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- ItemThin film models for an active gel(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Kitavtsev, Georgy; Münch, Andreas; Wagner, BarbaraIn this study we present a free-boundary problem for an active liquid crystal based on the Beris-Edwards theory that uses a tensorial order parameter and includes active contributions to the stress tensor to analyse the rich defect structure observed in applications such as the Adenosinetriphosphate (ATP) driven motion of a thin film of an actin filament network. The small aspect ratio of the film geometry allows for an asymptotic approximation of the free-boundary problem in the limit of weak elasticity of the network and strong active terms. The new thin film model captures the defect dynamics in the bulk as well as wall defects and thus presents a significant extension of previous models based on the Leslie-Erickson-Parodi theory. Analytic expressions are derived that reveal the interplay of anchoring conditions, film thickness and active terms and their control of transitions of flow structure.
- ItemTwo-phase flow model for concentrated suspensions(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Ahnert, Tobias; Münch, Andreas; Wagner, BarbaraA new two-phase model is derived that make use of a constitutive law combining non-Brownian suspension with granular rheology, that was recently proposed by Boyer et al. [PRL, 107(18),188301 (2011)]. It is shown that for the simple channel flow geometry, the stress model naturally exhibits a Bingham type flow property with an unyielded finite-size zone in the center of the channel. As the volume fraction of the solid phase is increased, the various transitions in the flow fields are discussed using phase space methods for a boundary value problem, that is derived from the full model. The predictions of this analysis is then compared to the direct finite-element numerical solutions of the full model.
- ItemSurface induced phase separation of a swelling hydrogel(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Hennessy, Matthew G.; Münch, Andreas; Wagner, BarbaraWe present a formulation of the free boundary problem for a hydrogel that accounts for the interfacial free energy and finite strain due to the large deformation of the polymer network during solvent transport across the free boundary. For the geometry of an initially dry layer fixed at a rigid substrate, our model predicts a phase transition when a critical value of the solvent concentration has been reached near the free boundary. A one-dimensional case study shows that depending on the flux rate at the free boundary an initial saturation front is followed by spinodal decomposition of the hydrogel and the formation of an interfacial front that moves through the layer. Moreover, increasing the shear modulus of the elastic network delays or even suppresses phase separation.
- ItemSpin coating of an evaporating polymer solution(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Münch, Andreas; Please, Colin P.; Wagner, BarbaraWe consider a mathematical model of spin coating of a single polymer blended in a solvent. The model describes the one-dimensional development of the thin layer of the mixture as the layer thins due to flow created by a balance of viscous forces and centrifugal forces and due to evaporation of the solvent. In the model both the diffusivity of the solvent in the polymer and the viscosity of the mixture are very rapidly varying functions of the solvent volume fraction. Guided by numerical solutions an asymptotic analysis reveals a number of different possible behaviours of the thinning layer dependent on the nondimensional parameters describing the system. The main practical interest is in controlling the appearance and development of a ``skin'' on the polymer where the solvent concentration reduces rapidly on the outer surface leaving the bulk of the layer still with high concentrations of solvent. The critical parameters controlling this behaviour are found to be eps the ratio of the diffusion to advection time scales, delta the ratio of the evaporation to advection time scales and exp(-gamma), the ratio of the diffusivity of the initial mixture and the pure polymer. In particular, our analysis shows that for very small evaporation with delta ll exp(-3/(4gamma)) eps^3/4 skin formation can be prevented
- ItemThin-film models for viscoelastic liquid bi-layers(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Jachalski, Sebastian; Münch, Andreas; Wagner, BarbaraIn this work we consider a two-layer system of viscoelastic liquids of corotational Jeffreys type dewetting from a Newtonian liquid substrates. We derive conditions that allow for the first time the asymptotically consistent reduction of the free boundary problem for the two-layer system to a system of coupled thin-film equations that incorporate the full nonlinear viscoelastic rheology. We show that these conditions are controlled by the order of magnitude of the viscosity ratio of the liquid layers and their thickness ratio. For pure Newtonian flow, these conditions lead to a thin-film model that couples a layer with a parabolic flow field to a layer described by elongational flow. For this system we establish asymptotic regimes that relate the viscosity ratio to a corresponding apparent slip. We then use numerical simulations to discuss the characteristic morphological and dynamical properties of viscoelastic films of corotational Jeffreys type dewetting from a solid as well as liquid substrate.
