On some smooth symmetric transonic flows with nonzero angular velocity and vorticity

Author:

Weng Shangkun1,Xin Zhouping2,Yuan Hongwei2

Affiliation:

1. School of Mathematics and Statistics, Wuhan University, Wuhan, Hubei Province, 430072, P. R. China

2. Department of Mathematics and The Institute of Mathematical Sciences, The Chinese University of Hong Kong, Shatin, NT, Hong Kong

Abstract

This paper concerns the structural stability of smooth cylindrically symmetric transonic flows in a concentric cylinder. Both cylindrical and axi-symmetric perturbations are considered. The governing system here is of mixed elliptic–hyperbolic and changes type and the suitable formulation of boundary conditions at the boundaries is of great importance. First, we establish the existence and uniqueness of smooth cylindrical transonic spiral solutions with nonzero angular velocity and vorticity which are close to the background transonic flow with small perturbations of the Bernoulli’s function and the entropy at the outer cylinder and the flow angles at both the inner and outer cylinders independent of the symmetric axis, and it is shown that in this case, the sonic points of the flow are nonexceptional and noncharacteristically degenerate, and form a cylindrical surface. Second, we also prove the existence and uniqueness of axi-symmetric smooth transonic rotational flows which are adjacent to the background transonic flow, whose sonic points form an axi-symmetric surface. The key elements in our analysis are to utilize the deformation-curl decomposition for the steady Euler system to deal with the hyperbolicity in subsonic regions and to find an appropriate multiplier for the linearized second-order mixed type equations which are crucial to identify the suitable boundary conditions and to yield the important basic energy estimates.

Funder

National Natural Science Foundation of China

Zheng Ge Ru Foundation, Hong Kong RGC Earmarked Research

Guangdong Basic and Applied Basic Research Foundation

Publisher

World Scientific Pub Co Pte Ltd

Subject

Applied Mathematics,Modeling and Simulation

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