Some generalizations of the Burgers and the Oldroyd equations for compressible fluids

What is it about?

The paper develops some generalizations of the Burgers and the Oldroyd equations for the dynamics of fluids. To account for compressibility, in addition to viscosity, first the Oldroyd derivative is replaced with the Truesdell derivative. Consequently, the two equations can be given a linear form within the Lagrangian formulation. Furthermore, possible anisotropies are modelled by replacing some (scalar) coefficients with tensors. To emphasize the compressibility property, generalizations are established to allow for non-zero longitudinal viscosity. Next the thermodynamic consistency is investigated by regarding both types of equations as rate equations, of second order and first order. The requirements on the parameters entering the two equations are derived while the linearity of the two equations allow the free energy potential be quadratic. The Oldroyd equation is found to be compatible via appropriate restrictions of the tensor coefficients, through different pairs of free energy and entropy production.

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Why is it important?

The thermodynamic consistency with Clausius-Duhem inequality is investigated on the basis of the constitutive property of the entropy production. In connection with a given rate equation of the Oldroyd type we prove that the free energy and the entropy production need not be unique.