Shear dispersion along circular pipes is affected by bends, but the torsion of the pipe is negligible

Abstract

The flow of a viscous fluid along a curving pipe of fixed radius is driven by a pressure gradient. For a generally curving pipe it is the fluid flux which is constant along the pipe, and so I extend fluid flow solutions of Dean [W. R. Dean, Phil. Mag., 5 (1928), pp. 673-695] and Topakoglu [H. C. Topakoglu J. Math. Mech., 16 (1967), pp. 1321-1338] which assume constant pressure gradient. When the pipe is straight, the fluid adopts the parabolic velocity profile of Poiseuille flow; the spread of any contaminant along the pipe is then described by the shear dispersion model of Taylor [G. I. Taylor Proc. Roy. Soc. London A, 219 (1953), pp. 186-203] and its refinements. However, two conflicting effects occur in a generally curving pipe: viscosity skews the velocity profile which enhances the shear dispersion, whereas in faster flow centrifugal effects establish secondary flows that reduce the shear dispersion. The two opposing effects cancel at a Reynolds number of about 15. Interestingly, the torsion of the pipe seems to have very little effect upon the flow or the dispersion; the curvature is by far the dominant influence. Last, curvature and torsion in the fluid flow significantly enhance the upstream tails of concentration profiles in qualitative agreement with observations of dispersion in river flow. © 2004 Society for Industrial and Applied Mathematics.