Hydrodynamics, wall-slip, and normal-stress differences in rarefied granular Poiseuille flow

Hydrodynamic fields, macroscopic boundary conditions, and non-Newtonian rheology of the acceleration-driven Poiseuille flow of a dilute granular gas are probed using “direct simulation Monte Carlo” method for a range of Knudsen numbers ($\text{Kn}$, the ratio between the mean free path and the macroscopic length), spanning the rarefied regime of slip and transitional flows. It is shown that the “dissipation-induced clustering” (for $1-e_n>0$, where $e_n$ is the restitution coefficient), leading to inhomogeneous density profiles along the transverse direction, competes with “rarefaction-induced declustering” (for $\text{Kn}>0$) phenomenon, leaving seemingly “anomalous” footprints on several hydrodynamic and rheological quantities; one example is the well-known rarefaction-induced temperature bimodality, which could also result from inelastic dissipation that dominates in the continuum limit ($\text{Kn}\to 0$) as found recently [Alam et al., J. Fluid Mech. 782, 99 (2015)]. The simulation data on the slip velocity and the temperature slip are contrasted with well-established boundary conditions for molecular gases. A modified Maxwell-Navier-type boundary condition is found to hold in granular Poiseuille flow, with the velocity slip length following a power-law relation with Knudsen number ${\text{Kn}}^{\delta }$, with $\delta \approx 0.95$, for $\text{Kn}\le 0.1$. Transverse profiles of both first $\left[{\mathcal{N}}_1\left(y\right)\right]$ and second $\left[{\mathcal{N}}_2\left(y\right)\right]$ normal stress differences seem to correlate well with respective density profiles at small $\text{Kn}$; their centerline values $[{\mathcal{N}}_1\left(0\right)$ and ${\mathcal{N}}_2\left(0\right)]$ can be of “odd” sign with respect to their counterparts in molecular gases. The phase diagrams are constructed in the ($\text{Kn},1-e_n$) plane that demarcates the regions of influence of inelasticity and rarefaction, which compete with each other resulting in the sign change of both ${\mathcal{N}}_1\left(0\right)$ and ${\mathcal{N}}_2\left(0\right)$. The results on normal stress differences are rationalized via a comparison with a Burnett-order theory [Sela and Goldhirsch,  J. Fluid Mech. 361, 41 (1998)], which is able to predict their correct behavior at small values of the Knudsen number. Lastly, the Knudsen paradox and its dependence on inelasticity are analyzed and contrasted with related recent works.

Figure 1

Left: Schematic showing the acceleration-driven Poiseuille flow of hard spheres, with $a$ representing the acceleration (directed along the $x$ direction); two impenetrable-walls are placed at $y=\pm L_y/2$ and both $x$ and $z$ directions are periodic. Top-right: Typical temperature profile of “bimodal” shape having a local minimum at $y=0$ and two maxima $T\left(y\right)=T_{max}$ at $y=\pm y_s$. Bottom-right: Typical variation of the flow-rate, $Q=\int \rho \left(y\right)u_x\left(y\right)dy$, with Knudsen number $\text{Kn}$, with $Q=Q_{min}$ (“Knudsen minimum”) at $\text{Kn}=O\left(1\right)$; here $\rho \left(y\right)$ and $u_x\left(y\right)$ represent the hydrodynamic profiles of the density and streamwise velocity, respectively, across the channel-width.

Figure 2

Profiles of (a) streamwise velocity and (b) temperature and their comparisons with Mansour et al. [13] for a molecular gas at a Knudsen number of $\text{Kn}=0.1$; the dimensionless acceleration is set to $\widehat{a}=0.3$ as in Ref. [13]. Note that the temperature has been rescaled by $T_w$ and the velocity by $\sqrt{2k_B{T}_w/m}$.

Figure 3

Effect of restitution coefficient ($e_n$) on density profiles for (a) $\text{Kn}=0.05$ and (b) $\text{Kn}=1$; the dimensionless acceleration is $\widehat{a}=0.5$.

