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

2021 ◽  
Vol 121 (2) ◽  
pp. 195-217 ◽  
Author(s):  
Takayuki Kobayashi ◽  
Kazuyuki Tsuda
Keyword(s):  
Phase Transition ◽  
Global Existence ◽  
Critical Case ◽  
Monotone Function ◽  
Diffuse Interface ◽  
Navier Stokes ◽  
Small Data ◽  
Two Phase ◽  
Global Existence Of Solutions ◽  
Constant State

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.

Application of Interface-Tracking Method Based on Phase-Field Model to Numerical Analysis of Isothermal and Thermal Two-Phase Flows

2007 ◽  
Author(s):  
Naoki Takada
Keyword(s):  
Phase Field ◽  
Density Ratio ◽  
Phase Field Model ◽  
Field Method ◽  
Diffuse Interface ◽  
Phase Field Method ◽  
Navier Stokes ◽  
Interface Tracking ◽  
Two Phase Flows ◽  
Two Phase

For interface-tracking simulation of two-phase flows in various micro-fluidics devices, the applicability of two versions of Navier-Stokes phase-field method (NS-PFM) was examined, combining NS equations for a continuous fluid with a diffuse-interface model based on the van der Waals-Cahn-Hilliard free-energy theory. Through the numerical simulations, the following major findings were obtained: (1) The first version of NS-PFM gives good predictions of interfacial shapes and motions in an incompressible, isothermal two-phase fluid with high density ratio on solid surface with heterogeneous wettability. (2) The second version successfully captures liquid-vapor motions with heat and mass transfer across interfaces in phase change of a non-ideal fluid around the critical point.

scholarly journals Global existence of weak solutions for a nonlocal model for two-phase flows of incompressible fluids with unmatched densities

2016 ◽  
Vol 26 (10) ◽  
pp. 1955-1993 ◽  
Author(s):  
Sergio Frigeri
Keyword(s):  
Weak Solutions ◽  
Type System ◽  
Diffuse Interface ◽  
Navier Stokes ◽  
Diffuse Interface Model ◽  
Two Phase ◽  
Existence Of Weak Solutions ◽  
Cahn Hilliard Equation ◽  
The One ◽  
Three Space

We consider a diffuse interface model for an incompressible isothermal mixture of two viscous Newtonian fluids with different densities in a bounded domain in two or three space dimensions. The model is the nonlocal version of the one recently derived by Abels, Garcke and Grün and consists in a Navier–Stokes type system coupled with a convective nonlocal Cahn–Hilliard equation. The density of the mixture depends on an order parameter. For this nonlocal system we prove existence of global dissipative weak solutions for the case of singular double-well potentials and non-degenerate mobilities. To this goal we devise an approach which is completely independent of the one employed by Abels, Depner and Garcke to establish existence of weak solutions for the local Abels et al. model.

scholarly journals ON APPROXIMATE SOLUTIONS OF SEMILINEAR EVOLUTION EQUATIONS II: GENERALIZATIONS, AND APPLICATIONS TO NAVIER–STOKES EQUATIONS

2008 ◽  
Vol 20 (06) ◽  
pp. 625-706 ◽  
Author(s):  
CARLO MOROSI ◽  
LIVIO PIZZOCCHERO
Keyword(s):  
Exact Solution ◽  
Global Existence ◽  
Exponential Decay ◽  
Evolution Equations ◽  
Stokes Equations ◽  
Approximate Solutions ◽  
Navier Stokes ◽  
Small Data ◽  
Navier Stokes Equations ◽  
Semilinear Evolution Equations

In our previous paper [12], a general framework was outlined to treat the approximate solutions of semilinear evolution equations; more precisely, a scheme was presented to infer from an approximate solution the existence (local or global in time) of an exact solution, and to estimate their distance. In the first half of the present work, the abstract framework of [12] is extended, so as to be applicable to evolutionary PDEs whose nonlinearities contain derivatives in the space variables. In the second half of the paper, this extended framework is applied to the incompressible Navier–Stokes equations, on a torus Td of any dimension. In this way, a number of results are obtained in the setting of the Sobolev spaces ℍn(Td), choosing the approximate solutions in a number of different ways. With the simplest choices we recover local existence of the exact solution for arbitrary data and external forces, as well as global existence for small data and forces. With the supplementary assumption of exponential decay in time for the forces, the same decay law is derived for the exact solution with small (zero mean) data and forces. The interval of existence for arbitrary data, the upper bounds on data and forces for global existence, and all estimates on the exponential decay of the exact solution are derived in a fully quantitative way (i.e. giving the values of all the necessary constants; this makes a difference with most of the previous literature). Next, the Galerkin approximate solutions are considered and precise, still quantitative estimates are derived for their ℍn distance from the exact solution; these are global in time for small data and forces (with exponential time decay of the above distance, if the forces decay similarly).

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