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- ItemSignatures of slip in dewetting polymer films(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Peschka, Dirk; Haefner, Sabrina; Jacobs, Karin; Münch, Andreas; Wagner, BarbaraThin liquid polymer films on hydrophobic substrates are susceptable to rupture and formation of holes, which in turn initiate a complex dewetting process that eventually evolves into characteristic stationary droplet patterns. Experimental and theoretical studies suggest that the specific type of droplet pattern largely depends on the nature of the polymer-substrate boundary condition. To follow the morphological evolution numerically over long time scales and for the multiple length scales involved has so far been a major challenge. In this study a highly adaptive finite-element based numerical scheme is presented that allows for large-scale simulations to follow the evolution of the dewetting process deep into the nonlinear regime of the model equations, capturing the complex dynamics including shedding of droplets. In addition, the numerical results predict the previouly unknown shedding of satellite droplets during the destabilisation of liquid ridges, that form during the late stages of the dewetting process. While the formation of satellite droplets is well-known in the context of elongating fluid filaments and jets, we show here that for dewetting liquid ridges this property can be dramatically altered by the interfacial condition between polymer and substrate, namely slip.
- ItemAn active poroelastic model for mechanochemical patterns in protoplasmic droplets of Physarum polycephalum(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Radszuweit, Markus; Engel, Harald; Bär, MarkusMotivated by recent experimental studies, we derive and analyze a two-dimensional model for the contraction patterns observed in protoplasmic droplets of Physarum polycephalum. The model couples a description of an active poroelastic two-phase medium with equations describing the spatiotemporal dynamics of the intracellular free calcium concentration. The poroelastic medium is assumed to consist of an active viscoelastic solid representing the cytoskeleton and a viscous fluid describing the cytosol. The equations for the poroelastic medium are obtained from continuum force balance and include the relevant mechanical fields and an incompressibility condition for the two-phase medium. The reaction-diffusion equations for the calcium dynamics in the protoplasm of Physarum are extended by advective transport due to the flow of the cytosol generated by mechanical stress. Moreover, we assume that the active tension in the solid cytoskeleton is regulated by the calcium concentration in the fluid phase at the same location, which introduces a mechanochemical coupling. A linear stability analysis of the homogeneous state without deformation and cytosolic flows exhibits an oscillatory Turing instability for a large enough mechanochemical coupling strength. Numerical simulations of the model equations reproduce a large variety of wave patterns, including traveling and standing waves, turbulent patterns, rotating spirals and antiphase oscillations in line with experimental observations of contraction patterns in the protoplasmic droplets.
- ItemCoherence-incoherence patterns in a ring of non-locally coupled phase oscillators(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Omel'chenko, OlehWe consider a paradigmatic spatially extended model of non-locally coupled phase oscillators which are uniformly distributed within a one-dimensional interval and interact depending on the distance between their sites modulo periodic boundary conditions. This model can display peculiar spatio-temporal patterns consisting of alternating patches with synchronized (coherent) or irregular (incoherent) oscillator dynamics, hence the name coherence-incoherence pattern, or chimera state. For such patterns we formulate a general bifurcation analysis scheme based on a hierarchy of continuum limit equations. This gives us possibility to classify known coherence-incoherence patterns and to suggest directions for searching new ones
- ItemDynamic formation of oriented patches in chondrocyte cell cultures(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Grote, Marcus; Palumberi, Viviana; Wagner, Barbara; Barbero, Andrea; Martin, IvanGrowth factors have a significant impact not only on the growth dynamics but also on the phenotype of chondrocytes (Barbero et al. , J. Cell. Phys. 204, pp. 830-838, 2005). In particular, as chondrocyte populations approach confluence, the cells tend to align and form coherent patches. Starting from a mathematical model for fibroblast populations at equilibrium (Mogilner et al., Physica D 89, pp. 346-367, 1996), a dynamic continuum model with logistic growth is developed. Both linear stability analysis and numerical solutions of the time-dependent nonlinear integro-partial differential equation are used to identify the key parameters that lead to pattern formation in the model. The numerical results are compared quantitatively to experimental data by extracting statistical information on orientation, density and patch size through Gabor filters.
- ItemThe mathematics behind chimera states(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Omelchenko, Oleh E.Chimera states are self-organized spatiotemporal patterns of coexisting coherence and incoherence. We give an overview of the main mathematical methods used in studies of chimera states, focusing on chimera states in spatially extended coupled oscillator systems. We discuss the continuum limit approach to these states, Ott-Antonsen manifold reduction, finite size chimera states, control of chimera states and the influence of system design on the type of chimera state that is observed.