Three Dimensional Spontaneous Flow Transition in Active Polar Fluids

Abstract

Active liquid crystals exhibit spontaneous flow when sufficient active stress is generated by internal molecular mechanisms. This is also referred to as active Fréedericksz transition. We show this transition in three dimensions and study its dependence on the boundary conditions. Using nonlinear numerical solutions and linear perturbation analysis, we find an out-of-plane transition under extensile active stress for perpendicular polarity anchoring at the boundary, whereas parallel anchoring permits both in-plane flow under contractile stress and out-of-plane wrinkling under extensile stress.Active fluids are out-of-equilibrium materials driven by energy injection at the microscopic scale [1, 2]. Active materials can have a polar or nematic alignment symmetry of the orientation vector field, and the constituents of the material allow it to generate contractile or extensile active stress. Prominent examples of active fluids are found in living matter across scales, from the cytoskeleton [3, 4] and tissues [5][6][7] to collective behavior in flocks [8]. The hydrodynamic theory of incompressible active polar fluids describes the dynamics of such active liquid crystals at long wavelengths. A key behavior of active polar fluids is their ability to generate spontaneous flow under confinement and sufficient active stress. It has been shown in two dimensions (2D) that spontaneous flow can emerge due to a Fréedericksz-type transition first observed in passive liquid crystals [9]. The passive Fréedericksz transition describes the change of an isotropic state to an anisotropic state under the influence of external electric or magnetic fields [10]. In active liquid crystals, the transition is driven by active molecular processes causing spontaneous material flow [11][12][13].Recent works suggest such instabilities to also exist in three-dimensional (3D) active polar fluids [14]. For example, an extensile active fluid was found to exhibit a bending instability in a simplified model [15], and flow-aligning active fluids were found to display coherent motion in 3D channels upon increased activity [16]. Furthermore, It has been shown that 3D contractile active fluids dampen out-of-plane perturbations, whereas extensile fluids amplify them in the absence of boundary effects [17]. 3D active fluids under confinement behave fundamentally different from their 2D counterparts. Notably, they exhibit flow due to buckling under extensile active stress, which is not possible in 2D for rigid boundaries [18]. Experimentally, this instability has been observed in microtubule assays capable of generating extensile active stresses [18,19].Here, we consider the active Ericksen-Leslie hydrodynamic model and show that the instability exists in 3D. We consider a thick active polar film, which is the 3D extension of a Fréedericksz cell, with anchoring of the polarity and stress-free boundary conditions on the walls. We show how the steady-state spontaneous flow depends on the polarity boundary conditions and the sign of active stress. We derive a...