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

This paper presents an exact analytical solution and numerical simulations to characterize the hydrodynamic interactions and propulsion performance of two collinearly swimming squirmers in both Newtonian and shear-thinning fluids, revealing specific co-swimming configurations with identical velocities and establishing quantitative benchmarks for future many-body studies.

The Gist

Imagine a microscopic world where tiny, invisible swimmers (like bacteria or tiny robots) are trying to move through fluids. In our everyday world, if you push a boat, it moves forward. But in this microscopic world, the water feels like thick honey, and there's no momentum to carry you forward. If you stop paddling, you stop instantly. This is the world of "microswimmers."

This paper is like a detailed traffic report for two of these tiny swimmers trying to swim in a straight line, one right behind the other. The researchers wanted to figure out: How do they affect each other's speed? Does it matter if they are "pushers" or "pullers"? And what happens if the water they are swimming in is weird and sticky (like mucus or blood)?

Here is the breakdown of their findings using simple analogies:

1. The Swimmers: Pushers vs. Pullers

Think of these swimmers as having two different "styles" of paddling:

• The Puller (like a rower): Imagine a swimmer who pulls the water toward them from the front and pushes it out the back. They are like a person rowing a boat toward the shore.
• The Pusher (like a propeller): Imagine a swimmer who pushes water away from them at the back and pulls it in from the front. They are like a boat with a propeller pushing them forward.
• The Neutral: A swimmer who doesn't really push or pull the water much, just glides.

2. The Big Discovery: The "Perfect Dance" (Co-Swimming)

The most surprising thing the researchers found is that sometimes, two swimmers can lock into a perfect rhythm where they both swim at the exact same speed, no matter how close or far apart they are. They don't need to hold hands or be tied together; the water itself forces them to match speeds.

• The "Tailgater" Effect: If a Puller is in front and a Pusher is behind, they become a super-team. The Puller pulls the water in, and the Pusher pushes the water out. Their movements actually help each other, making them swim faster than they would alone. It's like a cyclist drafting behind another cyclist, but in reverse—their combined wake actually gives them a boost!
• The "Traffic Jam" Effect: If you flip them around (Pusher in front, Puller behind), they get in each other's way. The Pusher in front creates a wake that the Puller behind has to fight against. They slow down significantly and waste more energy.

3. The "Thickening" Fluid (Shear-Thinning)

Now, imagine these swimmers aren't in water, but in something like ketchup, mucus, or blood. These fluids are "shear-thinning."

• The Analogy: Think of ketchup in a bottle. When it sits still, it's thick and hard to move. But if you shake it or squeeze it hard (apply force), it suddenly becomes runny and flows easily.
• The Result: When the swimmers move, they create friction that "shakes" the fluid around them, making it thinner (less sticky) right where they are swimming.
  • Good News: Because the fluid gets thinner around them, it takes less energy for them to swim. It's like running on a track that turns into a soft, slippery slide right under your feet.
  • Mixed News: Even though it's easier to move (less energy), they don't necessarily get faster. In fact, the complex way the fluid thins out can sometimes slow them down a bit, depending on their "dance" (who is in front and who is behind).

4. How They Did It

The researchers used two methods to solve this puzzle:

1. Math Magic: They wrote down a perfect, exact mathematical formula (like a recipe) that describes exactly how the water moves around two spheres. This is rare because usually, you have to guess or approximate.
2. Computer Simulation: They built a virtual world in a computer to watch the swimmers move and confirmed that their math was 100% correct.

Why Does This Matter?

This isn't just about math; it's about the future of medicine and technology.

• Drug Delivery: Imagine tiny robots swimming through your blood (which is shear-thinning) to deliver medicine to a tumor. Knowing how they interact with each other helps us design swarms of robots that work together efficiently instead of getting in each other's way.
• Understanding Life: It helps us understand how bacteria in your gut or mucus in your lungs move and interact.

In a nutshell: Two tiny swimmers can either be a high-speed team or a slow-motion traffic jam, depending on their style and order. If they swim in "ketchup-like" fluids, they save energy, but the fluid's weird behavior adds a new layer of complexity to their journey. The researchers mapped out all these possibilities so we can better understand the microscopic world.

Technical Summary

1. Problem Statement

The paper investigates the hydrodynamic interactions between two spherical microswimmers (modeled as "squirmers") swimming along their common line of centers (collinear configuration). The study addresses two primary regimes:

• Newtonian Fluids: Revisiting the canonical problem to derive exact analytical solutions and understand the interplay between swimmer types (pullers, pushers, neutrals) and separation distances.
• Shear-Thinning Fluids: Extending the analysis to non-Newtonian fluids (mimicking biological environments like blood or mucus) to determine how viscosity dependent on shear rate alters propulsion speed and energetic cost.

The core challenge is to move beyond integral methods (like the reciprocal theorem) that provide global quantities (speed/force) but lack local flow field details, and to understand how pairwise interactions evolve in complex rheological environments.

