External force-induced focus pattern of a flexible filament in a viscous fluid

In order to study the mechanism of wall attraction of micro-swimmers, this paper presents a numerical study on the external force-induced focus of a flexible filament in a quiescent viscous fluid. In the numerical model, the fluid is solved by the lattice Boltzmann method (LBM) and the moving filament is modeled with the immersed boundary (IB). The filament is parallel and adjacent to a near-wall initially. A periodic driving force is applied on it to induce a vibration motion which could lead to the wall attraction phenomenon of the vibrating filament. Furthermore, by adjusting the mean value of the periodic driving force, we can get a stable vibrating mode of the filament, which is called as focus pattern. The pattern is a special state of the wall attraction, which provides a convenient way to investigate the law of hydrodynamic balance. Simulations are conducted by varying the following parameters, including the boundary setting of the flow field, the magnitude, period and waveform of the external driving force, the bending modulus of the filament and the fluid viscosity. The laws of the focus pattern are discussed in detail. The results indicate that all the above parameters can affect the focus position remarkably. By analyzing the outputs of each item of the Navier–Stokes momentum conservation equation, we found that the focus is mainly caused by the balance of the joint force on the filament and the pressure effect from the flow. This finding may provide a novel way to uncover the hydrodynamic mechanism for wall attraction of waving propulsion.