Rotational dynamics of helical flagella in viscoelastic fluids

The motility of bacteria in viscoelastic fluids plays a crucial role in diverse biological processes such as infection and biofilm formation. In such environments, the rotation of long helical flagella is often employed as a primary means of bacterial propulsion. In this paper, we present a general version of the immersed boundary method incorporating the finitely extensible nonlinear elastic (FENE) constitutive equations for viscoelastic fluids. We first verify that the method solves correctly the rotational dynamics of a helical flagellum in a FENE fluid by conducting a convergence study. We then investigate the hydrodynamic interactions of single and multiple flagella in Newtonian and FENE viscoelastic fluids. For a single flagellum at the same solvent viscosity, the fluid flux generated by the flagellum is higher in FENE fluid than in Newtonian fluid for nearly identical rotation rates, implying enhanced bacterial swimming speeds in a viscoelastic fluid. The dependence of rotational dynamics on applied motor torque, relaxation time, and polymer viscosity is systematically analyzed. For multiple flagella, the time required for bundle formation is shown to increase with hook rigidity, initial separation, and the number of flagella. In the case of two flagella rotating counterclockwise under different applied torques, bundling occurs only when the torque difference is within a limited range, producing a bundle whose axis is tilted from the flagellar axes. We further examine conditions leading to bundling failure, providing insights into the mechanical and hydrodynamic factors governing flagellar coordination.