Revision of the modified moment method and a differential form for the compensated part of entropy

Abstract

The modified moment method is revised so that the consistency condition is replaced by a differential equation for a dissipation function. This revision given this by a differential form for the compensated part of entropy which is given in terms of substantial usage derivatives of conserved and nonconserved macroscopic variables and derivatives of the former with respect to the latter. In this method the distribution function and the entropy are determined in functionals of the macroscopic variables when the conservation laws and the evolution equations are solved subject to initial and boundary condition and the differential equations for the compensated part of entropy is integrated. The internal consistency of the method is proven. A couple of methods are discussed which allows to determine the distribution function approximately. One of the method is discussed in detail and the results agree with the lowest-order solutions for the consistency condition for the non-equilibrium part of the distribution function in the original version of the modified moment method. Therefore the present revision does not affect the evolution equation using each conditions in the dissipation terms in the previous applications of the modified moment method.