Navier Stokes Korteweg Equations Research Articles

Non-existence of classical solutions with finite energy to the Cauchy problem of the Navier–Stokes–Korteweg equations

This paper concerns the compressible Navier–Stokes–Korteweg equations. Based on previous work [Li et al., Arch. Ration. Mech. Anal. 232, 557–590 (2019)], we prove that the classical solution with finite energy does not exist in the inhomogeneous Sobolev space for any short time under some natural assumptions on initial data near the vacuum.

Global existence of weak solutions to the Navier–Stokes–Korteweg equations

In this paper we consider the Navier-Stokes-Korteweg equations for a viscous compressible fluid with capillarity effects in three space dimensions. We prove global existence of finite energy weak solutions for large initial data. Contrary to previous results regarding this system, vacuum regions are allowed in the definition of weak solutions and no additional damping terms are considered. The convergence of the approximating solutions is obtained by introducing suitable truncations in the momentum equations 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. Moreover, the approximating solutions enjoy only a limited amount of regularity, and the derivation of the truncations of the velocity and the density is performed by a suitable regularization procedure.

On the Exponential Decay for Compressible Navier–Stokes–Korteweg Equations with a Drag Term

In this paper, we consider global weak solutions to compressible Navier–Stokes–Korteweg equations with density dependent viscosities, in a periodic domain $$\Omega = \mathbb T^3$$ , with a linear drag term with respect to the velocity. The main result concerns the exponential decay to equilibrium of such solutions using log-sobolev type inequalities. In order to show such a result, the starting point is a global weak-entropy solutions definition, introduced in D. Bresch, A. Vasseur and C. Yu (Global existence of entropy-weak solutions to the compressible Navier–Stokes equations with non-linear density dependent viscosities. arXiv:1905.02701 , 2019). Assuming extra assumptions on the shear viscosity when the density is close to vacuum and when the density tends to infinity, we conclude the exponential decay to equilibrium. Note that our result covers the quantum Navier–Stokes system with a drag term.

Lattice-Boltzmann model for van der Waals fluids with liquid-vapor phase transition

A lattice Boltzmann model (LBM) for two-phase flows with liquid-vapor phase transition based on a dynamic van der Waals theory [Phys. Rev. Lett. 94, 054501 (2005)] is proposed. The proposed model consists of two lattice Boltzmann equations (LBE): one for the Navier-Stokes-Korteweg (NSK) equations and the other for the temperature equation. In the thermal LBE, the equilibrium distribution function is redesigned by introducing a reference temperature, which is used to reduce the numerical errors of velocity divergence in the thermal LBE. A free-energy-based LBE is developed for the hydrodynamic equations and a novel force term is used to correctly recover the NSK equations. Several numerical simulations, including the liquid-vapor coexistence curve, phase separation, stationary droplet, droplet on partially wetting surface, droplet evaporation and bubble nucleate and departure, are conducted to validate the capability and performance of the present model. The numerical results of the proposed model are found to be in excellent agreement with the results of theoretical and/or the hybrid method. It is also shown that numerical stability and accuracy of the present models can be greatly improved by adjusting the reference temperature. The present models provide an effective predictive tool for two-phase flows involving phase change.

Stability of the planar rarefaction wave to three-dimensional Navier–Stokes–Korteweg equations of compressible fluids

This study is concerned with the large time behaviour of the three-dimensional isentropic compressible Navier–Stokes–Korteweg equations, which are used to model viscous and compressible fluids with internal capillarity. Based on the fact that the rarefaction wave is nonlinearly stable to the one-dimensional isentropic compressible Navier–Stokes–Korteweg equations, the planar rarefaction wave to the three-dimensional isentropic compressible Navier–Stokes–Korteweg equations is first constructed. Then it is shown that the planar rarefaction wave is asymptotically stable in the case that the initial data are a suitably small perturbation of the planar rarefaction wave and the strength of the rarefaction wave is small. The proof is based on the delicate energy method. The result indicate that the planar rarefaction wave of the inviscid Euler system is stable for the three-dimensional isentropic compressible fluids with physical viscosities and internal capillarity.

Blow up of classical solutions to the barotropic compressible fluid models of Korteweg type in bounded domains

In this paper, we study the multi-dimensional barotropic compressible Navier–Stokes–Korteweg equations with degenerate coefficients of viscosities and capillary in bounded domains. It is shown that any classical solutions to the initial–boundary-value problem will blow up in finite time, when the initial density admits an isolated mass group. Moreover, a direct application of our result (μ(ρ)=μρ,λ=0,α=0) shows that the weak solution obtained by Bresch et al. (2003) cannot be improved to a classical one as long as there is an isolated mass group initially.

Zero-viscosity–capillarity limit toward rarefaction wave with vacuum for the Navier–Stokes–Korteweg equations of compressible fluids

We shall consider the one-dimensional Navier–Stokes–Korteweg (NSK) equations that are used to model compressible fluids with internal capillarity. Formally, the NSK equations converge, as the viscosity and capillarity vanish, to the corresponding Euler equations, and we do justify this for the case that the Euler equations have a rarefaction wave with one-side vacuum state. To prove this result, we first construct a sequence of approximate solutions to the NSK equations and then show that the sequence converges, as both viscosity and capillarity tend simultaneously to zero, to this rarefaction wave. The uniform convergence rates are also obtained. The key ingredients of our proof are the re-scaling technique and energy estimates.

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.

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.

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.

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.

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.

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].

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.

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.