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- Title
Riemann problem solutions for a balance law under Dirac-Delta source with a discontinuous flux.
- Authors
Abreu, Eduardo; Matos, Vitor; Pérez, John; Rodríguez-Bermúdez, Panters
- Abstract
We study the Riemann problem for a new model on immiscible vertical two-phase flow under point injection. The point injection is modeled by a Dirac δ -source term as well as by a spatially discontinuous flux function, which defines two fluxes, one on each side of the discontinuity. The solutions comprise up to three wave groups: downward waves, a stationary shock and upward waves. Owning the interplay between the Dirac δ -source and the discontinuous flux, there is no standard entropy condition for the stationary shock (flux's connections). An entropy condition was deduced based on impinging characteristics and perturbation of the constant solution. This condition leads to shocks that do not satisfy the classical Lax's conditions — even for arbitrarily small shocks — and may also have no viscous profile. The Rankine–Hugoniot condition — embedding the Dirac δ -source — and the entropy condition are geometrically represented by "Flux Projections" that support the analytical method proposed in this paper. We then obtain a ℒ loc 1 analytic solution for all initial value problems. We verify the entropy condition using a Lagrangian–Eulerian scheme recently introduced in the literature, which is based in the new concept of no-flow curves. Analytic and numerical solutions fit.
- Subjects
RIEMANN-Hilbert problems; INITIAL value problems; EULERIAN graphs; TWO-phase flow; DISCONTINUOUS functions; SHOCK waves
- Publication
Journal of Hyperbolic Differential Equations, 2024, Vol 21, Issue 1, p1
- ISSN
0219-8916
- Publication type
Article
- DOI
10.1142/S0219891624500012