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The analysis and boundary optimal control of the nonlinear transport of gas on a network of pipelines is considered. The evolution of the gas distribution on a given pipe is modeled by an isothermal semilinear compressible Euler system in one space dimension. On the network, solutions satisfying (at nodes) the Kirchhoff flux continuity conditions are shown to exist in a neighborhood of an equilibrium state. The associated nonlinear optimization problem then aims at steering such dynamics to a given target distribution by means of suitable (network) boundary controls while keeping the distribution within given (state) constraints. The existence of local optimal controls is established and a corresponding Karush–Kuhn–Tucker (KKT) stationarity system with an almost surely non-singular Lagrange multiplier is derived. [ABSTRACT FROM AUTHOR]

We present a theory for the three-dimensional evolution of tubes with expandable walls conveying fluid. Our theory can accommodate arbitrary deformations of the tube, arbitrary elasticity of the walls, and both compressible and incompressible flows inside the tube. We also present the theory of propagation of shock waves in such tubes and derive the conservation laws and Rankine-Hugoniot conditions in arbitrary spatial configuration of the tubes and compute several examples of particular solutions. The theory is derived from a variational treatment of Cosserat rod theory extended to incorporate expandable walls and moving flow inside the tube. The results presented here are useful for biological flows and industrial applications involving high-speed motion of gas in flexible tubes. [ABSTRACT FROM AUTHOR]

In this paper, we present an overview of the entropy production in fluid dynamics in a systematic way. First of all, we clarify a rigorous derivation of the incompressible limit for the Navier-Stokes-Fourier system of equations based on the asymptotic analysis, which is a very well known mathematical technique used to derive macroscopic limits of kinetic equations (Chapman-Enskog expansion and Hilbert expansion are popular methodologies). This allows to overcome the theoretical limits of assuming that the material derivative of the density simply vanishes. Moreover, we show that the fundamental Gibbs relation in classical thermodynamics can be applied to non-equilibrium flows for generalizing the entropy and for expressing the second law of thermodynamics in case of both incompressible and compressible flows. This is consistent with the thermodynamics of irreversible processes and it is an essential condition for the design and optimization of fluid flow devices. Summarizing a theoretical framework valid at different regimes (both incompressible and compressible) sheds light on entropy production in fluid mechanics, with broad implications in applied mechanics. [ABSTRACT FROM AUTHOR]

We consider the Euler system made of three conservation laws modeling one-dimensional, inviscid, compressible fluid flows. Considering first a general equation of state, we reformulate the standard condition that the specific entropy be increasing at a shock, The new formulation turns out to be easier to check in concrete examples when searching for admissible shock waves. Then, restricting attention to van der Waals fluids, we first determine regions in the phase space in which the system is hyperbolic or elliptic, or fails to be genuinely nonlinear. Second, based on our reformulation of the entropy condition, we provide a complete description of all admissible shock waves, classified in two distinct categories: the compressive shocks satisfying standard (Liu, Lax) entropy criteria, and undercompressive shocks violating these criteria and requiring a kinetic relation. [ABSTRACT FROM AUTHOR]

With increasing engineering demands, there need high order accurate schemes embedded with precise physical information in order to capture delicate small scale structures and strong waves with correct "physics". There are two families of high order methods: One is the method of line, relying on the Runge-Kutta (R-K) time-stepping. The building block is the Riemann solution labeled as the solution element "1". Each step in R-K just has first order accuracy. In order to derive a fourth order accuracy scheme in time, one needs four stages labeled as " 1⊙1⊙1⊙1=4". The other is the one-stage Lax-Wendroff (LW) type method, which is more compact but is complicated to design numerical fluxes and hard to use when applied to highly nonlinear problems. In recent years, the pair of solution element and dynamics element, labeled as "2", are taken as the building block. The direct adoption of the dynamics implies the inherent temporal-spatial coupling. With this type of building blocks, a family of two-stage fourth order accurate schemes, labeled as " 2⊙2=4", are designed for the computation of compressible fluid flows. The resulting schemes are compact, robust and efficient. This paper contributes to elucidate how and why high order accurate schemes should be so designed. To some extent, the " 2⊙2=4" algorithm extracts the advantages of the method of line and one-stage LW method. As a core part, the pair "2" is expounded and LW solver is revisited. The generalized Riemann problem (GRP) solver, as the discontinuous and nonlinear version of LW flow solver, and the gas kinetic scheme (GKS) solver, the microscopic LW solver, are all reviewed. The compact Hermite-type data reconstruction and high order approximation of boundary conditions are proposed. Besides, the computational performance and prospective discussions are presented. [ABSTRACT FROM AUTHOR]