Abstract
We study the existence of weak solutions to the two-phase fluid model with congestion constraint. The model encompasses the flow in the uncongested regime (compressible) and the congested one (incompressible) with the free boundary separating the two phases. The congested regime appears when the density in the uncongested regime ϱ(t,x) achieves a threshold value ϱ∗(t,x) that describes the comfort zone of individuals. This quantity is prescribed initially and transported along with the flow. We prove that this system can be approximated by the fully compressible Navier–Stokes system with a singular pressure, supplemented with transport equation for the congestion density. We also present the application of this approximation for the purposes of numerical simulations in the one-dimensional domain.
Highlights
Our aim is to analyse the free-boundary two-phase fluid system that could be used to model the congestions in the large group of individuals in a bounded area
In this paper we present, as far as we know, the first mathematical result for the fluid model that incorporates various sizes of the individuals/particles and their inhomogeneities
The main result of this paper is the existence of solutions to the system (1) under the aforementioned assumptions on the constitutive relations and the initial condition, in the sense of the following definition
Summary
Our aim is to analyse the free-boundary two-phase fluid system that could be used to model the congestions in the large group of individuals in a bounded area. The main result of this paper is the existence of solutions to the system (1) under the aforementioned assumptions on the constitutive relations and the initial condition, in the sense of the following definition. The novelty of this paper is the proof of the fact that the same relation can be obtained for system with two densities, and that the final relation (1 − Z)π = 0 can be identified with (ρ∗ − ρ)π = 0 For this we need to prove that various weak formulations of the limit system are equivalent, and that the formal derivation of (8a) by ρ∗ leading to equation for Z can be inverted and made rigorous.
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