Exact Riemann solutions for the drift-flux equations of two-phase flow under gravity
type: Item
instance of: scholarly article
title: Exact Riemann solutions for the drift-flux equations of two-phase flow under gravity
author: Chun Shen
author: M. N. Sun
published in: Journal of Differential Equations
publication date: 2022-02-10T00:00:00Z
review text: The authors consider the system of one-dimensional drift-flux equations describing a two-phase flow and written as: \((\alpha_{g}\rho_{g})_{t}+(\alpha_{g}\rho_{g}u_{g})_{x}=0\), \((\alpha_{l}\rho_{l})_{t}+(\alpha_{l}\rho_{l}u_{l})_{x}=0\), \((\alpha_{g}\rho_{g}u_{g}+\alpha_{l}\rho_{l}u_{l})_{t}+(\alpha_{g}\rho_{g}u_{g}^{2}+\alpha_{l}\rho_{l}u_{l}^{2}+p)_{x}=-q\), where \(\alpha_{g}\), \(\alpha_{l}\in \lbrack 0,1]\) are the volume fractions of gas and liquid which satisfy \(\alpha_{g}+\alpha_{l}=1\), \(\rho_{g}\) the gas density, \(u_{g}\) the gas velocity, \(\rho_{l}\) the liquid density, \(u_{l}\) the liquid velocity, \(p\) the pressure for both gas and liquid, and \(q\) the external forces such as gravity and friction. Considering a 1D inviscid, compressible and isentropic liquid-gas two-phase flow on an inclined pipeline under gravity, which leads to special expressions for the external forces \(q\) and pressure \(p\), the authors simplify the above system as the following one written in a conservative form as: \(m_{t}+(m(v-\mu t))_{x}=0\), \(n_{t}+(n(v-\mu t))_{x}=0\), \(((m+n)v)_{t}+((m+n)v(v-\mu t)+(\gamma -1)(m+n)^{\gamma })_{x}=0\), where \(m\) and \(n\) are the gas and liquid masses defined as \(m=\alpha_{g}\rho_{g}\), \(n=\alpha_{l}\rho_{l}\), and \(v\) is the velocity \(v(x,t)=u(x,t)+\mu t\). Riemann initial conditions \((m,n,v)(x,t=0)=(m_{-},n_{-},u_{-})\), \(x<0\), \((m,n,v)(x,t=0)=(m_{+},n_{+},u_{+})\), \(x>0\), where \(m_{-},n_{-},m_{+},n_{+}>0\) are naturally deduced from physical considerations are imposed. The authors rewrite this last problem in a quasi-linear form for which they derive the associated characteristic equation: \((m+n)(\lambda -v+\mu t)((\lambda -v+\mu t)^{2}-(\gamma -1)\gamma (m+n)^{\gamma -1})=0\) and the three eigenvalues. They draw computations to determine and analyze the properties of the Riemann solutions to the last problem. The first main result proves that for any given Riemann initial conditions, there exists a unique sol…
reviewed by: Alain Brillard
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 76T10
Mathematics Subject Classification ID: 76L05
Mathematics Subject Classification ID: 76P05
Mathematics Subject Classification ID: 76N15
Mathematics Subject Classification ID: 35L65
Mathematics Subject Classification ID: 35L67
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 7471746
zbMATH Keywords: Riemann problem
zbMATH Keywords: drift-flux equations
zbMATH Keywords: two-phase flow
zbMATH Keywords: Riemann solution
zbMATH Keywords: existence and uniqueness
zbMATH Keywords: vacuum state
zbMATH Keywords: generalized delta shock wave solution
zbMATH Keywords: Rankine-Hugoniot conditions
MaRDI profile type: Publication
full work available at URL: j.jde.2022.01.009
OpenAlex ID: W4205253156
cites work: The Riemann problem for the Baer-Nunziato two-phase flow model
cites work: Q4333357
cites work: Sticky particle dynamics with interactions
cites work: Q4490854
cites work: Q4040818
cites work: Formation of $\delta$-Shocks and Vacuum States in the Vanishing Pressure Limit of Solutions to the Euler Equations for Isentropic Fluids
cites work: Concentration and cavitation in the vanishing pressure limit of solutions to the Euler equations for nonisentropic fluids
cites work: Definition and weak stability of nonconservative products
cites work: Delta shock wave formation in the case of triangular hyperbolic system of conservation laws
