Study on the hydrodynamics of the helix with non-uniform helical radius in viscoelastic fluid at low Reynolds number

Bacteria exhibit swimming behavior through the rotation of helical flagellar filaments, a common mode of propulsion for artificial micro-swimmers as well. However, most studies have focused on helical filaments with a uniform radius, leaving limited research on filaments with nonuniform radii. This paper presents a simulation of the hydrodynamics of a rotating helical filament with a nonuniform radius. The degree of nonuniformity is characterized by a radius ratio,β. Using the OpenFOAM platform, a simulation module for non-Newtonian Stokes flow was developed and integrated. The kinematics of the filament were predefined by fixing either the pitch or the helical angle, in addition to the helical radius. Thrust acting on the filament were analyzed with regards to the helical radius ratio, rotation speed, and fluid viscoelasticity. Certain modifications were made to apply the resistive force theory (RFT) to filaments with nonuniform helical radii. Results revealed that when the pitch was fixed, the thrust of the helix varied non-monotonically with the radius ratio, with excessive ratios leading to increased resistance. In contrast, helices with fixed helical angles exhibited an increasing thrust with a higher radius ratio. The thrust was greater in Newtonian fluids compared to non-Newtonian fluids, and the propulsion of the helix was influenced by the Deborah number. As the Deborah number approached zero, the thrust of viscoelastic swimming speed approached that of viscous swimming speed. For small, β(,β=1 and 4), the ratio of the thrust in viscoelastic fluid to the thrust in viscous fluid was always less than unity. At, β = 9, elasticity hindered propulsion at low De (De<1.5), while the opposite was observed at high De (De>1.5). A new approach was proposed to evaluate the impact of viscoelasticity on the helix, it was expected that the new calculation method can reduce the computational effort effectively.