Global bounded variation solutions describing Fanno-Rayleigh fluid flows in nozzles

In this paper, we investigate the initial-boundary value problem of compressible Euler equations including friction and heating that model the transonic Fanno-Rayleigh flows through symmetric variable area nozzles. In particular, the case of contracting nozzles is considered. A new version of a generalized Glimm scheme (GGS) is presented for establishing the global existence of entropy solutions with bounded variation. Modified Riemann and boundary Riemann solutions are applied to design this GGS, which is constructed using the contraction matrices acting on the homogeneous Riemann (or boundary-Riemann) solutions. The extended Glimm-Goodman's type of wave interaction estimates are investigated to determine the stability of the scheme and the positivity of gas velocity that results in the existence of the weak solution. The limit of approximation solutions serves as an entropy solution. Moreover, a quantitative relation between the shape of the nozzle, friction, and heat is proposed for the global existence result in the contracting nozzle. Numerical simulations of the contraction-expansion and expansion-contraction nozzles are presented to validate the scheme.