^{1,a)}, Craig White

^{1,b)}, Thomas J. Scanlon

^{1,c)}, Yonghao Zhang

^{1,d)}and Jason M. Reese

^{1,e)}

### Abstract

The gas flow between two concentric rotating cylinders is considered in order to investigate non-equilibrium effects associated with the Knudsen layers over curved surfaces. We investigate the nonlinear flow physics in the near-wall regions using a new power-law (PL) wall-scaling approach. This PL model incorporates Knudsen layer effects in near-wall regions by taking into account the boundary limiting effects on the molecular free paths. We also report new direct simulation Monte Carlo results covering a wide range of Knudsen numbers and accommodation coefficients, and for various outer-to-inner cylinder radius ratios. Our simulation data are compared with both the classical slip flow theory and the PL model, and we find that non-equilibrium effects are not only dependent on Knudsen number and accommodation coefficient but are also significantly affected by the surface curvature. The relative merits and limitations of both theoretical models are explored with respect to rarefaction and curvature effects. The PL model is able to capture some of the nonlinear trends associated with Knudsen layers up to the early transition flow regime. The present study also illuminates the limitations of classical slip flow theory even in the early slip flow regime for higher curvature test cases, although the model does exhibit good agreement throughout the slip flow regime for lower curvature cases. Torque and velocity profile comparisons also convey that a good prediction of integral flow properties does not necessarily guarantee the accuracy of the theoretical model used, and it is important to demonstrate that field variables are also predicted satisfactorily.

The research leading to these results has received funding from the European Community's Seventh Framework Programme FP7/2007-2013 under grant agreement ITN GASMEMS (Grant Agreement No. 215504). The author C.W. gratefully acknowledges funding from the James Weir Foundation. The author J.M.R. gratefully acknowledges funding through EPSRC Programme Grant EP/I011927/1. Our calculations were performed on the 1100 core HPC Facility of the Faculty of Engineering at the University of Strathclyde. The authors thank the reviewers for their useful comments.

I. INTRODUCTION

II. DSMC SIMULATIONS

III. POWER-LAW WALL SCALING MODEL

A. Effective mean free path

B. Shear flow between two parallel plates

IV. CYLINDRICAL COUETTE FLOW

A. Governing equations

V. RESULTS AND DISCUSSION

VI. CONCLUSIONS

### Key Topics

- Couette flows
- 20.0
- Slip flows
- 15.0
- Torque
- 12.0
- Slip boundary effects
- 11.0
- Navier Stokes equations
- 10.0

## Figures

Contours of velocity calculated by dsmcFoam for a typical concentric cylindrical Couette flow case.

Contours of velocity calculated by dsmcFoam for a typical concentric cylindrical Couette flow case.

Comparison of DSMC tangential velocity profiles from dsmcFoam and Ref. 3 .

Schematic of Couette flow between concentric rotating cylinders.

Schematic of Couette flow between concentric rotating cylinders.

Normalised half channel velocity profiles for planar Couette flow at various Knudsen numbers. Comparison of our power-law (PL) model results with the DSMC data, and the R26 and R13 moment equations. 46

Normalised half channel velocity profiles for planar Couette flow at various Knudsen numbers. Comparison of our power-law (PL) model results with the DSMC data, and the R26 and R13 moment equations. 46

Variation of the non-dimensional velocity [U* = u ϕ/(ω1 R 1)] with normalised radial distance for cylindrical Couette flow with σ1 = σ2 = σ. Comparison of PL model results against (a) DSMC data and (b) the classical slip solution (Yuhong et al. 5 ). The results are presented for and R 2/R 1 = 5/3.

Variation of the non-dimensional velocity [U* = u ϕ/(ω1 R 1)] with normalised radial distance for cylindrical Couette flow with σ1 = σ2 = σ. Comparison of PL model results against (a) DSMC data and (b) the classical slip solution (Yuhong et al. 5 ). The results are presented for and R 2/R 1 = 5/3.

Variation of the non-dimensional velocity [U* = u ϕ/(ω1 R 1)] with normalised radial distance for cylindrical Couette flow with σ1 = σ2 = σ. Comparison of PL model results against (a) DSMC data and (b) the classical slip solution. 5 The results are presented for and R 2/R 1 = 5/3.

Variation of the non-dimensional velocity [U* = u ϕ/(ω1 R 1)] with normalised radial distance for cylindrical Couette flow with σ1 = σ2 = σ. Comparison of PL model results against (a) DSMC data and (b) the classical slip solution. 5 The results are presented for and R 2/R 1 = 5/3.

Variation of the non-dimensional velocity [U* = u ϕ/(ω1 R 1)] with normalised radial distance for cylindrical Couette flow with σ1 = σ2 = σ. Comparison of PL model results against (a) DSMC data and (b) the classical slip solution. 5 The results are presented for and R 2/R 1 = 5/3.

Effect of the power-law (PL) exponent n on the cylindrical Couette flow velocity profiles. Variation of the non-dimensional velocity [U* = u ϕ/(ω1 R 1)] with normalised radial distance for σ1 = 1.0 and (a) σ2 = 1.0, (b) σ2 = 0.4, and (c) σ2 = 0.15, for and R 2/R 1 = 5/3.

Effect of the power-law (PL) exponent n on the cylindrical Couette flow velocity profiles. Variation of the non-dimensional velocity [U* = u ϕ/(ω1 R 1)] with normalised radial distance for σ1 = 1.0 and (a) σ2 = 1.0, (b) σ2 = 0.4, and (c) σ2 = 0.15, for and R 2/R 1 = 5/3.

Variation of the non-dimensional velocity [U* = u ϕ/(ω1 R 1)] with normalised radial distance for cylindrical Couette flow with σ1 = σ2 = σ. Comparison of PL model results against DSMC data. The results are presented for and R 2/R 1 = 6/5 (top left), 2 (top right), 3 (bottom left), and 5 (bottom right).

Variation of the non-dimensional velocity [U* = u ϕ/(ω1 R 1)] with normalised radial distance for cylindrical Couette flow with σ1 = σ2 = σ. Comparison of PL model results against DSMC data. The results are presented for and R 2/R 1 = 6/5 (top left), 2 (top right), 3 (bottom left), and 5 (bottom right).

Variation of the non-dimensional velocity [U* = u ϕ/(ω1 R 1)] with normalised radial distance for cylindrical Couette flow with σ1 = σ2 = σ. Comparison of PL model results against DSMC data. The results are presented for and R 2/R 1 = 6/5 (top left), 2 (top right), 3 (bottom left), and 5 (bottom right).

Variation of normalised torque Γ exerted on the rotating inner cylinder with Knudsen number ( ). Our DSMC data are compared with both the slip and the PL models. The data are obtained for a specific case, where the accommodation coefficients of both the inner and outer cylinders are unity (σ1 = σ2 = 1) and R 2/R 1 = 6/5 (top left), 5/3 (top right), 3 (bottom left), and 5 (bottom right).

Variation of normalised torque Γ exerted on the rotating inner cylinder with Knudsen number ( ). Our DSMC data are compared with both the slip and the PL models. The data are obtained for a specific case, where the accommodation coefficients of both the inner and outer cylinders are unity (σ1 = σ2 = 1) and R 2/R 1 = 6/5 (top left), 5/3 (top right), 3 (bottom left), and 5 (bottom right).

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