Navier Stokes Korteweg Research Articles

Asymptotic behavior of solutions for the 1-D isentropic Navier–Stokes–Korteweg equations with free boundary

In this paper, we consider the problem with a gas–gas free boundary for the one dimensional isentropic compressible Navier–Stokes–Korteweg system. For shock wave, asymptotic profile of the problem is shown to be a shifted viscous shock profile, which is suitably away from the boundary, and prove that if the initial data around the shifted viscous shock profile and its strength are sufficiently small, then the problem has a unique global strong solution, which tends to the shifted viscous shock profile as time goes to infinity. Also, we show the asymptotic stability toward rarefaction wave without the smallness on the strength if the initial data around the rarefaction wave are sufficiently small.

High friction limit for Euler–Korteweg and Navier–Stokes–Korteweg models via relative entropy approach

The aim of this paper is to investigate the singular relaxation limits for the Euler–Korteweg and the Navier–Stokes–Korteweg system in the high friction regime. We shall prove that the viscosity term is present only in higher orders in the proposed scaling and therefore it does not affect the limiting dynamics, and the two models share the same equilibrium equation. The analysis of the limit is carried out using the relative entropy techniques in the framework of weak, finite energy solutions of the relaxation models converging toward smooth solutions of the equilibrium. The results proved here take advantage of the enlarged formulation of the models in terms of the drift velocity introduced in [6], generalizing in this way the ones proved in [15] for the Euler–Korteweg model.

Asymptotic stability of the stationary solution to an out-flow problem for the Navier–Stokes–Korteweg equations of compressible fluids

We shall investigate the large-time behavior of solutions to an out-flow problem in one-dimensional half space for the Navier–Stokes–Korteweg equations which models compressible fluids with internal capillarity. Applying the center manifold theory, we prove the existence of the stationary solution under a smallness condition on the boundary data and a proper relation between the capillary coefficient κ and Mach number M+ at far field. Then making use of the energy method and the inequality of Poincaré type, we show that the stationary solution is asymptotically stable under smallness assumptions on the boundary data and the initial perturbation in Sobolev space.

Lower bound of decay rate for higher-order derivatives of solution to the compressible fluid models of Korteweg type

This paper concerns the lower bound decay rate of global solution for compressible Navier–Stokes–Korteweg system in three-dimensional whole space under the H^{4}times H^{3} framework. At first, the lower bound of decay rate for the global solution converging to constant equilibrium state (1, 0) in L^2-norm is (1+t)^{-frac{3}{4}} if the initial data satisfy some low-frequency assumption additionally. Furthermore, we also show that the lower bound of the k(kin [1, 3])th-order spatial derivatives of solution converging to zero in L^2-norm is (1+t)^{-frac{3+2k}{4}}. Finally, it is proved that the lower bound of decay rate for the time derivatives of density and velocity converging to zero in L^2-norm is (1+t)^{-frac{5}{4}}.

A class of global large solutions to the compressible Navier–Stokes–Korteweg system in critical Besov spaces

In this paper, we consider the Cauchy problem for the compressible Navier–Stokes–Korteweg system in critical Besov spaces. The global solutions are established under a nonlinear smallness assumption on the initial data, whether the sound speed is positive or equal to zero. Furthermore, we explain that this kind of nonlinear smallness condition is large in the sense that we can construct an example of initial data satisfying it, even though each component of the initial velocity can be arbitrarily large in $$\dot{B}^{\frac{n}{p}-1}_{p,1}$$.

Strong solutions to compressible–incompressible two-phase flows with phase transitions

We consider a free boundary problem of compressible–incompressible two-phase flows with phase transitions in general domains of N-dimensional Euclidean space (e.g. whole space; half-spaces; bounded domains; exterior domains). The compressible fluid and the incompressible fluid are separated by either compact or non-compact sharp moving interface, and the surface tension is taken into account. In our model, the compressible fluid and incompressible fluid are occupied by the Navier–Stokes–Korteweg equations and the Navier–Stokes equations, respectively. This paper shows that for given T>0 the problem admits a unique strong solution on (0,T) in the maximal Lp−Lq regularity class provided the initial data are small in their natural norms.

Global existence and time decay estimate of solutions to the compressible Navier–Stokes–Korteweg system under critical condition

In this research, we study the global existence of solutions to the compressible Navier–Stokes–Korteweg system around a constant state. This system describes liquid-vapor type two-phase flow with a phase transition with diffuse interface. Previous works assume that pressure is a monotone function for change of density similarly to the usual compressible Navier–Stokes system. On the other hand, due to phase transition the pressure is in fact non-monotone function, and the linearized system loses symmetry in a critical case such that the derivative of pressure is 0 at the given constant state. We show that global L 2 solutions are available for the critical case of small data, whose momentum is in its derivative form, and obtain parabolic type decay rate of the solutions. This is proved based on the decomposition of solutions to a low frequency part and a high frequency part.

