Scale dependence and non-isochoric effects on the thermohydrodynamics of rarefied Poiseuille flow

The plane Poiseuille flow of a rarefied gas in a finite length channel, driven by an axial pressure gradient, is analysed numerically to probe (i) the role of ‘dilatation’ ( = ∇ · u≠ 0 ) on its thermohydrodynamics as well as to clarify (ii) the possible equivalence with its well-studied ‘dilatation-free’ or ‘isochoric’ ( Dρ / Dt=0 ) counterpart driven by a constant acceleration. Focussing on the mass flow rate M(Kn) , which is an invariant quantity for both pressure-driven and acceleration-driven Poiseuille flows, it is shown that while M~ log Kn at Kn 1 in the acceleration-driven case, the mass flow saturates to a constant value M~ Kn⁰ at Kn 1 in the pressure-driven case due to the finite length ( Lₓ<∞ ) of the channel. The latter result agrees with prior theory and recent experiments, and holds irrespective of the magnitude of the axial pressure gradient ( Gₚ ). The pressure-dilatation cooling ( ₚ=-p <0 ) is shown to be responsible for the absence of the bimodal shape of the temperature profile in the pressure-driven Poiseuille flow. The dilatation-driven reduction of the shear viscosity and the odd signs of two normal stress differences ( N₁ and N₂ ) in the pressure-driven flow in comparison with those in its acceleration-driven counterpart are explained from the Burnett-order constitutive relations for the stress tensor. While both N₁ and N₂ appear at the Burnett order O(Kn²) in the acceleration-driven flow, the leading term in N₁ scales as (μ/p) due to the non-zero dilatation in the pressure-driven Poiseuille flow which confirms that the two flows are not equivalent even at the Navier–Stokes–Fourier order O(Kn) . The heat-flow rate ( Qqₓ=∫ qₓ(x,y) \, d y ) of the tangential heat flux is found to be negative (i.e. directed against the axial pressure gradient), in contrast to its positive asymptotic value (at Kn 1 ) in the acceleration-driven flow. Similar to the scale-dependence of the mass flow rate, Qqₓ(Kn, Lₓ) is found to saturate to a constant value at Kn 1 in finite length channels. The double-well shape of the qₓ(y) -profile in the near-continuum limit agrees well with predictions from a generalized Fourier law. On the whole, the dilatation-driven signatures (such as the pressure-dilatation work and the ‘normal’ shear-rate differences) are shown to be the progenitor for the observed differences between the two flows with regard to (i) the hydrodynamic fields, (ii) the rheology and (iii) the flow-induced heat transfer.