Cerebrospinal fluid fills the brain's ventricles and flows through them in patterns driven by cardiovascular pulsations, choroid plexus secretion, and — in simpler organisms like zebrafish — the beating of motile cilia. The bulk flow is slow and simple. But the transport of molecules, waste products, and signaling substances through the ventricles depends not on the bulk flow but on coherent structures within it: vortices, barriers, and channels that guide or trap material.
Finite-time Lyapunov exponent (FTLE) fields computed from the flow reveal these structures. The FTLE measures how quickly nearby fluid particles separate — high FTLE ridges are barriers that material cannot cross, and the boundaries between coherent flow regions. In the brain ventricles, these barriers organize the flow into compartments that would be invisible in a standard velocity field plot.
An important technical detail: the Stokes equations (viscosity-dominated, no inertia) suffice for predicting the bulk flow velocity and pressure. But the Navier-Stokes equations (including inertia) are necessary to capture the coherent structures. The inertial terms are small in the ventricles — the Reynolds number is low — but even small inertia changes the fine structure of the flow enough to alter where the Lagrangian barriers form. The coherent transport structures are more sensitive to inertia than the bulk flow is.
This means that simplifying the physics (dropping the inertial terms) gives the right answer for the wrong question. If you want to know how much fluid flows through the ventricle, Stokes is fine. If you want to know where a drug molecule deposited in the ventricle will end up, you need Navier-Stokes. The transport question is harder than the flow question, even though both describe the same fluid.
(arXiv:2603.18849)