Lectures
K. R. Rajagopal
A thermodynamical framework for chemically reacting systems
In this talk I present a very general thermodynamic framework that is capable of describing a large class of bodies undergoing entropy producing processes. Attention will be focused on bodies that are undergoing chemical reactions and the framework takes stoichiometry into account. As a special sub-case, we describe the response of viscoelastic materials that undergo chemical reactions. One of the quintessential features of this framework is that the second law of thermodynamics is formulated by introducing the Gibbs potential, which is the natural way to study problems involving chemical reactions. The Gibbs potential based formulation also naturally leads to implicit constitutive equations for the stress tensor. The assumption of maximization of the rate of entropy production due to dissipation, heat conduction, and chemical reactions is invoked to determine an equation for the evolution of the natural configuration, the heat flux vector and a new set of equations for the evolution of the concentration of the chemical constituents. To determine the efficacy of the framework with regard to chemical reactions we consider the reactions occurring during vulcanization of rubber. The theoretically predicted distribution of mono,di and polysulfidic cross-links by using the framework agree reasonably well with available experimental data.
Warning: Professor Rajagopal will deliver the lecture on Monday, April 2, 2012 in Nečas Seminar on Continuum Mechanics. His lecture is not scheduled as a part of the workshop on Saturday, March 31, 2012.
Miroslav Bulíček
Regularity for systems of PDEs arising in continuum thermodynamics
For linear systems of PDEs, the usual result that can be established is the so-called maximal regularity, which means that the solution is smooth provided that the data are smooth. On the other hand if we switch to the nonlinear word of PDEs no such result is true in general and one needs to discuss carefully the structure of possible nonlinearities. In particular, in continuum mechanics/thermodynamics, the resulting system of PDEs may be "very far" from being linear and therefore the key-role in the regularity theory plays the structural assumptions on the nonlinear terms. Furthermore, to prove at least certain regularity of a solution is essential to justify rigorously the derivation of a model (balance equations) and also to prove the convergence of a numerical approximation to a solution. We give a short overview of available results (mainly for flows of incompressible fluids), the challenging open problems and discuss how one can overcome the difficulties coming from the presence of certain nonlinearities, namely: a) convective term; b) nonlinear relation between the Cauchy stress and the velocity gradient; c) presence of the critical term in the heat equation.
Jens Frehse
The Prandtl--Reuss law for mixtures
We consider the analogue of the Prandtl Reuss law for a mixture of a hard and soft material. This model was developed by Rajagopal and his co-authors. The purpose of this model is to try to have a physical explanation of elastic plastic deformation with hardening. The role of the internal parameters is replaced by the partial stress given by the hard material.
We prove that the partial strains and stresses have the same degree of regularity in Lebesgue-,Sobolev-, and Besov-spaces which are known in the classical case.
Jaroslav Hron
On numerical solution of implicitly constituted fluid flow
In this contribution we investigate the advantages and problems arising in numerical solution of flows when the material constitutive relation is given in implicit form. The differences to standard finite element discretization of Navier-Stokes equations will be discussed with respect to the selection of the mixed finite element spaces, combined with the efficient solution of the resulting algebraic nonlinear and linear systems. Some effects are demonstrated on a simple example of Bingham-type fluid.
Jan Kratochvíl
Crystal plasticity treated as a quasi-static material flow through adjustable crystal lattice
Considering high pressure torsion experiments as a motivation, plastic behavior of crystalline solids is treated as a highly viscous material flow through an adjustable crystal lattice. Instead of the traditional decomposition rule considering the deformation gradient as a product of the elastic and plastic parts, the proposed model is based on its rate form: the velocity gradient consists of the lattice velocity gradient and the sum of the velocity gradients corresponding to the slip rates of individual slip systems; the slips themselves are not defined in the model. The geometrical changes caused by material flow and the slips can be specified a posteriori. The lattice space is treated as a solid, its distortions are measured with respect to a lattice reference configuration. In a rigid plastic approximation adopted the lattice distortions are reduced to rotations. Constitutive equations incorporate non-local hardening caused by close range interactions among dislocations.
