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In the present contribution, we investigate first-order nonlinear systems ofpartial differential equations which are constituted of two parts: a system ofconservation laws and non-conservative first order terms. Whereas the theory offirst-order systems of conservation laws is well established and the conditionsfor the existence of supplementary conservation laws, and more specifically ofan entropy supplementary conservation law for smooth solutions, well known,there exists so far no general extension to obtain such supplementaryconservation laws when non-conservative terms are present. We propose aframework in order to extend the existing theory and show that the presence ofnon-conservative terms somewhat complexifies the problem since numerouscombinations of the conservative and non-conservative terms can lead to asupplementary conservation law. We then identify a restricted framework inorder to design and analyze physical models of complex fluid flows by means ofcomputer algebra and thus obtain the entire ensemble of possible combination ofconservative and non-conservative terms with the objective of obtainingspecifically an entropy supplementary conservation law. The theory as well asdeveloped computer algebra tool are then applied to a Baer-Nunziato two-phaseflow model and to a multicomponent plasma fluid model. The first one is afirst-order fluid model, with non-conservative terms impacting on the linearlydegenerate field and requires a closure since there is no way to deriveinterfacial quantities from averaging principles and we need guidance in orderto close the pressure and velocity of the interface and the thermodynamics ofthe mixture. The second one involves first order terms for the heavy speciescoupled to second order terms for the electrons, the non-conservative termsimpact the genuinely nonlinear fields and the model can be rigorously derivedfrom kinetic theory. We show how the theory allows to recover the wholespectrum of closures obtained so far in the literature for the two-phase flowsystem as well as conditions when one aims at extending the thermodynamics andalso applies to the plasma case, where we recover the usual entropysupplementary equation, thus assessing the effectiveness and scope of theproposed theory.

communications in mathematical sciences

Entropy supplementary conservation law for non-linear systems of PDEs  with non-conservative terms: application to the modelling and analysis of  complex fluid flows using computer algebra