Leading-order Stokes flows near a corner

Abstract

Singular solutions of the Stokes equations play important roles in a variety of fluid dynamics problems. They allow the calculation of exact flows, are the basis of the boundary integral methods used in numerical computations, and can be exploited to derive asymptotic flows in a wide range of physical problems. The most fundamental singular solution is the flow's Green function due to a point force, termed the Stokeslet. Its expression is classical both in free space and near a flat surface. Motivated by problems in biological physics occurring near corners, we derive in this paper the asymptotic behaviour for the Stokeslet both near and far from a corner geometry by using complex analysis on a known double integral solution for corner flows. We investigate all possible orientations of the point force relative to the corner and all corner geometries from acute to obtuse. The case of salient corners is also addressed for point forces aligned with both walls. We use experiments on beads sedimenting in corn syrup to qualitatively test the applicability of our results. The final results and scaling laws will allow to address the role of hydrodynamic interactions in problems from colloidal science to microfluidics and biological physics.

Stokesian motion of a spherical particle near a right corner made by tangentially moving walls

“…As the flow field for dihedral corners (without particles) does not experience abrupt changes of its topology upon a smooth variation of the dihedral angle, we expect our results are robust with respect to small deviations of the corner angle from the considered in this study. This expectation is supported by the results of Dauparas & Lauga (2018), who found that the forces and torques on a particle far from the corner vary smoothly with the dihedral angle.…”

“…Cox & Mason 1971), and (ii) perfectly smooth symmetric surfaces (planar, cylindrical or spherical) of the interacting solid bodies. A step towards more general geometries has recently been made by Dauparas & Lauga (2018), who investigated the Stokes flow past a sphere moving far from a stationary dihedral corner of arbitrary angle. Romanò, des Boscs & Kuhlmann (2020 a ) further extended the results of Dauparas & Lauga (2018) considering a particle near a right dihedral corner between a stationary and a tangentially sliding wall, i.e.…”

Engineering reconfigurable flow patterns via surface-driven light-controlled active matter

“…The topological equivalence between a sphere and a box dictates that these flow structures should be compatible. We do note, however, that corner effects can give rise to eddies that are unique to the geometry of a box [48]. Most interestingly, the b 20 and c 20 modes have built-in defects analogous to topologies proposed earlier: the former consists of a line of head-on defects at the equator, while the latter consists of two oppositely rotating hemispheres, giving rise to a shear defect along the equator.…”

Reconstructing the neutrally-buoyant particle flow near a singular corner

“…To date, several modeling approaches have been proposed *Corresponding author. E-mail address: francesco.romano@ensam.eu (Francesco Romanò) Executive Editor: Cristian Marchioli for particles in unbounded flows [13][14][15] and complementary exact [16,17], numerical [18,19], asymptotic [20][21][22][23],…”