A direct theory of viscous fluid flow in pipes. I: Basic general developments

This paper is concerned w ith the construction by a direct approach of a fairly general nonlinear theory of viscous fluid flow in both straight and curved pipes. First, a new procedure is used to establish a 1-1 correspondence between the lagrangian and eulerian form ulations of the integral balance laws of a general thermom echanical theory of directed (or Cosserat ) curve in the presence of a number of directors. (Such correspondence between lagrangian and eulerian formulations was previously possible in the special case of two directors w ith the resulting theory appropriate, for example, only for free surface jets.) The utility of the basic theory, which is valid for both compressible and incompressible linear viscous fluids, is then demonstrated with reference to the unsteady flow of an incompressible viscous fluid in straight pipes of varying elliptical cross section. A general solution is obtained for unsteady flow in straight pipes of elliptical cross section, and is applied to the cases in which a swirling motion is superposed on a uniform axial flow and a flow which is symmetric about the m ajor and minor axes of the elliptical cross section.