Asymptotic theory of hydrodynamic interactions between slender filaments

Hydrodynamic interactions are important in biophysics research because they influence both the collective and the individual behavior of micro-organisms and self-propelled particles. For instance, hydrodynamic interactions at the microswimmer level determine the attraction or repulsion between individuals, and hence the rise of collective behavior. Meanwhile, hydrodynamic interactions between swimming appendages (e.g., cilia and flagella) influence the emergence of swimming gaits, synchronized bundles, and metachronal waves, and hence the propulsive capacity of the individual swimmer. In this study we address the issue of hydrodynamic interactions between slender filaments separated by a distance larger than their contour length $(d>L)$ by means of asymptotic calculations and numerical simulations. We first derive analytical expressions for the extended resistance matrix of two arbitrarily shaped rigid filaments, as a series expansion in inverse powers of $d/L>1$. The coefficients in our asymptotic series expansion are then evaluated using two well-established methods for slender filaments, resistive-force theory (RFT) and slender-body theory (SBT), and our asymptotic theory is verified using numerical simulations based on SBT for the case of two parallel helical filaments. The theory is able to capture the qualitative features of the interactions in the regime $d/L>1$, which opens the path to a deeper physical understanding of hydrodynamically governed phenomena such as interfilament synchronization and multiflagellar propulsion. To demonstrate the usefulness of our results, we next apply our theory to the case of two helical filaments rotating side-by-side, where we quantify the dependence of all forces and torques on the distance and phase difference between the helices. Using our understanding of pairwise interactions, we then provide physical intuition for the case of a circular array of rotating helices. Our theoretical results will be useful for the study of hydrodynamic effects that emerge between interacting bacterial flagella, nodal cilia, and slender microswimmers, both artificial and biological.

Figure 1

Geometrical setup of the problem. (a) Two rigid filaments of dimensionless contour length $L=2$ interact with each other hydrodynamically as they move through a viscous fluid. Our asymptotic theory is valid for sufficiently large interfilament separation, $d>L$, and in the limit of small filament thickness, $\epsilon \ll 1$. We identify three useful coordinate systems: the laboratory frame (green), the interaction frame for a pair of filaments (blue), and the body frame for an individual filament (black). (b) Parameters describing the geometry of a helical filament, which we will use for the validation and application of our asymptotic theory.

Figure 2

Relative error in (a) helix dynamics and (b) helix kinematics, as defined in Eqs. (75) and (76), respectively, with $p=2$. As we increase the helix separation $d$, the asymptotic theory with SBT coefficients converges to the numerical solution, and the error decays as expected with each higher order included in the theory. Parameter specification: helices have configurations $({\theta }_1,{\chi }_1,{\varphi }_1)=(0,0,\pi /6)$ and $({\theta }_2,{\chi }_2,{\varphi }_2)=(0,0,2\pi /3)$, and $N=2.75$ helical turns. Helix angle $\psi =0.5$ rad and filament slenderness $\epsilon ={10}^{-2}$ are representative of bacterial flagella.

Figure 3

Comparison between computations and the asymptotic theory with RFT/SBT coefficients, by means of the time evolution of forces and torques induced by the second (rightmost) filament on the first (leftmost). The helices are vertical ($\theta =0$) and rotate with constant angular velocity $\mathrm{\Omega }{\mathbf{e}}_z$. We fix the phase difference $\mathrm{\Delta }\varphi =\pi /2$ between them, and a horizontal distance equal to the integrated filament length (a)–(f) or ten times larger (g)–(l). The helix angle $\psi =0.5043$ rad, and filament slenderness $\epsilon =0.0038$, were chosen as representative of bacterial flagella. The helices have $N=2.5$ helical turns.