Martin Lanzendörfer
Incompressible piezoviscous fluids: first steps, a long ways to go
In certain situations (hydrodynamic lubrication flow being one example) a fluid may be considered incompressible while its viscosity grows rapidly with pressure. The talk will survey recent results on the mathematical description of the flow, both concerning the constitution of the governing PDEs and their well-posedness, marking some interesting differences from other fluid models. We will present numerical simulations of the planar steady flow, motivated by the lubrication problems, noting some specific features. Finally, we will document that both the theoretical and numerical results available have certain limitations, leaving the mathematical basement of some important engineering approaches questionable.
František Maršík
Thermodynamic stability condition -- Couette flow application
Equations of conservation of mass, momentum and total energy in the classical continuum mechanics of fluids and solids are usually formulated as the balance laws of the corresponding quantities. II. Law of thermodynamics is understood as the balance of entropy. The direct correlation between classical mechanics of material points (Lagrange principle of classical mechanics) and classical continuum mechanics can be established when the existence of a trajectory and a friction force are added. The conservation of the total enthalpy for fluid systems is derived and consequently with the analogy with the balance of total energy in continuum mechanics, the thermodynamic stability conditions are established. The important role of the total enthalpy follows from the variational analysis. Moreover, the thermodynamic criterion of stability based on the total enthalpy is formulated and the comparison with the Rayleigh theory is shown. This theory was verified for all modifications of the Couette flow, even for a solid body rotation, where the Rayleigh condition fails. It was shown that the solid body vortex is at the margin of the stability, which is experimentally observed. Analogously, the potential vortex is by thermodynamic criterion stable, however by Rayleigh criteria is on the onset of stability.
Milan Pokorný
On steady compressible Navier--Stokes--Fourier system
This lecture is meant as an overview of the results concerning the system of partial differential equations describing steady flow of Newtonian compressible heat conducting fluid obtained in the last five years together with Piotr B. Mucha, Antonín Novotný, Šárka Nečasová and Ondřej Kreml. More precisely, we study $$ \begin{array}{c} \mbox{div}\, (\varrho \mathbf{u}) = 0, \\ \mbox{div}\, (\varrho \mathbf{u} \otimes \mathbf{u}) - \mbox{div}\, \mathbf{S} + \nabla p = \varrho \mathbf{f}, \\ \mbox{div}\, (\varrho E \mathbf{u}) = \varrho \mathbf{f} \cdot \mathbf{u} - \mbox{div}\, (p \mathbf{u}) + \mbox{div}\, (\mathbf{S} \mathbf{u}) -\mbox{div}\, \mathbf{q} \end{array} $$ with $\varrho$ the density, $\mathbf{u}$ the velocity field, $\mathbf{S}$ the stress tensor, $p$ the pressure, $\mathbf{f}$ the given volume force, $\mathbf{q}$ the heat flux and the total energy $E= \frac 12 |\mathbf{u}|^2 + e$ with $e$ the internal energy. We consider the pressure law of the form $p(\varrho, \vartheta) \sim \varrho^{\gamma} + \varrho \vartheta$. In dependence on the value of $\gamma>1$, the form of the viscosity coefficients (temperature dependent or not) and the boundary conditions for temperature and velocity we show the existence of a solution to this system. We get existence of a weak solution even for the formulation in the internal energy for $\gamma$ large, in other cases only existence of a weak solution for the formulation in the total energy and for $\gamma$ small sometimes only existence of a variational entropy solution, i.e. we replace the weak formulation of total energy balance by the entropy inequality and by the global total energy balance. The solutions to the system are constructed for arbitrarily large sufficiently regular data.
Dalibor Pražák
Simple mechanical oscillators with implicit constitutive relations
Mechanical oscillations of a point mass $m$ attached to some general "material" are described by $mx'' + F_v = F(t)$, where $x=x(t)$ is the unknown displacement, $F=F(t)$ is given external force and $F_v$ is the internal force of the studied material. Usually one can express $F_v$ explicitly in terms of $x$ and $x'$, thus closing the system to obtain a standard 2nd order ODE. In this talk, we will discuss several physically meaningful examples (Coulomb friction, most notably) where $F_v$ is only implicitly related to $x$, $x'$. We will focus on the mathematical analysis of the resulting system (existence, uniqueness, stability).
