Explosive instabilities for a generalized second grade fluid

We consider a general model for a non-Newtonian fluid of second grade which is capable of describing shear thinning and shear thickening effects. The model incorporates parameters m and n   which are varied to account for which effect, shear thinning or thickening, is desired in the tangential and normal shear rate directions, although here we restrict attention to the shear thickening case. By constructing a generalized energy-like equation and manipulating this we show that the solution may exhibit finite-time blow up if the initial data are regular enough for the solution to exist for a sufficiently long time. The nature of the blow up, or nonexistence, behavior depends critically on the normal stress coefficients α1α1 and α2α2, and on the parameters m and n.