In the slip flow regime, a larger Kn_{e} has a flatter velocity profile and hence smaller MFR (see the first row in the inset of figure 9). Near the free-molecular flow regime, the slip velocity U_{x}(y=±1/2) is proportional to √(T(y=±1/2)), and since √(T (y = ±1/2)/F) decreases as the normalized external acceleration F increases, the normalized velocity slip V_{x}(y=±1/2) = U_{x}(y=±1/2)/F decreases as F increases (see the second row in the inset of figure 9), and so the MFR decreases because the normalized density profile is nearly unitary across the channel when Kn is large. Similar effects of the external force on the MFR are also observed at other values of L/σ and α.

## Conclusions

Through both numerical solution of the generalized Enskog equation and analytical approaches, we have investigated the force-driven Poiseuille flow of a gas between two parallel plates. The dilute-to-dense gas exhibited new flow physics due to the competition of three characteristic length scales: the mean free path, the channel width and the molecular diameter. For elastic collisions in the hard-disc gas, we found the following. (i) In the slip flow regime, the normalized MFR becomes smaller as the confinement (i.e. L/σ) becomes tighter, for a fixed Knudsen number.

In the limit of L/σ → ∞, the MFR approaches that of the Boltzmann equation (in which the binary collisions are localized in space). When L/σ is fixed, the variation of the MFR with the Knudsen number is not monotonic. As the Knudsen number decreases from 0.1, the MFR first increases and then decreases; the maximum MFR occurs when the average solid fraction is approximately 0.3. We explained this exotic behaviour using the Navier–Stokes equation with a first-order velocity slip boundary condition. (ii) In the free-molecular flow regime, for a fixed Knudsen number, the MFR increases as L/σ is reduced, but in the limit L/σ → ∞ the MFR is reduced to that of the Boltzmann equation. Our simple treatment of the average collision frequency accurately captures the influence of the tight wall confinement. (iii) In the transitional flow regime, for a fixed Knudsen number, the variation of the MFR with L/σ is not monotonic, and the minimum MFR is achieved at L/σ ≈ 2–3.

When the collisions between the hard discs are inelastic, we found that the MFR increases as the restitution coefficient decreases due to the increase of the velocity slip and the decrease of the effective Knudsen number. We also proposed a simple formula to predict the anomalous velocity slip (which decreases as the Knudsen number increases). This simple formula and numerical solutions of the generalized Enskog equation also showed that the slip velocity could remain constant with varying Knudsen number at appropriate values of L/σ and the restitution coefficient. Finally, we showed that the normalized MFR reduces as the normalized external acceleration increases. Although we have only considered the diffuse boundary condition, the use of other momentum accommodation coefficients yield qualitatively the same results.

This research sheds new light on the influence of tight confinement on the mass flow rate of dense gases, and indicates that the MFR for Poiseuille flow obtained from the Boltzmann equation is not accurate for dense gases. For practical application to predicting the permeability of ultra-tight shale strata, more work needs to be done; for instance, to include the mean-field term so that a realistic equation of state for the shale gas is recovered, and to properly deal with the gas–wall interactions. Also, the inclusion of a mean-field term will enable us to study the flow dynamics of dense charged grains.

**Acknowledgements**

This work is financially supported by the UK’s Engineering and Physical Sciences Research Council (EPSRC) under grants EP/M021475/1, EP/L00030X/1, EP/K038621/1, EP/I011927/1 and EP/N016602/1. H.L. gratefully acknowledges the financial support of the ‘Thousand Talents Program’ for Distinguished Young Scholars and the National Natural Science Foundation of China under grant no. 51506168.

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This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited. doi:10.1017/jfm.2016.173