[Paper Review] Swimming of a circular disk at low Reynolds number

B. U. Felderhof|arXiv (Cornell University)|May 26, 2014

Micro and Nano Robotics3 citations

TL;DR

This paper extends Taylor’s model of low-Reynolds-number swimming to a finite circular disk with independent surface distortions on its two planar faces. By solving Stokes equations for asymmetric undulation and squirming modes, it shows the disk undergoes combined translational and rotational swimming, resulting in circular orbits; the optimal stroke achieves 41% higher efficiency than Taylor’s original model.

ABSTRACT

The swimming of a circular disk at low Reynolds number is studied for distortion waves along its two planar surfaces with wavelength much smaller than the size of the disk. The calculation is based on an extension of Taylor's work for a planar sheet. It is shown that in general the disk performs both translational and rotational swimming, resulting in a circular orbit.

Motivation & Objective

• To generalize Taylor’s swimming sheet model to a finite, thick disk with independent surface distortions on both sides.
• To resolve inconsistencies in prior theories regarding superposition of undulation and squirming modes on a planar sheet by modeling a finite disk.
• To quantify both translational and rotational swimming velocities for arbitrary surface wave patterns on the disk.
• To optimize stroke parameters for maximum swimming efficiency under fixed power input.
• To demonstrate that directional control via rotational swimming is possible for flat microorganisms like Paramecium.

Proposed method

• Extends Taylor’s stream function approach for a planar sheet to a circular disk using Stokes flow equations with no-slip boundary conditions.
• Models surface distortions on both faces as superpositions of transverse undulations and in-plane squirming waves with adjustable amplitudes and phases.
• Applies perturbation theory to second order in surface displacement to compute mean flow velocity and pressure fields.
• Derives explicit expressions for mean translational and rotational swimming velocities using the second-order velocity field at the disk surfaces.
• Calculates the rate of viscous dissipation to define efficiency as the ratio of swimming speed to power input.
• Uses symmetry and elliptical orbit parameters (Stokes parameters) to optimize stroke patterns for maximal efficiency.

Experimental results

Research questions

• RQ1 How does the swimming behavior of a circular disk differ from that of a planar sheet when both surfaces undergo independent surface distortions?
• RQ2 What causes the discrepancy in prior calculations of swimming velocity for superposed undulation and squirming modes on a planar sheet?
• RQ3 Can a finite disk exhibit both translational and rotational swimming simultaneously, and what determines the resulting trajectory?
• RQ4 What is the maximum achievable swimming efficiency for a disk with symmetric and asymmetric strokes?
• RQ5 How do phase shifts and amplitude ratios between upper and lower surface waves affect the net swimming motion?

Key findings

• The disk performs both translational and rotational swimming when surface distortions on the two sides are asymmetric, leading to a net circular orbit.
• The mean translational swimming velocity is given by a formula involving amplitudes and phase differences of surface waves on both sides, with a dependence on the sign and magnitude of wave components.
• The mean rotational swimming velocity arises due to a torque imbalance caused by differing second-order flow velocities on the two surfaces, requiring a fluid rotation at infinity to satisfy no-slip conditions.
• For the optimal stroke with $ A = /pm B $, $ heta = rac{ u}{2} $, and $ C = A $, $ D = B $, $ eta = 0 $, $ u = - heta $, the efficiency reaches $ oxed{rac{1}{2} ext{max}} $, which is $ oxed{rac{1}{2} ext{max}} $, a factor of $ oxed{rac{1}{2} ext{max}} $ higher than Taylor’s model.
• The symmetric squeezing mode ($ C = -A, D = -B, eta = 0, u = heta $) yields zero rotational swimming velocity, consistent with Blake’s earlier result.
• The model resolves a long-standing inconsistency in prior theories by showing that the second-order flow velocity is discontinuous across the sheet unless a rotational component is included, which is naturally accounted for in the finite disk geometry.