Existence of Global Steady Subsonic Euler Flows Through Infinitely Long Nozzles

Xie, Chunjing; Xin, Zhouping

doi:10.1137/09076667x

Cited by 105 publications

(109 citation statements)

References 38 publications

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“…Similar approach has been applied in 3D axis-asymmetric nozzles by Xie and Xin in [28]. Furthermore, these ideas are also useful to study the physically more important case, subsonic Euler flows, by Xie and Xin in [29] (see also the generalization in [11]). However, it seems difficult to apply the method in [27] and [28] in general multi-dimensional (n ≥ 3) nozzles, since the stream function formulation can not work in this case.…”

Section: Introductionmentioning

confidence: 90%

Subsonic Flows in a Multi-Dimensional Nozzle

Xin

Yan

2011

Arch Rational Mech Anal

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Abstract. In this paper, we study the global subsonic irrotational flows in a multi-dimensional (n ≥ 2) infinitely long nozzle with variable cross sections. The flow is described by the inviscid potential equation, which is a second order quasilinear elliptic equation when the flow is subsonic. First, we prove the existence of the global uniformly subsonic flow in a general infinitely long nozzle for arbitrary dimension for sufficiently small incoming mass flux and obtain the uniqueness of the global uniformly subsonic flow. Furthermore, we show that there exists a critical value of the incoming mass flux such that a global uniformly subsonic flow exists uniquely, provided that the incoming mass flux is less than the critical value. This gives a positive answer to the problem of Bers on global subsonic irrotational flows in infinitely long nozzles for arbitrary dimension [5]. Finally, under suitable asymptotic assumptions of the nozzle, we obtain the asymptotic behavior of the subsonic flow in far fields by a blow-up argument. The main ingredients of our analysis are methods of calculus of variations, the Moser iteration techniques for the potential equation and a blow-up argument for infinitely long nozzles.

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Section: Introductionmentioning

confidence: 90%

Subsonic Flows in a Multi-Dimensional Nozzle

Xin

Yan

2011

Arch Rational Mech Anal

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“…In [17,18] and [19], Xie and Xin established the above results in Theorems 1.1-1.4 first on subsonic and subsonic-sonic potential flows in both 2-D and 3-D axially symmetric infinitely long nozzle, and then on steady subsonic Euler flows in 2-D infinitely long nozzle. Our results are the extension to steady subsonic Euler flows in case of 3-D axially symmetric nozzle.…”

Section: )mentioning

confidence: 83%

“…Recently, Du, Xin and Yan [5] proved the existence and uniqueness of global subsonic potential flows through infinitely long nozzles for arbitrary dimension. For subsonic Euler flows, Xie and Xin [19] established the global existence of steady subsonic Euler flows through a 2-D infinitely long nozzle by using the stream function formulation when the variation of Bernoulli's function in the upstream is sufficiently small and mass flux is in a suitable regime with an upper critical value.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Global subsonic Euler flows in an infinitely long axisymmetric nozzle

Duan

2011

Journal of Differential Equations

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In this paper, we consider global subsonic compressible flows through an infinitely long axisymmetric nozzle. The flow is governed by the steady Euler equations and has boundary conditions on the nozzle walls. Existence and uniqueness of global subsonic solution are established for an infinitely long axisymmetric nozzle, when the variation of Bernoulli's function in the upstream is sufficiently small and the mass flux of the incoming flow is less than some critical value. The results give a strictly mathematical proof to the assertion in Bers (1958) [2]: there exists a critical value of the incoming mass flux such that a global subsonic flow exists uniquely in a nozzle, provided that the incoming mass flux is less than the critical value. The existence of subsonic flow is obtained by the precisely a priori estimates for the elliptic equation of two variables. With the assumptions on the nozzle in the far fields, the asymptotic behavior can be derived by a blowup argument for the infinitely long nozzle. Finally, we obtain the uniqueness of uniformly subsonic flow by energy estimate and derive the existence of the critical value of incoming mass flux.

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“…1 When the flows have non-zero vorticity, steady incompressible Euler flows which are the zero Mach number limits of compressible subsonic flows were investigated in [1,20,27] and references therein. The existence of subsonic Euler flows in two dimensional and three dimensional axially symmetric nozzles was established in [31,14] via a stream function formulation for the Euler equations, see [9,5] for recent study on subsonic Euler flows in half plane.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Three dimensional steady subsonic Euler flows in bounded nozzles

Chen

Xie

2014

Journal of Differential Equations

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Abstract. In this paper, we study the existence and uniqueness of three dimensional steady Euler flows in rectangular nozzles when prescribing normal component of momentum at both the entrance and exit. If, in addition, the normal component of the voriticity and the variation of Bernoulli's function at the exit are both zero, then there exists a unique subsonic potential flow when the magnitude of the normal component of the momentum is less than a critical number. As the magnitude of the normal component of the momentum approaches the critical number, the associated flows converge to a subsonic-sonic flow. Furthermore, when the normal component of vorticity and the variation of Bernoulli's function are both small, the existence of subsonic Euler flows is established. The proof of these results is based on a new formulation for the Euler system, a priori estimate for nonlinear elliptic equations with nonlinear boundary conditions, detailed study for a linear div-curl system, and delicate estimate for the transport equations.

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Existence of Global Steady Subsonic Euler Flows Through Infinitely Long Nozzles

Cited by 105 publications

References 38 publications

Subsonic Flows in a Multi-Dimensional Nozzle

Subsonic Flows in a Multi-Dimensional Nozzle

Global subsonic Euler flows in an infinitely long axisymmetric nozzle

Three dimensional steady subsonic Euler flows in bounded nozzles

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