Stationary solutions to outflow problem for 1-D compressible fluid models of Korteweg type: Existence, stability and convergence rate

In this paper, we are concerned with the outflow problem in the half line (0,∞) to the isothermal compressible Navier–Stokes–Korteweg system with a nonlinear boundary condition for vanishing capillary tensor at x=0. We first give some necessary and sufficient conditions for the existence of the stationary solutions with the aid of center manifold theory. We also show the stability of the stationary solutions under smallness assumptions on the initial perturbation in the Sobolev space, by employing an energy method. Moreover, the convergence rate of the solution toward the stationary solutions is obtained, provided that the initial perturbations belong to the weighted Sobolev spaces.

Low Mach number limit of the three-dimensional full compressible Navier–Stokes–Korteweg equations

In this paper, we justify the low Mach number limit for the three-dimensional full compressible Navier–Stokes–Korteweg equations rigorously within the framework of smooth solution. Under the assumptions of small density and temperature perturbation, we show that for sufficiently small Mach number, the initial-value problem of the three-dimensional full compressible Navier–Stokes–Korteweg equations admits a unique smooth solution on the time interval where the smooth solution of the corresponding incompressible Navier–Stokes equations exists. Moreover, we obtain the convergence of smooth solutions for the full compressible Navier–Stokes–Korteweg equations toward those for the incompressible Navier–Stokes equations with a convergence rate.

A high-order finite volume method with improved isotherms reconstruction for the computation of multiphase flows using the Navier–Stokes–Korteweg equations

In this work we solve the Navier–Stokes–Korteweg (NSK) equations to simulate a two-phase fluid with phase change. We use these equations on a diffuse interface approach, where the properties of the fluid vary continuously across the interface that separates the different phases. The model is able to describe the behavior of both phases with the same set of equations, and it is also able to handle problems with great changes in the topology of the problem. However, high-order derivatives are present in NSK equations, which is a difficulty for the design of a numerical method to solve the problem. Here, we propose the use of a high-order Finite Volume method with Moving Least Squares approximations to handle high-order derivatives and solve the NSK equations. Moreover, a new methodology to obtain accurate equations of state is presented. In this method, we use any accurate equation of state for the pure phases. Under the saturation curve, a B-spline reconstruction fulfilling a given set of thermodynamic criteria is performed. The new EOS can be used for computations using diffuse interface modeling. Several numerical examples to show the accuracy of the new approach are presented.

Binary-fluid–solid interaction based on the Navier–Stokes–Korteweg equations

We consider a computational model for binary-fluid–solid interaction based on an arbitrary Lagrangian–Eulerian formulation of the Navier–Stokes–Korteweg equations, and we assess the predictive capabilities of this model. Due to the presence of two distinct fluid components, the stress tensor in the binary-fluid exhibits a capillary component in addition to the pressure and viscous-stress components. The distinct fluid–solid surface energies of the fluid components moreover lead to preferential wetting at the solid substrate. Compared to conventional FSI problems, the dynamic condition coupling the binary-fluid and solid subsystems incorporates an additional term associated with the binary-fluid–solid surface tension. We consider a formulation of the Navier–Stokes–Korteweg equations in which the free energy associated with the standard van-der Waals equation of state is replaced by a polynomial double-well function to provide better control over the diffuse-interface thickness and the surface tension. For the solid subsystem, we regard a standard hyperelastic model. We explore the main properties of the binary-fluid–solid interaction problem and establish a dissipation relation for the aggregated system. In addition, we present numerical results based on a fully monolithic approach to the complete nonlinear system. To validate the computational model, we consider the elasto-capillary interaction of a sessile droplet on a soft solid substrate and compare the numerical results with a corresponding solid model with fabricated fluid loads and with experimental data.

On the compactness of weak solutions to the Navier–Stokes–Korteweg equations for capillary fluids

In this paper we consider the Navier–Stokes–Korteweg equations for a viscous compressible fluid with capillarity effects in three space dimensions. We prove compactness of finite energy weak solutions for large initial data. Incontrast with previous results regarding this system, vacuum regions are allowed in the definition of weak solutions and no additional damping terms are considered. The compactness is obtained by introducing suitable truncations of the velocity field and the mass density at different scales and use only the a priori bounds obtained by the energy and the BD entropy.

