J zyxwvutsrqponmlk Fluid Jlech. zyxwvutsrqpon (1988), aol. zyxwvutsrq 193, pp 347-367 Printed in Great zyxwvutsr Britain 347 The effect of secondary motion on axial transport in oscillatory tube flow By T. J. PEDLEYT AND R. D. KAMMS t Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 SEW, UK 1 Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA (Received 25 May 1987 and in revised form 7 March 1988) In oscillatory flows through systems of branched or curved tubes, Taylor dispersion is modified both by the oscillation and by the induced secondary motions. As a model for this process, we examine axial transport in an annular region containing an oscillatory axial and steady secondary (circumferential) flow. Two complementary approaches are used: an asymptotic analysis for an annulus with a narrow gap (6) and for large values of the secondary flow PBclet number (P); and a numerical solution for arbitrary values of zyxwvu 6 and P. The results exhibit a form of resonance when the secondary-flow time equals the oscillation period, giving rise to a prominent maximum in the transport rate. This observation is consistent with preliminary numerical results for oscillatory flow in a curved tube, and can be explained physically. 1. Introduction Taylor (1953) first analysed the mixing of a bolus of marked fluid in steady laminar and turbulent flow in long straight tubes, far from the ends and from the site at which the bolus was introduced. In those cases he showed that the longitudinal spreading could be described as a diffusive process, with an effective diffusivity proportional to the square of a typical axial velocity difference between different fluid elements, multiplied by the timescale for an individual solute molecule to sample all cross- stream locations at which different velocities exist. The more vigorous the mass transfer laterally (by diffusion, turbulence, etc.), the more restricted is the axial dispersion of the bolus. Subsequent workers have analysed the corresponding process for time-dependent laminar flow in a straight tube (Chatwin 1975; Watson 1983), and the results have been successfully tested experimentally (Joshi et al. 1983). In many applications, however, the tubes are curved or branched, resulting in vigorous secondary motions which are likely to be the dominant source of the lateral mixing involved in Taylor dispersion. One such application of particular interest involves the transport of gas through the airways of the lung during artificial ventilation with small-volume, high-frequency oscillation (Drazen, Kamm zyxw & Slutsky 1984). Others include dispersion in rivers or estuaries or in piping systems which typically involve bends and branching. Despite its importance, there has been relatively little work on the effects of seconda,ry motions on Taylor dispersion. Erdogan & Chatwin (1967) and Nunge, Lin 8: Gill (1972) studied the problem of longitudinal dispersion in steady curved-tube zy 12 FLM 193