Cancellation of vorticity in steady-state non-isentropic flows of complex fluids

In steady-state non-isentropic flows of perfect fluids, there is always thermodynamic generation of vorticity when the difference between the product of the temperature with the gradient of the entropy and the gradient of total enthalpy is different from zero. We note that this property does not hold in general for complex fluids for which the prominent influence of the material substructure on the gross motion may cancel the thermodynamic vorticity. We indicate the explicit condition for this cancellation (topological transition from vortex sheet to shear flow) for general complex fluids described by coarse-grained order parameters and extended forms of Ginzburg-Landau energies. As a prominent sample case we treat first Korteweg's fluid, used commonly as a model of capillary motion or phase transitions characterized by diffused interfaces. Then we discuss general complex fluids. We show also that, when the entropy and the total enthalpy are constant throughout the flow, vorticity may be generated by the inhomogeneous character of the distribution of material substructures, and indicate the explicit condition for such a generation. We discuss also some aspects of unsteady motion and show that in two-dimensional flows of incompressible perfect complex fluids the vorticity is in general not conserved, due to a mechanism of transfer of energy between different levels.