltzmann equation is required to properly describe the behavior of the flow. In this work a third order quadrature-based moment method is used to find a numerical solution of the Boltzmann equation, both with the Bhatnagar-Gross-Krook (BGK) collision operator (Fox, 2008) and with the Boltzmann hard-sphere collision integral (Vedula and Fox, 2008). The velocity profiles and the mass flow rates for flows in microchannels with Knudsen numbers between 0.04 and 25 in the case of the Poiseuille flow, and with Knudsen numbers between 0.1 and 1 in the case of the Couette flow are reported and compared with Discrete Simulation Monte Carlo (DSMC) data taken from the literature (Kim at al, 2008). The quadrature-based moment method is shown to capture correctly the wall slip velocity at a function of the Knudsen number up to unity at a fraction of the computational cost of DSMC. For larger Kn, the third-order moment method is in qualitative agreement with DSMC, but a higher-order moment method would be required for quantitative agreement.
Fox, R. O., A quadrature based third-order moment method for dilute gas-particle flows, Journal of Computational Physics, 227, 6313 ? 6350, 2008.
Kim, S. H., Pitsch, H., Boyd, I. D., Slip velocity and Knudsen layer in the lattice Boltzmann method for microscale flows, Phisical Review E, 77, 026704, 2008.
Vedula, P., Fox, R. O., A quadrature-based method of moments for solution of the collisional Boltzmann equation, Journal of Statistical Physics, Submitted, 2008.