On the modeling of compressible viscous fluids via Burgers and Oldroyd equations

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 modeled by replacing some (scalar) coefficients with tensors. To emphasize the compressibility property, generalizations are established to allow for nonzero 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.