Narrow Results

This paper is concerned with the circumstances under which the dissipative character of a one-dimensional scalar conservation law may be described by a formalism strictly analogous to that arising naturally in the dynamics of nonlinearly elastic materials. It is shown that this occurs if and only if the entropy density, entropy flux pair associated with the conservation law takes a particular form. We compare the admissibility condition associated with this special entropy with other admissibility criteria such as those of Lax, Oleinik and regularization theory. Using the special entropy, we consider the Riemann problem for an example in which genuine nonlinearity fails and a kinetic relation is needed to determine a unique solution.

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.

The authors consider the Euler equations for a compressible fluid in one space dimension when the equation of state of the fluid does not fulfill standard convexity assumption and viscosity and capillarity effects are taken into account. A typical example of nonconvex constitutive equation for fluids is Van der Waals' equation. The first order terms of these partial differential equations form a nonlinear system of mixed (hyperbolic -elliptic) type. For a class of nonconvex equations of state, an existence theorem of traveling waves solutions with arbitrary large amplitude is established here. The autors distinguish between classical (compressive) and nonclassical (undercompressive) traveling waves. The latter do not fulfill Lax shock inequalities, and are characterized by the so-called kinetic relation, whose properties are investigated in this paper.

The motion of individual twin boundaries in ferromagnetic Ni–Mn–Ga alloys is the basic mechanism by which large reversible strains are generated. The resulting mechanical response makes these alloys potential candidates for magneto-mechanical actuation applications. In this work, we characterize the relations between the magnetically induced strain response of Ni–Mn–Ga single crystals and their internal twin boundary arrangements and kinetics. The macroscopic response is measured under constant magnetic fields, while the kinetics of individual twin boundaries is measured by the pulsed magnetic field method. By testing two different crystals with different twinning microstructures, we show that the main characteristics of the macroscopic response can be controlled by the number of mobile twin boundaries. In addition, we show that the magnitudes of controlling kinetic parameters of different twin boundaries vary with their location within the sample.