We study a binary mixture of compressible viscous fluids modelled by the Navier-Stokes-Allen-Cahn system with isentropic or ideal gas law. We propose a finite volume method for the approximation of the system based on upwinding and artificial diffusion approaches. We prove the entropy stability of the numerical method and present several numerical experiments to support the theory.

In this article, we consider the initial value problem which is obtained after a space discretization (with space step $h$) of the equations governing the solidification process of a multicomponent alloy. We propose a numerical scheme to solve numerically this initial value problem. We prove an error estimate which is not affected by the step size $h$ chosen in the space discretization. Consequently, our scheme provides global convergence without any stability condition between $h$ and the time step size...

In this article, we consider the initial value problem which is obtained after a space discretization (with space step h) of the equations governing the solidification process of a multicomponent alloy. We propose a numerical scheme to solve numerically this initial value problem. We prove an error estimate which is not affected by the step size h chosen in the space discretization. Consequently, our scheme provides global convergence without any stability condition between h and the time...

We present in this paper a pressure correction scheme for the drift-flux model combining finite element and finite volume discretizations, which is shown to enjoy essential stability features of the continuous problem: the scheme is conservative, the unknowns are kept within their physical bounds and, in the homogeneous case (i.e. when the drift velocity vanishes), the discrete entropy of the system decreases; in addition, when using for the drift velocity a closure law which takes the form of...

The aim of the paper is an analytical and numerical approach to the pseudo-compositional black-oil model for simulating a 3-D isothermal constrained polyphasic flow in porous media, taking into account realistic boundary conditions. The handling of the component conservation laws leads to a strongly coupled system including parabolic quasilinear degenerated equations and first-order hyperbolic inequalities: the introduction of unilateral problems arises from the nature of the thermodynamical equilibrium...

The two-phase free boundary value problem for the Navier-Stokes system is considered in a situation where the initial interface is close to a halfplane. We extract the boundary symbol which is crucial for the dynamics of the free boundary and present an analysis of this symbol. Of particular interest are its singularities and zeros which lead to refined mapping properties of the corresponding operator.

We have developed a multiphase flow code that has been applied to study the behavior of non-aqueous phase liquids (NAPL) in the subsurface. We describe model formulation, discretization, and use the model for numerical investigation of sensitivity of the NAPL plume with respect to capillary parameters of the soil. In this paper the soil is assumed to be spatially homogeneous. A 2-D reference problem has been chosen and has been recomputed repeatedly with modified parameters of the Brooks–Corey capillary...

This paper concerns numerical methods for two-phase flows. The governing equations are the compressible 2-velocity, 2-pressure flow model. Pressure and velocity relaxation are included as source terms. Results obtained by a Godunov-type central scheme and a Roe-type upwind scheme are presented. Issues of preservation of pressure equilibrium, and positivity of the partial densities are addressed.

This paper concerns numerical methods for two-phase flows. The governing equations are the compressible 2-velocity, 2-pressure flow model. Pressure and velocity relaxation are included as source terms. Results obtained by a Godunov-type central scheme and a Roe-type upwind scheme are presented. Issues of preservation of pressure equilibrium, and positivity of the partial densities are addressed.

The paper proposes a general framework allowing the analysis of wetting problems in the situation when interfacial tensions depend on external fields. An equation predicting apparent contact angles of sessile droplets deposited on rough surfaces in the presence of external fields is derived. The problem of wetting is discussed in the framework of the variational approach. Derivation of a general equation generalizing the Cassie and Wenzel approaches...

A nonlinear differential system for describing an air-water system in groundwater hydrology is given. The system is written in a fractional flow formulation, i.e., in terms of a saturation and a global pressure. A continuous-time version of the finite element method is developed and analyzed for the approximation of the saturation and pressure. The saturation equation is treated by a Galerkin finite element method, while the pressure equation is treated by a mixed finite element method. The analysis...

We introduce a new condition which extends the definition of sticky particle dynamics to the case of discontinuous initial velocities $u_0$ with negative jumps. We show the existence of a stochastic process and a forward flow $\phi$ satisfying $X_{s+t}=\phi (X_s,t,P_s,u_s)$ and $\mathrm{d}{X}_t=\mathrm{E}[u_0\left(X_0\right)/X_t]\mathrm{d}t$, where $P_s=P{X}_s^{-1}$ is the law of $X_s$ and $u_s\left(x\right)=\mathrm{E}[u_0\left(X_0\right)/X_s=x]$ is the velocity of particle $x$ at time $s\ge 0$. Results on the flow characterization and Lipschitz continuity are also given.Moreover, the map $(x,t)\mapsto M(x,t):=P(X_t\le x)$ is the entropy solution of a scalar conservation law ${\partial }_{t}M+{\partial }_x\left(A\left(M\right)\right)=0$ where the flux $A$ represents the particles...

We consider a dense granular shear flow in a two-dimensional system. Granular systems (composed of a large number of macroscopic particles) are far from equilibrium due to inelastic collisions between particles: an external driving is needed to maintain the motion of particles. Theoretical description of driven granular media is especially challenging for dense granular flows. This paper focuses on a gravity-driven dense granular Poiseuille flow...

In this work, we are interested in two different diffusion models for multicomponent mixtures. We numerically recover experimental results underlining the inadequacy of the usual Fick diffusion model, and the importance of using the Maxwell-Stefan model in various situations. This model nonlinearly couples the mole fractions and the fluxes of each component of the mixture. We then consider a subregion of the lower part of the lung, in which we compare...