Stability and Nonexistence Results in the Generalized Theory of a Fluid of Second Grade

A model is studied of a generalized fluid of second grade which was proposed by Man and Sun (J. Glaciol.33 (1987), 268-273) in their analysis of glacier flow. We prove two global nonexistence results for suitable boundary-initial value problems, under certain conditions on the constitutive coefficients, the spatial region being a general three-dimensional one. A generalized energy analysis is found to be necessary and is. therefore, developed to investigate nonlinear stability for the Bénard problem utilizing the constitutive theory of Man an Sun while allowing the viscosity to be a linear function of temperature.