Consider gas in a one-dimensional container (e.g., a long thin tube). Assume that the fluid is
inviscid (i.e., it shows no
viscosity effects as for example friction with the tube walls). Furthermore, assume that there is no heat transfer by conduction or radiation and that gravitational acceleration can be neglected. Such a system can be described by the following system of
conservation laws, known as the 1D
Euler equations, that in
conservation form is: {{NumBlk|:| \frac{\partial\rho}{\partial t} \;\; = -\frac{\partial}{\partial x}\left(\rho u\right)|}} {{NumBlk|:| \frac{\partial}{\partial t}(\rho u) \, = -\frac{\partial}{\partial x}\left(\rho u^2 + p\right)|}} {{NumBlk|:| \frac{\partial}{\partial t}\left(E^t\right) = -\frac{\partial}{\partial x}\left[u\left(E^t + p\right)\right],|}} where • \rho, fluid
mass density, • u, fluid
velocity, • e, specific
internal energy of the fluid, • p, fluid
pressure, and • E^t = \rho e + \rho\tfrac{1}{2}u^2, is the total energy density of the fluid, [J/m3], while
e is its specific internal energy Assume further that the gas is calorically ideal and that therefore a polytropic
equation-of-state of the simple form is valid, where \gamma is the constant ratio of specific heats c_p/c_v. This quantity also appears as the
polytropic exponent of the
polytropic process described by {{NumBlk|:| \frac{p}{\rho^\gamma} = \text{constant}.|}} For an extensive list of
compressible flow equations, etc., refer to
NACA Report 1135 (1953). Note: For a calorically ideal gas \gamma is a constant and for a thermally ideal gas \gamma is a function of temperature. In the latter case, the dependence of pressure on mass density and internal energy might differ from that given by equation ().
The jump condition Before proceeding further it is necessary to introduce the concept of a
jump condition – a condition that holds at a discontinuity or abrupt change. Consider a 1D situation where there is a jump in the scalar conserved physical quantity w, which is governed by integral conservation law {{NumBlk|:| \frac{d}{dt} \int_{x_1}^{x_2}w \, dx = -f(w)\Big|_{x_1}^{x_2}|}} for any x_1, x_2, x_1, and, therefore, by
partial differential equation {{NumBlk|:| \frac{\partial w}{\partial t} + \frac{\partial}{\partial x}f\left(w\right)=0|}} for smooth solutions. Let the solution exhibit a jump (or shock) at x=x_{s}(t), where x_{1} and x_{s}(t) , then {{NumBlk|:| \frac{d}{dt}\left[ \left(\int_{x_1}^{x_s(t)}w \, dx + \int_{x_s(t)}^{x_2} w\,dx \right) \right] = -\int_{x_1}^{x_2}\frac{\partial}{\partial x}f\left(w\right)\,dx|}} {{NumBlk|:|\therefore w_1\frac{dx_s}{dt} - w_2\frac{dx_s}{dt} + \int_{x_1}^{x_s(t)} w_t \, dx + \int_{x_s(t)}^{x_2}w_t \, dx = -f(w)\Big|_{x_1}^{x_2}|}} The subscripts 1 and 2 indicate conditions
just upstream and
just downstream of the jump respectively, i.e. w_1 = \lim_{\epsilon \to 0^{+}} w\left(x_s - \epsilon\right) and {{nowrap|w_2 = \lim_{\epsilon \to 0^{+}} w\left(x_s + \epsilon\right).}} \therefore is the
therefore sign. Note, to arrive at equation () we have used the fact that dx_1/dt = 0 and dx_2/dt = 0. Now, let x_1 \to x_s(t) - \epsilon and x_2 \to x_s(t) + \epsilon, when we have \int_{x_1}^{x_s(t)-\epsilon} w_t \, dx \to 0 and \int_{x_s(t)+\epsilon}^{x_2} w_t \, dx \to 0, and in the limit where we have defined u_s = dx_s(t)/dt (the system
characteristic or
shock speed), which by simple division is given by {{NumBlk|:| u_s = \frac{f\left(w_1\right) - f\left(w_2\right)}{w_1 - w_2}.|}} Equation () represents the jump condition for conservation law (). A shock situation arises in a system where its
characteristics intersect, and under these conditions a requirement for a unique single-valued solution is that the solution should satisfy the
admissibility condition or
entropy condition. For physically real applications this means that the solution should satisfy the
Lax entropy condition where f'\left(w_1\right) and f'\left(w_2\right) represent
characteristic speeds at upstream and downstream conditions respectively.
Shock condition In the case of the hyperbolic conservation law (), we have seen that the shock speed can be obtained by simple division. However, for the 1D Euler equations (), () and (), we have the vector state variable \begin{bmatrix}\rho & \rho u & E\end{bmatrix}^\mathsf{T} and the jump conditions become {{NumBlk|:| u_s\left(E_2 - E_1 \right) = \left[\rho_2 u_2 \left(e_2 + \frac{1}{2} u_2^2 + \frac{p_2}{\rho_2}\right)\right] - \left[\rho_1 u_1 \left(e_1 + \frac{1}{2} u_1^2 + \frac{p_1}{\rho_1}\right)\right].|}} Equations (), () and () are known as the
Rankine–Hugoniot conditions for the Euler equations and are derived by enforcing the conservation laws in integral form over a
control volume that includes the shock. For this situation u_s cannot be obtained by simple division. However, it can be shown by transforming the problem to a moving co-ordinate system (setting u_s' := u_s - u_1, u'_1 := 0, u'_2 := u_2 - u_1 to remove u_1) and some algebraic manipulation (involving the elimination of u'_2 from the transformed equation () using the transformed equation ()), that the shock speed is given by {{NumBlk|:| u_s = u_1 + c_1 \sqrt{1 + \frac{\gamma+1}{2\gamma} \left( \frac{p_2}{p_1} - 1\right)},|}} where c_{1}=\sqrt{\gamma p_{1}/\rho_{1}} is the speed of sound in the fluid at upstream conditions. == Shock Hugoniot and Rayleigh line in solids ==