2019-07-01, Freistühler, Heinrich, Kotschote, Matthias

The Navier-Stokes-Allen-Cahn (NSAC), the Navier-Stokes-Cahn-Hilliard (NSCH), and the Navier-Stokes-Korteweg (NSK) equations have been used in the literature to model the dynamics of two-phase fluids. In their previous article Phase-field and Korteweg-type models for the time-dependent flow of compressible two-phase fluids, Arch. Rational Mech. Anal. 224 (2017), 1-20, the authors showed that both NSAC and NSCH reduce to versions of NSK, when one makes the (unphysical) assumption that microforces are absent. The present paper shows that the same reduction property holds without that assumption.

2017-03, Kotschote, Matthias

This is the first part of two papers whose purpose is to investigate stability of travelling wave solutions to the so-called Navier–Stokes–Allen–Cahn system. This set of equations is a combination of the Navier–Stokes equations for compressible fluids supplemented with a phase field description of Allen–Cahn type. The main part of this work deals with studying the problem obtained by linearizing the NSAC system around so-called standing waves. The main results are (1) local well-posedness of the linearized equations and (2) a detailed description of the point and essential spectrum. As a by-product, we obtain analyticity of the associated semigroup.

2015, Kotschote, Matthias, Zacher, Rico

In this paper we investigate the compressible Navier–Stokes–Cahn–Hilliard equations (the so-called NSCH model) derived by Lowengrub and Truskinovsky. This model describes the flow of a binary compressible mixture; the fluids are supposed to be macroscopically immiscible, but partial mixing is permitted leading to narrow transition layers. The internal structure and macroscopic dynamics of these layers are induced by a Cahn–Hilliard law that the mixing ratio satisfies. The PDE constitute a strongly coupled hyperbolic–parabolic system. We establish a local existence and uniqueness result for strong solutions.

2013, Freistühler, Heinrich, Kotschote, Matthias

2017-07-14, Freistühler, Heinrich, Kotschote, Matthias

Aiming at an understanding of the nonlinear stability of moving fluidic phase boundaries, this project has provided (a) solution theories for three basic systems of nonlinear partial differential equations that model two-phase fluid flow as well as (b) results on the existence of corresponding traveling waves and the spectrum of the operators that result from linearizing the PDEs about these waves.

The three models are the Navier-Stokes-Korteweg (NSK), the Navier-Stokes-Allen-Cahn (NSAC), and the Navier-Stokes-Cahn-Hilliard (NSCH) equations. For compressible NSAC and NSCH, the theories of strong solutions obtained seem to be the first ones in the literature. For NSK, a new theory of strong solutions has been developed, which in particular provides an alternative to the ‘quasi-incompressible’ approach that Abels et al. pursue for NSCH in the case of two separately incompressible phases of different density.

While for NSK the existence of traveling waves representing phase boundaries was known before, corresponding results have been newly obtained for NSAC and NSCH. For all three contexts, the project has established the spectral stability of these traveling waves. Viscous shock waves providing useful heuristic inspiration, fluidic interfaces corresponding to phase boundaries turn out to have their own flavour.

2016-06-22, Kotschote, Matthias

The framework of this article is the compressible Navier-Stokes-Cahn- Hilliard system for the dynamics of a fluid whose two phases are macroscopically immiscible; partial mixing is permitted leading to narrow transition layers. This so-called NSCH model was originally derived by Lowengrub and Truskinowsky [7], but only for the isothermal case. The purpose of this work is to present the non-isothermal version as well as a well-posedness result. The PDEs constitute a strongly coupled hyperbolicparabolic system.

2014, Kotschote, Matthias

In this paper we establish global existence and uniqueness of strong solutions to the non-isothermal compressible Navier–Stokes equations in bounded domains. The initial data have to be near equilibria that may be non-constant due to considering large external forces. We are able to show exponential stability of equilibria in the phase space and, above all, to study the problem in Eulerian coordinates. The latter seems to be a novelty, since in works by other authors, global strong L _p -solutions have been investigated only in Lagrangian coordinates; Eulerian coordinates are even declared as impossible to deal with. The proof is based on a careful derivation and study of the associated linear problem.

2017-04, Freistühler, Heinrich, Kotschote, Matthias

Various versions of the Navier–Stokes–Allen–Cahn (NSAC), the Navier–Stokes–Cahn–Hilliard (NSCH), and the Navier–Stokes–Korteweg (NSK) equations have been used in the literature to model the dynamics of two-phase fluids. One main purpose of this paper consists in (re-)deriving NSAC, NSCH and NSK from first principles, in the spirit of rational mechanics, for fluids of very general constitutive laws. For NSAC, this deduction confirms and extends a proposal of Blesgen. Regarding NSCH, it continues work of Lowengrub and Truskinovsky and provides the apparently first justified formulation in the non-isothermal case. For NSK, it yields a most natural correction to the formulation by Dunn and Serrin. The paper uniformly recovers as examples various classes of fluids, distinguished according to whether none, one, or both of the phases are compressible, and according to the nature of their co-existence. The latter is captured not only by the mixing energy, but also by a ‘mixing rule’—a constitutive law that characterizes the type of the mixing. A second main purpose of the paper is to communicate the apparently new observation that in the case of two immiscible incompressible phases of different temperature-independent specific volumes, NSAC reduces literally to NSK. This finding may be considered as an independent justification of NSK. An analogous fact is shown for NSCH, which under the same assumption reduces to a new non-local version of NSK.

2015, Freistühler, Heinrich, Kotschote, Matthias

One purpose of this paper on the Navier-Stokes-Allen-Cahn (NSAC), the Navier-Stokes-Cahn-Hilliard (NSCH), and the Navier-Stokes-Korteweg (NSK) equations consists in surveying solution theories that one of the authors, M. K., has developed for these three evolutionary systems of partial differential equations. All three theories start from a Helmholtz free energy description of the compressible two-phase fluids whose dynamics they describe in various ways. While a diphasic fluid composed from two constituents of individually constant density is still compressible as long as these two densities are different from each other, the abovementioned solution theories for NSAC and NSCH do not apply in this “quasi-incompressible” case, as the Helmholtz-energy framework degenerates. The second purpose of the paper is to present an observation made by both authors together that shows how to fill these gaps. As ‘by-products’ one obtains (a) in the case that the phases can transform into each other, a justification of NSK, and (b) in the case that they cannot, a new Korteweg type system with non-local ‘viscosity’.

2014, Kotschote, Matthias

In this paper, we investigate isothermal and non-isothermal models of capillary compressible fluids as derived by J. E. Dunn and J. Serrin (1985). We establish global existence and uniqueness for initial data near equilibria, and show exponential stability of equilibrias in the phase space. The proof is based on maximal L_p-regularity results for the associated linear problem.