A mathematical model for the motion of a micro-robot consisting of three spheres linked with axially aligned retractable arms in Stokes flow that gives expressions for mean distance moved, mean drift velocity and energy efficiency of the non-reciprocal cyclic swimming motion

Abstract

The problem studied was the non-reciprocal cyclic swimming motion of three spheres linked with axially aligned retractable arms in Stokes flow. The arms are assumed to be able to retract at a steady speed to half their length, and then at a later time in the motion extend at the same steady speed back to their original length. It is important in the field of medical biology, because this motion could enable a micro-robotic device to swim within the arterial and cellular fluid to perform medical biological procedures such as surgery or drug delivery. The method used was the development of a theoretical mathematical model in low Reynolds number Stokes flow, that takes the mathematical expression for steady motion of a sphere, and from this obtains the motion for three spheres by assuming leading order interactions between the spheres. This gives an interaction matrix from which the important results of the study are obtained which are exact mathematical expressions for the mean distance travelled, mean drift velocity and energy efficiency of the motion. These are that the mean distance travelled is 3a0ln(9/8), the mean drift velocity is (3a0/2A)ln(9/8), and the energy efficiency of the motion is 9a02/(4A2)[ln(9/8)]2, where a0 is the sphere radius and A is the arm length. The conclusions to be drawn from the results are that the most efficient design has the largest ratio for sphere radius in relation to arm length. The novelty of the work is that it gives exact mathematical expressions for distance travelled, velocity and energy efficiency not given previously in the literature.