82 research outputs found
The reflection of an ionized shock wave
In a previous paper we studied the thermodynamic and kinetic theory for an
ionized gas, in one space dimension; in this paper we provide an application of
those results to the reflection of a shock wave in an electromagnetic shock
tube. Under some reasonable limitations, which fully agree with experimental
data, we prove that both the incident and the reflected shock waves satisfy the
Lax entropy conditions; this result holds even outside genuinely nonlinear
regions, which are present in the model. We show that the temperature increases
in a significant way behind the incident shock front but the degree of
ionization does not undergo a similar growth. On the contrary, the degree of
ionization increases substantially behind the reflected shock front. We explain
these phenomena by means of the concavity of the Hugoniot loci. Therefore, our
results not only fit perfectly but explain what was remarked in experiments.Comment: 16 page
On a model of multiphase flow
We consider a hyperbolic system of three conservation laws in one space
variable. The system is a model for fluid flow allowing phase transitions; in
this case the state variables are the specific volume, the velocity and the
mass density fraction of the vapor in the fluid. For a class of initial data
having large total variation we prove the global existence of solutions to the
Cauchy problem.Comment: 32 pages. Revised and corrected versio
Global existence of solutions for a multi-phase flow: a bubble in a liquid tube and related cases
In this paper we study the problem of the global existence (in time) of weak,
entropic solutions to a system of three hyperbolic conservation laws, in one
space dimension, for large initial data. The system models the dynamics of
phase transitions in an isothermal fluid; in Lagrangian coordinates, the phase
interfaces are represented as stationary contact discontinuities. We focus on
the persistence of solutions consisting in three bulk phases separated by two
interfaces. Under some stability conditions on the phase configuration and by a
suitable front tracking algorithm we show that, if the BV-norm of the initial
data is less than an explicit (large) threshold, then the Cauchy problem has
global solutions
Coupling conditions for isothermal gas flow and applications to valves
We consider an isothermal gas flowing through a straight pipe and study the
effects of a two-way electronic valve on the flow. The valve is either open or
closed according to the pressure gradient and is assumed to act without any
time or reaction delay. We first give a notion of coupling solution for the
corresponding Riemann problem; then, we highlight and investigate several
important properties for the solver, such as coherence, consistence, continuity
on initial data and invariant domains. In particular, the notion of coherence
introduced here is new and related to commuting behaviors of valves. We provide
explicit conditions on the initial data in order that each of these properties
is satisfied. The modeling we propose can be easily extended to a very wide
class of valves
Viscous profiles in models of collective movement with negative diffusivity
In this paper, we consider an advection\u2013diffusion equation, in one space dimension, whose diffusivity can be negative. Such equations arise in particular in the modeling of vehicular traffic flows or crowds dynamics, where a negative diffusivity simulates aggregation phenomena. We focus on traveling-wave solutions that connect two states whose diffusivity has different signs; under some geometric conditions, we prove the existence, uniqueness (in a suitable class of solutions avoiding plateaus) and sharpness of the corresponding profiles. Such results are then extended to the case of end states where the diffusivity is positive, but it becomes negative in some interval between them. Also the vanishing viscosity limit is considered. At last, we provide and discuss several examples of diffusivities that change sign and show that our conditions are satisfied for a large class of them in correspondence of real data
Global existence of solutions for a multi-phase flow: a drop in a gas-tube
In this paper we study the flow of an inviscid fluid composed by three
different phases. The model is a simple hyperbolic system of three conservation
laws, in Lagrangian coordinates, where the phase interfaces are stationary. Our
main result concerns the global existence of weak entropic solutions to the
initial-value problem for large initial data
Global weak solutions for a model of two-phase flow with a single interface
We consider a simple nonlinear hyperbolic system modeling the flow of an
inviscid fluid. The model includes as state variable the mass density fraction
of the vapor in the fluid and then phase transitions can be taken into
consideration; moreover, phase interfaces are contact discontinuities for the
system. We focus on the special case of initial data consisting of two
different phases separated by an interface. We find explicit bounds on the
(possibly large) initial data in order that weak entropic solutions exist for
all times. The proof exploits a carefully tailored version of the front
tracking scheme
Sonic Hyperbolic Phase Transitions and Chapman-Jouguet Detonations
AbstractWe prove that the Cauchy problem for an n×n system of strictly hyperbolic conservation laws in one space dimension admits a weak global solution also in presence of sonic phase boundaries. Applications to Chapman–Jouguet detonations, liquid–vapor transitions and elastodynamics are considered
Coherence of coupling Riemann solvers for gas flows through flux-maximizing valves
In this paper we propose a model, based on the strictly hyperbolic system of isothermal Euler equations , for the gas flow in a straight pipe with a valve. We are then faced with an initial value problem with coupling conditions at the valve position. The valves under consideration are requested to maximize the flux; moreover, the flow is imposed to occur within prescribed bounds of pressure and flow. The issue is the mathematical characterization of the coherence of the corresponding coupling Riemann solvers; this property is related to the phenomenon of chattering, the rapid switch on and off of the valve. Within this framework we describe three kinds of valves, which differ for their action; two of them lead to a coherent solver, the third one does not. Proofs involve geometric and analytic properties of the Lax curves
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