Navier Stokes Korteweg Research Articles

Global well-posedness and long-time dynamics for a higher order quasi-geostrophic type equation

In this paper we study a higher order viscous quasi-geostrophic type equation. This equation was derived in [11] as the limit dynamics of a singularly perturbed Navier–Stokes–Korteweg system with Coriolis force, when the Mach, Rossby and Weber numbers go to zero at the same rate.The scope of the present paper is twofold. First of all, we investigate well-posedness of such a model on the whole space R2: we prove that it is well-posed in Hs for any s≥3, globally in time. Interestingly enough, we show that this equation owns two levels of energy estimates, for which one gets existence and uniqueness of weak solutions with different regularities (namely, H3 and H4 regularities); this fact can be viewed as a remainder of the so called BD-entropy structure of the original system.In the second part of the paper we investigate the long-time behavior of these solutions. We show that they converge to the solution of the corresponding linear parabolic type equation, with same initial datum and external force. Our proof is based on dispersive estimates both for the solutions to the linear and non-linear problems.

Global classical solutions to the 3D Navier–Stokes–Korteweg equations with small initial energy

In this paper, we investigate the global well-posedness of classical solutions to three-dimensional Cauchy problem of the compressible Navier–Stokes type system with a Korteweg stress tensor under the condition that the initial energy is small. This result improves previous results obtained by Hattori–Li in [H. Hattori and D. Li, Solutions for two dimensional system for materials of Korteweg type, SIAM J. Math. Anal. 25 (1994) 85–98; H. Hattori and D. Li. Global solutions of a high-dimensional system for Korteweg materials. J. Math. Anal. Appl. 198 (1996) 84–97.], where the existence of the classical solution is established for initial data close to an equilibrium in some Sobolev space [Formula: see text].

The least-squares spectral element method for phase-field models for isothermal fluid mixture

The phase-field approach has been regarded as a powerful method in numerically handling the interface dynamics in multiphase flow in several scientific and engineering applications. For an isothermal fluid mixture, the Navier–Stokes–Korteweg equation and the Navier–Stokes–Cahn–Hilliard equation have represented two major branches of the phase-field methods. We present a general discretization formulation for these two equations and conduct a comparison study of them. The formulation using a least-squares spectral element method is implemented by adopting a time-stepping procedure, a high-order continuity approximation and an element-by-element solver technique. To describe the same fluid mixtures by the isothermal Navier–Stokes–Korteweg and the Navier–Stokes–Cahn–Hilliard equations, we suggest a non-dimensionalization with the same dimensionless quantities. Numerical experiments are conducted to verify the spectral/hp least-squares formulation for the isothermal Navier–Stokes–Korteweg model. Besides, the equilibrium state of the van der Waals fluid model is calculated both analytically and numerically. Through spontaneous decomposition example, the isothermal Navier–Stokes–Korteweg system and the Navier–Stokes–Cahn–Hilliard system are compared in terms of the equilibrium pressure and the energy minimizing process. As a general example, the coalescence of two liquid droplets is studied with our solver for the isothermal Navier–Stokes–Korteweg system. The minimum discretization levels for space and time are investigated and a parametric study on Weber number is carried out.

An asymptotic-preserving method for a relaxation of the Navier–Stokes–Korteweg equations

The Navier–Stokes–Korteweg (NSK) equations are a classical diffuse-interface model for compressible two-phase flows. As direct numerical simulations based on the NSK system are quite expensive and in some cases even impossible, we consider a relaxation of the NSK system, for which robust numerical methods can be designed. However, time steps for explicit numerical schemes depend on the relaxation parameter and therefore numerical simulations in the relaxation limit are very inefficient. To overcome this restriction, we propose an implicit–explicit asymptotic-preserving finite volume method. We prove that the new scheme provides a consistent discretization of the NSK system in the relaxation limit and demonstrate that it is capable of accurately and efficiently computing numerical solutions of problems with realistic density ratios and small interfacial widths.

Phase-Field and Korteweg-Type Models for the Time-Dependent Flow of Compressible Two-Phase Fluids

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.

Vanishing capillarity limit of the compressible non-isentropic Navier–Stokes–Korteweg system to Navier–Stokes system

In this paper, we consider the three dimensional compressible nonisentropic fluid model of Korteweg type. We mainly present the vanishing capillarity limit of smooth solution to the initial value problem. Based on the uniform estimates with respect to the capillary coefficient κ, we show the unique smooth solutions of the three dimensional Korteweg model converges globally in time to the smooth solution of the three dimensional compressible nonisentropic Navier–Stokes system, as κ tends to zero. Also, we give the convergence rate estimates in H2-norm with respect to κ.

On the compressible Navier-Stokes-Korteweg equations

In this paper, we consider compressible Navier-Stokes-Korteweg (N-S-K) equations with more general pressure laws, that is the pressure $P$ is non-monotone. We prove the stability of weak solutions in the periodic domain $\Omega=\mathbb{T}^{N}$, when $N = 2,3$. Utilizing an interesting Sobolev inequality to tackle the complicated Korteweg term, we obtain the global existence of weak solutions in one dimensional case. Moreover, when the initial data is compactly supported in the whole space $\mathbb{R}$, we prove the compressible N-S-K equations will blow-up in finite time.

