Passing through a shock front moving with a speed
, the physical variables
,
, and
change abruptly.
Since the basic equations of hydrodynamics is unchanged after chosing a system moving
,
the continuity equation
![\begin{displaymath}
\frac{\partial \rho}{\partial t}+\frac{\partial \rho u}{\partial x}=0
\end{displaymath}](img1621.png) |
(A.26) |
gives an equation for a steady state as
![\begin{displaymath}
\frac{\partial \rho u}{\partial x}=0.
\end{displaymath}](img1622.png) |
(A.27) |
Considering a region containing the shock front at
which extends from
to
and integrating the above equation, we get
![\begin{displaymath}
\int_{x_s-\Delta x}^{x_s+\Delta x}\frac{d \rho u}{d x}dx=
\left[\rho u\right]_{x_s-\Delta x}^{x_s+\Delta x}.
\end{displaymath}](img1626.png) |
(A.28) |
Thus, we obtain the jump condition conserning the mass coservation as
![\begin{displaymath}
\rho_1 u_1=\rho_2 u_2,
\end{displaymath}](img1627.png) |
(A.29) |
where the quatities with suffix 1 are for preshock and those with suffix 2 are for postshock.
Equation of motion for steady state
![\begin{displaymath}
\rho u\frac{\partial u}{\partial x} =-\frac{\partial p}{\partial x},
\end{displaymath}](img1628.png) |
(A.30) |
gives
![\begin{displaymath}
p_1+\rho_1 u_1^2=p_2+\rho_2 u_2^2,
\end{displaymath}](img1629.png) |
(A.31) |
where we used equation(A.29).
Subsections
Kohji Tomisaka
2012-10-03