The existence and monotonicity of a three-dimensional transonic shock in a finite nozzle with axisymmetric exit pressure

Li, Jun; Zhou, Xin; Yin, Huicheng

doi:10.2140/pjm.2010.247.109

Cited by 19 publications

(10 citation statements)

References 31 publications

(53 reference statements)

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“…Moreover, interesting decompositions for hyperbolic and elliptic modes of 3-D Euler system are also presented by [6] and [7]. There are also some interesting results in a 2-D de Laval nozzles [16] and a conic divergent nozzle [13][14][15]. For subsonic flow in an infinite nozzle, one may refer to [9] and the reference therein.…”

Section: Introduction and Main Resultsmentioning

confidence: 90%

“…Compared with the results in [11][12][13], we do not need to require that the diverging part of the nozzle wall changes slowly. The key ingredient in the analysis of [11][12][13] is to establish the monotonic property of the shock position along the nozzle wall with respect to the exit pressure so that one can avoid the difficulties caused by the unknown position of the shock.…”

Section: Remark 11mentioning

confidence: 99%

“…The key ingredient in the analysis of [11][12][13] is to establish the monotonic property of the shock position along the nozzle wall with respect to the exit pressure so that one can avoid the difficulties caused by the unknown position of the shock. After some modification of the new elaborate scheme developed in [14], we can determine the shock position together with the solution in each iteration step.…”

Section: Remark 11mentioning

confidence: 99%

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Transonic shock in finitely long nozzle with porous medium boundary condition

Duan

Weng

2011

Journal of Differential Equations

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In this paper, we investigate the 2-D Euler equations with complex boundary conditions. For the entrance and exit of finitely long nozzle, we use the supersonic incoming flow condition and the end pressure condition, respectively. In addition, for the nozzle wall, the lower one is solid and the upper one is permitted to have leakage. We establish the well-posedness if appropriate seepage discharge is given. We also obtain the shock position by solving a free boundary problem without assuming the shock must go through a fixed point.

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Section: Introduction and Main Resultsmentioning

confidence: 90%

Section: Remark 11mentioning

confidence: 99%

Section: Remark 11mentioning

confidence: 99%

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Transonic shock in finitely long nozzle with porous medium boundary condition

Duan

Weng

2011

Journal of Differential Equations

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“…Transonic flow without liquid-gas phase transition has been studied by many authors; see the recent articles [3,5,4,23,35,36]. Among them [35] dealt with subsonic and subsonic-to-sonic flows through infinitely long nozzles.…”

Section: Introductionmentioning

confidence: 99%

Transonic Evaporation Waves in a Spherically Symmetric Nozzle

Lin¹,

Wechselberger²

2014

SIAM J. Math. Anal.

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This paper studies the liquid-to-vapor phase transition in a cone-shaped nozzle. Using the geometric method presented in [P. Szmolyan and M. Wechselberger, we further develop results on subsonic and supersonic evaporation waves in [H. Fan and X.-B. Lin, SIAM J. Math. Anal., 44 (2012), pp. 405-436] to transonic waves. It is known that transonic waves do not exist if restricted solely to the slow system on the slow manifolds [H. Fan and X.-B. Lin, SIAM J. Math. Anal., 44 (2012), pp. 405-436]. Thus we consider the existence of transonic waves that include layer solutions of the fast system that cross or connect to the sonic surface. In particular, we are able to show the existence and uniqueness of evaporation waves that cross from supersonic to subsonic regions and evaporation waves that connect from the subsonic region to the sonic surface and then continue onto the supersonic branch via the slow flow. Introduction.In this paper, we investigate the liquid-to-vapor phase transition in a spherically symmetric cone-shaped nozzle. Subsonic evaporation processes, such as fuel injection into a combustion engine, show up in many important engineering problems and are studied in many research laboratories around the world. The ring formation observed in a shock tube experiment [33] is an example of a supersonic process, as argued by Fan in [11]. Our focus is on transonic evaporation waves. While we are not aware of any experiments where transonic evaporation waves have been observed, we expect that our theoretical results might be useful in some physical and engineering process when the speed of the fluid in a nozzle approaches sonic speed.The study of nozzle flows was pioneered by Courant and Friedrichs [6] and Liu [26]. Transonic flow without liquid-gas phase transition has been studied by many authors; see the recent articles [3,5,4,23,35,36]. Among them [35] dealt with subsonic and subsonic-to-sonic flows through infinitely long nozzles. For transonic flows modeled by reaction-diffusion equations, Liu and coworkers [17,18,27] considered one-dimensional standing waves for a simplified model of gas flows in a nozzle with general variable cross-sectional area a(x). To the best of our knowledge, transonic flows that include a reaction-diffusion equation describing evaporation or condensation have not been rigorously analyzed mathematically.Liquid-vapor phase transitions have been much studied using a van der Waals pressure function [15]. The van der Waals model requires detailed modeling of the evaporation of individual droplets [8,30] or of the nucleation process by which vapor condenses. Figure 1(a) shows the graph of pressure p as a function of specific volume v

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“…The studies of nozzle flow were pioneered by Courant and Friedrichs [10] and Liu [32]. Many authors have studied the transonic flows in nozzles; see the recent articles [5,7,8,29,42,43] and references cited therein. Hong, Hsu, and Liu [23,24] and Liu and Oh [25] used the dynamical systems approach to study one-dimensional standing wave solutions of gas flows in a nozzle with variable cross-sectional area a(x).…”

Section: Introductionmentioning

confidence: 99%

Standing Waves for Phase Transitions in a Spherically Symmetric Nozzle

Fan¹,

Lin²

2012

SIAM J. Math. Anal.

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We study the existence of standing waves for liquid/vapor phase transition in a spherically symmetric nozzle. The system is singularly perturbed and the solution consists of an internal layer where the liquid quickly becomes vapor. Using methods from dynamical systems theory, we prove the existence of the internal layer as a heteroclinic orbit connecting the liquid state to the vapor state. The heteroclinic orbit is reproduced numerically and is also shown numerically to be a transversal heteroclinic orbit. The proof of the existence of an exact standing wave solution near the singular limit is based on the geometric singular perturbation theory and is outlined in the paper.

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The existence and monotonicity of a three-dimensional transonic shock in a finite nozzle with axisymmetric exit pressure

Cited by 19 publications

References 31 publications

Transonic shock in finitely long nozzle with porous medium boundary condition

Transonic shock in finitely long nozzle with porous medium boundary condition

Transonic Evaporation Waves in a Spherically Symmetric Nozzle

Standing Waves for Phase Transitions in a Spherically Symmetric Nozzle

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