Green function and singularities in Stokes flow confined by cylindrical walls

TL;DR

This work develops a bi-invariant Green function for Stokes flow in domains bounded by cylindrical walls (internal, external, and annular) and derives a complete set of higher-order singularities in confinement. By employing a bitensorial representation and cylindrical-harmonic expansions, the Green function is written as G^b_beta(x, xi) = S^b_beta(x, xi) + W^b_beta(x, xi), with higher-order singularities obtained through differentiation at the pole and, for Source-type singularities, via a reciprocal formulation involving the Sourcelet M^b. The authors explicitly construct the Stokeslet, Stokeslet dipole, Couplet, Stresslet, Sourcelet, and Sourcelet dipole in cylindrical coordinates, and they demonstrate how these singularities govern sedimentation and microswimmer-wall interactions in annular geometries and reduce to familiar planar-wall limits when curvature vanishes. The resulting framework provides a unified, scalable method to quantify wall effects on colloids in cylindrical confinement, enabling accurate predictions of drag, trapping, and propulsion near curved boundaries.

Abstract

In this article, the Green function for the Stokes flow in the interior, exterior, and annular regions bounded by cylindrical walls is derived as a function of the pole position and expressed invariantly both at the field and pole points. Specifically, the Green function is obtained using a cylindrical harmonic expansion of the Stokes flow within the bitensorial formulation. This formulation allows us to obtain higher-order singularities within the same domains, such as the confined Couplet and Stresslet, by simply differentiating the Green function at its pole. Moreover, the confined Sourcelet and its associated multipoles are derived from the Green function through a new method that enforces the reciprocal properties of the Stokes flow. The resulting singularities are then employed to address hydrodynamic problems involving active and passive colloids interacting with cylindrical walls, such as sedimenting particles in the annular cylindrical region and the attractive or repulsive hydrodynamic forces exerted by the cylindrical boundaries on a microswimmer.

Figures (13)

• Figure 1: Schematic representation of the geometry of the system consisting in the annular region between an internal cylinder with radius $R_i$ and an external cylinder with radius $R_o$. In panel (a), the absolute Cartesian coordinate system $(Y_1,Y_2,Y_3)$ and the absolute cylindrical coordinate system $(R,\Phi,Y_3)$ are represented; the position of the field point ${\pmb x}$ and of the pole point ${\pmb \xi}$ are indicated by blue and red arrows respectively. In panel (b) (top view), the absolute and local coordinate systems are represented projected on a plane normal to the axis $Y_3$. Green arrows represent a bi-vector (e.g. a Stokes Green function) with origin at the point ${\pmb \xi}$ and ${\pmb x}$.
• Figure 2: Streamlines (white solid lines) and intensity of the velocity field in the annular region between two concentric cylinder with $R_i=1$ and $R_o=2$ due to a Stokeslet (red arrows in the figure) in the radial direction (panels (a)-(c)) and in the angular direction (panel (d)-(f)). All the data are reported for $x^3=\xi^3$ and $\xi^2=0$. The radial position of the Stokeslet is $\xi^1=1.25$ in panels (a) and (d), $\xi^1=1.5$ in panels (b) and (e), and $\xi^1=1.75$ in panels (c) and (f). Henceforth, we adopt $|\overline{G}_{i j}| = \sqrt{|\overline{G}_{1 j}|^2 + |\overline{G}_{2 j}|^2 + |\overline{G}_{3 j}|^2}$ to indicate the norm of the vector with component $\overline{G}_{i j}$ at the field point.
• Figure 3: Contour plot of the velocity field induced by a Stokeslet (red circled dots in the figure) in the fluid bounded by cylindrical walls for $x^3=\xi^3$. In panel (a)-(c) the fluid is confined between two concentric cylinders with internal radius $R_i=1$ and external radius $R_o=2$ and the radial position of the Stokeslet are $\xi^1=1.25$ (panel (a)), $\xi^1=1.5$ (panel (b)), $\xi^1=1.75$ (panel (c)). In panel (d)-(f) the fluid is confined an external cylinder with radius $R_o=2$, and the radial position of the Stokeslet are $\xi^1=0$ (panel (d)), $\xi^1=1$ (panel (e)), $\xi^1=1.5$ (panel (f)).
• Figure 4: Streamlines at the plane $x^3=0$ of a Stokeslet (red arrows) with pole at $\xi^3=0$, $\xi^2=0$ and $\xi^1=2$ (panel (a)) or $\xi^1=1.5$ (panel (b)) near a cylindrical or spherical obstacle. Solid blue lines represent the streamlines in the case the obstacle is a cylinder with radius $R_i=1$ and black dashed-dotted lines represents streamlines in the case the obstacle is a sphere with the same radius.
• Figure 5: Streamlines (withe solid lines) and intensity of the norm of the velocity field at $x^3=\xi^3$ for a Stokeslet placed at $\xi^1=1.5$ in a fluid domain bounded by two concentric cylinders with an external radius $R_o=1$ and an internal radius $R_i=0.1$ (panel (a)), $R_i=0.01$ (panel (b)), $R_i=0.001$ (panel (c)). In panel (d), the solution for the fluid bounded by a single external cylinder with radius $R_o=1$ is depicted. The intensity of the norm of the velocity field is subdivided in three zone: (i) the yellow zone corresponds to an intense flow where $|\overline{G}_{a\, 1}|\, > \, 10$, (ii) the turquoise zone corresponds to a moderate flow where $1\, < \,|\overline{G}_{a\, 1}|\, \leq \, 10$, (iii) the blue zone corresponds to a weak flow where $|\overline{G}_{a\, 1}|\, < \, 1$.