A positivity-preserving central-upwind scheme for isentropic two-phase flows through deviated pipes

Directional drilling in oil and gas extraction can encounter difficulties such as accumulation of solids in deviated pipes. Motivated by such phenomenon, we consider a model for isentropic two-phase flows through deviated pipes. The system of partial differential equations is aimed at simulating the dynamics between a particle bed and a gas phase. The pipe can be either horizontal or vertically deviated where the effects of gravity are incorporated. Furthermore, the acceleration or deceleration due to friction between phases is investigated and spectral properties of the hyperbolic system of balance laws are described. The existence and characterization of steady states under appropriate conditions is analyzed. A new type of steady states arises when a balance between gas and solid phases results in a non-uniform solid particle bed and vanishing solid velocity. This state corresponds to an accumulation of sedimented solids. A central-upwind scheme that preserves the positivity of the gas and solid densities and volume fractions is presented. Including an application of the model to an analysis of accumulation of solids, a variety of numerical tests is presented to show the merits of the scheme.

Approximation of generalized Riemann solutions to compressible Euler-Poisson equations of isothermal flows in spherically symmetric space-times

In this paper, we consider the compressible Euler-Poisson system in spherically symmetric space-times. This system, which describes the conservation of mass and momentum of physical quantity with attracting gravitational potential, can be written as a $3\times 3$ mixed-system of partial differential systems or a $2\times 2$ hyperbolic system of balance laws with $global$ source. We show that, by the equation for the conservation of mass, Euler-Poisson equations can be transformed into a standard $3\times 3$ hyperbolic system of balance laws with $local$ source. The generalized approximate solutions to the Riemann problem of Euler-Poisson equations, which is the building block of generalized Glimm scheme for solving initial-boundary value problems, are provided as the superposition of Lax's type weak solutions of the associated homogeneous conservation laws and the perturbation terms solved by the linearized hyperbolic system with coefficients depending on such Lax solutions.