cites work: Dynamics of propagation and interaction of \(\delta\)-shock waves in conservation law systems
cites work: Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics
cites work: Hybrid flux-splitting schemes for a common two-fluid model
cites work: On the Wave Structure of Two‐Phase Flow Models
cites work: Global existence of weak solutions for a viscous two-phase model
cites work: On global solutions to the viscous liquid–gas model with unconstrained transition to single-phase flow
cites work: Non-classical shallow water flows
cites work: The approximate Riemann solver of Roe applied to a drift-flux two-phase flow model
cites work: The vanishing pressure limits of Riemann solutions to the Chaplygin gas equations with a source term
cites work: A global unique solvability of entropic weak solution to the one-dimensional pressureless Euler system with a flocking dissipation
cites work: The Riemann problem for a weakly hyperbolic two-phase flow model of a dispersed phase in a carrier fluid
cites work: Weak Solution to Pressureless Type System
cites work: Nonlinear stability and existence of vortex sheets for inviscid liquid-gas two-phase flow
cites work: Well posedness for pressureless flow
cites work: A trajectory map for the pressureless Euler equations
cites work: Lagrangian Coordinates for the Sticky Particle System
cites work: Concentration of mass in the pressureless limit of Euler equations for power law
cites work: Eulerian droplet model: delta-shock waves and solution of the Riemann problem
cites work: On the Riemann Problem Simulation for the Drift-Flux Equations of Two-Phase Flows
cites work: Note on the compressible Euler equations with zero temperature
cites work: On the wave interactions in the drift-flux equations of two-phase flows
cites work: DELTA SHOCK WAVES AS A LIMIT OF SHOCK WAVES
cites work: Shadow waves: Entropies and interactions for delta and singular shocks
cites work: \(\delta ^{\prime }\)-shock waves as a new type of solutions to systems of conservation laws
cites work: High Mach number limit of one-dimensional piston problem for non-isentropic compressible Euler equations: Polytropic gas
cites work: Rectilinear vortex sheets of inviscid liquid-gas two-phase flow: linear stability
cites work: Multiplication of distributions and a nonlinear model in elastodynamics
cites work: Delta shock waves in the shallow water system
cites work: Q4222728
cites work: The Riemann problem for the pressureless Euler system with the Coulomb-like friction term
cites work: The singular limits of solutions to the Riemann problem for the liquid–gas two-phase isentropic flow model
cites work: The asymptotic limits of Riemann solutions for the isentropic drift‐flux model of compressible two‐phase flows
cites work: Formation of delta shocks and vacuum states in the vanishing pressure limit of Riemann solutions to the perturbed Aw-Rascle model
cites work: The vanishing adiabatic exponent limits of Riemann solutions to the isentropic Euler equations for power law with a Coulomb-like friction term
cites work: The Riemann problem for the transportation equations in gas dynamics
cites work: Q4314474
cites work: Concentration and cavitation phenomena of Riemann solutions for the isentropic Euler system with the logarithmic equation of state
cites work: Delta-shock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws
cites work: Flux approximation to the isentropic relativistic Euler equations
cites work: The Riemann problem for a hyperbolic model of two-phase flow in conservative form
cites work: Numerical study of wave propagation in compressible two-phase flow
cites work: Riemann problems for a class of coupled hyperbolic systems of conservation laws with a source term

Exact Riemann solutions for the drift-flux equations of two-phase flow under gravity (DOI)
DOI: 10.1016/J.JDE.2022.01.009