The role of fluid flow in the dynamics of active nematic defects

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.

Flow around topological defects in active nematic films

“…The slow 1/r-decay term in equation (3.17) is independent of viscosity η and identical to the one derived in Ref. [24] in the frictiondominated regime. Corrections due to viscosity give rise to faster 1/r 3 decay.…”

“…As for equation (3.17) the 1/r term here was also obtained in [24]. The vorticity related to this velocity is…”

“…As anticipated from dimensional analysis, v a x ∼ (α 0 /Γ ξ ), in the overdamped limit where dissipation is controlled only by frictional drag [23,24,26]. In the underdamped limit, where the effect of drag is much smaller than viscous dissipation, hydrodynamic lengthscale becomes important in screening the divergence of the self-propulsion speed with system size, such that v a x scales instead as v a…”

“…The dependence of the scaling function F on ζ is plotted in figure 4 and its asymptotic scaling at ζ≫1 as F∼ζ−1 is included as the dotted line. We can discuss the implications of these results better when we use dimensional quantities and write the asymptotic behaviour of the self-propulsion speed as vxa≈{π4α0Γℓd=π4α0ℓdη,ζ≫1α0Γξ,ζfalse→0.As anticipated from dimensional analysis, vxa∼false(α0/Γξfalse), in the overdamped limit where dissipation is controlled only by frictional drag [23,24,26]. In the underdamped limit, where the effect of drag is much smaller than viscous dissipation, hydrodynamic lengthscale becomes important in screening the divergence of the self-propulsion speed with system size, such that vxa scales instead as vxa∼α0/ηΓ.…”

“…The flow generated by defects and the resulting propulsive speed of the +1/2 also vary depending on the dissipative processes at play in the system and the role of fluid incompressibility. Specifically, important differences exist between ‘dry’ systems, where dissipation is dominated by friction Γ with a substrate or an external medium [23,24] and ‘wet’ systems, where dissipation is mainly controlled by viscosity η, resulting in long-range hydrodynamic effects [14,17,24,25]. In incompressible wet systems, activity is also a source of pressure gradients, which in turn contribute alongside with the nematic distortion to the self-motility of positive defects.…”