Hydrodynamic interactions in polar liquid crystals on evolving surfaces

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

A finite element approach for vector- and tensor-valued surface PDEs

We derive a Cartesian componentwise description of the covariant derivative of tangential tensor fields of any degree on general manifolds. This allows to reformulate any vector- and tensor-valued surface PDE in a form suitable to be solved by established tools for scalar-valued surface PDEs. We consider piecewise linear Lagrange surface finite elements on triangulated surfaces and validate the approach by a vector- and a tensor-valued surface Helmholtz problem on an ellipsoid. We experimentally show optimal (linear) order of convergence for these problems. The full functionality is demonstrated by solving a surface Landau-de Gennes problem on the Stanford bunny. All tools required to apply this approach to other vector- and tensor-valued surface PDEs are provided

Nematic liquid crystals on curved surfaces - a thin film limit

We consider a thin film limit of a Landau-de Gennes Q-tensor model. In the limiting process we observe a continuous transition where the normal and tangential parts of the Q-tensor decouple and various intrinsic and extrinsic contributions emerge. Main properties of the thin film model, like uniaxiality and parameter phase space, are preserved in the limiting process. For the derived surface Landau-de Gennes model, we consider an L2-gradient flow. The resulting tensor-valued surface partial differential equation is numerically solved to demonstrate realizations of the tight coupling of elastic and bulk free energy with geometric properties.Comment: 20 pages, 4 figure

A stable backbone for the fungi

Fungi are abundant in the biosphere. They have fascinated mankind as far as written history goes and have considerably influenced our culture. In biotechnology, cell biology, genetics, and life sciences in general fungi constitute relevant model organisms. Once the phylogenetic relationships of fungi are stably resolved individual results from fungal research can be combined into a holistic picture of biology. However, and despite recent progress, the backbone of the fungal phylogeny is not yet fully resolved. Especially the early evolutionary history of fungi and the order or below-order relationships within the ascomycetes remain uncertain. Here we present the first phylogenomic study for a eukaryotic kingdom that merges all publicly available fungal genomes and expressed sequence tags (EST) to build a data set comprising 128 genes and 146 taxa. The resulting tree provides a stable phylogenetic backbone for the fungi. Moreover, we present the first formal supertree based on 161 fungal taxa and 128 gene trees. The combined evidences from the trees support the deep-level stability of the fungal groups towards a comprehensive natural system of the fungi. They indicate that the classification of the fungi, especially their alliance with the Microsporidia, requires careful revision. Our analysis is also an inventory of present day sequence information for the fungi. It provides insights into which phylogenenetic conclusions can and which cannot be drawn from the current data and may serve as a guide to direct further sequencing initiatives. Together with a comprehensive animal phylogeny, we provide the second of three pillars to understand the evolution of the multicellular eukaryotic kingdoms, fungi, metazoa, and plants, in the past 1.6 billion years

Active nematodynamics on deformable surfaces

We consider active nematodynamics on deformable surfaces. Based on a thermodynamically consistent surface Beris-Edwards model we add nematic activity and focus on the emerging additional coupling mechanism between the nematic field, the flow field and the curved surface. We analyse the impact of the active nematic force at topological defects. Under the presence of curvature all defects become active and contribute not only tangential forces but also normal forces. This confirms the proposed role of topological defects in surface evolution and provides the basis for a dynamic description of morphogenetic processes

Beris-Edwards Models on Evolving Surfaces: A Lagrange-D'Alembert Approach

Using the Lagrange-D'Alembert principle we develop thermodynamically consistent surface Beris-Edwards models. These models couple viscous inextensible surface flow with a Landau-de Gennes-Helfrich energy and consider the simultaneous relaxation of the surface Q-tensor field and the surface, by taking hydrodynamics of the surface into account. We consider different formulations, a general model with three-dimensional surface Q-tensor dynamics and possible constraints incorporated by Lagrange multipliers and a surface conforming model with tangential anchoring of the surface Q-tensor field and possible additional constraints. In addition to different treatments of the surface Q-tensor, which introduces different coupling mechanisms with the geometric properties of the surface, we also consider different time derivatives to account for different physical interpretations of surface nematics. We relate the derived models to established models in simplified situations, compare the different formulations with respect to numerical realizations and mention potential applications in biology.Comment: 52 page

A numerical approach for fluid deformable surfaces

Fluid deformable surfaces show a solid-fluid duality which establishes a tight interplay between tangential flow and surface deformation. We derive the governing equations as a thin film limit and provide a general numerical approach for their solution. The simulation results demonstrate the rich dynamics resulting from this interplay, where in the presence of curvature any shape change is accompanied by a tangential flow and, vice versa, the surface deforms due to tangential flow. However, they also show that the only possible stable stationary state in the considered setting is a sphere with zero velocity

Properties of surface Landau-de Gennes Q-tensor models

Uniaxial nematic liquid crystals whose molecular orientation is subjected to a tangential anchoring on a curved surface offer a non trivial interplay between the geometry and the topology of the surface and the orientational degree of freedom. We consider a general thin film limit of a Landau-de Gennes Q-tensor model which retains the characteristics of the 3D model. From this, previously proposed surface models follow as special cases. We compare fundamental properties, such as alignment of the orientational degrees of freedom with principle curvature lines, order parameter symmetry and phase transition type for these models, and suggest experiments to identify proper model assumptions