Global subsonic and subsonic-sonic flows through infinitely long axially symmetric nozzles

Xie, Chunjing; Xin, Zhouping

doi:10.1016/j.jde.2010.02.007

Cited by 94 publications

(72 citation statements)

References 10 publications

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“…So one can obtain the boundary L ∞ estimate of the gradient of the stream function, ∇ψ, by constructing proper barrier functions and the standard comparison principle for subsolution to second order elliptic equation. 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]).…”

Section: Introductionmentioning

confidence: 89%

“…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. Thus, we have to consider a different approach from that in [27] to treat the subsonic problem in multi-dimension case.…”

Section: Introductionmentioning

confidence: 99%

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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: 89%

Section: Introductionmentioning

confidence: 99%

Subsonic Flows in a Multi-Dimensional Nozzle

Xin

Yan

2011

Arch Rational Mech Anal

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“…Subsonic potential flows around a body were studied extensively by Shiffman [26], Bers [2,3], Finn, Gilbarg [17,18], and Dong [13], et al The existence of subsonic potential flows in multidimensional infinitely long nozzles was achieved in [29,30,15]. The subsonic-sonic flow as a limit of subsonic flows were studied in [8,29,30,21] via compensated compactness method. Subsonic flows with a sonic boundary were constructed in [28].…”

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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“…Since the equations of uniform subsonic flow possess ellipticity, its solutions have extra-smoothness to those related to transonic flow or supersonic flow. There are a large of literatures on the smooth uniform subsonic solutions, for instance, see [5], [29], [32], [47], [48] for two dimensional flow and [24], [25], [26], [30], [44], [51], [52], [53] for three dimensional flow. Among them, Frankl and Keldysh [32] obtained the first result about the subsonic flow past a two dimensional finite body (or airfoil).…”

Section: Introductionmentioning

confidence: 99%

“…In fact, the authors [14] first applied the momentum equations to reduce the support of the corresponding Young measure to two points, then the irrotational equation and the mass equation are used to deduce the Young measure to a Dirac measure. Xie and Xin [52] investigated the sonic-subsonic limit for the three-dimensional axis-symmetric flow(it is similar to the two dimensional case) through an axis-symmetric nozzle.…”

Section: Introductionmentioning

confidence: 99%

On multi-dimensional sonic-subsonic flow

Huang

Wang

Yong

2011

Acta Mathematica Scientia

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In this paper, a compensated compactness framework is established for sonic-subsonic approximate solutions to the n-dimensional(n ≥ 2) Euler equations for steady irrotational flow that may contain stagnation points. This compactness framework holds provided that the approximate solutions are uniformly bounded and satisfy H −1 loc (Ω) compactness conditions. As illustration, we show the existence of sonic-subsonic weak solution to n-dimensional(n ≥ 2) Euler equations for steady irrotational flow past obstacles or through an infinitely long nozzle. This is the first result concerning the sonic-subsonic limit for n-dimension(n ≥ 3).

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Global subsonic and subsonic-sonic flows through infinitely long axially symmetric nozzles

Cited by 94 publications

References 10 publications

Subsonic Flows in a Multi-Dimensional Nozzle

Subsonic Flows in a Multi-Dimensional Nozzle

Three dimensional steady subsonic Euler flows in bounded nozzles

On multi-dimensional sonic-subsonic flow

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