I’m writing this post to help me learn some basics of shocks in fluid dynamics. It’s based on the book The Physics of Fluids and Plasmas by Arnab Rai Choudhuri.
In fluid dynamics, a shock wave is a region where fluid dynamical variables vary rapidly. Those fluid variables are density, pressure, and velocity.
We can write an expression of them that treats the shock as a mathematical discontinuity using conservation of mass, momentum, and energy on either side of the location of the shock:
\( \rho_1 v_1 = \rho_2 v_2 \) (mass)
\( p_1 +\rho_1v_1^2 = p_2 + \rho_2v_2^2 \) (momentum)
\( \frac{1}{2}v_1^2+\frac{\gamma p_1}{(\gamma-1)\rho_1} = \frac{1}{2}v_2^2+\frac{\gamma p_2}{(\gamma-1)\rho_2} \) (energy)
In the energy equation, the total energy is the kinetic energy plus the enthalpy (for a perfect gas, i.e. an non-interacting gas with a cosntant heat capacity) for each side of the shock.
These equations can be combined to write the density to the right of the shock ($\rho_2$) as a function of the left-side fluid variables:
\( \frac{\rho_2}{\rho_1}=\frac{(\gamma+1)M^2}{2+(\gamma-1)M^2} \)
wher $M$ is the Mach number, which gives a measure of how quickly the shock is moving compared to the local sound speed of the medium through which it’s traveling. It’s defined as
\( M=\frac{v_1}{\sqrt{\gamma p_1/\rho_1}}=\frac{v_1}{c_{s,1}} \).
If $M>1$, then the shock is traveling faster than the local sound speed. If it’s not greater than one, then it is possible for acoustic waves to travel faster than the wavefront and dissipate the shock. Also, if the mach number is greater than one, then that tells us that $\rho_2/\rho_1>1$. In other words, the downstream medium (i.e. $\rho_2$) is compressed by the shock wave. The faster the wave is traveling, the more that medium gets compressed. We can write down a similar relation for pressure:
\( \frac{p_2}{p_1}=\frac{2\gamma^2-(\gamma-1)}{\gamma+1} \).
Equations that relate fluid variables before and after a shock are called Rankine-Hugoniot relations. These apply for shock waves that conserve their energy, so radiative shock waves are a separate discussion.
Spherical Blast Waves
Shocks originating from an exposion that propagates outward (such as supernovae) are spherical blast waves. The common formalism for these types of shocks relies on the assumption that they can be treated in a self-similar manner. This means that they can be described by initial conditions that expand outward uniformly over time. This allows us to solve for quantities like density, velocity, etc. within the sphere of the blast wave over time. This self-similar approximation is commonly referred to as a Sedov-Taylor solution.
Assume that an explosion has some energy $E$ triggers a spherical blast wave that propagates in an ambient medium of density $\rho_1$. The pressure of this medium is negligible compared to that of the blast wave. $\lambda$ is a scale parameter that quanitifes that size of the blast wave a certain time after the explosion. Relating these, we have:
\( \lambda=(Et^2/\rho_1)^{1/5} \).
We can also define a dimesnionless parameter that quantifies the distance of a shell of the explosion from its center:
\( \xi=r/\lambda=r\Big(\frac{\rho_1}{Et^2}\Big)^{1/5} \).
The shock front is located at $\xi_0$ and the shock has radius:
\( r_s(t)=\xi_0\Bigg(\frac{Et^2}{\rho_1\Bigg)^{1/5} \).
We can also take the derivative of this expression to find the velocity. These two relations give us an idea of how the size and velocity of a spherical blast scale with time:
\( \boxed{r_s\propto t^{2/5}} \,\,\,\, \text{and} \,\,\,\, \boxed{v_s\propto t^{-3/5}} \).
These can be used to get an idea of how a $1~M_\odot$ supernova with an initial velocity of $10^4~\text{km}\text{s}^{-1}$ evolves as it propagates into a $10^{-24}~\text{g}\,\text{cm}^{-3}$ ISM using coefficients of $0.3~\text{pc}$ and $10^5~\text{km}\text{s}^{-1}$ (with $t$ in years). Radiative losses will start to matter after $10^5~\text{yr}$.
We can also use Rankine-Hugoniot relations to see how density, velocity, and pressure of a spherical blast wave evolve over time. There’s a derivation in Choudhuri that uses these conditions and the continuity equation, Euler equation, and adiabatic equation adapted to a spherically symmetric adiabatic gas flow to solve for these quanitites.