Some Flows of Fluids of Grade 2

Abstract

The incompressible fluid of grade 2 is defined by the constitutive relation (6.1-17), in which μ, α¹, and α₂ are constants, and μ, the shear viscosity, is positive. The sign of the coefficient α₁ has important effects on the nature of the solutions. If the constitutive relation is taken as defining a particular fluid, just as the Navier-Stokes fluid is almost always regarded, other restrictions (for example, those implied by thermodynamics) lead to the conclusion that α₁ ≥ 0, α₁ +α₂ = 0. However, strong sentiments have been espoused for assuming that α₁ > 0 when this constitutive relation is regarded as a second-order approximation in the sense of retarded motions (6.1-9). A critical discussion of the relevant issues can be found in the recent review article by Dunn and Rajagopal.1 In the following purely mechanical treatment we do not impose any of these adscititious inequalities. We instead emphasize the effects that the sign of α₁ has upon the phenomena associated with fluids of grade 2.

We note that μ/ǀα₁ǀ has the dimension of time. Thus the fluid of grade 2 may be expected to show evidence of having a time scale proper to itself. The special flows we study now will reveal some effects of the existence of this proper time. Of course these effects must vanish when α₁ = α₂ = 0, for then the fluid of grade 2 reduces to the Navier-Stokes fluid, with which no constitutive parameter bearing the dimension of time can be associated.