On the thermodynamics, stability and hierarchy of entropy functions in fluid flow

Symmetric forms of systems of equations describing physical phenomena present exceptional desirable properties such as symmetric coefficient matrices, well posedness of the Cauchy initial value problem, finite wave propagation speeds and a link to nonlinear stability principles. Therefore, they should be used when possible to design better numerical methods. In this paper such symmetric forms are reviewed for various models of fluid flow, and a hierarchy of generalized entropy functions is presented. Standard concepts are extended to the diffusive terms.