Compressible to incompressible transition
Phenomenology
The example of car traffic: The typical phenomenology is that of a traffic jam: in the congested phase, the vehicle density is maximal, corresponding to the minimal allowed distance between two vehicles, while in the free flow phase, the vehicle density is below this threshold.
The left picture shows four snapshots of the vehicle density on a road section, computed from a hyperbolic model with density constraint. The initial density is uniform but the initial velocity is random. Many congestions are rapidly created but aggregation occurs until a single congestion is formed.
These systems can be modeled on the macroscopic scale by fluid models with density constraints.
Pedestrian crowds or mammal herds are two-dimensional analogs of traffic jams. A fish school or a tumor may also, to some extent, be viewed as three dimensional analogs.
Fluid models with maximal density constraint and compressible to incompressible transition
In these flulids, the density of individuals cannot exceed a maximal density threshold corresponding to congestion (individuals are in contact with each other). When the density is below this threshold, the ensemble of individuals viewed as a fluid is compressible. When the density reaches the threshold, the fluid becomes incompressible (see in the pictures below typical examples of a compressible regime (left) and an incompressible one (right) within a sheep herd ; pictures courtesy of G. Théraulaz).
Because of the incompressibility of the flow in the congested phase, acoustic perturbations propagate at infinite speed.
The boundary of a congested region is the locus of the transition. Interface conditions between these two fluid regimes must be defined. They depend on microscopic information about the agents' interactions which can be partly retrieved from the singular pressure model.
Singular pressure model
Fluid model with maximal density constraint are limits of standard compressible models without involving a singular pressure-density relation as shown in the figure below (the maximal density is set to 1): the parameter eps controls the pressure stiffness. In the limit eps -> 0, the constrained model is obtained.
By analyzing this asymptotic limit, the interface conditions between the uncongested region (density < 1) and the congested one (density = 1) can be derived (see also Bouchut et al (J. Nonlinear Sci. 22 (1994), pp. 171-190) ). This asymptotic limit also provides an efficient numerical method for the simulation of fluid models with maximal density constraints, thanks to Asymptotic-Preserving schemes.
This procedure has been applied to the gas dynamics equations, to the Aw-Rascle model of car traffic (and has produced the traffic jam pictures above) to macroscopic models of herding behaviour and of crowds.
Supply chains
In economic circuits, such as supply chains, the constraint acts on the flux.
It induces concentrations rather than congestions, as can be seen on the pictures, which compare the outcome of a microscopic model (a Discrete Event Simulator or DES) and that of a macroscopic model.
The picture represents the density of parts in the production line as a function of the production stage (left axis) and of time (right axis) ; left: Discrete system (Discrete Event Simulator) ; right: macroscopic model).
