Exact solutions for shock waves in dilute gases

In 1922 Becker found an exact solution for shock waves in gases using the Navier–Stokes–Fourier constitutive equations for a Prandtl number of value 3/4 with constant transport coefficients. His analysis has been extended to study some cases where an implicit solution can be found in an exact way. In this work we consider this problem for the so-called soft-spheres model in which the viscosity and thermal conductivity are proportional to a power of the temperature $\eta ,\kappa \propto T^{\sigma }$. In particular, we give implicit exact solutions for the Maxwell model ($\sigma =1$), hard spheres ($\sigma =1/2$), and when $\sigma$ (the viscosity index) is a natural number.

Figure 1

(a) Explicit and implicit solutions for $\text{Pr}=3/4, \sigma =0$, and ${\tau }_0=1/5$. Reduced velocity vs the reduced distance, $v\left(s\right)$ vs $s$. Solid line: Explicit solution given by Eq. (45); open circles: implicit solution given by Eq. (42); solid circles: numerical solution to Eqs. (15) and (16); dashed line: solution with the plus sign in Eq. (43). (b) Implicit and numerical solutions for $\text{Pr}=3/4, \sigma =1$, and ${\tau }_0=1/5$. Velocity vs reduced distance, $v\left(s\right)$ vs $s$. Open circles: implicit solution given by Eq. (47); solid circles: numerical solution to Eqs. (15) and (16); solid line: numerical solution to Eq. (24) using the initial condition $v\left(0\right)=2/3$. (c) Implicit and numerical solutions for $\text{Pr}=3/4, \sigma =1/2$, and ${\tau }_0=1/5$. Velocity vs reduced distance, $v\left(s\right)$ vs $s$. Open circles: implicit solution given by Eq. (47); solid circles: numerical solution to Eqs. (15) and (16) with initial condition $v\left(0\right)=2/3$ and $\tau \left(0\right)=14/45$; solid line: numerical solution to Eq. (24) using the initial condition $v\left(0\right)=2/3$. (d) Normalized density profiles, ${\rho }_n$ vs $s$, for ${\tau }_0=1/5$. Solid line: Becker solution; long dashed line: hard-sphere model; dashed line: Maxwell model; dotted line: $\sigma =2$.

Figure 2

(a) Normalized density profiles for $M=8, {\rho }_n$ vs $s_A$, obtained by solving Eq. (24) with initial condition $v\left(0\right)=134/323$ for different values of $\sigma$. Circles: Experiments by Steinhilper [50]; solid line: solution for $\sigma =0.92$; dotted line: solution for $\sigma =1/2$; dashed line: solution for $\sigma =1$; open circles: ab initio calculations. (b) Orbits, $\tau$ vs $v$, for $M=8$ and different Prandtl numbers. Except for $\text{Pr}=3/4$ the orbits were obtained by solving Eqs. (10) with ${\mathrm{\Xi }}_1=1$ and ${\mathrm{\Xi }}_2=15/\left(8\text{Pr}\right)$. Solid line: $\text{Pr}=3/4$, see Eq. (21); dotted line: $\text{Pr}=2/3$; long dashed line: $\text{Pr}=1/10$; dashed line: $\text{Pr}=2$; circles: direct simulation Monte Carlo method for $\sigma =0.68$.

Figure 3

Entropy change for $M=\sqrt{3}$ (${\tau }_0=1/5$) and different values for the viscosity index. Solid line: Explicit solution for $\sigma =0$; dashed line: solution to Eq. (24) for $\sigma =1/2$; long dashed line: solution to Eq. (24) for $\sigma =1$; pointed line: solution to Eq. (24) for $\sigma =3/2$. Its maximum value is the same for all values of $\sigma$ and it is given as $\mathrm{\Delta }{S}_{\text{max}}^{*}=0.261$, however the $s_A-$ values where they occur are: $s_A=-0.277(\sigma =0),s_A=-0.343(\sigma =1/2),s_A=-0.424(\sigma =1.0),s_A=-0.524(\sigma =3/2)$.