In this paper we describe an electrical network, whose current evolution does agree with a Korteweg–de Vries equation. Our aim is to prepare pupils to understand the analytical aspects of nonlinear and dispersive phenomena, which very often are neglected in high-school and graduate textbooks. Some historical remarks introduce the topic and a bibliography is provided.
We consider system of hyperbolic balance laws governing relativistic two-fluid flow in which entropy is produced only by disequilibrium between the temperatures of the fluids. We compare two such models: one in which thermal equilibrium is attained through a relaxation procedure, and a fully relaxed model. We describe how the relaxation procedure may be made consistent with the second law of thermodynamics. The wave velocities for both models are obtained and compared: the mixture hydrodynamical velocity of the relaxed system is always less than the hydrodynamical velocity of the relaxation system.
A simple interface-capturing approach is developed in order to deduce the relativistic fluid equations for a two-component mixture, using a stiffened gas equation of state. The two species are assumed to be at thermal equilibrium and the total pressure of the mixture is expressed in terms of the pressures of the two components by Dalton's law. Moreover, weak discontinuity waves compatible with such a fluid are examined.
In the present paper a model for a compressible relativistic fluid 2-component mixture with different temperature, and even different pressure laws, is obtained. Total pressure is then the sum of the two independent pressures, which are supposed to be described by an evolution equation formally analogous to the usual one governing the pressure of a simple fluid. The deduced system is able to describe different complex situations, like plasma physics where the temperature of the electrons must be different from the temperatures of the other species. Weak discontinuities compatible with such kind of mixtures are also studied. In the isentropic case the governing equations are put into symmetric hyperbolic form in which the dependent variables are the two pressures plus a generalized velocity vector defined by the fluid's unit 4-velocity divided pointwise by the fluid's index.
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