Lattice Boltzmann simulations of pressure-driven flows in microchannels using Navier–Maxwell slip boundary conditions

Abstract

We present lattice Boltzmann simulations of rarefied flows driven by pressure drops along two-dimensional microchannels. Rarefied effects lead to non-zero cross-channel velocities, and nonlinear variations in the pressure along the channel. Both effects are absent in flows driven by uniform body forces. We obtain second-order accuracy for the two components of velocity and the pressure relative to asymptotic solutions of the compressible Navier–Stokes equations with slip boundary conditions. Since the common lattice Boltzmann formulations cannot capture Knudsen boundary layers, we replace the usual discrete analogs of the specular and diffuse reflection conditions from continuous kinetic theory with a moment-based implementation of the first-order Navier–Maxwell slip boundary conditions that relate the tangential velocity to the strain rate at the boundary. We use these conditions to solve for the unknown distribution functions that propagate into the domain across the boundary. We achieve second-order accuracy by reformulating these conditions for the second set of distribution functions that arise in the derivation of the lattice Boltzmann method by an integration along characteristics. Our moment formalism is also valuable for analysing the existing boundary conditions. It reveals the origin of numerical slip in the bounce-back and other common boundary conditions that impose conditions on the higher moments, not on the local tangential velocity itself.

Analysis and accurate numerical solutions of the integral equation derived from the linearized BGKW equation for the steady Couette flow

“…In other words, one cannot accurately capture the Knudsen layer without resolving the endpoint singularities [32]. Thus one cannot hope to capture the Knudsen layer or its manifestation a priori by using the lattice Boltzmann equation [50][51][52][53] or similar schemes, which is a special case of the linearized BGKW equation in the diffusive limit [54] and is incapable to capture any Knudsen-number effects beyond the Navier-Stokes equations [55][56][57].…”

Lattice Boltzmann Formulation for Linear Viscoelastic Fluids Using an Abstract Second Stress

“…The first two conditions (9.4a) and (9.4b) impose no-flux and no-slip boundary conditions, and the third boundary condition (9.4c) on the tangential stress has a natural physical interpretation, unlike the alternatives involving higher moments [5,6,75] …”

“…We use the approach of Wagner and Yeomans [93] and Bennett [5,6,75] to impose boundary conditions on the hydrodynamic moments u x , u y , and Π xx ,…”

Local boundary reflections in lattice Boltzmann schemes: Spurious boundary layers and their impact on the velocity, diffusion and dispersion

“…The Knudsen boundary layers are usually considered in the context of the rarefied gas dynamics as boundary conditions for non-equilibrium flows [1]. A very comprehensive critical review on the efforts devoted to appropriate LBM boundary conditions can be found in works [2,3]. The objective of this work is quite different.…”