Vít Průša
Implicitly constituted materials with fading memory
The notion of simple fluid, thus a fluid given by the constitutive relation in the form $$ \mathbb{T} = -p\mathbb{I} + \mathfrak{F}_{s=0}^{+\infty} (\mathbb{C}_t(t-s) - \mathbb{I}), $$ where $\mathbb{T}$ is the Cauchy stress tensor, $\mathbb{C}$ is the relative right Cauchy--Green tensor, $p$ is the "pressure" and $\mathfrak{F}$ is a functional acting on the history of the relative Cauchy--Green tensor, has played a fundamental role in the development of the theory of constitutive relations. Although many fluid models, in particular the so called differential type models, are special instances of this general constitutive relation, the constitutive relation for simple fluid is not a sufficiently general form of a constitutive relation for fluids. For example, some viscoelastic fluid models (in particular some more complicated rate type models) do not fit into the framework of simple fluid.
By appealing to a variant of the well-known approximation theorem for "slow" histories by Colleman and Noll (1960), it will be shown that both the rate type models and differential type models can be obtained as special instances of the implicit constitutive relation in the form $$ \mathfrak{G}_{s=0}^{+\infty}(\mathbb{T}(t-s), \mathbb{C}_t(t-s) - \mathbb{I}) = \mathbb{0}. $$ The procedure of approximating the behaviour of the functional above for "slow" histories leads to a hierarchy of implicit models of decreasing complexity, and can be seen as an example of an effective model reduction procedure.
Eduard Rohan
Modeling double porosity media using hierarchical homogenization
Models of fluid saturated porous media (FSPM) are widely used in geomechanics, civil engineering and biomechanics; in the last application, FSPM models can approximate bone mechanics, or tissue perfusion, to name a few examples. Asymptotic analysis of PDEs with strongly oscillating coefficients forms a mathematically sound basis for modeling complicated interactions in heterogeneous materials with respect to their microstructure. Assuming scale separation, this approach can be adapted for simultaneous modeling of materials on the micro-, meso- and macroscopic scales. In the lecture, various models of FSPM will be presented which were obtained using hierarchical homogenization, or using homogenization of PDEs with scale-dependent coefficients. Also different origins of the fading memory effects observed at the macroscopic scale will be discussed.
Tomáš Roubíček
Modelling of phase transformations in magnetostrictive materials like NiMnGa
Magnetostrictive materials such as NiMnGa exhibit mutually coupled martensitic and ferro/paramagnetic phase transformations. Moreover, such materials are electrically conductive, which leads to other coupling through induced magnetic field and produced Joule heat. After presentation of these phenomena, the thermodynamically-consistent model in terms of small-strain and eddy-current approximations will be formulated. Existence of weak solutions to such a coupled system of momentum equilibrium, Landau-Lifshitz-Gilbert equation, heat equation, and parabolic Maxwell system will be proved by limit passage in a carefully designed semi-implicit regularized scheme.
Ondřej Souček
A constitutive model for non-reacting binary mixtures
We present a derivation of a constitutive model for non-reacting binary mixtures with components of the same temperature. We show that employing carefully the balance laws of mass and momenta for the constituents and the mixture as a whole yields certain a-priori information which immediately implies natural "minimal" constitutive ansatz for the free energy, entropy and the rate of entropy production. The identification of the constitutive response via the procedure of maximization of the rate of entropy production then allows for comparison with the traditionally considered models and recovers all the sought interaction phenomena in terms of the resulting interaction forces, mixture stress tensor and entropy and heat fluxes.
Zdeněk Strakoš
Challenges in analysis of algebraic iterative solvers
The current state-of-the art of iterative solvers is the outcome of the tremendous algorithmic development over the last few decades and of investigations of how to solve given problems. In this contribution we focus more on the dual question why things do or do not work. In particular, we will pose and discuss open questions such as what does the spectral information tell us about the behaviour of iterative methods in general and Krylov subspace methods in particular, whether it is useful to view Krylov subspace methods as matching moments model reduction, and whether the algebraic errors can be easily included into locally efficient and fully computable a-posteriori error bounds for adaptive PDE solvers.