Swimming speeds of filaments in nonlinearly viscoelastic fluids

Many microorganisms swim through gels and non-Newtonian fluids in their natural environments. In this paper, we focus on microorganisms which use flagella for propulsion. We address how swimming velocities are affected in nonlinearly viscoelastic fluids by examining the problem of an infinitely long cylinder with arbitrary beating motion in the Oldroyd-B fluid. We solve for the swimming velocity in the limit in which deflections of the cylinder from its straight configuration are small relative to the radius of the cylinder and the wavelength of the deflections; furthermore, the radius of the cylinder is small compared to the wavelength of deflections. We find that swimming velocities are diminished by nonlinear viscoelastic effects. We apply these results to examine what types of swimming motions can produce net translation in a nonlinear fluid, comparing to the Newtonian case, for which Purcell’s “scallop” theorem describes how time-reversibility constrains which swimming motions are effective. We find that the leading order violation of the scallop theorem occurs for reciprocal motions in which the backward and forward strokes occur at different rates.

💡 Research Summary

The paper investigates how the swimming speed of flagellated microorganisms is altered when they move through a non‑linearly viscoelastic fluid, using the Oldroyd‑B constitutive model as a representative of such media. The authors consider an infinitely long cylindrical filament whose centerline executes a prescribed transverse wave of small amplitude. Two geometric small‑parameter limits are imposed: the filament radius a is much smaller than the wavelength λ of the deformation, and the transverse displacement ξ is small compared with both a and λ. Under these assumptions a slender‑body approximation together with a regular perturbation expansion in the amplitude ε is justified.

The Oldroyd‑B model introduces three material parameters: the solvent viscosity ηₛ, the polymeric contribution ηₚ, and the relaxation time λ. The stress tensor obeys τ + λ ∇τ = ηₛ γ̇ + ηₚ γ̇ + λ ηₚ ∇γ̇, where γ̇ is the rate‑of‑strain tensor and ∇ denotes the upper‑convected derivative. The first two terms are linear in the strain rate, while the last term is quadratic and therefore responsible for genuine non‑linear viscoelastic effects.

The governing equations (incompressibility and momentum balance with the Oldroyd‑B stress) are solved in cylindrical coordinates. At O(ε) the solution reproduces the classic result for a waving filament in a linear viscoelastic fluid (identical to the Oldroyd‑A or Maxwell limit). At O(ε²) the quadratic term λ ηₚ ∇γ̇ contributes, generating corrections that depend on the square of the local strain rate. By imposing no‑slip on the filament surface and vanishing velocity at infinity, the authors obtain closed‑form expressions for the flow field using Fourier decomposition of the prescribed wave.

The central quantitative outcome is an expression for the time‑averaged swimming speed U:

 U = U₀