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arxiv: 2604.03875 · v1 · submitted 2026-04-04 · ⚛️ physics.flu-dyn

Collinear Swimming of a Squirmer Pair in Newtonian and Shear-Thinning Fluids

Chih-Tang Liao , Ali G\"urb\"uz , Victor Bueno Garcia , Yuan-Nan Young , Devanayagam Palaniappan , On Shun Pak This is my paper

Pith reviewed 2026-05-13 17:03 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords squirmer pairhydrodynamic interactionsNewtonian fluidsshear-thinning fluidsStokes flowco-swimmingmicroswimmerspropulsion
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0 comments X

The pith

Two squirmers swimming collinearly can maintain identical velocities over a range of separations in Newtonian fluids.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives an exact closed-form solution for the Stokes flow around two collinear squirmers in Newtonian fluids, complementing previous reciprocal theorem approaches. This solution, validated by finite element simulations, identifies configurations where the two squirmers achieve the same swimming speed. The analysis extends to shear-thinning fluids to assess how non-Newtonian effects influence the propulsion speed and energy cost of these co-swimming pairs. These results provide quantitative benchmarks for understanding pairwise interactions that underpin collective microswimmer dynamics.

Core claim

For the Newtonian case, an exact closed-form solution for the axisymmetric Stokes flow generated by the interacting pair reveals co-swimming configurations in which the two squirmers develop identical velocities over a range of separations. Symmetry arguments rationalize these behaviors, and propulsion performance is quantified in terms of speed and energetic cost. In shear-thinning fluids, the non-Newtonian rheology modifies these propulsion characteristics.

What carries the argument

Exact closed-form solution for the axisymmetric Stokes flow around two collinear squirmers, which provides direct access to the flow details and enables identification of co-swimming states via symmetry.

If this is right

  • The two squirmers can co-swim with identical velocities at various separations.
  • Propulsion performance is characterized by specific speeds and energetic costs in Newtonian fluids.
  • Shear-thinning fluids alter the velocity and energy requirements for co-swimming.
  • These pairwise interactions serve as building blocks for many-body microswimmer dynamics.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Such co-swimming states may stabilize larger groups of microswimmers in biological fluids.
  • Future studies could explore non-axisymmetric configurations or other rheological behaviors like viscoelasticity.
  • The benchmarks can inform models of bacterial colonies or sperm motility in complex media.

Load-bearing premise

The squirmer model with prescribed surface velocity accurately captures the propulsion mechanism of real microswimmers, and the flow remains axisymmetric Stokes flow at all separations considered.

What would settle it

Experimental observation that the two squirmers have different swimming speeds at separations predicted to yield identical velocities, or detection of non-axisymmetric flow patterns.

Figures

Figures reproduced from arXiv: 2604.03875 by Ali G\"urb\"uz, Chih-Tang Liao, Devanayagam Palaniappan, On Shun Pak, Victor Bueno Garcia, Yuan-Nan Young.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Schematic and notation of the problem setup, where two collinear squirmers are shown. [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Comparisons between analytical and numerical solutions in terms of (a) the swimming [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Illustration of symmetry arguments for (a) puller–pusher and (b) puller–neutral config [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Flow fields around the co-swimming pairs: (a) puller-pusher with [ [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. The viscosity map around the co-swimming squirmers at different Carreau number [PITH_FULL_IMAGE:figures/full_fig_p020_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. The power dissipation of co-swimming squirmers in a shear-thinning fluid, [PITH_FULL_IMAGE:figures/full_fig_p022_10.png] view at source ↗
read the original abstract

Pairwise hydrodynamic interactions of microswimmers form the fundamental building blocks for understanding their more complex collective behaviors. In this work, we revisit the canonical problem of two interacting squirmers swimming along their common line of centers in both Newtonian and shear-thinning fluids. For the Newtonian case, we first derive an exact, closed-form solution for the axisymmetric Stokes flow generated by the interacting pair, thereby complementing prior analyses based on the reciprocal theorem approach by providing direct access to the detailed knowledge of the flow around the swimmers. The analytical solution is then used to cross-validate numerical simulations based on the finite element method. The combined theoretical and numerical investigation reveals co-swimming configurations in which the two squirmers develop identical velocities over a range of separations. We rationalize these behaviors through symmetry arguments and quantify their propulsion performance in terms of the speed and energetic cost of swimming. Furthermore, motivated by the prevalence of shear-thinning biological fluids encountered by microswimmers, we examine how this ubiquitous non-Newtonian rheological behavior modifies the propulsion characteristics of these co-swimming pairs. Taken together, our results establish quantitative benchmarks for interacting squirmers in both Newtonian and shear-thinning fluids, laying the groundwork for future studies of many-body dynamics of microswimmers in complex fluid environments.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper derives an exact closed-form solution for the axisymmetric Stokes flow generated by two collinear squirmers in a Newtonian fluid, cross-validates it against finite-element simulations, identifies co-swimming configurations in which both particles acquire identical velocities over a range of separations via symmetry arguments, quantifies the associated swimming speed and energetic cost, and extends the same numerical framework to shear-thinning fluids to examine how non-Newtonian rheology modifies the propulsion characteristics of these pairs.

