A general existence result for isothermal two-phase flows with phase transition

Liquid–vapor flows with phase transitions have a wide range of applications. Isothermal two-phase flows described by a single set of isothermal Euler equations, where the mass transfer is modeled by a kinetic relation, have been investigated analytically in [M. Hantke, W. Dreyer and G. Warnecke, Exact solutions to the Riemann problem for compressible isothermal Euler equations for two-phase flows with and without phase transition, Quart. Appl. Math. 71(3) (2013) 509–540]. This work was restricted to liquid water and its vapor modeled by linear equations of state. The focus of this work lies on the generalization of the primary results to arbitrary substances, arbitrary equations of state and thus a more general kinetic relation. We prove existence and uniqueness results for Riemann problems. In particular, nucleation and cavitation are discussed.

Tensional Relation

This chapter argues that the kinetic relation between different aesthetic forms is defined by a “relational image.” Relation, however, is not a distinct autonomous flow that simply occurs between contrasting aesthetic forms. Kinesthetic relations permeate, define, and order the forms themselves. Abstract and concrete forms rise to dominance in antiquity, but the order and relation of these forms to one another only becomes the primary focus of art during the medieval and early modern periods. This chapter offers first a preliminary and more general definition of kinetic relation, which is then historically developed in the next two chapters. In short, kinesthetic relations or linkages keep distinct aesthetic forms or fields of images both together and apart—distinct, contrasted, and yet moving together in ordered correlation. Relation is present in all works of art, but during the medieval and early modern period, relation becomes one of the most primary and constitutive features of the historical aesthetic field. The goal of this chapter is to prepare a description of the conceptual and kinetic features that define this period of the image: tensional motion, illumination, and contrast.

A gradient system with a wiggly energy and relaxed EDP-convergence

For gradient systems depending on a microstructure, it is desirable to derive a macroscopic gradient structure describing the effective behavior of the microscopic scale on the macroscopic evolution. We introduce a notion of evolutionary Gamma-convergence that relates the microscopic energy and the microscopic dissipation potential with their macroscopic limits via Gamma-convergence. This new notion generalizes the concept of EDP-convergence, which was introduced in [26], and is now called relaxed EDP-convergence. Both notions are based on De Giorgi’s energy-dissipation principle (EDP), however the special structure of the dissipation functional in terms of the primal and dual dissipation potential is, in general, not preserved under Gamma-convergence. By using suitable tiltings we study the kinetic relation directly and, thus, are able to derive a unique macroscopic dissipation potential. The wiggly-energy model of Abeyaratne-Chu-James (1996) serves as a prototypical example where this nontrivial limit passage can be fully analyzed.

Pinning of interfaces in a random elastic medium and logarithmic lattice embeddings in percolation

For a model of a driven interface in an elastic medium with random obstacles we prove the existence of a stationary positive supersolution at non-vanishing driving force. This shows the emergence of a rate-independent hysteresis through the interaction of the interface with the obstacles despite a linear (force = velocity) microscopic kinetic relation. We also prove a percolation result, namely, the possibility to embed the graph of an only logarithmically growing function in a next-nearest neighbour site percolation cluster at a non-trivial percolation threshold.

Numerical methods with controlled dissipation for small-scale dependent shocks

We provide a ‘user guide’ to the literature of the past twenty years concerning the modelling and approximation of discontinuous solutions to nonlinear hyperbolic systems that admitsmall-scale dependentshock waves. We cover several classes of problems and solutions: nonclassical undercompressive shocks, hyperbolic systems in nonconservative form, and boundary layer problems. We review the relevant models arising in continuum physics and describe the numerical methods that have been proposed to capture small-scale dependent solutions. In agreement with general well-posedness theory, small-scale dependent solutions are characterized by akinetic relation, a family of paths, or anadmissible boundary set. We provide a review of numerical methods (front-tracking schemes, finite difference schemes, finite volume schemes), which, at the discrete level, reproduce the effect of the physically meaningful dissipation mechanisms of interest in the applications. An essential role is played by theequivalent equationassociated with discrete schemes, which is found to be relevant even for solutions containing shock waves.

Why many theories of shock waves are necessary: kinetic relations for non-conservative systems

For a class of non-conservative hyperbolic systems of partial differential equations endowed with a strictly convex mathematical entropy, we formulate the initial-value problem by supplementing the equations with a kinetic relation prescribing the rate of entropy dissipation across shock waves. Our condition can be regarded as a generalization to non-conservative systems of a similar concept introduced by Abeyaratne, Knowles and Truskinovsky for subsonic phase transitions and by LeFloch for non-classical undercompressive shocks to nonlinear hyperbolic systems. The proposed kinetic relation for non-conservative systems turns out to be equivalent, for the class of systems under consideration at least, to Dal Maso, LeFloch and Murat's definition based on a prescribed family of Lipschitz continuous paths. In agreement with previous theories, the kinetic relation should be derived from a phase-plane analysis of travelling-wave solutions associated with an augmented version of the non-conservative system. We illustrate with several examples that non-conservative systems arising in the applications fit in our framework, and for a typical model of turbulent fluid dynamics we provide a detailed analysis of the existence and properties of travelling waves which yields the corresponding kinetic function.

Role of the Kinetic Relation in a Phase Transition-Based Model for Mechanical Unfolding in Macromolecules

We present applications of a model developed to describe unfolding in macromolecules under an axial force. We show how different experimentally observed force-extension behaviors can be reproduced within a common theoretical framework. We propose that the unfolding occurs via the motion of a folded/unfolded interface along the length of the molecule. The molecules are modeled as one-dimensional continua capable of existing in two metastable states under an applied tension. The interface separates these two metastable states and represents a jump in stretch, which is related to applied force by the worm-like-chain relation. The mechanics of the interface are governed by the Abeyaratne-Knowles theory of phase transitions. The thermodynamic driving force controls the motion of the interface via an equation called the kinetic relation. By choosing an appropriate kinetic relation for the unfolding conditions and the macro-molecule under consideration, we have been able to generate a variety of unfolding processes in macromolecules.