Internal energy in compressible Poiseuille flow

We analyze a compressible Poiseuille flow of ideal gas in a plane channel. We provide the form of internal energy $U$ for a nonequilibrium stationary state that includes viscous dissipation and pressure work. We demonstrate that $U$ depends strongly on the ratio $\mathrm{\Delta }p/p_0$, where $\mathrm{\Delta }p$ is the pressure difference between inlet and outlet and ${\mathrm{p}}_0$ is the outlet's pressure. In addition, $U$ depends on two other variables: the channel aspect ratio and the parameter equivalent to Reynolds number. The stored internal energy, $\mathrm{\Delta }U=U-U_0$, is small compared to the internal energy $U_0$ of the equilibrium state for a moderate range of values of $\mathrm{\Delta }p/p_0$. However, $\mathrm{\Delta }U$ can become large for big $\mathrm{\Delta }p$ or close to vacuum conditions at the outlet ($p_0\approx 0\mathrm{Pa}$).

Figure 1

Geometry and boundary conditions (BCs) of the modeled system. Flow profile at the entrance is fully developed with the average pressure set to the value ${\mathrm{p}}_{\mathrm{av}}$. $L_y$ and $L_x$ denote width and length of the channel, respectively. Inlet and the channel's sidewalls are thermostated at $T=T_0$. We applied zero heat flux BC across the outlet $(J_{\mathrm{qout}}=0)$.

Figure 2

(a) Velocity in the flow direction and (b) the temperature profile calculated using material parameters for He. The values of other parameters are: $L_x=0.5\mathrm{m}, L_y=0.025\mathrm{m}, T_0=300\mathrm{K}, \mu =2.0048\times {10}^{–5}\mathrm{Pa}\mathrm{s}, k=0.152\mathrm{W}{\mathrm{m}}^{–1}{\mathrm{K}}^{–1}, p_0=1\mathrm{atm}, \mathrm{\Delta }p=0.5\mathrm{Pa}$. The profiles correspond to a fully developed flow and temperature profiles and are compared to the analytical solution adopted from Ref. [29]. The maximal absolute errors equal to (a) ${\left|u\right|}_{\mathrm{err}} =4.5\times {10}^{–5}\mathrm{m}{\mathrm{s}}^{–1}$ and (b) $|T-T_0{|}_{\mathrm{err}}=3.5\times {10}^{-8}$ K.

Figure 3

Dependence of $\mathrm{\Delta }U/U_0$ on $\frac{\mathrm{\Delta }p}{p_0}$ obtained from numerical model and its linear fit (equation and coefficients shown in the legend). Each point on the plot corresponds to a single simulation with a unique set of parameters varied arbitrarily within the following ranges: $\mathrm{\Delta }p$ from 0.001 Pa to 0.08 Pa, $p_0$ from 0.2 atm to 2 atm, $T_0$ from 50 to 2342 K, $L_x$ from 0.5 to 8.4 m and $L_y$ from 0.05 to 0.1 m. The exact algorithm of selecting parameter values is described in the SM.

Figure 4

The results of simulations with $\frac{\mathrm{\Delta }p}{p_0}$ fixed at $2.5\times {10}^{–7}$ while the values of all the other parameters were changed in every simulation and varied as follows: $\mathrm{\Delta }p$ from 0.001Pa to 0.08 Pa, $p_0$ from 0.2 to 2 atm, $T_0$ from 50 to 2342 K, $L_x$ from 0.5 to 8.4 m and $L_y$ from 0.05 to 0.1 m. $\frac{\mathrm{\Delta }U}{U_0}-\frac{1}{2}\frac{\mathrm{\Delta }p}{p_0}$ is presented as a function of two arbitrarily selected parameters: (a) $T_0$ and (b) $p_0$. Additionally, the separate 11 line segments were color-coded and enumerated. The range of parameters for each segment is given in Table III in the SM.

Figure 5

(a) $\frac{\mathrm{\Delta }U}{U_0}$ and (b) $\frac{\mathrm{\Delta }U}{U_0}-\frac{1}{2}\frac{\mathrm{\Delta }p}{p_0}$ in function of $\mathrm{\Delta }p/p_0$. $\mathrm{\Delta }p$ was changed from 0.005 Pa to 0.163 Pa. The values of other simulation parameters are constant and equal to $p_0=1\mathrm{atm}, T_0=600\mathrm{K}, L_x=0.5\mathrm{m}, L_y=0.05\mathrm{m}$. The red line in is the best quadratic fit with the equation and coefficients shown in the legend.

Figure 6

$\mathrm{\Delta }U/U_0$ in the function of (a) $\mathrm{\Delta }p/p_0$ and (b) $\mathrm{\Delta }p$. The values of other dimensionless parameters are constant and equal to $\frac{p_0{L}_x}{\mu }\sqrt{\frac{M}{R{T}_0}}=3\times {10}^6, \frac{L_y}{L_x}=0.1, \mathrm{Pr}=0.67$. The red line in (a) is the best linear fit with the equation and coefficients shown in the legend. Note that $\mathrm{\Delta }p$ is not a proper variable for $\mathrm{\Delta }U/U_0$.

Figure 7

$\frac{\mathrm{\Delta }U}{U_0}-\frac{1}{2}\frac{\mathrm{\Delta }p}{p_0}$ in the function of (a) $\frac{L_y}{L_x}$ and (b) $L_x$. The values of other dimensionless parameters are constant and equal to $\frac{\mathrm{\Delta }p}{p_0}=2.5\times {10}^{–7}, \frac{p_0{L}_x}{\mu }\sqrt{\frac{M}{R{T}_0}} =3\times {10}^6, \mathrm{Pr}=0.67$. The solid red and dashed blue lines in (a) are the best fit of a second and a fourth-degree polynomial function. The exact equations and polynomial coefficients are given in Table IV in the SM. Note that $L_x$ is not a proper variable for $\mathrm{\Delta }U/U_0$.

Figure 8

$\frac{\mathrm{\Delta }U}{U_0}-\frac{1}{2}\frac{\mathrm{\Delta }p}{p_0}$ in the function of (a) $\frac{p_0{L}_x}{\mu }\sqrt{\frac{M}{R{T}_0}}$ and (b) $p_0$. The values of other dimensionless parameters are constant and equal to $\frac{\mathrm{\Delta }p}{p_0}=2.47\times {10}^{–7}, \frac{L_y}{L_x}=0.1, \mathrm{Pr}=0.67$. The solid red and dashed blue lines in (a) are the best fit of a second and a fourth-degree polynomial function. The exact equations and polynomial coefficients are given in Table IV in the SM. Note that $p_0$ is not a proper variable for $\mathrm{\Delta }U/U_0$.

Figure 9

$\frac{\mathrm{\Delta }U}{U_0}-\frac{1}{2}\frac{\mathrm{\Delta }p}{p_0}$ in the function of $\mathrm{Pr}$ for $\frac{\mathrm{\Delta }p}{p_0}$ fixed at (a) $2.5\times {10}^{–7}$ and (b) $2.5\times {10}^{–4}$. The values of other dimensionless parameters are constant the same for both cases: $\frac{p_0{L}_x}{\mu }\sqrt{\frac{M}{R{T}_0}}=3\times {10}^6, \frac{L_y}{L_x}=0.1$.