Abstract
We adapt the Halperin–Mazenko formalism to analyze two-dimensional active nematics coupled to a generic fluid flow. The governing hydrodynamic equations lead to evolution laws for nematic topological defects and their corresponding density fields. We find that ±1/2 defects are propelled by the local fluid flow and by the nematic orientation coupled with the flow shear rate. In the overdamped and compressible limit, we recover the previously obtained active self-propulsion of the +1/2 defects. Non-local hydrodynamic effects are primarily significant for incompressible flows, for which it is not possible to eliminate the fluid velocity in favor of the local defect polarization alone. For the case of two defects with opposite charge, the non-local hydrodynamic interaction is mediated by non-reciprocal pressure-gradient forces. Finally, we derive continuum equations for a defect gas coupled to an arbitrary (compressible or incompressible) fluid flow.
Highlights
Nematic order has been widely observed in two-dimensional (2D) realizations of active matter [1, 2], from vertically vibrated rods [3] to mixtures of cytoskeletal filaments and associated motor proteins [4, 5, 6], bacterial suspensions [7, 8, 9], cell sheets [10, 11, 12, 13], and even developing organisms [14]
We find that ±1/2 defects are propelled by the local fluid flow and by the nematic orientation coupled with the flow shear rate
Nematic order has only quasi-long-range falloff in 2D active systems [15, 16, 17], as it does in equilibrium, and is destroyed by active stresses that drive spontaneous flows accompanied by the proliferation of topological defects [4]
Summary
Nematic order has been widely observed in two-dimensional (2D) realizations of active matter [1, 2], from vertically vibrated rods [3] to mixtures of cytoskeletal filaments and associated motor proteins [4, 5, 6], bacterial suspensions [7, 8, 9], cell sheets [10, 11, 12, 13], and even developing organisms [14]. Important corrections to the effective dynamics arise from the fact that the phase field induced at the core of a defect by all the others depends on all the defect velocities, as well as on their past history of accelerations, but including such effects remains a formidable challenge These previous works all treat the case of an active nematic on a substrate, where friction dominates over viscous dissipation and the compressible flow ‡ is slaved to the gradient of the active stress. For the first time, that nonlocal hydrodynamic effects become important for multi-defect configurations and yield a finite mobility for the −1/2 defects We demonstrate this explicitly by calculating the pressure-gradient forces on a defect dipole and showing that incompressibility gives rise to hydrodynamic interactions that renormalize the propulsion of positive defects and mediate non-reciprocal active defect-defect interactions additional to the familiar Coulomb force. The Appendices detail analytical computations of the flow field induced by a variety of specific defect configurations
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