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Some remarks on boundary conditions for flows of fluids of the differential type. (English) Zbl 0706.76008

Continuum mechanics and its applications, Proc. Conf., Burnaby/Can. 1988, 935-942 (1989).
[For the entire collection see Zbl 0706.00028.]
In the last four decades a number of constitutive equations have been proposed to characterize the non-Newtonian behaviour of fluids. Amongst these the class that has gained considerable support is the fluid of differential type of complexity n. We elaborate some of the above remarks, with specific examples, within the context of an incompressible fluid of second grade. We consider the flow of a non-Newtonian fluid between two coaxial circular cylinders with suction. In the first example we only consider the creeping flow about rotating cylinders. We find that Tanner’s theorem is not valid in this case. In the second example we consider the steady radial flow in the cylindrical annulus. Here we show that one needs an additional boundary condition to have a meaningful solution and that different boundary condition assumptions can lead to altogether different behaviour. In non-Newtonian fluids, in almost all situations, the equations to be solved become highly nonlinear and these are usually solved by employing perturbation procedures. In the final part of the note we discuss how the boundary condition problem, if not handled carefully, can lead to singular perturbation problems.

MSC:

76A10 Viscoelastic fluids

Citations:

Zbl 0706.00028