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Smooth helically symmetric transonic flows with nonzero vorticity in a concentric cylinder

Yi Ke, Shangkun Weng

Abstract

This paper concerns the structural stability of smooth cylindrical symmetric transonic flows in a concentric cylinder under helically symmetric perturbation of suitable boundary conditions. The deformation-curl decomposition developed by the second author and his collaborator is utilized to effectively decouple the elliptic-hyperbolic mixed structure in the steady compressible Euler equation. A key parameter in the helical symmetry is the step (denoted by $σ$), which denotes the magnitude of the translation along the symmetry axis after rotating one full turn. It is shown that the step determines the type of the first order partial differential system satisfied by the radial and vertical velocity. There exists a critical number $σ_{*}$ depending only on the background transonic flows, such that if $0<σ<σ_{*}$, one can prove the existence and uniqueness of smooth helically symmetric transonic flows with nonzero vorticity.

Smooth helically symmetric transonic flows with nonzero vorticity in a concentric cylinder

Abstract

This paper concerns the structural stability of smooth cylindrical symmetric transonic flows in a concentric cylinder under helically symmetric perturbation of suitable boundary conditions. The deformation-curl decomposition developed by the second author and his collaborator is utilized to effectively decouple the elliptic-hyperbolic mixed structure in the steady compressible Euler equation. A key parameter in the helical symmetry is the step (denoted by ), which denotes the magnitude of the translation along the symmetry axis after rotating one full turn. It is shown that the step determines the type of the first order partial differential system satisfied by the radial and vertical velocity. There exists a critical number depending only on the background transonic flows, such that if , one can prove the existence and uniqueness of smooth helically symmetric transonic flows with nonzero vorticity.
Paper Structure (3 sections, 5 theorems, 67 equations)

This paper contains 3 sections, 5 theorems, 67 equations.

Table of Contents

  1. Introduction and main results
  2. Some properties of the background flow
  3. Smooth helically symmetric transonic flows with small nonzero vorticity

Key Result

Proposition 1.1

Suppose that the incoming flow is subsonic, i.e. $A_0\gamma \rho_{0}^{\gamma-1}>U_{10}^{2}+U_{20}^{2}$. Then there exist constants $0<r^{\#}<r_c<r_1$, where $r^{\#}$ depends only on $r_1$, $\gamma$ and the incoming flow at $r_1$, such that if $r^{\#}<r_0<r_c$, there exists a unique smooth irrotational transonic spiral flow $(\bar{U}_1,\bar{U}_2,\bar{\rho},\bar{A})$ to 1.3-1.4 in $[r_0,r_1]$ with

Theorems & Definitions (6)

  • Proposition 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Proposition 2.1
  • proof
  • Lemma 3.1