Swimming of an inertial squirmer array in a Newtonian fluid

Abstract

An immersed boundary-lattice Boltzmann method is employed to simulate a squirmer (a classical self-propelled model) array swimming in a Newtonian fluid. The swimming Reynolds number Res is set in the range 0.05 ≤  Res ≤ 5 to study three typical arrays (i.e., the two-squirmer, triangular-squirmer, and quadrilateral-squirmer arrays) in their swimming speed, their power expenditure ( P), and their hydrodynamic efficiency ( η). Our results show that the two-pusher array with a smaller ds (the distance between the squirmers) yields a slower speed in contrast to the two-puller array, where a smaller ds yields a faster speed at Res ≥ 1 (“pusher” is propelled from the rear and “puller” from the front). The regular triangular-pusher (triangular-puller) array with θ = −60° (the included angle between the squirmers) swims faster (slower) than that with θ = 60°; the quadrilateral-pusher (quadrilateral-puller) array with model 2 swims faster (slower) than model 1 (the models are to be defined later). It is also found that a two-puller array with a larger ds is more likely to become unstable than that with a smaller ds. The triangular-puller array with θ = 60° is more likely to become unstable than that with θ = 60°; the quadrilateral-puller array with model 1 becomes unstable easier than that with model 2. In addition, a larger ds generally results in a less energy expenditure. A faster squirmer array yields a higher η, except for two extraordinarily puller arrays. A quantitative relation for η with ReU > 1 is obtained approximately, in that the increasing ratio of η is proportional to an exponent of the motion Reynolds number ReU.