The steady laminar flow of an elastico-viscous liquid in a curved pipe of varying elliptic cross section

Abstract

The fully developed steady laminar flow of an idealized elastico-viscous liquid through a curved tube with elliptic cross section is studied. The axes of the ellipse are placed in an arbitrary position with respect to the radius of curvature, and the cross-sectional area varies slowly with the longitudinal distance. A perturbation scheme in terms of Dean number [1, 2] and a nonuniform curvature parameter is applied to determine the effects of 1. (i) the nonuniformity of the curvature 2. (ii) the shape of the cross section of the curved pipe 3. (iii) the arbitrary positioning of the elliptic cross section relative to the radius of curvature of the curved pipe. It is shown that in a tube of increasing curvature, there will be delay in setting up a secondary motion. The point of maximum shear stress varies with the cross section under consideration. The flow of Newtonian fluid in a curved tube of elliptic cross section is discussed, as a particular case.