Gas-dynamic equations for spatially inhomogeneous gas mixtures with internal degrees of freedom. III. Renormalized reaction rates

In previous papers (Kolesnichenko and Gorbachev, 2010, 2013) the general approach for solving kinetic equations for gas mixtures with internal degrees of freedom and for obtaining corresponding gas-dynamic equations was develope. In paper (Kolesnichenko and Gorbachev, 2013) general expressions for the reaction rates were derived for zero-order (Euler) gas-dynamic equations for a one-temperature case. The contributions of two different types of non-equilibrium effect were considered: spatially homogeneous and spatially inhomogeneous (the last represented by the terms proportional to velocity divergency) (Kolesnichenko and Gorbachev, 2010, 2013). General expressions were obtained for reaction rates containing integrals, with the functions which satisfy their corresponding integral equations. In the present article studies concerning derivation of the expressions for reaction rates are continued and a form, named renormalized, is obtained for those reaction rates. In contrast to the previous result (Kolesnichenko and Gorbachev, 2013)the factor which has an explicit rational-function dependence on the species densities can be separated within the renormalized expression for the reaction rate. The procedure of solving integral equations using an expansion over various complete sets of polynomials is discussed. One-reaction case is considered. The expressions derived for reaction rates are analyzed.