On The Free-Energy And Stability Of Non-Linear Fluids

Abstract

For any incompressible fluid whose stress is a frame indifferent function of the velocity gradient and the material time derivative of the velocity gradient, i.e., for any Rivlin_Ericksen fluid of complexity 2, we show that thermodynamics implies that the first normal stress difference of viscometric flows must be nonpositive for small enough shearings unless a certain very special degeneracy occurs. More precisely, we show that the Clausius_Duhem inequality, together with the postulate that the Helmholtz free energy has a minimum in equilibrium, suffices to ensure that, except for a very special subclass, every Rivlin_Ericksen fluid of complexity 2 has a negative first normal stress difference for all small enough shearings in any viscometric flow. Our results significantly extend a similar analysis given by Dunn and Fosdick in 1974 for those special Rivlin_Ericksen fluids of complexity 2 known as second grade fluids. In addition, they direct attention at a new class of complexity 2 fluids that have been little explored by theorists or experimenters. Furthermore, we study in detail the implications of our thermodynamic postulates for a certain subclass of these complexity 2 fluids that is more general than either second grade fluids or generalized Newtonian fluids. We find that for the fluids in this class the first normal stress difference may change sign as the shearing changes, and we find an interesting linkage between such sign alterations and potential local instabilities in the flow field. Finally, we examine the global stability of the rest state for our fluids and show that if the free energy has a strict, gobal minimum in equilibrium, then our fluids are better behaved than any Navier_Stokes fluid, since not only does the kinetic energy of any disturbance decay in mean but so too does a certain positive definite function of the stretching tensor.