Figure 4

Variations of (a) $\rho \left(0\right)$ and (b) ${\rho }_{\text{fw}}$ with Knudsen number $\text{Kn}$ for different $e_n$; variations of (c) $\rho \left(0\right)$ and (d) ${\rho }_{\text{fw}}$ with restitution coefficient $e_n$ for different $\text{Kn}$. Other parameters are as described in the caption of Fig. 3.

Figure 5

Profiles of local Knudsen number ${\text{Kn}}_y$ for different $e_n$: the global Knudsen number is (a) $\text{Kn}=0.05$ and (b) $\text{Kn}=1$. Other parameters are as described in the caption of Fig. 3.

Figure 6

Temperature profiles, $T/T\left(0\right)$, for (a) $\text{Kn}=0.05$ and (b) $\text{Kn}=1$. Parameter values are the same as described in the caption of Fig. 3.

Figure 7

Variations of excess temperature $\mathrm{\Delta }T$, Eq. (22), with (a) Knudsen number for different $e_n$ and (b) restitution coefficient $e_n$ for different $\text{Kn}$. Other parameters are as described in the caption of Fig. 6.

Figure 8

Phase diagram of temperature bimodality in the $(\text{Kn},1-e_n)$ plane for $\widehat{a}=0.5$; the black triangles represent data for $e_n=0.99,0.995,0.999$ with $\widehat{a}=5$. The blue-circled line delineates the regions of (i) dissipation-driven and (ii) rarefaction-driven bimodal temperature profiles; the region to the left of red-squared line represents the region of “unimodal” temperature with $T_{max}=T\left(0\right)$, i.e., $\mathrm{\Delta }T=0$. The “unimodal-to-bimodal” transition occurring along two arrows is ascertained in Fig. 9.

Figure 9

“Unimodal-to-bimodal” transition of temperature profiles at (a) $\text{Kn}=0.03$ (with increasing $1-e_n$, i.e., “dissipation-driven” bimodality) and (b) $e_n=0.9999$ (with increasing $\text{Kn}$, i.e., “rarefaction-driven” bimodality). In each panel, the temperature has been scaled by its centerline value; other parameters are as described in the caption of Fig. 8.

Figure 10

(a), (b) Streamwise velocity profiles at (a) $\text{Kn}=0.05$ and (b) $\text{Kn}=1$ for different values of $e_n$. (c), (d) Variations of slip velocity with (c) $\text{Kn}$ and (d) $e_n$ for $\widehat{a}=0.5$.

Figure 11

(a) Validity of Maxwell-Navier velocity-slip condition: dimensionless slip-length $\zeta =u_s/u_w^{\prime }$, Eq. (25), versus local Knudsen number ${\text{Kn}}_{\mathrm{fw}}$ in linear scale (main panel) and logarithmic scale (inset). (b) Variation of near-wall velocity gradient $u_w^{\prime }={(du/dy)}_{\text{fw}}$ with $\text{Kn}$ for different $e_n$.

Figure 12

(a), (b) Variations of temperature slip $T_s$, Eq. (27), with (a) $\text{Kn}$ and (b) $e_n$; in panel (b), $T_s$ has been scaled by its value for $T_s(e_n=1)$. (c) Variation of dimensionless $T_w^{\prime }={(dT/dy)}_{\text{fw}}$ with $\text{Kn}$ for different $e_n$; inset shows the variation of temperature slip-length ${\zeta }_T$, Eq. (29), with local Knudsen number ${\text{Kn}}_{\text{fw}}$. (d) Temperature profiles for different $\text{Kn}$ with $e_n=1$, confirming the sign-change of $T_w^{\prime }$ with increasing $\text{Kn}$; see text for details. Other parameters are as described in the caption of Fig. 11.

