Blast Dynamics in a Dissipative Gas

The blast caused by an intense explosion has been extensively studied in conservative fluids, where the Taylor–von Neumann–Sedov hydrodynamic solution is a prototypical example of self-similarity driven by conservation laws. In dissipative media, however, energy conservation is violated, yet a distinctive self-similar solution appears. It hinges on the decoupling of random and coherent motion permitted by a broad class of dissipative mechanisms. This enforces a peculiar layered structure in the shock, for which we derive the full hydrodynamic solution, validated by a microscopic approach based on molecular dynamics simulations. We predict and evidence a succession of temporal regimes, as well as a long-time corrugation instability, also self-similar, which disrupts the blast boundary. These generic results may apply from astrophysical systems to granular gases, and invite further cross-fertilization between microscopic and hydrodynamic approaches of shock waves.

Figure 1

Section of the blast wave: cartoon (top) and hard sphere MD simulation (bottom). Dissipation causes particles to accrete into a dense, hollow shell. Equations (5)–(7) reveal its layered structure. First, the shock front where particles from the blast collide and mix with those at rest, creating incoherent motion. Then, a fixed-width cooling region (CR) where further compression occurs as temperature is dissipated, and finally a self-similar cold fluid region where velocities are well aligned, around a central cavity. This stands in stark contrast with the elastic case, where the bulk of the blast is comprised of a single, entirely self-similar region.

Figure 2

Hydrodynamic profiles from MD simulations for dense conservative (crosses) and dissipative (circles) fluids with ${\varphi }_{\mathrm{rest}}=0.3$, and analytical solutions (solid lines). Profiles are rescaled by the Rankine-Hugoniot boundary value (see Ref. [29]), from left to right: $n(\lambda )/n_{\mathrm{RH}}$, $u(\lambda )/u_{\mathrm{RH}}$, and $\mathrm{\Theta }(\lambda )/{\mathrm{\Theta }}_{\mathrm{RH}}$ with $\lambda =r/R(t)$. The conservative solution is partitioned into the gas at rest, the shock front (shaded area), and the self-similar bulk. This is the first validation of a TvNS-like solution in a dense fluid. For any dissipation, i.e., $\alpha <1$ (here $\alpha =0.8$), the bulk structure becomes threefold, as shown by dashed vertical lines: the cooling region between the front and $R_c\approx 0.93R$; the maximally dense cold fluid down to $R_i\approx 0.78R$; and the central cavity.

Figure 3

Left: Shock radius by angular sector $R(\theta ,t)$ at successive times (see Ref. [29] for its definition). The line thickness represents the mean free path in the shock front, or about half its average width $w\approx 0.01$. Right: Corrugation width $\delta R\sim t^{\delta +s}$ with theoretical exponent $s\simeq 0.3$ (solid line) and numerical validation for $\alpha =0.3$ (circles) and $\alpha =0.8$ (crosses). Inset: Real part of the dispersion relation $s(k)$ for the unstable mode, crossing the marginal stability line $\Re (s)=0$ (dashed line), with a plateau at $\Re (s)\approx 0.3$.