Figure 4

Comparison between computations and the asymptotic theory with SBT coefficients to $O\left(d^{-1}\right)$ and $O\left(d^{-2}\right)$, by means of the time evolution of forces and torques induced by the second (rightmost) filament on the first (leftmost). The helices are vertical ($\theta =0$) and rotate with constant angular velocity $\mathrm{\Omega }{\mathbf{e}}_z$. We impose the phase difference $\mathrm{\Delta }\varphi =\pi /2$ between them, and a horizontal distance equal to the integrated filament length (a)–(f) or ten times larger (g)–(l). The helix angle $\psi =0.5043$ rad, and filament slenderness $\epsilon =0.0038$, were chosen as representative of bacterial flagella. The helices have $N=2.5$ helical turns.

Figure 5

Average forces and torques exerted by the leftmost helix due to the presence of a second parallel helix rotating at a distance $d$ to the right, with fixed phase difference $\mathrm{\Delta }\varphi =\pi /4$. The data points come from SBT simulations including HIs. The power law triangles indicate that the average forces and torques along the axis of the helix (c) and (f) are an $O\left(d^{-1}\right)$ effect, while the other forces and torques (a), (b), (d), and (e) are an $O\left(d^{-2}\right)$ effect. Simulation parameters: $\psi =0.5043$ rad, $\epsilon =0.0038$, $N=2.5$ helical turns.

Figure 6

Variance over time in the forces and torques exerted by the leftmost helix due to the presence of a second parallel helix rotating at a distance $d$ to the right, with fixed phase difference $\mathrm{\Delta }\varphi =\pi /4$. The data points come from SBT simulations including HIs. The power law triangles indicate that the variances in force and torque along the axis of the helix (c) and (f) are an $O\left(d^{-2}\right)$ effect, while the other forces and torques (a), (b), (d), and (e) are an $O\left(d^{-1}\right)$ effect. Simulation parameters: $\psi =0.5043$ rad, $\epsilon =0.0038$, $N=2.5$ helical turns.

Figure 7

Average forces (a), (c), and (e) and torques (b), (d), and (f) due to HIs between the helices, as a function of the phase difference between filaments. The helix angle $\psi =0.5043$ rad and filament slenderness $\epsilon =0.0038$ were chosen as representative of bacterial flagella. The helices have $N=2.5$ helical turns.

Figure 8

Variance in forces (a), (b), and (e) and torques (c), (d), and (f) due to HIs between the helices, as a function of the phase difference between filaments. The helix angle $\psi =0.5043$ rad and filament slenderness $\epsilon =0.0038$ were chosen as representative of bacterial flagella. The helices have $N=2.5$ helical turns.

Figure 9

(Not to scale.) Physical mechanism for the reduction in pumping force due to HIs. Top panels illustrate the local velocity of the filament relative to the surrounding fluid. Lower panels show the periodic force density along the filament, rendered at points along a horizontal projection of the centerline. The total force and torque exerted by the helical pump are obtained by integrating the force density around the circle as many times as needed. (a) Due to the anisotropic drag on the slender filament, a rotating helix exerts a net force along its axis of rotation ${\mathbf{e}}_3^{\left(1\right)}$. If the helix does not have an integer number of turns, there is also a net component of the force along the ${\mathbf{e}}_2^{\left(1\right)}$ direction, due to a “surplus” of filament on one side (indicated by a thick orange arc on the circular projection of the centerline). (b) Changes to the force density along the second filament due to the ${\mathbf{e}}_3^{\left(1\right)}$ component of the force exerted by the first filament on the fluid. (c) Likewise for the ${\mathbf{e}}_2^{\left(1\right)}$ component of the force.

Figure 10

Basic principles of HIs between helical pumps. (a) Minimal setup with two helical pumps rotating with constant angular velocity around their axes. (b) There is no net attraction or repulsion between the two rotating helices (cf. symmetry arguments for zero phase difference in Ref. [43]), but rather a sideways migration whose sign depends on the phase difference. (c) There is a persistent (i.e., independent of phase difference) swirling effect in the same direction as the rotation of the helices. (d) A ring of helical pumps would initially experience counterclockwise swirling (due to the forces $-\langle F_y\rangle$ exerted by the fluid) and outward splaying of the tips (due to the torques $-\langle T_y\rangle$ exerted by the fluid).