- ItemImpact of interfacial slip on the stability of liquid two-layer polymer films(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Jachalski, Sebastian; Peschka, Dirk; Münch, Andreas; Wagner, BarbaraIn this study systems of coupled thin-film models for two immiscible liquid polymer layers on a solid substrate that account for interfacial slip and intermolecular forces are derived. On the scale of tens to hundred nanometers such two-layer systems are susceptable to instability and may rupture and dewet. The stability of the two-layer system and its significant dependence on the order of magnitude of slip is investigated via these thin-film models. With no-slip at both, the liquid-liquid and liquid-solid interface and polymer layers of comparable thickness, the dispersion relation typically shows two local maxima, one in the long-wave regime and the other at moderate wavenumbers. The former is associated with perturbations that mainly affect the gas-liquid interface and the latter with higher relative perturbation amplitudes at the liquid-liquid interface. Slip at the liquid-liquid interface generally favors the former perturbations. However, when the liquid-liquid and the liquidsolid interface exhibit large slip, the maxima shift to small wavenumbers for increasing slip and hence may significantly change the spinodal patterns.
- ItemAnisotropic surface energy formulations and their effect on stability of a growing thin film(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Korzec, Maciek D.; Münch, Andreas; Wagner, BarbaraIn this paper we revisit models for the description of the evolution of crystalline films with anisotropic surface energies. We prove equivalences of symmetry properties of anisotropic surface energy models commonly used in the literature. Then we systematically develop a framework for the derivation of surface diffusion models for the self-assembly of quantum dots during Stranski-Krastanov growth that include surface energies also with large anisotropy as well as the effect of wetting energy, elastic energy and a randomly perturbed atomic deposition flux. A linear stability analysis for the resulting sixth-order semilinear evolution equation for the thin film surface shows that that the new model allows for large anisotropy and gives rise to the formation of anisotropic quantum dots. The nonlinear three-dimensional evolution is investigated via numerical solutions. These suggest that increasing anisotropy stabilizes the faceted surfaces and may lead to a dramatic slow-down of the coarsening of the dots.
- ItemImpact of slippage on the morphology and stability of a dewetting rim(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Münch, Andreas; Wagner, BarbaraIn this study lubrication theory is used to describe the stability and morphology of the rim that forms as a thin polymer film dewets from a hydrophobized silicon wafer. Thin film equations are derived from the governing hydrodynamic equations for the polymer to enable the systematic mathematical and numerical analysis of the properties of the solutions for different regimes of slippage and for a range of time scales. Dewetting rates and the cross sectional profiles of the evolving rims are derived for these models and compared to experimental results. Experiments also show that the rim is typically unstable in the spanwise direction and develops thicker and thinner parts that may grow into ``fingers''. Linear stability analysis as well as nonlinear numerical solutions are presented to investigate shape and growth rate of the rim instability. It is demonstrated that the difference in morphology and the rate at which the instability develops can be directly attributed to the magnitude of slippage. Finally, a derivation is given for the dominant wavelength of the bulges along the unstable rim.
- ItemA thin film model for corotational Jeffreys fluids under strong slip(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Münch, Andreas; Wagner, B.; Rauscher, M.; Blossey, R.We derive a thin film model for viscoelastic liquids under strong slip which obey the stress tensor dynamics of corotational Jeffreys fluids.
- ItemIntermediate-asymptotic structure of a dewetting rim with strong slip(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Evans, Peter L.; King, John R.; Münch, AndreasWhen a thin viscous liquid film dewets, it typically forms a rim which spreads outwards, leaving behind a growing dry region. We consider the dewetting behaviour of a film, when there is strong slip at a liquid-substrate interface. The film can be modelled by two coupled partial differential equations (PDEs) describing the film thickness and velocity. Using asymptotic methods, we describe the structure of the rim as it evolves in time, and the rate of dewetting, in the limit of large slip lengths. An inner region emerges, closest to the dewetted region, where surface tension is important; in an outer region, three subregions develop. This asymptotic description is compared with numerical solutions of the full system of PDEs.