October 6-7, 2011
We study a coupled system of Navier-Stokes/Mullins-Sekerka type, which describes the two-phase flow of two immiscible, viscous, incompressible fluids with same densities in a bounded domain. In contrast to classical sharp interface models for such flows, diffusion of mass particles is taken into account for the evolution of the interface. The system is motivated by a formal asymptotic limit of a diffuse interface model of Navier-Stokes/Cahn-Hilliard type. In this presentation we will discuss the existence of strong solutions locally in time. Since the system is decoupled in highest order, the coupled system can be treated as a perturbation of the classical Mullins-Sekerka equation. Finally, we will discuss dynamic stability of spheres.
TBA
We show the existence of weak solutions for a diffuse interface model for incompressible two phase flows with different densities. The model was derived recently by Abels, Garcke and Grün and is the first one for different densities which allows for an energy estimate, so that we speak of a thermodynamical consistent model. We give a straightforward definition of weak solutions and show existence for all times. To this end, we introduce an implicit time discretization, which respects the energy estimate. With the help of a result from Abels and Wilke about the subgradient of a certain energy we can use the Leray-Schauder fix point argument to get discrete solutions. Finally we use different compactness arguments to get convergence in time.
We present phase-field models for electrowetting with electrolyte solutions for the case that the liquids involved have different mass densities. The model has the following features: solenoidal velocity field, arbitrary number ionic species, Henry's law respected for ion densities, contact-angle hysteresis respected, individual masses conserved, Dirichlet boundary conditions for the electrostatic potential.
We consider the Prandtl-Reuss problem modelling elastic-perfect-plastic deformation with loading and related hardening problems. We prove that the stress velocities and internal variables have fractional derivatives in time direction of order 1/2 in the sense of a certain Besov space. In the case of kinematic hardening we succeed to have the analogue result also in space direction, this includes the strain velocities. This seems to be the first "regularity" result for the velocities of stress and strain. A Prandtl-Reuss-model for mixtures of elastic-plastic materials due to Kratovil-Malek-Rajagopal-Srinivasa is discussed where the known Prandtl-Reuss-regularity-results continue to hold.
A new diffuse interface model for a two-phase flow of two incompressible fluids with different densities is introduced using methods from rational continuum mechanics. The model fulfills local and global dissipation inequalities and is frame indifferent. Moreover, it is generalized to situations with a soluble species. Using the method of matched asymptotic expansions we derive various sharp interface models in the limit when the interfacial thickness tends to zero. Depending on the scaling of the mobility in the diffusion equation we either derive classical sharp interface models or models where bulk or surface diffusion is possible in the limit. In the latter case a new term resulting from surface diffusion appears in the momentum balance at the interface. Finally, we show that all sharp interface models fulfill natural energy inequalities.
We prove existence of weak solutions to a novel two-phase model for various electrokinetic phenomena, including in particular dynamic electrowetting with electrolytes. The model is thermodynamically consistent and allows for contact angle hysteresis. It combines Navier-Stokes and Cahn-Hilliard type-phase-field equations with Nernst-Planck equations for ion density-evolution and with an elliptic transmission problem for the electrostatic potential. An interesting feature is the strong coupling between the equations for electrostatic potential and for phase-field and ion densities -- the field intensity enters the chemical potential quadratically. As ion densities are a priori only bounded in $L\log{L}$-Orlicz-classes, a new iteration method is proposed to establish higher regularity and integrability of these quantities. This is joint work with M. Fontelos (Madrid) and S. Jörres (Erlangen).
We study multiphase and multifluid systems using the assumption of maximum rate of entropy production by Rajagopal and Srinivasa. We will show that commonly used models such as the Cahn-Hilliard, Allen-Cahn and Korteweg models are but special cases within this general framework. Based on the global rate of entropy production, a generalization of Rajagopal's and Srinivasa's method will help us to derive thermodynamically consistent boundary conditions. We will see that this new ansatz is thermodynamically consistent, transparent, effective and easily extendable.
We present a model for liquid-solid phase transitions in substances, where the specific volume in the solid phase is higher than in the liquid phase (e.g., water and ice), taking into account both mechanical and thermal effects. Our goal is to explain the occurrence of high stresses during the solidification process, and their influence on the deformation of the container. Existence, uniqueness, and asymptotic convergence to equilibria are established for several different material properties of the boundary (joint work with Elisabetta Rocca and Juergen Sprekels).
TBA
We will discuss our work in progress focused on the analysis of some approaches used for description of non-interacting binary mixtures. We will show that using detailed balance equations for the components of the mixture, one can obtain restrictions on the form of the internal energy, entropy and entropy production for the whole mixture, and seamlessly derive generalizations of the classical relations, namely of the Fick's law of diffusion. Such a generalization is based on a well justified evolution equation for the diffusive flux that, close to the equilibrium, reduces to the classical Fick's law.
Our DT-MRI visualization algorithm based on anisotropic texture diffusion relies on the problem for the Allen-Cahn equation with a space-dependent anisotropic diffusion operator. To preserve its anisotropic properties in the discretized version of the problem, an appropriate numerical treatment is necessary, reducing the isotropic numerical diffusion. The first part of this contribution is concerned with the design and investigation of the finite volume scheme with multipoint flux approximation. Its desirable properties are investigated by means of our technique based on total variation measurement. The second part presents the recent achievements in applying the same scheme to the phase field model of dendritic crystal growth.
We consider diffusive interface models describing flows which undergo a phase change. The equations consist of the compressible Navier-Stokes system coupled with an Allen-Cahn equation (phase field equation), and are based on an energetic variational formulation. We concentrate on a model where a jump in the mass density on the free boundary in the sharp interface limit arise. Here, the central point is the evaluation of the transition profiles in the interface region connecting the both bulk regions. In this talk we show the existence of such transition profiles for a special class of given double well potentials $W$.