Helmut Abels

THERMODYNAMICALLY CONSISTENT, FRAME INDIFFERENT DIFFUSE INTERFACE MODELS FOR INCOMPRESSIBLE TWO-PHASE FLOWS WITH DIFFERENT DENSITIES

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.

On Generalized Solutions of Two-Phase Flows for Viscous Incompressible Fluids

We discuss the existence of generalized solutions of the flow of two immiscible, incompressible, viscous Newtonian and non-Newtonian fluids with and without surface tension in a domain R d , d = 2, 3. In the case without surface tension, the existence of weak solutions is shown, but little is known about the interface between both fluids. If surface tension is present, the energy estimate gives an a priori bound on the (d 1)-dimensional Hausdorff measure of the interface, but the existence of weak solutions is open. This might be due to possible oscillation and concentration effects of the interface related to instabilities of the interface as for example fingering, emulsification or just cancellation of area, when two parts of the interface meet. Nevertheless we will show the existence of so-called measure-valued varifold solutions, where the interface is modeled by an oriented general varifold V(t) which is a non-negative measure on ◊ S d 1 , where S d 1 is the unit sphere in R d . Moreover, it is shown that measure-valued varifold solutions are weak solutions if an energy equality

Existence of weak solutions for a non-classical sharp interface model for a two-phase flow of viscous, incompressible fluids

Abstract We introduce a new sharp interface model for the flow of two immiscible, viscous, incompressible fluids. In contrast to classical models for two-phase flows we prescribe an evolution law for the interfaces that takes diffusional effects into account. This leads to a coupled system of Navier–Stokes and Mullins–Sekerka type parts that coincides with the asymptotic limit of a diffuse interface model. We prove the long-time existence of weak solutions, which is an open problem for the classical two-phase model. We show that the phase interfaces have in almost all points a generalized mean curvature.

On an incompressible Navier-Stokes/Cahn-Hilliard system with degenerate mobility

Abstract We prove existence of weak solutions for a diffuse interface model for the flow of two viscous incompressible Newtonian fluids in a bounded domain by allowing for a degenerate mobility. The model has been developed by Abels, Garcke and Grun for fluids with different densities and leads to a solenoidal velocity field. It is given by a non-homogeneous Navier–Stokes system with a modified convective term coupled to a Cahn–Hilliard system, such that an energy estimate is fulfilled which follows from the fact that the model is thermodynamically consistent.

Well-Posedness of a Fully Coupled Navier--Stokes/Q-tensor System with Inhomogeneous Boundary Data

We prove short-time well-posedness and existence of global weak solutions of the Beris--Edwards model for nematic liquid crystals in the case of a bounded domain with inhomogeneous mixed Dirichlet and Neumann boundary conditions. The system consists of the Navier--Stokes equations coupled with an evolution equation for the

Strong well-posedness of a diffuse interface model for a viscous, quasi-incompressible two-phase flow

Q

On the trace space of a Sobolev space with a radial weight

-tensor. The solutions possess higher regularity in time of order one compared to the class of weak solutions with finite energy. This regularity is enough to obtain Lipschitz continuity of the nonlinear terms in the corresponding function spaces. Therefore the well-posedness is shown with the aid of the contraction mapping principle using that the linearized system is an isomorphism between the associated function spaces.

Large Time Existence for Thin Vibrating Plates

In this article we discuss the existence of strong solutions locally in time for a model of a binary mixture of viscous incompressible fluids in a bounded domain. The model was derived by Lowengrub and Truskinovski. It is used to describe a diffuse interface model for a two-phase flow of two viscous incompressible Newtonian fluids with different densities. The fluids are macroscopically immiscible but partially mix in a small interfacial region. The model leads to a system of Navier–Stokes/Cahn–Hilliard type. Using a suitable result on maximal

Pseudodifferential Boundary Value Problems with Non-Smooth Coefficients

L^2

On sharp interface limits for diffuse interface models for two-phase flows

-regularity for the linearized system, the existence of strong solutions is shown with the aid of the contraction mapping principle. The analysis shows that in the case of different densities the system is coupled in highest order and the principal part of the linearized system is of very different structure compared to the case of same densities. The linear system is solved with the aid of a general result on an abstract damped wave equation by Chen and Triggiani.