A Blow-up Criterion for the Density-Dependent Navier–Stokes–Korteweg Equations in Dimension Two

This paper proves a blow-up criterion for the strong solutions with vacuum to the density-dependent Navier–Stokes–Korteweg equations over a bounded smooth domain in \(\mathbb{R}^{2}\), which only in terms of the density.

Asymptotic Profiles and Convergence Rates of the Linearized Compressible Navier–Stokes– Korteweg System

In this paper, we consider the initial value problem for the linearized compressible Navier–Stokes–Korteweg system. Asymptotic profiles and convergence rates are established by Fourier splitting frequency technique. Moreover, some applications of asymptotic profile and convergence rates are exhibited.

Asymptotic stability of viscous shock profiles for the 1D compressible Navier–Stokes–Korteweg system with boundary effect

This paper is concerned with the time-asymptotic behavior of strong solutions to an initial-boundary value problem of the compressible Navier-Stokes-Korteweg system on the half line $\mathbb{R}^+$. The asymptotic profile of the problem is shown to be a shifted viscous shock profile, which is suitably away from the boundary. Moreover, we prove that if the initial data around the shifted viscous shock profile and the strength of the shifted viscous shock profile are sufficiently small, then the problem has a unique global strong solution, which tends to the shifted viscous shock profile as time goes to infinity. The analysis is based on the elementary $L^2$-energy method and the key point is to deal with the boundary estimates.

Existence of strong solutions to the stationary compressible Navier–Stokes–Korteweg equations with large external force

In this study, we show the existence of strong solution to the boundary value problem of the steady compressible Navier–Stokes–Korteweg equation with large external forces in bounded domain, provided that the Mach number is appropriately small. Moreover, the Mach number limit of the strong solutions is rigorously verified. The main idea in the proof is to split the original equation into two parts: (i) a system of stationary incompressible Navier–Stokes–Korteweg equations with large forces, (ii) a system of stationary compressible Navier–Stokes–Korteweg equations with small forces. Introducing the “modified pressures”, (i) is reduced to a system of stationary incompressible Navier–Stokes equations with large forces coupled with an elliptic equation, (ii) is reduced to a system of stationary compressible Navier–Stokes equations with small forces coupled with an elliptic equation. Based on the known results for linear incompressible Navier–Stokes equation, linear transport equation and elliptic equation, we establish uniformity in the Mach number a priori estimates. Further, using the Schauder fixed point theorem, we present the existence of a strong solution. At the same time, from the uniform a priori estimates, we show the zero Mach number limit of the strong solutions, which converge to the solutions of the corresponding incompressible Navier–Stokes–Korteweg equations.

Simulations of droplet collisions with a Diffuse Interface Model near the critical point

In the present study a method is presented for simulations of droplet collisions in three spatial dimensions under non-isothermal conditions using a Diffuse Interface Model, which is based on the non-isothermal Navier–Stokes–Korteweg equations combined with the Van der Waals equation of state. This particular Diffuse Interface Model can only be used for a single component fluid and the temperature needs to be close to the critical point of the fluid to prevent the interfacial thickness from being too small compared to the size of the computational domain. Results are produced with a numerical method based on the finite-volume approach in combination with a Total Variation Diminishing time-integration scheme. Simulations of droplet collisions show the existence of multiple collision regimes for which the energy transfer and dissipation processes during the collisions are studied in detail.

Phase-field descriptions of two-phase compressible fluid flow: Interstitial working and a reduction to Korteweg theory

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.

Global well-posedness of the 3D non-isothermal compressible fluid model of Korteweg type

The aim of this work is to study the global existence and large-time behavior of solutions to the non-isothermal model of capillary compressible fluids derived by Dunn and Serrin (1985). It is proved that the three-dimensional non-isothermal compressible Navier–Stokes–Korteweg system admits a unique global classical solution, provided that the initial energy is suitably small. This result improves previous results obtained by Hattori and Li (1996), where the existence of global classical solutions is established for the initial data close to an equilibrium in some Sobolev space Hs(s≥3). Furthermore, the large time behavior of the solution is also investigated.

Thermal two-phase flow with a phase-field method

Phase field methods have become of great interest for the simulation of droplet and bubble dynamics, moving free interfaces and more recently phase change phenomena. One example is the Navier–Stokes–Korteweg (NSK) equations. With a particular focus on the van der Waals fluid, we present the numerical solution of the NSK system with the energy equation included. The least-squares spectral element formulation with a time-stepping procedure, a high-order continuity approximation and an element-by-element technique is implemented to provide a general and robust solver for the thermal NSK equations. A convergence analysis is conducted to verify our solver. Two numerical examples regarding phase transitions of a droplet and thermocapillary convection are provided to validate our solver.