Spectral methods for wedge and corner flows: The Fourier-Kontorovich-Lebedev integral transform

Abstract

Understanding fluid flow in wedge-shaped geometries is essential for predicting hydrodynamic interactions in confined systems, such as microfluidic devices and near-corner transport phenomena. This article reviews analytical methods and techniques for addressing wedge problems in low-Reynolds-number hydrodynamics, focusing on solutions of the Stokes equations for a point force (Stokeslet) and a point torque (rotlet). The formulation is based on the Papkovich-Neuber representation, which uses four harmonic functions to characterize the fluid flow. A concise overview of the Fourier-Kontorovich-Lebedev (FKL) transform method is provided, highlighting key properties and steps employed in deriving these solutions. This offers a versatile framework for predicting particle dynamics in wedge confinements and for designing microfluidic systems with corner geometries.

Figures (1)

• Figure 1: Schematic of the system representing a fluid confined within a wedge-shaped domain bounded by two planar walls. The geometry is described in cylindrical coordinates $\bm{ r } = (r, \theta, z)$, with $r$ the radial distance from the wedge apex, $\theta$ the polar angle from the centerline between the walls, and $z$ the axial direction along the wedge edge. No-slip boundary conditions (zero velocity) are applied at $\theta = \pm \alpha$, and a flow singularity, either a point force or a point torque, is located at $\bm{ r }_0 = (\rho, \beta, 0)$