1,720,972 research outputs found
On the Shock Wave Discontinuities in Grad Hierarchy for a Binary Mixture of Inert Gases
No full textThe shock wave problem is investigated for a binary mixture of inert gases described at each closure level within the hierarchy of Grad 13–moment system. The analysis focuses on the occurrence of singularities as the Mach number increases: their compatibility with jump discontinuities is examined through a geometric approach, and confirmed by stability arguments
The evaporation–condensation problem for a binary mixture of rarefied gases
Full text linkHalf space problems of evaporation and condensation for a binary mixture of inert gases are investigated when the dynamics is governed by a system of Navier–Stokes equations, obtained as hydrodynamic limit of a BGK-type description with dominant elastic collisions. Typical methods of qualitative theory of dynamical systems are used to investigate the one-dimensional stationary problem and to classify the solutions both in subsonic and supersonic cases. Numerical results for a mixture of noble gases are presented; the shock wave structure, representing transition between a subsonic and a supersonic steady flow in thermodynamic equilibrium, and the occurrence of under- and overshoots are discussed
On the shock thickness for a binary gas mixture
No full textWe discuss the structure of the shock wave solution for a system of Navier–Stokes equations, obtained as hydrodynamic limit of a BGK description of the dynamics of monoatomic gases at kinetic level. We investigate first the thickness of the transition region of the shock profile for a monoatomic gas, for varying Mach number and different physical options for the viscosity coefficient. The analysis is then extended to a binary gas mixture. Some numerical results for noble gases are presented and discussed
A mixed Boltzmann–BGK model for inert gas mixtures
Full text linkWe propose a mixed Boltzmann-BGK model for mixtures of monatomic gases, where some kinds of collisions are described by bi-species Boltzmann operators and the others by the binary BGK terms given in [Bobylev et al., Kinetic and Related Models 11 (2018)], that is the relaxation model for mixtures with the closest structure to the Boltzmann one. At first, we assume that collisions occurring within the same species (intra-species) are modelled by Boltzmann operators, while interactions between different constituents (inter-species) are described by BGK terms. This option allows us to rigorously derive hydrodynamic equations not only in the classical collision dominated regime, but also in situations with intra-species collisions playing the dominant role (as in mixtures with very disparate particle masses). Then, we present a general form of this mixed Boltzmann-BGK model, characterized by further parameters allowing us to select which binary interactions have to be described by Boltzmann integrals or by BGK operators. We prove that this model preserves conservations of global momentum and energy, positivity of all temperatures and the validity of Boltzmann H-theorem, allowing us to conclude that the unique admissible equilibrium state is the expected Maxwellian distribution with all species sharing a common mean velocity and a common temperature
Macroscopic equations for inert gas mixtures in different hydrodynamic regimes
Full text linkStarting from a BGK model for gas mixtures involving sums of relaxation operators, we formally derive Euler and Navier–Stokes equations in different regimes, in the asymptotic limit for proper Knudsen number, with explicit computation of the transport coefficients of viscosity and thermal conductivity. First, we consider a regime dominated by the whole collision phenomena; then, we focus on the case of ε−mixtures of heavy and light species, assuming accordingly that the collisions within each component constitute the dominant process. In this latter case we show that the two-scale collision regime leads to a multi-velocity and multi-temperature hydrodynamic description, which emphasizes the distinctive features of each constituent
A new mixed Boltzmann-BGK model for mixtures solved with an IMEX finite volume scheme on unstructured meshes
No full textIn this work, we consider a novel model for a binary mixture of inert gases. The model, which preserves the structure of the original Boltzmann equations, combines integro-differential collision operators with BGK relaxation terms in each kinetic equation: the first involving only collisions among particles of the same species, while the second ones taking into account the inter–species interactions. We prove consistency of the model: conservation properties, positivity of all temperatures, H-theorem and convergence to a global equilibrium in the shape of a global Maxwell distribution. We also derive hydrodynamic equations under different collisional regimes. In a second part, to numerically solve the governing equations, we introduce a class of time and space