2023 | Book

# The Riemann Problem in Continuum Physics

Authors: Philippe G. LeFloch, Mai Duc Thanh

Publisher: Springer International Publishing

Book Series : Applied Mathematical Sciences

2023 | Book

Authors: Philippe G. LeFloch, Mai Duc Thanh

Publisher: Springer International Publishing

Book Series : Applied Mathematical Sciences

This monograph provides a comprehensive study of the Riemann problem for systems of conservation laws arising in continuum physics. It presents the state-of-the-art on the dynamics of compressible fluids and mixtures that undergo phase changes, while remaining accessible to applied mathematicians and engineers interested in shock waves, phase boundary propagation, and nozzle flows. A large selection of nonlinear hyperbolic systems is treated here, including the Saint-Venant, van der Waals, and Baer-Nunziato models. A central theme is the role of the kinetic relation for the selection of under-compressible interfaces in complex fluid flows. This book is recommended to graduate students and researchers who seek new mathematical perspectives on shock waves and phase dynamics.

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Abstract

In the realm of physical phenomena, one often encounters events such as gas pipe explosions, or the potential rupture of a dam in a hydroelectric plant. The common scenario is the breakdown of the fluid flow or material flow under pressure. Our interest lies in the mathematical analysis of such phenomena, which naturally leads us to the study of the Riemann problem in continuum physics. We distinguish between several class of nonlinear waves (shock wave, rarefaction wave, moving interface) and we introduce, and thoroughly investigate, the notion of a kinetic relation for the selection of undercompressive shock or interfaces.

Abstract

We begin by introducing a list of models that describe complex flows in continuum physics, with the material being a gas, liquid, or solid. The governing equations of fluid flows are stated, consisting of balance laws for mass, momentum, and energy of the unknown fields. These equations express fundamental laws of continuum physics. Throughout this presentation, we discuss basic concepts of thermodynamics and provide examples of constitutive equations that describe the internal properties of materials. We present specific physical models for gases, liquids, and solids, including the system of fluid dynamics in Eulerian and Lagrangian coordinates. Additionally, we introduce a viscous-capillary model for fluid dynamics or elastodynamics, a model of fluid flow in a nozzle with variable cross-section, and the shallow water equations with variable topography in both two and one dimensions.

Abstract

In this chapter, we introduce the fundamental concepts of the theory of systems of balance laws, which can be expressed in either conservative or nonconservative form. We provide several examples to illustrate these concepts. We define elementary waves, including shock waves, contact discontinuities, and rarefaction waves, and compare various admissibility criteria that can be applied at discontinuities.

Abstract

For an ideal fluid, the quantities pv and \(\varepsilon \)—as functions of (v, T)—depend only on T. In a polytropic fluid, the adiabatic exponent is constant.

Abstract

This chapter focuses on the Riemann problem for fluids with a general equation of state (EOS), covering both convex and nonconvex EOSs. We begin with the simple case of isentropic van der Waals fluids, for which the Riemann problem admits a unique solution for large data. We then present non-isentropic fluids with a general EOS, relying solely on the requirement that the Grüneisen coefficient is positive. Examples of EOSs that satisfy this assumption will be provided.

Abstract

This chapter serves as an example of how a kinetic relation can be used to implement Lax’s shock inequalities. Specifically, we consider a model from elastodynamics, given by a p-system consisting of conservation laws for mass and momentum. In the conservation of momentum, the stress is given as an increasing and concave-convex function of the strain, making the system strictly hyperbolic with two characteristic fields that are not genuinely nonlinear. Lax’s shock inequalities can only handle limited initial data, but a global construction for a Riemann solver can be achieved by following Wendroff’s idea, where the shock speed is allowed to coincide with the characteristic speed at the state(s) of the shock.

Abstract

This chapter deals with a hyperbolic-elliptic model of phase transition dynamics. The model is a typical p-system, where the stress can be decreased as a function of the strain in a certain interval. Consequently, this causes the system to become elliptic in the corresponding region. The elliptic region separates the phase domain into two regions, called the phases, in which the system is strictly hyperbolic. The system admits undercompressive subsonic phase boundaries to be characterized via a kinetic relation. The dynamics of phase boundaries in solids undergoing phase transformations play an important role in many applications of material science. It is interesting to note that the Riemann problem may admit more than one solution. In fact, there can be two distinct solutions.

Abstract

In this chapter, we consider the Riemann problem for the model of an isentropic fluid in a nozzle with a discontinuous cross-sectional area. The modeling for fluid flows in a nozzle with variable cross-section is given in Chapter 2. For simplicity, we assume that the fluid is isentropic and ideal. Unlike the models considered in previous chapters, this model contains a source term involving the space partial derivative in nonconservative form in the equation for the balance of momentum. This nonconservative source term represents the influence of the nozzle’s geometry on the fluid’s dynamics. However, the model can be written as a nonconservative system of balance laws, allowing for a general definition of weak solutions in terms of nonconservative products. Studying such a system may provide better understanding of more complicated nonconservative models of multi-phase flows.

Abstract

As discussed earlier, the appearance of stationary contact discontinuities is significant in the previous chapter. When the fluid is assumed to be isentropic, the system of governing equations for gas dynamics consists of two conservation laws of mass and momentum. There is no linearly degenerate characteristic field in the isentropic gas dynamics equations. However, contact discontinuities are among the most basic topics in gas dynamics equations for non-isentropic fluids.

Abstract

In the previous two chapters, we considered models involving a compressible fluid, which flows due to changes in pressure. The pressure is a function of density, and the density in a compressible fluid flow depends on both spatial variables and time. This chapter deals with incompressible fluids, where the density is constant or considered to be constant. Specifically, we consider the Riemann problem for one-dimensional shallow water equations with discontinuous topography. Applications of this model can be seen in water flowing in a river, dam break analysis, open channel flows, etc.

Abstract

This chapter deals with the shallow water equations with variable topography and horizontal temperature gradient known as the Ripa system. This system was formulated to model ocean currents and can be derived from the Saint-Venant system of shallow water equations by taking into account horizontal water temperature fluctuations.

Abstract

In this chapter, we will investigate the Riemann problem for a model of two-phase flows. The model under study is obtained from the well-known Baer-Nunziato model, which is used for studying deflagration-to-detonation transition in porous energetic materials [33, 73]. In this model, it is observed that intergranular pores form an interconnected region occupied by gas.