Hydrodynamic flows induced by localized torques (rotlets) in wedge-shaped geometries

TL;DR

This work derives analytical flow fields for a localized torque (rotlet) in wedge-shaped, low-Reynolds-number confinement bounded by no-slip surfaces, using the Fourier–Kontorovich–Lebedev transform to manage the geometry. The authors split the solution into a free-space rotlet and a complementary image contribution, then transform back to real space to obtain the velocity field in the wedge; the approach yields a real-space velocity expressed as a sum of the unbounded rotlet and a ν-integral kernel involving Legendre functions. From the resulting Green's function, they extract leading-order hydrodynamic mobilities, revealing a nonzero translation–rotation coupling induced by the wedge and providing explicit expressions for the coupling matrix $\bm{A}$ and rotator mobilities $\bm{B}$, with planar-wall limits recovered when $\alpha=\pi/2$. The results offer analytical tools for predicting and controlling particle dynamics and mixing in microfluidic devices with wedge-like confinement, and they connect to classical results in limiting geometries (e.g., Sano–Hasimoto planar and Faxén-type parallel-wall limits).

Abstract

Wedge-shaped geometries in low-Reynolds-number flows are of increasing importance, for instance, in the design of microfluidic devices. The corresponding Green's functions describing the induced flow in response to a locally applied force were derived some time ago. To achieve a complete characterization of particle motion at low Reynolds numbers, we derive the flow response to locally applied torques. This is accomplished through a direct calculation based on the Fourier-Kontorovich-Lebedev transform. We then illustrate the resulting flow fields, highlighting their structure, key features, and dependence on the geometry and orientation of the applied torque. Based on these solutions, we compute the corresponding hydrodynamic mobility tensor that couples torque and motion. Owing to the broken spatial symmetry imposed by the wedge-shaped confinement, a particle subjected to a torque will experience not only rotational motion but also translational motion. These results provide analytical tools relevant for predicting and controlling particle behavior in confined microfluidic environments.

Figures (6)

• Figure 1: Schematic illustration of the system setup. A point torque singularity is located at $\bm{r}_0 = (\rho, \beta, 0)$ in the cylindrical coordinate system $(r, \theta, z)$, acting within a viscous fluid confined by two planar boundaries that form a wedge with a straight edge aligned along the $z$-axis. The bounding surfaces are situated at $\theta = \pm \alpha$.
• Figure 2: Streamlines and quiver plots of the viscous flow field induced by an axially oriented rotlet singularity in a wedge-like confinement with no-slip boundary conditions and opening semi-angle $\alpha = \pi/6$. The rotlet is located at $\beta = \alpha/2$, see Fig. \ref{['fig:system-setup']}, at $z=0$ and points along the direction of the edge of the wedge (towards the reader). Results are presented in the radial–azimuthal plane, that is, in the plane normal to the edge of the wedge in (a) at height $z/\rho = 0.1$ and (b) height $z/\rho = 1$ above the plane of torque application. Conversely, the bottom row depicts the radial–axial plane containing the edge of the wedge for (c) $\theta = 0$ and (d) $\theta = -\beta$. The depicted scaled velocity field is defined as $\bm{v}^* = \left( \rho^2/ q_\parallel\right) \bm{v}$.
• Figure 3: Streamlines and quiver plots of the viscous flow field in analogy to Fig. \ref{['fig:axial']}, yet now induced by a radially oriented rotlet singularity pointing away from the tip of the wedge-like confinement. All other parameters are identical to those in Fig. \ref{['fig:axial']}.
• Figure 4: Similarly to Fig. \ref{['fig:axial']}, streamlines and quiver plots of the viscous flow field are evaluated, yet now induced by an azimuthally oriented rotlet singularity, pointing in a direction perpendicular to the edge of the wedge and to the radial direction. All other parameters are identical to those in Fig. \ref{['fig:axial']}.
• Figure 5: Variations of the leading-order corrections to the different components of the hydrodynamic coupling mobilities as functions of $\beta/\alpha$ for various values of the semi-opening angle $\alpha$ of the wedge. To leading order, the corrections vanish for $\alpha = \pi/2$ in panels (a) and (c), and for both $\alpha = 0$ and $\alpha = \pi/2$ in panels (b) and (d). Some curves are omitted to avoid overcrowding the figure.