Significance. If the results hold, the work supplies quantitative benchmarks for pairwise hydrodynamic interactions of microswimmers that are directly relevant to collective dynamics. The exact Newtonian solution, its cross-validation with FEM, and the parameter-free identification of co-swimming states constitute clear strengths; the extension to shear-thinning fluids addresses a biologically pertinent regime and thereby broadens the utility of the benchmarks.

major comments (2)
  1. [Analytical solution section] Analytical solution section: the manuscript states that an exact closed-form solution is obtained by direct solution of the biharmonic equation, yet the explicit series coefficients or stream-function expression (presumably in bispherical coordinates) are not reproduced in the main text or an appendix; without these, independent verification of the boundary conditions at both squirmer surfaces is difficult.
  2. [Shear-thinning results] Shear-thinning results: the modification of co-swimming velocities is reported for a non-Newtonian constitutive law, but the specific model (Carreau, power-law, etc.), the values of the dimensionless groups (e.g., Carreau number, power-law index), and any mesh-convergence or validation metrics beyond the Newtonian limit are not stated; these details are load-bearing for the claim that non-Newtonian effects systematically alter propulsion performance.
minor comments (2)
  1. [Figures] Figure captions: several panels comparing Newtonian and shear-thinning streamlines lack explicit indication of the squirmer type (pusher/puller) and the precise separation distances shown; adding these labels would improve clarity.
  2. [Introduction] Introduction: the contrast with prior reciprocal-theorem analyses is mentioned but could be sharpened by stating which flow quantities become newly accessible through the direct solution.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment of our work and the constructive comments. We address each major comment below.

read point-by-point responses

  1. Referee: [Analytical solution section] Analytical solution section: the manuscript states that an exact closed-form solution is obtained by direct solution of the biharmonic equation, yet the explicit series coefficients or stream-function expression (presumably in bispherical coordinates) are not reproduced in the main text or an appendix; without these, independent verification of the boundary conditions at both squirmer surfaces is difficult.

    Authors: We agree that the explicit form of the solution should be provided to enable independent verification. In the revised manuscript we have added the complete stream-function expression in bispherical coordinates together with the series coefficients to a new Appendix A. This addition directly allows readers to confirm satisfaction of the boundary conditions on both squirmer surfaces. revision: yes

  2. Referee: [Shear-thinning results] Shear-thinning results: the modification of co-swimming velocities is reported for a non-Newtonian constitutive law, but the specific model (Carreau, power-law, etc.), the values of the dimensionless groups (e.g., Carreau number, power-law index), and any mesh-convergence or validation metrics beyond the Newtonian limit are not stated; these details are load-bearing for the claim that non-Newtonian effects systematically alter propulsion performance.

    Authors: We acknowledge the omission of these essential details. The revised manuscript now explicitly states that the Carreau model is employed, provides the specific values of the power-law index and Carreau number used, and includes a new subsection on mesh-convergence studies together with validation metrics that extend the Newtonian cross-validation to the shear-thinning regime. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper derives an exact closed-form solution to the axisymmetric Stokes equations for the collinear squirmer pair via direct solution of the biharmonic equation in a suitable coordinate system, then cross-validates this parameter-free analytic result against independent FEM numerics. Co-swimming states with identical velocities are identified via symmetry arguments applied to the solved flow field. The shear-thinning extension applies the same validated numerical scheme to a non-Newtonian constitutive law. No load-bearing step reduces by construction to a fitted parameter, self-citation, or renamed input; the Newtonian solution is independent of the target co-swimming velocities and the modeling assumptions (squirmer BCs and axisymmetry) are satisfied identically by the problem setup.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard low-Reynolds-number Stokes approximation and the squirmer boundary condition; no new free parameters or invented entities are introduced beyond the usual squirmer modes and the power-law index for shear-thinning.

axioms (2)
  • domain assumption Low-Reynolds-number Stokes flow approximation
    Standard for microswimmer hydrodynamics; invoked throughout the Newtonian analysis.
  • domain assumption Squirmer model with prescribed tangential surface velocity
    Canonical representation of self-propulsion; used to set boundary conditions on each sphere.

pith-pipeline@v0.9.0 · 5553 in / 1314 out tokens · 36282 ms · 2026-05-13T17:03:21.717103+00:00 · methodology

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