Flow and transport in the spinal canal. Part 1: Eulerian velocity field

We have investigated the flow of cerebrospinal fluid in the subarachnoid space of the spinal canal. A major feature of spinal fluid flow is its oscillation, which is driven mainly by the intracranial pressure fluctuations that occur with each heart beat, resulting in an oscillatory velocity component with angular frequency ω, equal to that of the cardiac cycle. Because of the limited compliance of the subarachnoid space, the tidal volume is a a small fraction ε « 1 of the total CSF volume in the spinal canal. An asymptotic analysis in the limit ε « 1 accounting for the slenderness of the flow is employed to describe the Eulerian velocity field. The oscillatory motion is seen to be determined in the first approximation by a linear unsteady lubrication problem, which can be solved to give a closed-form solution, with the nonlinear terms associated with the convective acceleration and the deformation of the canal producing a small velocity correction with a steady-streaming component. The resulting magnitude of this steady velocity is a factor ε smaller than that of the pulsating flow, yielding associated residence times of order ε^-2  ω^-1 »  ω^-1, consistent with previous experimental observations of the bulk flow in the spinal canal.