Rayleigh--Taylor instability in a viscoelastic binary fluid

The effects of polymer additives on Rayleigh--Taylor (RT) instability of immiscible fluids is investigated using the Oldroyd-B viscoelastic model. Analytic results obtained exploiting the phase-field approach show that in polymer solution the growth rate of the instability speeds up with elasticity (but remains slower than in the pure solvent case). Numerical simulations of the viscoelastic binary fluid model confirm this picture

Phase separation in quasi incompressible fluids: Cahn-Hilliard model in the Cattaneo-Maxwell framework

In this paper we propose a mathematical model of phase separation for a quasi-incompressible binary mixture where the spinodal decomposition is induced by an heat flux governed by the Cattaneo-Maxwell equation. As usual, the phase separation is considered in the framework of phase field modeling so that the transition is described by an additional field, the concentration c. The evolution of concentration is described by the Cahn-Hilliard equation and in our model is coupled with the Navier-Stokes equation. Since thermal effect are included, the whole set of evolution equations is set up for the velocity, the concentration, the temperature and the heat flux. The model is compatible with thermodynamics and a maximum theorem holds.Comment: Submitted to ZAM

A phase-field model for liquid-vapor transitions

2Starting from the mesoscopic description of the state equations for the vapor and liquid pure phases of a single chemical species, we propose a phase-field model ruling the liquid-vapor phase transition. Two different phases are separated by a thin layer, rather than a sharp interface, where the phase-field changes abruptly from 0 to 1. All thermodynamic quantities are allowed to vary inside the transition layer, including the mass density. The approach is based on an extra entropy flux which is proved to be non vanishing inside the transition layer, only. Unlike classical phase-field models, the kinetic equation for the phase variable is obtained as a consequence of thermodynamic restrictions and it depends only on the rescaled free enthalpy. The system turns out to be thermodynamically consistent and accounts for both temperature and pressure variations during the evaporation process. Few commonly accepted assumptions allow us to obtain the explicit expression of the Gibbs free enthalpy and the Clausius-Clapeyron formula. As a consequence, the customary form of the vapor pressure curve is recovered.AMS Subject Classification:: 74A15, 74A50, 80A17, 80A22. 82C26openopenBERTI A; C. GIORGIBerti, Alessia; Giorgi, Claudi

A nonlocal phase-field model of Ginzburg-Landau-Korteweg fluids

A thermodynamic model of Korteweg fluids undergoing phase transition and/or phase separation is developed within the framework of weakly nonlocal thermodynamics. Compatibility with second law of thermodynamics is investigated by applying a generalized Liu procedure recently introduced in the literature. Possible forms of the free energy and of the stress tensor, which generalize some earlier ones proposed by several authors in the last decades, are carried out. Owing to the new procedure applied for exploiting the entropy principle, the thermodynamic potentials are allowed to depend on the whole set of variables spanning the state space, including the gradients of the unknown fields, without postulating neither the presence of an energy or entropy extra-flux, nor an additional balance law for microforce

Coupled diffusion and phase transition: Phase fields, constraints, and the Cahn–Hilliard equation

We develop a constrained theory for constituent migration in bodies with microstructure described by a scalar phase field. The distinguishing features of the theory stem from a systematic treatment and characterization of the reactions needed to maintain the internal constraint given by the coincidence of the mass fraction and the phase field. We also develop boundary conditions for situations in which the interface between the body and its environment is structureless and cannot support constituent transport. In addition to yielding a new derivation of the Cahn–Hilliard equation, the theory affords an interpretation of that equation as a limiting variant of an Allen–Cahn type diffusion system arising from the unconstrained theory obtained by considering the mass fraction and the phase field as independent quantities. We corroborate that interpretation with three-dimensional numerical simulations of a recently proposed benchmark problem

Modeling Multiphase Flow in Porous Media With an Application to Permafrost Soil

Inhalt dieser Arbeit ist die Herleitung von Zwei-Skalen-Modellen für Strömungen mehrere Flüssigkeiten und Phasen in porösen Medien. Dies wird mit Hilfe von Phasenfeld-Modellen und formaler asymptotischer Entwicklung erreicht. Die Gleichungen im Porenraum werden mit Hilfe der Vermutung von Rajagopal und Srinivasa über die maximale Entropieproduktionsrate hergeleitet. Diese Methode ermöglicht das Herleiten thermodynamisch konsistenter Modelle im Ramen der klassichen Kontinuumsmechanik, wobei einzig die konstituierenden Gleichungen für die Energie und die für die Entropieproduktionsrate vorgegeben werden müssen. Dieser Ansatz führt zu einem neuen Blickwinkel auf Phasenfeld Modelle und ermöglicht unter anderem die Herleitung der Cahn-Hilliard-Navier-Stokes Gleichungen, der Korteweg-Gleichungen und der Allen-Cahn Gleichung. Um auch thermodynamisch konsitente Randbedingungen zu finden, wurde die Methodik von Rajagopal und Srinivasa auf Randbedingungen verallgemeinert und auf Phasenfeld Modelle angewendet. Schließlich wurde die Methodik nochmals verallgemeinert um auch thermodynamisch konsistente Skalierungen der Phasenfeld Modelle finden zu können. Die Methodik wurde sowohl auf Wasser-Luft Strömungen in Porösen Medien angewendet als auch auf die Modellierung der oberen Schicht von Permafrost Böden, in denen Luft, Wasser, Eis und Dampf interagieren. Es wird gezeigt dass die resultierenden Zwei-Skalen-Modelle Verallgemeinerungen existierender makroskopischer Modelle sind, indem gezeigt wird, dass das makorskopische Verhalten der Lösungen der Zwei-Skalen-Modelle mit den makroskopischen Modellen übereinstimmt. Jedoch wird sich zeigen, dass die Zwei-Skalen-Modelle deutlich mehr Information enthalten und es ist denkbar, dass Simulation, die auf diesen Modellen basieren, in Zukunft genauere Ergebnisse liefern können als der herkömmliche makroskopische Ansatz