Programmable rheotaxis of magnetic rollers in time-varying fields

Magnetic microrobots capable of navigating complex fluid environments typically rely on real-time feedback to adjust external fields for propulsion and guidance. As an alternative, we explore the use of field-programmable rheotaxis, in which time-periodic magnetic fields drive directional migration of ferromagnetic particles in simple shear flows. Using a deterministic model that couples magnetic torques to hydrodynamic interactions near a surface, we show that the frequency, magnitude, and waveform of the applied field can encode diverse rheotactic behaviors-including downstream, upstream, and cross-stream migration relative to the flow. We analyze the mechanisms underlying these responses for canonical fields and use this understanding to design complex waveforms that optimize migration speed and direction. Our results reveal a tradeoff between performance and robustness: high-performance designs enable upstream motion but are sensitive to system parameters, whereas robust designs operate in the linear response regime with more modest performance gains. These findings establish a general strategy for programming flow-guided navigation in magnetic colloids and suggest routes toward self-guided microrobots that respond predictably to fluid environments without external feedback.