Mixed systems of conservation laws in industrial mathematical modelling motion
Surveys on Mathematics for Industry, 6, .
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Many mathematical models of evolutionary industrial processes may be written as N x N systems of conservation laws in terms of N independent variables comprising time and N-1 space variables. If such systems posses real and distinct eigenvalues, they are said to be strictly hyperbolic. For "mixed systems", however, the eigenvalues may be equal at points in phase space or even fail to be real, so that the problem has both hyperbolic and elliptic characteristics. In this case the system is ill-posed and requires the specification of boundary conditions that can violate causality. Mathematical models of physical processes that lead to mixed equations are discussed and reviewed, and some of the properties of mixed systems are compared to those of hyperbolic systems. The significance of prototype systems that have been proposed specifically to analyse such properties is considered, and attention is then turned to the archetypal mixed system; the two-phase flow equations. Possible resolutions of the two-phase flow dilema are compared, and a manner in which the modelling may be approached via a more general rational asymptotic scheme is indicated.
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