A Dynamical Systems Approach to Traveling Wave Solutions for Liquid/Vapor Phase Transition

Fan, Haitao; Lin, Xiao-Biao

doi:10.1007/978-1-4614-4523-4_3

Cited by 6 publications

(3 citation statements)

References 15 publications

(22 reference statements)

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“…The shooting technique has been an important method in proving the existence of traveling waves solutions, e.g., see [2,3,6,7,11,13] and the references therein. The method used in this paper is motivated by the techniques used there.…”

Section: Introductionmentioning

confidence: 99%

Traveling Wave Solutions for a Predator–Prey System With Sigmoidal Response Function

Lin

Weng

2011

J Dyn Diff Equat

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Abstract. We study the existence of traveling wave solutions for a diffusive predatorprey system. The system considered in this paper is governed by a Sigmoidal response function which is more general than those studied previously. Our method is an improvement to the original method introduced in the work of Dunbar [2,3]. A bounded Wazewski set is used in this work while unbounded Wazewski sets were used in [2,3].The existence of traveling wave solutions connecting two equilibria is established by using the original Wazewski's theorem which is much simpler than the extended version in Dunbar's work.

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Section: Introductionmentioning

confidence: 99%

Traveling Wave Solutions for a Predator–Prey System With Sigmoidal Response Function

Lin

Weng

2011

J Dyn Diff Equat

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“…Necessary and sufficient conditions for the existence of phase-changing traveling waves (liquid to vapor or vapor to liquid) in one-dimensional space were proved in [9,10,12]. Using dynamical systems methods, the proof of the existence of those one-dimensional waves was simplified in [13]. Let r be the radial coordinate of the nozzle.…”

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 existence and nonexistence of phase-changing traveling waves of various types were shown in [13], [15], and [17] for the isothermal case. The proof of the existence of these traveling waves was much simplified by Fan and Lin in [18]. Fan and Corli [9] showed the existence and uniqueness of the solution of Riemann problem for inviscid (1.7) with = γ = β = 0.…”

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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A Dynamical Systems Approach to Traveling Wave Solutions for Liquid/Vapor Phase Transition

Cited by 6 publications

References 15 publications

Traveling Wave Solutions for a Predator–Prey System With Sigmoidal Response Function

Traveling Wave Solutions for a Predator–Prey System With Sigmoidal Response Function

Transonic Evaporation Waves in a Spherically Symmetric Nozzle

Standing Waves for Phase Transitions in a Spherically Symmetric Nozzle

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