Some Flows of Particular Nonlinear Fluids

Abstract

The Principle of Local Action asserts that the stress at the body point Xis unaffected at the time t by the history of the motion at other body points except those in some arbitrarily small neighborhood of χ (X, t), but it allows influence to arbitrarily long-past time. Thus, in general, a material point may have an arbitrarily long memory. In viscometric flows (Chapter 5) and, more generally, in monotonous motions (4.2–3), any such memory is given scant opportunity to make itself known, and for this reason many special problems regarding such flows are amenable to an easy solution. There is a second way to find tractable problems: instead of specializing the motion, specialize the material. Because of the obvious difficulty introduced by long-range memory, it is natural to propose for study a class of materials in which the stress atXis affected by the history of the motion only within an arbitrarily short interval [t — δ, t] of preceeding time, where δ is some positive number. Materials of this kind have infinitesimal memory. The history of the motion before any given past time is irrelevant in determining the stress in such a material at the present time.