Geometric Mixing, Peristalsis, and the Geometric Phase of the Stomach

Simple analytic model for peristaltic flow and mixing

Peristaltic flows occur when fluid in a channel is driven by periodic, traveling wall deformations, as in industrial peristaltic pumps, urethras, stomachs, and cochleae. Peristaltic flows often vary periodically at every point in space but nonetheless cause net transport and mixing of solutes because of Lagrangian (Stokes) drift. Direct numerical simulation can predict peristaltic flows but is computationally expensive, particularly for determining functional relationships between drive parameters and transport or mixing. We present a simple analytic model of peristaltic flow that expresses flow velocity and drift velocity in terms of deformation speed and amplitude. The model extends beyond prior studies by allowing arbitrary wave forms via Fourier series. To validate our analytic model, we present experiments and simulations; both closely match the analytic model over a range of deformation speeds and amplitudes. We demonstrate the applicability of the model by quantifying variations in the thickness of the reflux region (where fluid drifts opposite the direction of travel of deformations) and by modeling mixing in the cochlea, which is promoted by peristaltic flow.

Mechanically facilitated micro-fluid mixing in the organ of Corti

The cochlea is filled with two lymphatic fluids. Homeostasis of the cochlear fluids is essential for healthy hearing. The sensory epithelium called the organ of Corti separates the two fluids. Corti fluid space, extracellular fluid space within the organ of Corti, looks like a slender micro-tube. Substantial potassium ions are constantly released into the Corti fluid by sensory receptor cells. Excess potassium ions in the Corti fluid are resorbed by supporting cells to maintain fluid homeostasis. Through computational simulations, we investigated fluid mixing within the Corti fluid space. Two assumptions were made: first, there exists a longitudinal gradient of potassium ion concentration; second, outer hair cell motility causes organ of Corti deformations that alter the cross-sectional area of the Corti fluid space. We hypothesized that mechanical agitations can accelerate longitudinal mixing of Corti fluid. Corti fluid motion was determined by solving the Navier–Stokes equations incorporating nonlinear advection term. Advection–diffusion equation determined the mixing dynamics. Simulating traveling boundary waves, we found that advection and diffusion caused comparable mixing when the wave amplitude and speed were 25 nm and 7 m/s, respectively. Higher-amplitude and faster waves caused stronger advection. When physiological traveling waves corresponding to 70 dB sound pressure level at 9 kHz were simulated, advection speed was as large as 1 mm/s in the region basal to the peak responding location. Such physiological agitation accelerated longitudinal mixing by more than an order of magnitude, compared to pure diffusion. Our results suggest that fluid motion due to outer hair cell motility can help maintain longitudinal homeostasis of the Corti fluid.

Geometric mixing

Mixing fluids often involves a periodic action, like stirring one's tea. But reciprocating motions in fluids at low Reynolds number, in Stokes flows where inertia is negligible, lead to periodic cycles of mixing and unmixing, because the physics, molecular diffusion excepted, is time reversible. So how can fluid be mixed in such circumstances? The answer involves a geometric phase. Geometric phases are found everywhere in physics as anholonomies, where after a closed circuit in the parameters, some system variables do not return to their original values. We discuss the geometric phase in fluid mixing: geometric mixing. This article is part of the theme issue 'Stokes at 200 (part 2)'.

The fluid mechanics of poohsticks

The year 2019 marked the bicentenary of George Gabriel Stokes, who in 1851 described the drag-Stokes drag-on a body moving immersed in a fluid, and 2020 is the centenary of Christopher Robin Milne, for whom the game of poohsticks was invented; his father A. A. Milne's The House at Pooh Corner, in which it was first described in print, appeared in 1928. So this is an apt moment to review the state of the art of the fluid mechanics of a solid body in a complex fluid flow, and one floating at the interface between two fluids in motion. Poohsticks pertains to the latter category, when the two fluids are water and air. This article is part of the theme issue 'Stokes at 200 (part 2)'.