Modal analysis and optimization of swimming active filaments

Abstract

Active flexible filaments form the classical continuum framework for modelling the locomotion of spermatozoa and algae driven by the periodic oscillation of flagella. This framework also applies to the locomotion of various artificial swimmers. Classical studies have quantified the relationship between internal forcing (localised or distributed internal moments or forces) and external output (filament shape and swimming speed). In this paper, we pose locomotion as a mathematical optimisation problem and demonstrate that the swimming of an isolated active filament can be accurately described and optimised using a small number of eigenmodes, significantly reducing computational complexity. In particular, we reveal that the motion of a filament with monophasic forcing, relevant to recently proposed artificial swimmers, is governed by exactly four forcing eigenmodes, only two of which are independent. We further present optimisations of such swimmers under various constraints.

Figures (5)

• Figure 1: Active filament. Parameterisation of a (dimensional) filament of total length $L$. Notation includes: arc length $0 \le s \le L$, tangent angle $\psi$, tangent vector $\mathbf{t}$, normal vector $\mathbf{n}$, tip position $\mathbf{X}$ and instantaneous tip velocity $\left(-U, V\right)$.
• Figure 2: Modal approach under travelling wave forcing. (a) Heatmap of the largest positive eigenvalue for various values of $Sp$ and $k$, with $k = 0$ (dashed red line) and the optimal $k \approx 0.72$ (dashed black line) indicated. Examples of spermatozoa swimming through in vitro fertilisation medium (red, $k = 1.5$, $Sp = 4$Smith_Gaffney_Gadêlha_Kapur_Kirkman‐Brown_2009Gadêlha_Gaffney_Smith_Kirkman-Brown_2010Gaffney_Gadêlha_Smith_Blake_Kirkman-Brown_2011) and a near-optimal active filament (solid pink, $k = 0.72$, $Sp = 3$) are also shown, as well as optimal swimming in the same direction as wave propagation (hollow pink, $k = -1.5$, $Sp = 3$). (b) Log-log plot (base $10$) showing the errors incurred as the truncation of Eq. \ref{['eq:sum']} is varied, for a near-optimal filament ($k = 0.72$, $Sp = 3$). Eigenvalues are arranged in order of decreasing magnitude, normalised relative to the largest/first eigenvalue, and their decay as $n$ increases is given (thick dashed black line). Also plotted is the relative error incurred by truncating Eq. \ref{['eq:sum']} at the n-th term, normalised relative to the exact swimming speed calculated using Eq. \ref{['eq:U_G_swim']}, for simple forcing functions $f(\xi) \equiv 1$ (red solid line), $f(\xi) = \sin\left(2\pi\xi\right)$ (blue) and $f(\xi) = \sin\left(8\pi\xi\right)$ (pink). (c) The eigenfunctions for the six eigenvalues with largest magnitudes, normalised using the fixed forcing magnitude constraint, Eq. \ref{['eq:ffmc']}. The first and sixth eigenvalues are positive (solid lines) whilst the others are negative (dashed lines).
• Figure 3: Eigenvalues and eigenfunctions for monophasic forcing. (a) The large (red) and small (blue) eigenvalues, $\lambda_+$ and $\lambda_-$ respectively, for monophasic forcing, $\phi \equiv 0$. Note that $\lambda_+$ is plotted against $10\lambda_-$. (b) Eigenfunctions corresponding to the large (red) and small (blue) eigenvalues, $g_+$ and $g_-$ respectively, normalised with the established fixed forcing magnitude condition, for the optimal sperm number $Sp = 4.7$.
• Figure 4: Symmetric and antisymmetric decomposition of eigenmodes. (a) Symmetric function $g_s$ (thick solid line) and antisymmetric function $g_a$ (thick dashed line); the lines $g = 0$ and $s = 0.5$ are shown as thin dashed lines. (b) Ratio $g_a(s)/g_s(s)$ used for optimisations.
• Figure 5: Artificial swimmers with continuous (a) and discrete (b) forcing. (a) Example artificial swimmers, with continuous, piecewise constant forcing, of progressively increasing speeds, optimised using Eq. \ref{['eq:int_speed']}. Solid lines represent the filaments at $t = 0$, and $t$ increases to $\pi$ for progressively fading lines. Speeds have been divided by $\lambda_+$ to produce the speed factor (SF) as a percentage. (i) Uniformly forced filament that cannot swim due to the scallop theorem. (ii) Simple swimmer with forcing in the front half only. (iii) Swimmer with frontal forcing occupying an optimal fraction of the filament. (iv) Swimmer with optimally chosen positive and negative forcing. An animated version of these swimmers is shown in Supplementary Video 1.(b) Example artificial swimmers for $M = 1, 2, 3, 5, 7, 9$ discrete actuators, each of strength $1/M$, with speed factors indicated. Minimum spacing of $0.1$ between actuators. An animated version of these swimmers is shown in Supplementary Video 2.