Oscillating flow around a circular cylindrical post confined between two parallel plates

This work is motivated by the interest in determining the effect of the micro-anatomy of the spinal subarachnoid space on the cerebrospinal fluid flow and on the associated transport of solutes. To that aim, we focus on a canonical model problem in which a circular post of radius 𝑎, confined between two parallel plates separated by a distance 2ℎ of order 𝑎, is subjected to an oscillatory flow. In particular, we are concerned with the steady, viscous, time-averaged flow that persists in the vicinity on the cylinder, when the stroke length of the oscillating flow, 𝑈∞/𝜔 is smaller or comparable to the radius of the post, 𝑎, where 𝑈∞ and 𝜔 are the mean velocity amplitude and the corresponding angular frequency, respectively. First, we analyze the asymptotic limit of small values of the stroke length, for a harmonic waveform, varying the aspect ratio of the post 𝜆 and the Womersley number 𝑀 = (a 2 𝜔/𝜈 )1/2. Firstorder steady-streaming corrections of the harmonic flow at leading order have been computed, together with associated Stokes-drift component, the sum of both yielding the mean Lagrangian velocity field. The confinement effect is seen to induce the three-dimensionality of flow. Moreover, similarly to the two-dimensional case, in the mid plane the time-averaged steady flow exhibits a recirculating vortex attached to the wall of the post that decreases as 𝑀 increases for low values of 𝑀. However, for values of 𝑀 larger than a critical one, 𝑀𝑐𝑟 (𝜆), that depends on 𝜆, a second, outer vortex is also formed. The dependence 𝑀𝑐𝑟 (𝜆) has been quantified in the range 𝜆 from 0.5 to ∞, where 𝑀𝑐𝑟 increases as 𝜆 decreases. The analysis has been corroborated experimentally for a fixed aspect ratio, 𝜆 = 2, and varying the Womersley number, the stroke length and the wave form of the oscillating flow. The experiments show similar flow patterns when the stoke length is increased from 𝜀 = 𝑈∞/𝜔𝑎 = 0.2 to 0.5, with more intense vortices at larger values of 𝜀. Consideration of an anharmonic oscillating flow shows the fort-and-aft symmetry of the steady flow is broken, with the formation of two vortices of different size when 𝑀 < 𝑀𝑐𝑟 and only one outer vortex in the systolic direction when 𝑀 > 𝑀𝑐𝑟 . Finally, the analysis is experimentally extended to consider an array of equally spaced posts, separated a semi-distance 𝑑 = 2𝑎 aligned with the flow axis of oscillation.