Controlled fluid transport by micro-rotors in a Stokes flow

Controlling the motion of fluid particles in low Reynolds number flows has become increasingly important in recent years, particularly in microrobotics and microfluidics. Microrotors and micropumps propelled by various mechanisms have been proposed as a useful means of transporting fluid particles or other submersed cargo in a microfluidic solution. In this work, we study the problem of steering an ensemble of fluid particles in a Stokes flow from an initial particle distribution to a target distribution, where the particles are advected by the flow field generated by a group of micro-rotors. In our formulation, the rotors are modeled as rotlets, the point-torque singularity solution of the Stokes equations and the particle distribution is described by a probability density function. We pose the density transport problem as an optimal control problem in terms of the moments of the particle distribution, which we solve using differential dynamic programming, an iterative trajectory optimization algorithm. We study cases of fixed rotors, where only the rotor strengths are controlled, as well as cases where both the strengths and translational velocities of the rotors are controlled. We analyze the benefits of using multiple rotors as well as the flow structures associated with the flow field generated by the optimal control.