Numerical study on the propulsion of a bacterial flagellum in a viscous fluid using an immersed boundary method

0045-7930/$ see front matter Crown Copyright 2 http://dx.doi.org/10.1016/j.compfluid.2012.03.012 ⇑ Corresponding author. Tel.: +82 51 200 7636; fax E-mail address: [email protected] (S. Kang). The propulsion of a bacterial flagellum in a viscous fluid has gained much attention in the field of biological fluid dynamics. Inspired by the bacterial propulsion, we present a three-dimensional computational model based on an immersed boundary (IB) method to study the propulsive and fluid dynamic features of a solid flexible flagellum in a viscous fluid driven at one side by an external torque. The helical flagellum is modelled by a series of triangular cross-sections with three IB points on each cross-section. Three types of elastic links are created to connect the IB points of one cross-section to the IB points of the next crosssection to obtain an elastic network model of the flagellum. An external torque is applied at the center of the first cross-section modelled as the flagellum motor. The elastic forces are computed based on elastic energy approach and the motor forces are obtained from the magnitude and direction of the applied torque. The Stokes equations governing the flow are solved on a staggered Cartesian grid system using the fractional-step based finite-volume method. The computational model is validated by comparing the swimming speed of the flagellum obtained from the numerical results with that of the existing numerical results. The interplay of propulsive and hydrodynamic features of a left handed helical flagellum driven by an external torque is well captured using the developed model. The effect of elasticity of various types of links on the swimming speed of the flagellum is investigated. It is revealed that among the three types of elastic links, the diagonal links have dominant role in the propulsion behavior of the flagellum. The effect of pitch on the propulsion speed of the flagellum is analyzed by computing its optimum value for a constant flagellum length based on the maximum swimming speed. Numerical simulations are performed to demonstrate the forward to backward swimming behavior of the flagellum and it is concluded that the backward swimming speed is higher than the forward swimming speed. The effect of the channel wall on the forward and backward propulsion of the flagellum is examined. It is observed that the flagellum swims faster in a channel than that in an unbounded fluid domain. Further, it is found that the swimming speed of the flagellum near to a channel wall is considerably higher than that away from the walls. Crown Copyright 2012 Published by Elsevier Ltd. All rights reserved.