The set of equations, which describe the concentration, velocity and temperature distributions downstream of the shock, are derived from the well-established conservation laws of mass, momentum and energy transfer.

The flow is assumed to be adiabatic; transport phenomena associated with mass diffusion thermal conduction and viscous effects are assumed to be negligible. Test times behind shock waves are typically on the order of a few hundred microseconds; hence, neglect of these transport processes is of little consequence. Initial conditions for the governing equations are derived from the Rankine-Hugoniot relations for flow across a normal shock. The conservation equations for one-dimensional flow through an arbitrarily assigned area profile are stated below:

(7–33)

(7–34)

(7–35)

(7–36)

Temperature is related to the specific enthalpy of the gas mixture through the relations:

(7–37)

and

(7–38)

The net molar production rate of each species due to chemical reaction is denoted by . A detailed description of this term is given in the Thermodynamic Data Format of the Chemkin Input Manual Input Manual. The equations of state relating the intensive thermodynamic properties is given by:

(7–39)

where the mixture molecular weight is determined from the local gas concentration via:

(7–40)

In the shock tube experiments, the usual measurable quantities are density, species concentration, velocity and temperature as functions of time. It is therefore desirable to have time as the independent variable and not distance. Employing the relation

(7–41)

differentiating Equation 7–37 , Equation 7–38 , Equation 7–39 , and Equation 7–40 , and combining the equations results in the following set of coupled, ordinary differential equations:

(7–42)

(7–43)

(7–44)

(7–45)

The time-histories of the measurable flow quantities should satisfy these relations. The distance of a fluid element from the shock, , follows from Equation 7–41 and is given by

(7–46)

These ODEs, Equation 7–42 , Equation 7–43 , Equation 7–44 and Equation 7–45 , are integrated along with Equation 7–46 , for distance from the shock, and Equation 7–4 for laboratory time, when gas-particle time is the independent variable. Values of the pressure, mean molecular weight and area as a function of gas-particle time are also determined.