Convection, stability and uniqueness for a fluid of third grade

The equations for a fluid of third grade derived by Fosdick and Rajagopal are first studied on an exterior region in three-dimensional space. A uniqueness theorem and a pointwise continuous dependence theorem (on the initial data) are proved. The conditions at infinity are weak, certainly L² integrability is not required. Then, the equations for the third grade fluid are adapted to the problem of thermal convection due to heating from below. It is shown that the linear instability problem reduces to that of a second grade fluid. Interestingly, a study of non-linear stability for the same problem reveals that the constitutive inequalities obtained by Fosdick and Rajagopal play a very important role; there may be stronger asymptotic stability than for a second grade fluid, although in certain cases the stability may be much weaker.