On the qualitative behaviour of incompressible two-phase flows with phase transitions: The case of equal densities

Abstract

The study of the basic model for incompressible two-phase flows with phase transitions in the case of equal densities, initiated in the paper Prüss, Shibata, Shimizu, and Simonett [16], is continued here with a stability analysis of equilibria and results on asymptotic behaviour of global solutions. The results parallel those for the thermodynamically consistent Stefan problem with surface tension obtained in Prüss, Simonett, and Zacher [19]. Mathematics Subject Classification (2010):Primary: 35R35, 35K55, 35B35, Secondary: 35Q30, 76D45, 80A22.

Qualitative Behaviour of Incompressible Two-Phase Flows with Phase Transitions: The Case of Non-Equal Densities

“…In particular, N S λ e k = 0 for each k, and N S λ g has mean value zero for each g ∈ L 2 (Σ). This result is proved in [27], Proposition 4.1 for the case of equal densities. It carries over directly to the case [[ρ]] = 0 considered here.…”

“…We emphasize that the major difference between the cases of equal or different densities lies in the occurrence of the so-called Stefan currents which are induced by the jump in the normal velocity across the interface in case ρ 1 = ρ 2 . If the densities are equal these are absent, this case which we call temperature-dominated has been analyzed in [22] and [27]. Here we are interested in the velocity-dominated case, i.e.…”

On Incompressible Two-Phase Flows with Phase Transitions and Variable Surface Tension

“…In the previous papers [12] and [16] we have mathematically analyzed the following problem with sharp interface:…”

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Our study of the basic model for incompressible two-phase flows with phase transitions consistent with thermodynamics [10], [11], [12], [16] is extended to the case of temperature-dependent surface tension. We prove well-posedness in an Lp-setting, study the stability of the equilibria of the problem, and show that a solution which does not develop singularities exists globally, and if its limit set contains a stable equilibrium it converges to this equilibrium in the natural state manifold for the problem as time goes to infinity.Mathematics Subject Classification (2010).

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Preface

“…This result turned out to have wide applications. Over the last 15 years, Jan has devoted his mathematical interests to the study of moving boundary problems in fluid flows and phase transitions [58,65,79,85,95,96,[98][99][100]103,104,106,108,111,113,115,118,120,122,123,129]. While processes with moving surfaces are omnipresent in nature, it turns out that their mathematical analysis poses great challenges.…”