high order finite volume schemes that are able to capture the behaviors of the different hydrodynamic limit models: the classical Euler equations as well as the multi-velocities and temperatures Euler system. The methods work by integrating the distribution functions over arbitrarily shaped and closed control volumes in 2D using Central Weighted ENO (CWENO) techniques and make use of spectral methods for the approximation of the Boltzmann integrals with high order Implicit-Explicit (IMEX) Runge Kutta schemes. For these methods, we prove accuracy and preservation of the discrete asymptotic states. In the numerical section we first show that the methods indeed possess the theoretical order of accuracy for different regimes and second we analyse their capacity in solving different two dimensional problems arising in kinetic theory. To speed up the computational time, all simulations are run with MPI parallelization on 64 cores, thus showing the potentiality of the proposed methods to be used for HPC (High Performance Computing) on massively parallel architectures
A new mixed Boltzmann-BGK model for mixtures solved with an IMEX finite volume scheme on unstructured meshes
Full text linkIn this work, we consider a novel model for a binary mixture of inert gases. The model, which preserves the structure of the original Boltzmann equations, combines integro-differential collision operators with BGK relaxation terms in each kinetic equation: the first involving only collisions among particles of the same species, while the second ones taking into account the inter–species interactions. We prove consistency of the model: conservation properties, positivity of all temperatures, H-theorem and convergence to a global equilibrium in the shape of a global Maxwell distribution. We also derive hydrodynamic equations under different collisional regimes. In a second part, to numerically solve the governing equations, we introduce a class of time and space high order finite volume schemes that are able to capture the behaviors of the different hydrodynamic limit models: the classical Euler equations as well as the multi-velocities and temperatures Euler system. The methods work by integrating the distribution functions over arbitrarily shaped and closed control volumes in 2D using Central Weighted ENO (CWENO) techniques and make use of spectral methods for the approximation of the Boltzmann integrals with high order Implicit-Explicit (IMEX) Runge Kutta schemes. For these methods, we prove accuracy and preservation of the discrete asymptotic states. In the numerical section we first show that the methods indeed possess the theoretical order of accuracy for different regimes and second we analyse their capacity in solving different two dimensional problems arising in kinetic theory. To speed up the computational time, all simulations are run with MPI parallelization on 64 cores, thus showing the potentiality of the proposed methods to be used for HPC (High Performance Computing) on massively parallel architectures
Boundary conditions for two-temperature Navier-Stokes equations for a polyatomic gas
Full text linkA polyatomic gas with slow relaxation of the internal modes in contact with a solid boundary is considered. In a previous paper [K. Aoki, Phys. Rev. E 102, 023104 (2020)10.1103/PhysRevE.102.023104], the two-Temperature Navier-Stokes system, i.e., a set of compressible Navier-Stokes equations with the translational and internal temperatures, was derived from the ellipsoidal-statistical (ES) model of the Boltzmann equation for a polyatomic gas under the assumption that the Knudsen number is small and the ratio of the collisional mean free time to the relaxation time of the internal modes is as small as the Knudsen number. In the present study, the appropriate boundary conditions for the two-Temperature Navier-Stokes system are derived by the analysis of the Knudsen layer on the basis of the ES model for a polyatomic gas and the Maxwell-Type diffuse-specular reflection condition on the boundary. The resulting boundary conditions, which are of the type of slip boundary conditions, are summarized, together with the two-Temperature Navier-Stokes equations, in a form that is applicable to practical applications immediately
Optimal control of leachate recirculation for anaerobic processes in landfills
Full text linkA mathematical model for the degradation of the organic fraction of solid waste in landfills, by means of an anaerobic bacterial population, is proposed. Additional phenomena, like hydrolysis of insoluble substrate and biomass decay, are taken into account. The evolution of the system is monitored by controlling the effects of leachate recirculation on the hydrolytic process. We investigate the optimal strategies to minimize substrate concentration and recirculation operation costs. Analytical and numerical results are presented and discussed for linear and quadratic cost functionals
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