Zero‐viscosity‐capillarity limit to rarefaction waves for the 1D compressible Navier–Stokes–Korteweg equations

In this paper, we study the zero viscosity and capillarity limit problem for the one‐dimensional compressible isentropic Navier–Stokes–Korteweg equations when the corresponding Euler equations have rarefaction wave solutions. In the case that either the effects of initial layer are ignored or the rarefaction waves are smooth, we prove that the solutions of the Navier–Stokes–Korteweg equation with centered rarefaction wave data exist for all time and converge to the centered rarefaction waves as the viscosity and capillarity number vanish, and we also obtain a rate of convergence, which is valid uniformly for all time. These results are showed by a scaling argument and elementary energy analysis. Copyright © 2016 John Wiley & Sons, Ltd.

Liquid-vapor transformations with surfactants. Phase-field model and Isogeometric Analysis

Surfactants are compounds that find energetically favorable to be located at the boundaries between fluids. They are able to modify the properties of those interfaces, for example, reducing surface tension. Here, we propose a new model for liquid–vapor flows with surfactants which captures the dynamics of the surfactant and accounts for phase transformations in the fluid. The aforementioned model is derived from a free energy functional by using a Coleman–Noll approach. The proposed theory emanates from the isothermal Navier–Stokes–Korteweg equations, which describe single-component two-phase flow and naturally allow for phase transformations. We believe that our model has significant potential to study the influence of surfactants in vaporization and condensation processes. From a numerical point of view, the proposed model poses significant challenges to existing discretization methods, including stiffness in space and time, internal and boundary layers as well as higher-order partial differential operators. To overcome these challenges we propose algorithms based on Isogeometric Analysis, which permit an accurate and efficient discretization. Finally, we illustrate the viability of the theoretical framework and the effectiveness of our algorithms by solving several numerical problems in two and three dimensions.

An h-adaptive local discontinuous Galerkin method for the Navier–Stokes–Korteweg equations

In this article, we develop a mesh adaptation algorithm for a local discontinuous Galerkin (LDG) discretization of the (non)-isothermal Navier–Stokes–Korteweg (NSK) equations modeling liquid–vapor flows with phase change. This work is a continuation of our previous research, where we proposed LDG discretizations for the (non)-isothermal NSK equations with a time-implicit Runge–Kutta method. To save computing time and to capture the thin interfaces more accurately, we extend the LDG discretization with a mesh adaptation method. Given the current adapted mesh, a criterion for selecting candidate elements for refinement and coarsening is adopted based on the locally largest value of the density gradient. A strategy to refine and coarsen the candidate elements is then provided. We emphasize that the adaptive LDG discretization is relatively simple and does not require additional stabilization. The use of a locally refined mesh in combination with an implicit Runge–Kutta time method is, however, non-trivial, but results in an efficient time integration method for the NSK equations. Computations, including cases with solid wall boundaries, are provided to demonstrate the accuracy, efficiency and capabilities of the adaptive LDG discretizations.

Models of Two-Phase Fluid Dynamics à la Allen-Cahn, Cahn-Hilliard, and ... Korteweg!

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’.

Highly rotating viscous compressible fluids in presence of capillarity effects

We study here a singular limit problem for a Navier–Stokes–Korteweg system with Coriolis force, in the domain \(\mathbb {R}^2\times \,]0,1[\,\) and for general ill-prepared initial data. Taking the Mach and the Rossby numbers proportional to a small parameter \(\varepsilon \rightarrow 0\), we perform the incompressible and high rotation limits simultaneously; moreover, we consider both the constant and vanishing capillarity regimes. In this last case, the limit problem is identified as a 2-D incompressible Navier–Stokes equation in the variables orthogonal to the rotation axis; if the capillarity is constant, instead, the limit equation slightly changes, keeping however a similar structure, due to the presence of an additional surface tension term. In the vanishing capillarity regime, various rates at which the capillarity coefficient goes to 0 are considered: in general, this produces an anisotropic scaling in the system. The proof of the results is based on suitable applications of the RAGE theorem, combined with microlocal symmetrization arguments.

Existence and Stability of Time Periodic Solution to the Compressible Navier–Stokes–Korteweg System on $${\mathbb{R}^3}$$ R 3

The compressible Navier–Stokes–Korteweg system is considered on \({\mathbb{R}^3}\) when the external force is periodic in the time variable. The existence of a time periodic solution is proved for a sufficiently small external force by using the time-T-map related to the linearized problem around the motionless state with constant density and absolute temperature. The spectral properties of the time-T-map is investigated by a potential theoretic method and an energy method in some weighted spaces. The stability of the time periodic solution is proved for sufficiently small initial perturbations. It is also shown that the \({L^\infty}\) norm of the perturbation decays as time goes to infinity.