Figure 13

Comparison between the present DSMC results (lines) with those of Uribe and Garcia (circles Ref. [14]) for (a) pressure $[p/p(0\left)\right]$ and (b) shear stress $[p_{xy}/p\left(0\right)]$ in the Poiseuille flow of a molecular gas at $\widehat{a}=0.3$ ($a=0.00016$) and $\text{Kn}=0.1$.

Figure 14

Effects of dissipation ($e_n$) on the profiles of (a), (b) pressure and (c), (d) shear stress for (a), (c) $\text{Kn}=0.05$ and (b), (d) $\text{Kn}=1$. The dimensionless acceleration is $\widehat{a}=0.5$.

Figure 15

(a) Effects of dissipation ($e_n$) on the profile of first $\left[{\mathcal{N}}_1\left(y\right)\right]$ normal stress difference at $\text{Kn}=0.05$; other parameters are as in Fig. 14. (b) Variations of the center-line value of ${\mathcal{N}}_1\left(0\right)$ with $\text{Kn}$ for different $e_n$. (c) Phase diagram in the ($\text{Kn},1-e_n$) plane delineating the regions of positive and negative ${\mathcal{N}}_1\left(0\right)$; the thick black contour represents ${\mathcal{N}}_1\left(0\right)=0$.

Figure 16

(a) Effect of dissipation ($e_n$) on the profile of second (${\mathcal{N}}_2$) normal-stress difference at $\text{Kn}=0.05$; other parameters are as described in the caption of Fig. 14. (b) Variations of ${\mathcal{N}}_2\left(0\right)$ with $\text{Kn}$ for different $e_n$. (c) Phase diagram in the ($\text{Kn},1-e_n$) plane delineating the regions of positive and negative ${\mathcal{N}}_2\left(0\right)$; the thick black contour represents ${\mathcal{N}}_2\left(0\right)=0$.

Figure 17

(a) Variation of $\chi$, Eq. (39), with $\text{Kn}$; left and right insets show the corresponding variations of ${\rho }^{\prime \prime }\left(0\right)$ and $T^{\prime \prime }\left(0\right)$. (b) Variations of ${\mathcal{N}}_1\left(0\right)$ and ${\mathcal{N}}_2\left(0\right)$ with $\text{Kn}$. For all cases, $e_n=0.95$ and $\widehat{a}=0.5$.

Figure 18

Same as Fig. 17, but for (a) $e_n=0.99$ and (b) $e_n=1$.

Figure 19

Nondimensional mass-flow rate vs. Knudsen number in the Poiseuille flow of a molecular gas. The circles and filled squares denote results from present DSMC code (with $\widehat{a}=0.5$) and Ref. [16], respectively; the dashed line is the numerical solution of the Boltzmann equation (Ohwada et al. [11]) taken from Fig. 2 of Ref. [16].

Figure 20

Variations of the flow rate with Knudsen number for different $e_n$ for dimensionless accelerations of (a) $\widehat{a}=0.5$ and (b) $\widehat{a}=10$. The inset in panel (a) is a zoom of its large $\text{Kn}$ variation.

Figure 21

Dissipation-dominated and rarefaction-dominated regimes in the ($\text{Kn},{\text{Kn}}_0/{\text{Kn}}_{\text{fw}}$) plane. In a dissipation-dominated regime the “local” Knudsen numbers $[{\text{Kn}}_0$ (filled symbols) and ${\text{Kn}}_{\text{fw}}$ (open symbols) that correspond to the gas-densities at the centerline and walls, respectively, of the channel] are different from “global” Knudsen number ($\text{Kn}$, based on preset mean density).

Figure 22

(a) Temporal evolution of total kinetic energy, $E={\sum }_{i}m{v}_i^2/2$ ($\times {10}^{-5}$), of all particles for $\text{Kn}=0.05$ and $\widehat{a}=0.5$; the time axis is scaled as $t\times {10}^5$. (b) Streamwise velocity and (c) temperature ($T/T_w$) profiles, averaged over different time windows for $e_n=1$.