Steady state solutions for a lubrication two-fluid flow

International audienceIn this paper, we describe possible solutions for a stationary flow of two superposed fluids between two close surfaces in relative motion. Physically, this study is within the lubrication framework, in which it is of interest to predict the relative positions of the lubricant and the air in the device. Mathematically, we observe that this problem corresponds to finding the interface between the two fluids, and we prove that it is equivalent to solve some polynomial equation. We solve this equation using an original method of polynomial resolution. First, we check that our results are consistent with previous work. Next, we use this solution to answer some physically relevant questions related to the lubrication setting. For instance, we obtain theoretical and numerical results enabling to predict the apparition of a full film with respect to physical parameters (fluxes, shear velocity, . . . ). In particular, we present a figure giving the number of stationary solutions depending on the physical parameters. Moreover, in the last part, we give some indications for a better understanding of the multi-fluid case

Diffusion models for mixtures using a stiff dissipative hyperbolic formalism

International audienceIn this article, we are interested in a system of fluid equations for mixtures with a stiff relaxation term of Maxwell-Stefan diffusion type. We use the formalism developed by Chen, Levermore, Liu in [4] to obtain a limit system of Fick type where the species velocities tend to align to a bulk velocity when the relaxation parameter remains small

The Maxwell-Stefan diffusion limit for a kinetic model of mixtures

International audienceWe consider the non-reactive fully elastic Boltzmann equation for mixtures. We deduce that, under the standard diffusive scaling, its limit for vanishing Mach and Knudsen numbers is the Maxwell-Stefan model for a multicomponent gaseous mixture

Viscoelastic fluids in thin domains: a mathematical proof

The present paper deals with non Newtonian viscoelastic flows of Oldroyd-B tye in thin domains. Such geometries arise for example in the context of lubrication. More precisely, we justify rigorously the asymptotic model obtained heuristically by proving the mathematical convergence of the Navier-Stokes/Oldroyd-B sytem towards the asymptotic model

Comparison of several complete cubic laws for two-phase flow models*

In the present paper, we investigate several cubic equations of state widely used in the literature, for which we are able to construct analytically the complete law. In order to describe two-phase flows, we use Maxwell's construction, which amounts to consider pure phases and a mixture zone at saturation. The parameters appearing in the different equations of state are fitted in order to be precise in the saturation zone at high pressures. The different laws are then compared in a large range of pressures, showing the best accuracy of Clausius equation of state

Energy method for the Boltzmann equation of monatomic gaseous mixtures

In this paper, we present an energy method for the system of Boltzmann equations in the multicomponent mixture case, based on a micro-macro decomposition. More precisely, the perturbation of a solution to the Bolzmann equation around a global equilibrium is decomposed into the sum of a macroscopic and a microscopic part, for which we obtain a priori estimates at both lower and higher orders. These estimates are obtained under a suitable smallness assumption. The assumption can be justified a posteriori in the higher-order case, leading to the closure of the corresponding estimate

Study of a low Mach nuclear core model for two-phase flows with phase transition I: stiffened gas law

International audienceIn this paper, we are interested in modelling the flow of the coolant (water) in a nuclear reactor core. To this end, we use a monodimensional low Mach number model coupled to the stiffened gas law. We take into account potential phase transitions by a single equation of state which describes both pure and mixture phases. In some particular cases, we give analytical steady and/or unsteady solutions which provide qualitative information about the flow. In the second part of the paper, we introduce two variants of a numerical scheme based on the method of characteristics to simulate this model. We study and verify numerically the properties of these schemes. We finally present numerical simulations of a loss of flow accident (LOFA) induced by a coolant pump trip event

2D numerical simulation of a low Mach nuclear core model with stiffened gas using FreeFem++

International audienceWe investigate a simplified model describing the evolution of the coolant within a nuclear reactor core (e.g. of PWR type). This model is named LMNC (for Low Mach Nuclear Core) and consists of the coupling between three equations of different types together with boundary conditions specific to the nuclear framework. After several articles dedicated to dimension 1, we present in this paper some monophasic two-dimensional numerical results when the fluid is modelled by the stiffened gas law describing the pure liquid phase. The underlying numerical strategy is based on the Finite-Element software FreeFem++

Numerical method for the 2D simulation of the respiration

International audienceIn this article we are interested in the simulation of the air flow in the bronchial tree. The model we use has already been described by Baffico, Grandmont and Maury and is based on a three part description of the respiratory tract. This model leads, after time discretization, to a Stokes system with non standard dissipative boundary conditions that cannot be easily and directly implemented in most FEM software, in particular in FreeFEM++. The objective is here to provide a new numerical method that could be implemented in any softwares. After describing the method, we illustrate it by two-dimensional simulations