Diffusioosmotic corner flows

TL;DR

This work addresses how diffusioosmotic slip on chemically active wedge walls generates corner eddies in Stokes flow, revealing Moffatt-like vortices arising from confinement and surface activity. The authors develop an exact 2D solution by solving the solute diffusion problem with various boundary conditions in a wedge and coupling it to the slip-driven Stokes flow via a biharmonic stream function, employing the Mellin transform to obtain analytic forms and residue-based inversions. They provide closed-form results for uniform wall coverage (single and double active walls) and detailed Mellin-transform solutions for active-absorbing and active-reflective wall configurations, plus Green’s-function methods for arbitrary wall activity patterns. The framework yields practical benchmarks and design principles for microscale mixing in dead-end pores and cornered microchannels, with slip-velocity magnitudes estimated around a few micrometers per second, and it generalizes to other phoretic mechanisms and finite geometries.

Abstract

We study flows generated within a two-dimensional corner by the chemical activity of the confining boundaries. Catalytic reactions at the surfaces induce diffusioosmotic motion of the viscous fluid throughout the domain. The presence of chemically active sectors can give rise to steady eddies reminiscent of classical Moffatt vortices, which are mechanically induced in similar confined geometries. In our approach, an exact analytical solution of the diffusion problem in a wedge geometry is derived and coupled to the diffusioosmotic slip-velocity formulation, yielding the stream function of associated Stokes flow. In selected limiting cases, simple closed-form expressions provide clear physical insight into the underlying mechanisms. Our results open new perspectives for the design of microscale mixing strategies in dead-end pores and cornered microfluidic channels, and offer benchmarks for numerical simulations of confined (diffusio)osmotic systems.

Figures (8)

• Figure 1: Geometry of the diffusioosmotic corner flow setup in polar coordinates $(\rho,\theta)$. In a wedge of opening angle $\alpha$, an active patch on the $\theta=0$ wall covering the radial section $\rho\in[a,b]$, releases solute (grey arrows) and generates an inhomogeneous concentration field which drives circulatory flow indicated by schematic streamlines.
• Figure 2: Diffusioosmotic flow induced by the activity of one phoretic wall in a wedge of angle $\alpha$ for (left to right) $\alpha =\{\pi/4,\pi/2,3\pi/4\}$. Flow streamlines are marked in white. The colour map indicates the total velocity magnitude. The emergent bulk flow remains comparable in magnitude to the driving slip flow on the active boundary, and decays rapidly close to the inert, no-slip wall.
• Figure 3: Diffusioosmotic flow induced by the activity of two phoretic walls in a wedge of angle $\alpha =\{\pi/4,\pi/2,3\pi/4\}$. The colour map indicates the total velocity magnitude. We note a strong flow close to the driving active boundaries, and a counterflow along the wedge bisector.
• Figure 4: Contours of integration for the evaluation of the inverse Mellin transform. The green contour is used for $a>\rho$, and the blue contour for $a\leq \rho$. Both integration contours are shifted by $\gamma$ along the real axis, and the poles of the integrand are denoted by $p_k$, where $k\in \mathbb{Z}$. To evaluate the integral, one takes the limit of the square side length approaching infinity. In the limit, contributions from the three dashed sides of each square contour vanish, and the desired integral along the imaginary axis can be evaluated using the method of residues.
• Figure 5: Solute concentration fields, (a)--(c), and isolines of the stream function $\psi$, (d)--(f), for an ideally absorptive wall at $\theta=\pi/6$, a catalytic (active) wall at $\theta=0$ with a catalytic sector at $(a,b)= \{ (1,3), (0,3), ({1},\infty)\}$ marked in red. Here, we assumed $A=1$ and the plotted radius of the wedge is $\rho<4$. The scale bar for the absolute concentration field, $|c|$, is common for plots (a), (b) and different for panel (c).