Catastrophic instabilities and related results in a fluid of third grade

Abstract

Thermodynamical considerations [1] have shown that the most general form for the stress constitutive relation of an incompressible fluid of grade three is T = − p1 + μA 1 + α 1A 2 + α 2A 2 1 + β(tr A 2 1)A 1, where A 1 and A 2 are the first two Rivlin-Ericksen tensors. In addition, the material parameters μ, α 1, α 2 and α were shown in [1] to be restricted by certain inequalities (see (1.7), (1.8)). Here we show that the condition α 1 <0, which is compatible with the Clausius-Duhem inequality but not with the free energy being a minimum in equilibrium, leads to behavior which may not be physically acceptable. An explicit solution is presented for the second grade fluid, for which β =0 and μ, α 1 and α 2 are arbitrary, which demonstrates that if μ #62; 0 and α 1 < 0 then a rotating vortex system may increase indefinitely in amplitude.