Hydrodynamic interactions between a point force and a slender filament

The Green's function of the incompressible Stokes equations, the stokeslet, represents the singular flow due to a point force. Its classical value in an unbounded fluid has been extended near surfaces of various shapes, including flat walls and spheres, and in most cases the presence of a surface leads to an advection flow induced at the location of the point force. In this paper, motivated by the biological transport of cargo along polymeric filaments inside eukaryotic cells, we investigate the reaction flow at the location of the point force due to a rigid slender filament located at a separation distance intermediate between the filament radius and its length (i.e., we compute the advection of the point force induced by the presence of the filament). An asymptotic analysis of the problem reveals that the leading-order approximation for the force distribution along the axis of the filament takes a form analogous to resistive-force theory but with drag coefficients that depend logarithmically on the distance between the point force and the filament. A comparison of our theoretical prediction with boundary element computations shows good agreement. We finally briefly extend the model to the case of curved filaments.

Figure 1

(a) Schematic illustration of our setup. A point force is located at a distance $d$ from a centerline of the cylinder of diameter $2a$ and length $2L$ and we derive the flow in the limit $a\ll d\ll L$. (b) Schematic illustration of the setup with lengths scaled by $L$ and with the regions relevant for the asymptotic analysis labeled. (c) Mesh used for the boundary element computations.

Figure 2

(a) Decay gradient of the reaction flow for various cylinder aspect ratios $L/a$ in the case of a point force pointing in the positive $x$ direction. The numerical results of boundary element method (BEM) computations are shown by symbols while the predictions from the standard RFT with drag coefficients independent of $d$ shown in Eq. (15) follow the appropriately colored solid lines. We plot our theoretical prediction in Eq. (16) as a solid black line while the infinite cylinder prediction from Ref. [16], abbreviated as AIH (1980) in the legend, is shown as a gray dashed line. Computations shown with filled symbols were performed with the same uniform size of surface elements while the ones represented by ($+$) and ($*$) have elements that are two and four times larger, respectively, in the direction of the axis of the cylinder. (b) The same data plotted against $lnd/L$. The RFT prediction lines from panel (a) all collapse onto the single solid blue line in panel (b).

Figure 3

(a) Setup for the calculation in the case of a curved filament. (b) Correction to the decay rate due to the curvature as a function of the separation $d/a$, for various relative curvatures $\alpha =L/R$. The point force is pointing in the positive $x$ direction. Symbols represent results of the BEM computations and solid lines follow the theoretical prediction from Eq. (20).

Figure 4

Validation of the BEM code on the case of a point force near a plane wall. We show the convergence towards the exact solution by plotting the RMS error in the cases of a point force parallel [(a), (c)] and perpendicular [(b), (d)] to an infinite no-slip wall. [(a), (b)] The size of the plane is fixed to $D=20h$ and the size of the discretizing elements is varied. [(c), (d)] The size of the elements is fixed but the size of the discretized part of the plane $D$ is varied.

Figure 5

Validation of the BEM code on the case of a point force near a rigid sphere. (a) An example of the discretization of the sphere. [(b), (c)] RMS error as a function of the number of triangles used to discretize the sphere, in the case of a point force being perpendicular (b) or parallel (c) to the surface of the sphere.

Figure 6

Relative error in computations of $dln|{\mathit{u}}_S^{\left(1\right)}|/d\lambda$ between results obtained with a mesh of pentagonal cross sections and meshes with triangular (a) and square (b) cross sections.