The Impact Of Accommodation Coefficient On Concentric Couette Flow

Rarefied Couette flow between concentric rotating cylinders is investigated using Maxwell’s slip-velocity boundary treatment. The study provides an independent reassessment of the velocity inversion process and demonstrates that the occurrence of an inverted velocity profile depends solely on the accommodation coefficient of the stationary outer cylinder. From the analysis, criteria are derived for the accommodation coefficient of the outer cylinder that can be used to predict whether an inverted, partially-inverted or normal velocity profile will occur. The present results are in close agreement with previous analytical solutions and in good qualitative agreement with available DSMC data. INTRODUCTION Couette flow between concentric rotating cylinders is a classical fluid dynamics problem that can be found in many textbooks. However, recent analytical and DSMC studies [1-3] have suggested that under certain conditions of rarefaction (in particular, when the accommodation coefficient is small), the velocity profile between the cylinders reverses direction so that the gas moves faster near the stationary outer cylinder. This anomalous behavior has been described as an ‘inverted velocity profile’ because the velocity of the gas increases with distance from the rotating inner cylinder. The effect is completely non-intuitive and contrary to the normal velocity profile expected within a cylindrical Couette flow. The phenomenon of velocity inversion was first predicted by Einzel, Panzer and Liu (EPL) [1] who recognized the importance of accounting for surface curvature and developed a generalized slip-boundary condition for flows over curved surfaces. EPL did not specifically consider a rarefied gas and instead analyzed the flow using the concept of a slip-length. They showed that the velocity profile within a concentric Couette flow would become inverted for large values of slip-length. Tibbs et al. [2] have subsequently recast EPL’s formulation so that it can be applied to a rarefied gas. This was achieved by defining the slip-length, 0 (2 / 1) a ζ = σ − λ where λ is the mean free path of the gas molecules, a ≈ 1.15 and σ is the tangential momentum accommodation coefficient (TMAC) which can vary from zero (for specular reflection) up to unity (for complete or diffuse accommodation). Tibbs et al. found that the velocity inversion process only occurs for small values of TMAC. They also presented results from a direct simulation Monte Carlo (DSMC) approach and showed that the velocity profiles predicted by the EPL formulation were in good agreement with the DSMC results. The phenomenon of velocity inversion has also been investigated by Aoki et al. [3] using two alternative approaches: an asymptotic analytical solution at low Knudsen numbers, and a direct numerical solution of the Boltzmann equation at higher Knudsen numbers. It was found that the occurrence of a velocity inversion depended upon the ratio of accommodation coefficient to Knudsen number. This paper provides an independent reassessment of the velocity inversion process using Maxwell’s original slipvelocity boundary condition [4]. Unlike previous investigations which have generally assumed the accommodation coefficients at the inner and outer cylinder walls are identical, the present study considers the influence of both the inner and outer accommodation coefficients. The analysis shows that the velocity inversion phenomenon is only dependent on the accommodation coefficient of the outer stationary cylinder. From the analysis, criteria are derived for the accommodation coefficient that can be used to predict whether an inverted, partially-inverted or normal velocity profile will occur.