Figure 11

Tests for a prolate spheroid. Verifying our implementation of SBT against the exact solution for spheroids [69]. In our implementation of SBT, we truncate the numerical solution to $N_{\mathrm{Legendre}}=5$ Legendre polynomial modes.

Figure 12

Tests for a semicircular filament. (a) Verifying our implementation of SBT (red circles) against Johnson's asymptotic solution (solid line) from Eq. (36) in Ref. [59]. The quantity being plotted is the vertical component of the force density exerted by a horizontal semicircular filament, with slenderness $\epsilon =0.1$, translating vertically as shown in the inset. (b) Verifying our implementation of SBT (red circles) against Johnson's SBT computations (solid line) and two asymptotic solutions (dashed lines) by Cox [57] and Johnson [59]. The quantity being plotted is the drag force per unit length on a semicircular filament, with slenderness $\epsilon =0.1$, translating along its axis of symmetry as shown in the inset. For both (a) and (b), we use a truncation of $N_{\mathrm{Legendre}}=10$ in our implementation of SBT. In (b), our results are overlaid onto Fig. 4 from Ref. [59]. Image reused with permission from the publisher. Copyright 1980 Cambridge University Press.

Figure 13

Tests for a helical filament. Verifying our implementation of SBT (red circles) against Lighthill's exact solution for infinite helices (dashed line) [58] and Johnson's SBT (solid line) [59]. The three data series represent the tangential ($f_s$), normal ($f_n$), and binormal ($f_b$) components of the force density along a helix (with a prolate spheroidal cross section) that is translating and rotating about its axis of symmetry such that the net force along the axis is zero. Helix parameters: (a) five helical turns, $\epsilon =0.0021$, $\psi =1.0039$ rad; (b) one helical turn, $\epsilon =0.0107$, $\psi =1.0039$ rad. In our implementation of SBT, we truncate the solution to $N_{\mathrm{Legendre}}=25$ Legendre polynomial modes. Our results are overlaid onto Fig. 6 from Ref. [59]. Image reused with permission from the publisher. Copyright 1980 Cambridge University Press.

Figure 14

Tests for cross-filament HIs. We compare the hydrodynamic resistance of two interacting helical filaments, having two helical turns each and placed head-to-head (H2H), with that of one helical filament that has four helical turns (ONE). In the limit $\delta \to 0$, the two problems are identical. We plot (on a log-linear scale) the relative errors in the coefficients of the resistance matrix $A_{33}=F_z/U_z$ (blue circles), $B_{33}=F_z/{\mathrm{\Omega }}_z$ (purple rightward triangles), $A_{11}=F_x/U_x$ (yellow upward triangles), and $D_{33}=T_z/{\mathrm{\Omega }}_z$ (orange squares). The relative error is defined as $|1-\text{H2H}/\text{ONE}|$ and decays with decreasing separation $\delta$, thus validating our implementation of cross-filament HIs. Helix parameters: $\psi =0.5043$ rad, $\epsilon =0.0024$ (for $N=4$ helical turns) and $\epsilon =0.0048$ (for $N=2$ helical turns).

Figure 15

Self-convergence test for a single helical filament. The percentage errors in $A_{33}=F_z/U_z$ (blue circles), $B_{33}=F_z/{\mathrm{\Omega }}_z$ (purple rightward triangles), $A_{11}=F_x/U_x$ (yellow upward triangles), and $D_{33}=T_z/{\mathrm{\Omega }}_z$ (orange squares) are calculated at each number of Legendre modes relative to the most accurate numerical solution available, in this case $N_{\mathrm{Legendre}}=20$. Helix parameters: $\psi =0.5043$ rad, $\epsilon =0.0038$, $N=4$ helical turns.