Numerical solution of Navier–Stokes–Korteweg systems by Local Discontinuous Galerkin methods in multiple space dimensions

Compressible liquid–vapor flow with phase transitions can be described by systems of Navier–Stokes–Korteweg type. They extend the Navier–Stokes equations by nonlinear higher-grade terms which take the form of either differential or nonlocal integral operators. A numerical approximation method on the basis of the Local Discontinuous Galerkin method in multiple space dimensions is suggested for isothermal flows. It relies on a specific discretization of a non-conservative formulation. To enhance the performance of the overall scheme two techniques are used: (i) local spatial adaptivity based on gradient indicators for the density and (ii) parallelism based on domain decomposition.The paper concludes with numerical experiments in two and three space dimensions. They show the reliability and efficiency of the proposed approach as well as they demonstrate the applicability of the models for several important phase transition phenomena.

Relaxation of the Navier–Stokes–Korteweg equations for compressible two‐phase flow with phase transition

SummaryThe Navier–Stokes–Korteweg (NSK) system is a classical diffuse‐interface model for compressible two‐phase flow. However, the direct numerical simulation based on the NSK system is quite expensive and in some cases even not possible. We propose a lower‐order relaxation of the NSK system with hyperbolic first‐order part. This allows applying numerical methods for hyperbolic conservation laws and removing some of the difficulties of the original NSK system. To illustrate the new ansatz, we first present a local discontinuous Galerkin method in one and two spatial dimensions. It is shown that we can compute initial boundary value problems with realistic density ratios and perform stable computations for small interfacial widths. Second, we show that it is possible to construct a semi‐discrete finite‐volume scheme that satisfies a discrete entropy inequality. Copyright © 2015 John Wiley & Sons, Ltd.

A local discontinuous Galerkin method for the (non)-isothermal Navier–Stokes–Korteweg equations

In this article, we develop a local discontinuous Galerkin (LDG) discretization of the (non)-isothermal Navier–Stokes–Korteweg (NSK) equations in conservative form. These equations are used to model the dynamics of a compressible fluid exhibiting liquid–vapor phase transitions. The NSK-equations are closed with a Van der Waals equation of state and contain third order nonlinear derivative terms. These contributions frequently cause standard numerical methods to violate the energy dissipation relation and require additional stabilization terms to prevent numerical instabilities. In order to address these problems we first develop an LDG method for the isothermal NSK equations using discontinuous finite element spaces combined with a time-implicit Runge–Kutta integration method. Next, we extend the LDG discretization to the non-isothermal NSK equations. An important feature of the LDG discretizations presented in this article is that they are relatively simple, robust and do not require special regularization terms. Finally, computational experiments are provided to demonstrate the capabilities, accuracy and stability of the LDG discretizations.

Long-time behavior of solution for the compressible Navier–Stokes–Korteweg equations in R3

In this paper, we study the long-time behavior of solution for the compressible Navier–Stokes–Korteweg equations in three-dimensional whole space. More precisely, we focus on establishing the optimal time decay rates for the higher-order spatial derivatives of density and velocity, which will improve the work of Wang and Tan (2011).

Quasi-Neutral Limit, Dispersion, and Oscillations for Korteweg-Type Fluids

In the setting of general initial data and whole space we perform a rigorous analysis of the quasineutral limit for a hydrodynamical model of a viscous plasma with capillarity tensor represented by the Navier Stokes Korteweg Poisson system. We shall provide a detailed mathematical description of the convergence process by analyzing the dispersion of the fast oscillating acoustic waves. However the standard acoustic wave analysis is not sufficient to control the high frequency oscillations in the electric field but it is necessary to estimates the dispersive properties induced by the capillarity terms. Therefore by using these additional estimates we will be able to control, via compensated compactness, the quadratic nonlinearity of the stiff electric force field. In conclusion, opposite to the zero capillarity case \cite{DM12} where persistent space localized time high frequency oscillations need to be taken into account, we show that as $\lambda \to 0$, the density fluctuation $\rho^{\lambda}-1$ converges strongly to zero and the fluids behaves according to an incompressible dynamics.

Local in time results for local and non-local capillary Navier–Stokes systems with large data

In this article we study three capillary compressible models (the classical local Navier–Stokes–Korteweg system and two non-local models) for large initial data, bounded away from zero, and with a reference pressure state ρ¯ which is not necessarily stable (P′(ρ¯) can be non-positive). We prove that these systems have a unique local in time solution and we study the convergence rate of the solutions of the non-local models towards the local Korteweg model. The results are given for constant viscous coefficients and we explain how to extend them for density dependant coefficients.

Energy consistent discontinuous Galerkin methods for the Navier–Stokes–Korteweg system

We design consistent discontinuous Galerkin finite element schemes for the approximation of the Euler-Korteweg and the Navier-Stokes-Korteweg systems. We show that the scheme for the Euler-Korteweg system is energy and mass conservative and that the scheme for the Navier-Stokes-Korteweg system is mass conservative and monotonically energy dissipative. In this case the dissipation is isolated to viscous effects, that is, there is no numerical dissipation. In this sense the methods is consistent with the energy dissipation of the continuous PDE systems.