A hybrid particle-mesh method for incompressible active polar viscous gels

Abstract

We present a hybrid particle-mesh method for numerically solving the hydrodynamic equations of incompressible active polar viscous gels. These equations model the dynamics of polar active agents, embedded in a viscous medium, in which stresses are induced through constant consumption of energy. The numerical method is based on Lagrangian particles and staggered Cartesian finite-difference meshes. We show that the method is second-order and first-order accurate with respect to grid and time-step sizes, respectively. Using the present method, we simulate the hydrodynamics in rectangular geometries, of a passive liquid crystal, of an active polar film and of active gels with topological defects in polarization. We show the emergence of spontaneous flow due to Fréedericksz transition, and transformation in the nature of topological defects by tuning the activity of the system.

Activity induces traveling waves, vortices and spatiotemporal chaos in a model actomyosin layer

“…We present the numerical solution to the hydrodynamic equation governing the model system 24 . We study the polarity and flow dynamics as activity induces larger contractile stresses in the active polar fluid model.…”

“…Such transitions have been observed experimentally in the organization of microtubules in vitro upon varying the concentration of motor proteins 21 . Numerical approaches have confirmed spontaneous flow transitions 22 23 and transitions between polar patterns 24 in active polar fluids. Numerical studies have also been used to find a rich variety of patterns in active nematic and polar fluids 25 26 27 .…”

“…With the given parametrization of the model and periodic boundary condition in the x-direction, we numerically solve the equations governing the hydrodynamics using a recently developed general hybrid particle-mesh method for incompressible active polar viscous gels 24 . The method imposes the unit vector polarity and incompressibility constraints exactly, and has shown to be consistent, stable and therefore convergent 24 . The initial condition and settings used for the numerical simulations are presented in Sec.…”

“…We numerically explore the spatiotemporal dynamics as a function of spatially-homogeneous activity of the system. We make use of a recently developed hybrid particle-mesh method to numerically solve the hydrodynamic equations of active polar fluids 24 . The numerical results show that the nonlinear dynamics as a function of activity depend on the nature of interaction between the polarity field and the local shear generated by the flow.…”

Computational approaches to substrate-based cell motility

“…Also, hybrid particle-mesh methods for solving the hydrodynamic equations of incompressible active polar viscous gels can be applied to describe moving deformable objects. 171 The efficiency can be possibly increased even further by applying the renormalization group approach to the phase-field method, as performed in a different context by Goldenfeld et al 172 Finally, one could develop an algorithm assigning a single phase field to several well-separated cells. As it is evident, e.g., from Figure 6, each migrating cell has only a few immediate neighbours with whom it interacts.…”