Effect of non-unity momentum and thermal accommodation coefficients on slip flow and heat transfer in an asymmetrically heated microchannel simulated by the thermal lattice Boltzmann method

The thermal lattice Boltzmann (TLBM) study of Zhang, Xie and Karimipour (Int. Commun. Heat Mass Transfer 118:104856, 2020) analyzed mixed-convection gas slip flow in a two-dimensional microchannel with asymmetric constant wall heat fluxes, but its boundary modeling assumed complete gas-surface accommodation (tangential momentum and thermal accommodation coefficients $\sigma_v=\sigma_t=1$). The open problem is to quantify how partial accommodation modifies the entrance-region flow development, the wall shear stress, and the local Nusselt number in this configuration. Later thermal-LBM microchannel studies still fix $\sigma_v=\sigma_t=1$, and accommodation-coefficient sensitivity has so far been quantified only outside LBM and in simpler or fully developed configurations; the problem therefore remains open.

Background

Gas flow through microchannels with characteristic heights of a few to hundreds of micrometers, as in micro-electro-mechanical systems and micro heat exchangers, is rarefied: the molecular mean free path is no longer negligible relative to the channel dimension. Rarefaction is measured by the Knudsen number $Kn=\lambda/H$ (mean free path over channel height); for $10^{-3}\lesssim Kn\lesssim 0.1$, the slip regime, the Navier-Stokes-Fourier equations remain usable but the walls require kinetic boundary conditions: Maxwell's velocity-slip condition and the Smoluchowski temperature-jump condition. In these, the tangential momentum accommodation coefficient $\sigma_v$ and the thermal accommodation coefficient $\sigma_t$ (each the fraction of impinging molecules re-emitted diffusely, i.e., fully equilibrated with the wall) enter through factors $(2-\sigma)/\sigma$ multiplying the near-wall velocity and temperature gradients. Measurements on real gas-surface pairs frequently give $\sigma_v$ below unity (Graur et al. 2009 report measured values below one for several gases in a plane microchannel), so assuming $\sigma_v=\sigma_t=1$ systematically biases predicted wall friction and heat transfer.

The lattice Boltzmann method (LBM) is a mesoscopic solver that evolves particle-distribution functions on a discrete lattice; in its double-population (thermal LBM, TLBM) form a second distribution carries internal energy, so hydrodynamics and heat transfer are simulated together. Imposing a prescribed wall heat flux, a Neumann-type thermal condition, in TLBM is awkward because the unknowns are distributions rather than temperature; closures such as the counter-slip scheme of D'Orazio and Succi (2004) or anti-bounce-back rules are known to be delicate with respect to stability and the relaxation time, which is why dedicated Neumann schemes for TLBM continue to appear (Lalami and Espili 2020; Tao et al. 2020; Martins et al. 2024). For rarefied microchannels the hydrodynamic wall is a kinetic rule, typically a bounce-back/specular-reflection mixture or a diffuse-scattering condition calibrated to $Kn$ (Tang, Tao and He 2005; Niu, Shu and Chew 2007; Verhaeghe, Luo and Blanpain 2009), and the accommodation coefficients enter only through the slip and temperature-jump relations imposed on top of it.

In this setting Zhang, Xie and Karimipour (2020) used a TLBM with slip and temperature-jump walls to analyze steady, laminar gas slip flow under mixed convection (combined forced and buoyancy-driven convection) in a two-dimensional horizontal microchannel whose top and bottom walls carry different constant heat fluxes (asymmetric heating), sweeping the wall heat-flux ratio, $Kn$, and the mixed-convection parameter, with particular attention to the entrance (developing) region. They found the thermal conversion in the inlet region extremely sensitive to all three factors and that buoyancy generates a vortex near the top wall at lower $Kn$. Their boundary modeling assumed full accommodation, $\sigma_v=\sigma_t=1$, so the sensitivity of the entrance-region flow, the wall shear stress, and the local Nusselt number $Nu$ (the dimensionless wall-to-bulk heat-transfer coefficient) to non-unity accommodation coefficients was left unquantified.

Problem Statement

For steady, laminar, rarefied gas flow in a two-dimensional parallel-plate microchannel in the slip regime, with asymmetric constant wall heat fluxes and mixed convection (the configuration of Zhang, Xie and Karimipour 2020, with its wall heat-flux ratio, Knudsen number, and mixed-convection parameter ranges), determine how partial gas-surface accommodation modifies the predictions obtained under the full-accommodation assumption $\sigma_v=\sigma_t=1$. Specifically, quantify the effect of the tangential momentum accommodation coefficient $\sigma_v$ and the thermal accommodation coefficient $\sigma_t$, each over the sub-unity range observed for real gas-surface pairs and varied both individually and jointly, on: (i) the hydrodynamic and thermal development of the entrance region; (ii) the wall shear stress, equivalently the friction coefficient, on each wall; and (iii) the local Nusselt number distribution on each wall. The determination is to be made with a thermal lattice Boltzmann method in which the kinetic wall treatment combines partial-accommodation Maxwell velocity-slip and Smoluchowski temperature-jump conditions with the constant-wall-heat-flux thermal closure of the source configuration. As a scientific consistency condition, in the limit $\sigma_v=\sigma_t\to 1$ the boundary treatment must reduce to the full-accommodation slip and temperature-jump behavior of the source configuration.

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