Zhou & Revzen show reciprocal motion can propel robots in Carreau-Yasuda fluids
A new paper mathematically proves that the Scallop Theorem can be violated in shear-thinning fluids.
A new paper by Yishun Zhou and Shai Revzen, published on arXiv (cs.RO/2606.00063), revisits one of the foundational constraints of low-Reynolds-number locomotion: Purcell's Scallop Theorem, which states that reciprocal (back-and-forth) shape changes cannot produce net displacement in a viscous fluid. The authors show that the linear motility maps that underpin this theorem actually hold for any power-law viscosity fluid (Ostwald–de Waele model), meaning the scallop theorem remains valid in many biological fluids within intermediate shear ranges where viscosity scales as a power of strain rate.
More striking, they demonstrate that the linear-in-velocity property can be broken in a more complex rheological model: the Carreau–Yasuda fluid. Using a toy 'inchworm' consisting of two unequal masses with different drag coefficients that perform strictly reciprocal motions, they achieve net forward motion. The direction of that motion even flips depending on the speed of the cycle. This result suggests that micro-robots operating in shear-thinning biological fluids (e.g., mucus, blood, cytoplasm) could exploit nonlinear drag to circumvent the scallop theorem—without needing non-reciprocal strokes. The work bridges geometric mechanics and non-Newtonian fluid dynamics, offering a new theoretical framework for designing untethered microswimmers.
- Linear motility maps extend to all power-law viscosity fluids (Ostwald–de Waele model), preserving the Scallop Theorem in those regimes.
- In Carreau–Yasuda fluids, a reciprocal-inchworm model with unequal masses and drag coefficients achieves net displacement, violating the classic scallop constraint.
- The direction of motion can be toggled by simply changing the speed of the reciprocal cycle, enabling novel control strategies for micro-robots.
Why It Matters
Opens a theoretical path for simple reciprocating micro-robots to swim in biological fluids where the scallop theorem was thought to apply.