The geometry of thresholdless active flow in nematic microfluidics

"Active nematics" are orientationally ordered but apolar fluids composed of interacting constituents individually powered by an internal source of energy. When activity exceeds a system-size dependent threshold, spatially uniform active apolar fluids undergo a hydrodynamic instability leading to spontaneous macroscopic fluid flow. Here, we show that a special class of spatially non-uniform configurations of such active apolar fluids display laminar (i.e., time-independent) flow even for arbitrarily small activity. We also show that two-dimensional active nematics confined on a surface of non-vanishing Gaussian curvature must necessarily experience a non-vanishing active force. This general conclusion follows from a key result of differential geometry: geodesics must converge or diverge on surfaces with non-zero Gaussian curvature. We derive the conditions under which such curvature-induced active forces generate "thresholdless flow" for two-dimensional curved shells. We then extend our analysis to bulk systems and show how to induce thresholdless active flow by controlling the curvature of confining surfaces, external fields, or both. The resulting laminar flow fields are determined analytically in three experimentally realizable configurations that exemplify this general phenomenon: i) toroidal shells with planar alignment, ii) a cylinder with non-planar boundary conditions, and iii) a "Frederiks cell" that functions like a pump without moving parts. Our work suggests a robust design strategy for active microfluidic chips and could be tested with the recently discovered "living liquid crystals".