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Fluid Dynamics

Fluid-dynamic approaches have been developed since the 1950's [2]. The road traffic is modeled as a one-dimensional compressible flow with density tex2html_wrap_inline540 with velocity v(x,t). The continuity equation reads,
The dynamics can be expressed by the Navier-Stokes equation [3], which states that the particle density times acceleration equals the sum of the acting forces,
where p is the local pressure of the vehicular gas, tex2html_wrap_inline546 is the viscosity and X represents the force caused by the interaction of the individual vehicles. The forces are derived from the density dependence of the velocity given by the fundamental diagram such as shown in Fig. 2,

Figure 3: Time evolution of a traffic jam showing the density tex2html_wrap_inline540 upper graph and tex2html_wrap_inline552 for a particular time (taken from [3]).  

where tex2html_wrap_inline554 is the relaxation time of the process. Kerner el al. [3, 4, 5] showed that this ansatz leads to the formation of stationary clusters (jams). Their form is shown in Fig. 3. The model was significantly improved by Helbing [6] who introduced the variance of the velocities as an additional space and time dependent variable. He showed the this approach leads to a more realistic behavior when jams develop.

Burkhard Militzer
Sat May 9 11:34:31 CDT 1998