Estimation of bulk viscosity of dilute gases using a nonequilibrium molecular dynamics approach

A method is proposed for the calculation of bulk viscosity, ${\mu }_b$, of dilute gases using nonequilibrium molecular dynamics (NEMD) simulations. The method uses measurement of mechanical ($P_{\mathrm{mech}}$) and thermodynamic pressure ($P_{\mathrm{thermo}}$) in a NEMD simulation of an expanding fluid and then relates the bulk viscosity to them by the relation ${\mu }_b=(P_{\mathrm{mech}}-P_{\mathrm{thermo}})/\mathbf{\nabla }·\vec{u}$, where $\mathbf{\nabla }·\vec{u}$ is the controlled rate of expansion of the fluid per unit volume. A special emphasis is given to the fundamental physical understanding of bulk viscosity in this work. The proposed method is demonstrated for the estimation of bulk viscosity of dilute nitrogen gas in the temperature range of 200–800 K. Variation of bulk viscosity with temperature is reported in the above-mentioned temperature range and is found to be in a reasonably good agreement with the data available in the literature. Furthermore, variation of bulk viscosity with pressure and volumetric expansion rate is obtained. A weak linear dependency of bulk viscosity with pressure is observed, which increases with temperature. However, no significant effect of the rate of volumetric expansion is observed until ${10}^8{\mathrm{s}}^{-1}$. Moreover, the effect of the direction of volumetric change (expansion vs. compression) is also demonstrated in this work.

Figure 1

Three layers of different velocity in a divergent flow field. Dots represent gas molecules and two vertical dashed lines are imaginary boundaries separating these three fluid layers.

Figure 2

Expansion of gas in a piston-cylinder arrangement.

Figure 3

Shear viscosity of nitrogen obtained from Green-Kubo formulation.

Figure 4

Variation of $(P_{\mathrm{thermo}}-P_{\mathrm{mech}})$ in an expansion and compression with constant $\mathbf{\nabla }·\vec{u}$. The system was first subjected to equilibration from $t=0$ to $t=5$ ns ($\mathbf{\nabla }·\vec{u}=0$), then expansion from $t=5$ to $t=20$ ns ($\mathbf{\nabla }·\vec{u}={10}^8{\mathrm{s}}^{-1}$), again equilibration from $t=20$ to $t=35$ ns ($\mathbf{\nabla }·\vec{u}=0$), compression from $t=35$ to 50 ns ($\mathbf{\nabla }·\vec{u}=-{10}^8{\mathrm{s}}^{-1}$), and finally again equilibration from $t=50$ to 55 ns.

Figure 5

Translational and rotational temperature variation. The inset shows enlarged view of region marked with dotted rectangle.

Figure 6

Time history of $\mathrm{\Delta }T/\mathrm{\Delta }{T}_0$. It should be noted that the $t=0$ ns of this figure corresponds to $t=50$ ns of Fig. 4.

Figure 7

Variation of nonequilibrium parameter, $\psi =(T_{\mathrm{trans}}-T_{\mathrm{rot}})/T_{\mathrm{eq}}$, during the simulation

Figure 8

Pressure vs. volume graph for simple expansion followed by a compression.

Figure 9

Contribution of translational kinetic energy ($P_{\text{kinetic}}$) and pairwise interaction ($P_{\text{pair}}$) in calculation of mechanical pressure

Figure 10

(a) Comparison of $P_{\mathrm{thermo}}-P_{\mathrm{mech}}$ vs. time for the two cases. In case 1, fluid is first expanded and then compressed, whereas in case 2, the fluid is first compressed and then expanded. To ease the comparison between these two cases, the expansion and compression parts of case 2 are separated and superimposed on the corresponding parts of case 1 curve. Panels (b) and (c) show the comparison of variation of $P_{\mathrm{thermo}}-P_{\mathrm{mech}}$ and ${\mu }_b$ vs. temperature during expansion and compression, respectively, for both the cases.

Figure 11

Variation of bulk viscosity, ${\mu }_b$, with temperature at pressure of 1 bar.

Figure 12

Variation of bulk viscosity with pressure.

Figure 13

Four different trajectories for variation of bulk viscosity with pressure at a constant temperature of 350 K in the plateau region of 1 to 2 bar (see Fig. 12).

Figure 14

Variation of bulk viscosity with $\mathbf{\nabla }·\vec{u}$ at 300 K.