Standing Waves for Phase Transitions in a Spherically Symmetric Nozzle

Fan, Haitao; Lin, Xiao-Biao

doi:10.1137/11082213x

Cited by 8 publications

(11 citation statements)

References 39 publications

(47 reference statements)

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“…Since ε is a small parameter, we will use singular perturbation techniques [14,16,19,34] to find standing waves for system (1.6). To convert the system into a fast-slow form, we introduce the new variables (2.1) m := ρu, θ := bρλ y , n := −mu − p + (u y + 2u/r), r = y.…”

Section: A Geometric Singular Perturbation Approachmentioning

confidence: 99%

“…The existence of nontransonic evaporation waves, i.e., subsonic-to-subsonic and supersonic-to-supersonic waves, has been proved in [14], where the transverse intersection of W u (S 0 ) and W s (S 1 ) for subsonic waves was checked numerically. Using the method presented in the proof of Theorem 3.3, it can also be proved rigorously.…”

Section: Stability Analysis Of the Layer Problemmentioning

confidence: 99%

“…A straightforward but rather lengthy calculation shows that = η 1 + η 2 . Fan and Lin in [14] studied the existence of nontransonic evaporation waves that are in either the subsonic or supersonic region for the entire evaporation process. In this paper we shall extend their results to evaporation waves that cross the sonic surface.…”

Section: Introductionmentioning

confidence: 99%

“…A key in our analysis is to identify system (1.6) as a singularly perturbed system. In section 2, we introduce appropriate slow and fast variables so that system (1.6) can be converted into a slow-fast dynamical system and studied by means of geometric singular perturbation theory [14,16,19,34].…”

Section: Introductionmentioning

confidence: 99%

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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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Section: A Geometric Singular Perturbation Approachmentioning

confidence: 99%

Section: Stability Analysis Of the Layer Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Transonic Evaporation Waves in a Spherically Symmetric Nozzle

Lin¹,

Wechselberger²

2014

SIAM J. Math. Anal.

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“…About the special case α ≡ 0, the Riemann problem was solved in [8] if τ = 0 while the relaxation limit τ → 0 was investigated in [2]; moreover, all possible traveling waves were characterized in [16]. When a diffusion term λ xx is added to the right side of the third equation in (1.1), traveling waves were obtained in [15,18,19]. Using these traveling waves, the system was found to have solutions exhibiting phenomena observed in actual experiments [17].…”

Section: Introductionmentioning

confidence: 99%

Phase transitions for laminar-turbulent flows in a pipeline or through porous media

Corli

Fan

2013

Communications in Information and Systems

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Liquid/vapor phase changes for a fluid flow through a porous medium or a pipeline are considered. In particular, the model covers both laminar and turbulent flows. The presence of both laminar and turbulent flows causes jump discontinuities in the friction coefficient. Classical trajectories of traveling waves terminate when they intersect the discontinuity. We construct traveling wave solutions by monotonically smoothing the discontinuity and then taking a limiting process. The limit is independent of the monotonepreserving smoothing. This uniqueness justifies the construction of the traveling wave via this smoothing and limiting approach. Existence of traveling waves is established in a wide range of situations; in particular, the end states may be formed either by pure phases or mixtures.

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Spatial Dynamics

Kuehn¹

2014

Applied Mathematical Sciences

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Standing Waves for Phase Transitions in a Spherically Symmetric Nozzle

Cited by 8 publications

References 39 publications

Transonic Evaporation Waves in a Spherically Symmetric Nozzle

Transonic Evaporation Waves in a Spherically Symmetric Nozzle

Phase transitions for laminar-turbulent flows in a pipeline or through porous media

Spatial Dynamics

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