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This paper investigates flow and transport in a slender wavy-walled vertical channel subject to a prescribed oscillatory pressure difference between its ends. When the ratio of the stroke length of the pulsatile flow to the channel wavelength is small, the resulting flow velocity is known to include a slow steady-streaming component resulting from the effect of the convective acceleration. Our study considers the additional effect of gravitational forces in configurations with a non-uniform density distribution. Specific attention is given to the slowly evolving buoyancy-modulated flow emerging after the deposition of a finite amount of solute whose density is different from that of the fluid contained in the channel, a relevant problem in connection with drug dispersion in intrathecal drug delivery (ITDD) processes, involving the injection of the drug into the cerebrospinal fluid that fills the spinal canal. It is shown that when the Richardson number is of order unity, the relevant limit in ITDD applications, the resulting buoyancy-induced velocities are comparable to those of steady streaming. As a consequence, the slow time-averaged Lagrangian motion of the fluid, involving the sum of the Stokes drift and the time-averaged Eulerian velocity, is intimately coupled with the transport of the solute, resulting in a slowly evolving problem that can be treated with two-time-scale methods. The asymptotic development leads to a time-averaged, nonlinear integro-differential transport equation that describes the slow dispersion of the solute, thereby circumventing the need to describe the small concentration fluctuations associated with the fast oscillatory motion. The ideas presented here can find application in developing reduced models for future quantitative analyses of drug dispersion in the spinal canal. 

This paper presents results of an experimental investigation of solute transport in a simplified model of the spinal canal. The work aims to provide increased understanding of the mechanisms responsible for drug dispersion in intrathecal drug delivery (ITDD) procedures. The model consists of an annular channel bounded externally by a rigid transparent tube of circular section, representing the dura mater, and internally by an eccentric cylindrical compliant insert, representing the spinal cord. The tube, closed at one end, is connected to a rigid acrylic reservoir, representing the cranial cavity. The system is filled with water, whose properties are almost identical to those of the cerebrospinal fluid. A programmable peristaltic pump is employed to generate oscillatory motion at frequencies that are representative of those induced by the cardiac and respiratory cycles. Laser induced fluorescence is used to characterize the dispersion of fluorescent dye along the canal and into the cranial cavity for different values of the relevant Womersley number and different eccentricities of the annular section. The present work corroborates experimentally, for the first time, the existence of a steady bulk flow, associated with the mean Lagrangian motion, which plays a key role in the transport of the solute along the spinal canal. The measurements of solute dispersion are found to be in excellent agreement with theoretical predictions obtained using a simplified transport equation derived earlier on the basis of a two-timescale asymptotic analysis. The experimental results underscore the importance of the eccentricity and its variations along the canal and identifies changes in the flow topology associated with differences in the Womersley number, with potential implications in guiding future designs of ITDD protocols.