2. Methodology

The authors employ a combined theoretical and numerical approach:

A. Theoretical Analysis (Newtonian Limit)

• Coordinate System: The problem is formulated using bispherical coordinates $(\xi, \eta, \phi)$, which naturally fit the geometry of two spheres.
• Governing Equations: The Stokes equations for incompressible, low-Reynolds-number flow are solved exactly. The velocity field is represented via a Stokes streamfunction $\psi$.
• Boundary Conditions: The squirmers are defined by surface tangential velocity distributions expanded into spherical harmonics (modes). The study focuses on the first two modes:
  • $B_1$: Source dipole (sets swimming speed).
  • $B_2$: Force dipole (stresslet; determines if the swimmer is a puller, pusher, or neutral).
• Solution: An exact, closed-form solution is derived for the streamfunction and the resulting swimming velocities ($U_\alpha, U_\beta$) by solving an infinite algebraic system for the coefficients. This complements previous reciprocal theorem approaches by providing the full flow field structure.

B. Numerical Simulations

• Solver: Finite Element Method (FEM) implemented in COMSOL Multiphysics.
• Domain: 2D axisymmetric domain with a large computational box ($501a \times 1014a$) to approximate an unbounded fluid.
• Validation: The numerical results are cross-validated against the derived exact analytical solutions for Newtonian fluids, showing excellent agreement for both swimming speeds and flow fields.
• Non-Newtonian Extension: For shear-thinning fluids, the Carreau constitutive model is used:
  $$\mu_s = \mu_\infty + (\mu_0 - \mu_\infty)(1 + \lambda_C^2 |\dot{\gamma}|^2)^{\frac{n-1}{2}}$$
  where $\mu_0$ and $\mu_\infty$ are zero/infinite shear viscosities, $\lambda_C$ is a characteristic time scale, and $n$ is the power-law index.

3. Key Contributions

1. Exact Analytical Solution: Derivation of a closed-form solution for the axisymmetric Stokes flow around two interacting squirmers. This provides direct access to local flow features and serves as a rigorous benchmark for numerical simulations.
2. Discovery of Co-Swimming States: Identification of specific configurations where two freely swimming (unconstrained) squirmers develop identical velocities over a wide range of separations. This occurs purely due to hydrodynamic interactions without mechanical linkage.
3. Symmetry Arguments: Rationalization of co-swimming states using kinematic reversibility and geometric symmetry arguments, explaining why certain pairings (e.g., puller-pusher) result in equal speeds while others do not.
4. Shear-Thinning Impact: First comprehensive study of how shear-thinning rheology modifies the propulsion of interacting pairs, revealing that while the magnitude of speed changes, the symmetry of co-swimming states is preserved.

4. Key Results

A. Newtonian Fluids

• Co-Swimming Configurations: Three specific pairings exhibit identical velocities ($U_\alpha = U_\beta$) at all separations:
  • Puller–Pusher (Leading Puller, Trailing Pusher)
  • Neutral–Neutral
  • Pusher–Puller (Leading Pusher, Trailing Puller)
  • Mechanism: This is a direct consequence of kinematic reversibility. For example, in a puller-pusher pair, the contractile flow of the leading puller and the extensile flow of the trailing pusher constructively interact to accelerate the pair.
• Speed Variations:
  • Puller-Pusher: Significant speed enhancement as separation decreases (constructive interference).
  • Pusher-Puller: Significant speed reduction (destructive interference).
  • Non-Co-Swimming Pairs: Exhibit non-monotonic speed variations and can even reverse direction at very small separations due to strong near-field hydrodynamic interactions.
• Energetics: The puller-pusher pair not only swims faster but also dissipates less power than isolated swimmers. Conversely, the pusher-puller pair swims slower and expends more energy.

B. Shear-Thinning Fluids

• Persistence of Symmetry: Despite the nonlinearity of the Carreau model, the velocity-reversal symmetry is preserved. Consequently, the co-swimming states persist; the two squirmers still attain identical speeds in the identified configurations.
• Speed Dependence on Carreau Number ($Cu$):
  • At low $Cu$ (Newtonian limit), speeds match Newtonian predictions.
  • As $Cu$ increases, speeds generally decrease, reaching a minimum around $Cu \sim O(1)–O(10)$ due to the formation of low-viscosity regions around the swimmers that alter stress distributions.
  • At very high $Cu$, speeds recover toward the Newtonian limit.
  • Exception: A modest speed enhancement is observed for the pusher-puller configuration at intermediate $Cu$, highlighting the complexity of spatially heterogeneous viscosity fields.
• Energetic Cost: Unlike the non-monotonic speed behavior, the power dissipation decreases monotonically with increasing shear thinning. The reduction in local viscosity in high-shear regions lowers the viscous stress required for propulsion, making swimming energetically cheaper regardless of the speed change.

5. Significance and Future Outlook

• Fundamental Physics: The work establishes quantitative benchmarks for interacting active matter in complex fluids, bridging the gap between analytical theory and numerical simulation.
• Biological Relevance: The findings are crucial for understanding microorganism locomotion in biological fluids (e.g., mucus, blood) where shear-thinning is prevalent. It suggests that collective behaviors (like clustering or synchronized swimming) may be robust to rheological changes, even if the absolute speed varies.
• Future Directions: The authors propose extending this framework to:
  • Many-body systems to study collective dynamics.
  • Viscoelastic fluids (combining shear-thinning with elasticity).
  • Swimmers of unequal sizes (size asymmetry).
  • Swimmers in externally imposed viscosity gradients (tactic behavior).

In summary, this paper provides a rigorous foundation for understanding how hydrodynamic interactions and non-Newtonian rheology jointly dictate the collective motion of microswimmers, revealing that specific pairwise configurations can self-organize into stable, co-swimming states with optimized energetic performance.