Hydrodynamic interactions in polar liquid crystals on evolving surfaces

Abstract

We consider the derivation and numerical solution of the flow of passive and active polar liquid crystals, whose molecular orientation is subjected to a tangential anchoring on an evolving curved surface. The underlying passive model is a simplified surface Ericksen-Leslie model, which is derived as a thin-film limit of the corresponding three-dimensional equations with appropriate boundary conditions. A finite element discretization is considered and the effect of hydrodynamics on the interplay of topology, geometric properties and defect dynamics is studied for this model on various stationary and evolving surfaces. Additionally, we consider an active model. We propose a surface formulation for an active polar viscous gel and exemplarily demonstrate the effect of the underlying curvature on the location of topological defects on a torus.

Linearly forced fluid flow on a rotating sphere

“…Conceptually, our analysis supports the view that non-equilibrium approaches can provide analytical insights into the dynamics of planetary flows (Delplace et al 2017) and atmospheres (Marston 2011(Marston , 2012. Moreover, in view of the recent successful application of phenomenological GNS models to active fluids (Dunkel et al 2013;S lomka & Dunkel 2017b), the results of this study can also help advance the understanding of active matter propagation on curved surfaces (Sanchez et al 2012;Sknepnek & Henkes 2015;Zhang et al 2016;Henkes et al 2018;Nitschke et al 2019) and in rotating frames (Löwen 2019).…”

A numerical approach for fluid deformable surfaces

“…Equations (2.3) and (2.4) are independent of . For given , these equations have also been previously derived by various approaches (see Arroyo & DeSimone (2009) (with corrected acceleration term Yavari, Ozakin & Sadik 2016) and Koba, Liu & Giga (2017), Jankuhn, Olshanskii & Reusken (2018), Miura (2018), Nitschke et al (2019 a )). For , also (2.5) is the same as the equation derived in Koba et al (2017), Jankuhn et al (2018).…”

“…Most work, including also numerical analysis, is concerned with the Stokes limit on stationary surfaces V = 0 (see e.g. Olshanskii et al (2018), Reusken (2020) Nitschke et al (2019a). The Stokes limit of (2.3)-(2.5) or (2.7) and (2.8) corresponds to the classical model of Scriven (1960) and resamples, if coupled with bulk flow, with the Boussinesq-Scriven boundary condition in multiphase flow problems (see e.g.…”

“…To numerically solve (2.3), (2.4) and (2.9), we consider a semi-implicit Euler time stepping scheme, a Chorin-like projection approach for (2.3) and (2.4), similar to Reuther & Voigt (2018 b ), Nitschke et al (2019 a ), evolution of geometric quantities and the generic finite element approach proposed in Nestler, Nitschke & Voigt (2019). The latter is based on a reformulation of all operators and quantities in Cartesian coordinates and penalization of normal components.…”

Fluid deformable surfaces

“…Based on Onsager's variational formalism Torres-Sánchez, Millán & Arroyo (2019a) derive a first approximation to the dynamics of lipid membranes by combining bending and surface hydrodynamics. Omitting inertial effects, the derived equations correspond to the force and torque balance equations and the constitutive 2 A. Voigt laws postulated in Salbreux & Jülicher (2017) as well as the thin-film limit of the corresponding bulk model equations in Nitschke, Reuther & Voigt (2019). The major contribution of Torres-Sánchez et al (2019a) is not only a systematic and transparent way to derive the governing equations in the fully nonlinear regime, but also a computational framework for these equations which goes beyond previously studied axisymmetric settings (Arroyo & DeSimone 2009).…”