Subsonic flows passing a duct for three-dimensional steady compressible Euler systems with friction

Abstract

This paper studies steady motion of gas in a rectilinear duct with square cross-sections,governed by the three-dimensional (3-d) non-isentropic compressible Euler equations with a friction term.Such flows are called Fanno flows in engineering. We construct respectively special subsonic flows, supersonic flows and transonic shocks in the duct.Since the 3-d steady compressible Euler equations are of quasi-linear hyperbolic-elliptic composite type for subsonic flows, and there is no generaltheory up to now, we formulate a boundary value problem arising from studies of transonic shocks, and prove the well-posednessof this problem by showing that the special subsonic flows constructed above are stable under small multi-dimensional perturbations.The proof depends on separation of the elliptic and hyperbolic parts in the Euler equations, and designation of a suitable nonlinear iteration scheme.Particularly, there are strong interactions between the elliptic part and the hyperbolic part due to the appearance of friction, and we deducea linear mixed boundary value problem of a second-order elliptic equation with an integral-type nonlocal term. Its well-posedness is establishedby applying methods of Fourier analysis and regularity theory of second-order elliptic equations.