Some Kinematical Results Concerning Steady Flows and Extensional Flows

Abstract

Vector fields used as local reference directions (i.e., base vectors) for the representation of tensorial physical quantities need not be directly associated with any curvilinear coordinate system. Such vector fields form an “anholonomic” system and their use in differential geometry has been expounded by mathematicians*. In classical fluid mechanics the significance of the streamline direction or the vorticity direction and their use as reference directions for tensorial physical quantities were long recognized. Besides, in certain classes of hydrodynamic flows there are material surfaces and one or more families of material lines whose tangent or normal vectors may be naturally included in a convected anholonomic reference system. However, because the constitutive equations of classical fluid mechanics are not history-dependent, there is no compelling need to trace the evolution of kinematical variables of a material element and, therefore, the use of convected anholonomic reference systems is not essential. On the other hand, the response functionals of rheological fluids with memory depend on the history of kinematical variables. The component forms of such relationships may be considerably simplified when an appropriate intrinsic reference system is used.