Remarks on DiPerna's Paper ``Convergence of the Viscosity Method for Isentropic Gas Dynamics'

Concerns have been voiced about the correctness of certain technical points in DiPerna's paper (Comm. Math. Phys.91 (1983), 1-30) related to the vacuum state. In this note, we provide clarifications. Our conclusion is that these concerns mainly arise from the statement of a lemma for constructing the viscous approximate solutions and some typos; however, the gap can be either fixed by correcting the statement of the lemma and the typos or bypassed by employing the finite difference methods.

     

Thermophysical properties of azomethine ether main-chain liquid crystalline polymers at pressures to 200 MPa

This article reports on an experimental investigation of the equation of state and the transition behavior of main-chain thermotropic liquid crystalline polymers over a wide temperature range, and at pressures to 200 MPa. The materials studied were a series of azomethine ether polymers. A varying number n (= 4, 7, 8, 9, 10 and 11) of methylene spacer units in the backbone provided systematic variation of the structure. Experimental techniques used included high-pressure dilatometry (PVT measurements) to 200 MPa, high-pressure differential thermal analysis, also to 200 MPa, and conventional (atmospheric-pressure) differential scanning calorimetry (DSC). The equation of state of the materials can be well represented by the Tait equation in distinct regions, separated by a glass transition, T_g(P), a first-order transition to a nematic state, T_k-n(P), and a first-order transition to an isotropic melt state T_c(P). The atmospheric pressure values of T_k-n and T_c decreased with increasing number of spacer units and showed a clear odd-even effect. T_g and T_k-n both increased with pressure. The pressure dependence of T_c could not be observed due to the onset of degradation in the same temperature region. On isobaric cooling at 3°C/min, the crystallization from the nematic state occurred a few tens of degrees below T_k-n. This supercooling was independent of pressure for some materials, while for others it increased with increasing pressure. The values of the enthalpy and entropy associated with the first-order transition into the nematic state were lower than those of typical isotropic polymers at their melting transitions. The transition enthalpy did not have any systematic variation with increasing number of spacer units. Values of the transition enthalpy calculated from the Ciapeyron equation did not always agree with the values measured by DSC. This may be due to the two-phase nature of the low-temperature state. At the transition to the isotropic state, the transition enthalpy at P = 0 decreased with n and showed an odd-even effect. © 1994 John Wiley & Sons, Inc.     

Partial molar isentropic compressibilities and volumes of selected electrolytes and nonelectrolytes in dimethylsulfoxide

Precision densities and sound velocities for solutions of selected univalent electrolytes and nonelectrolytes in DMSO have been measured at 25°C, and apparent molar isentropic compressibilities and volumes evaluated. The data were extrapolated to infinite dilution to obtain standard state partial molar quantities, K _s,2 ^° , and V ₂ ^° . Values of V ₂ ^° and K _s,2 ^° for alkali metal halides in DMSO are very similar to those in water. The results confirm conclusions derived from data in water and other nonaqueous solvents that K _s,2 ^° and V ₂ ^° for alkali metal halides are strongly dependent on solvent compressibility. K _s,2 ^° becomes more negative and V ₂ ^° decreases as solvent compressibility increases. Attempts to determine ionic K _s,2 ^° values suggest that a significant dissymmetry exists between ₄P⁺ and ₄B^– in DMSO, whereas in water and MeOH, these large ions appear to behave similarly. Ionic V ₂ ^° values support this conclusion. Steric hindrance in the DMSO molecule is believed to be responsible for this dissymmetry.     

Computer investigation of physical properties of water clusters. 1. Stability

The method of molecular dynamics is applied to investigate the stability of water clusters containing up to 90 molecules. With increasing size of aggregates, (H₂O)_n≥10 10, their thermal stability strengthens. Mechanical stability of great clusters keeps the quickly reached level, and the coefficient of dielectric stability passes through the maximum at n = 50.     

A characteristic-based method for incompressible flows

A new characteristic-based method for the solution of the 2D laminar incompressible Navier-Stokes equations is presented. For coupling the continuity and momentum equations, the artificial compressibility formulation is employed. The primitives variables (pressure and velocity components) are defined as functions of their values on the characteristics. The primitives variables on the characteristics are calculated by an upwind diffencing scheme based on the sign of the local eigenvalue of the Jacobian matrix of the convective fluxes. The upwind scheme uses interpolation formulae of third-order accuracy. The time discretization is obtained by the explicit Runge–Kutta method. Validation of the characteristic-based method is performed on two different cases: the flow in a simple cascade and the flow over a backwardfacing step.     

三维可压Euler方程简单波及双重波结构

本文研究了三维可压欧拉方程简单波解和双重波解的结构.给出了简单波的流动区域是被一族相互独立的平面所覆盖,沿着每个平面u,v,w从而p,ρ,c均为常数.双重波的流动区域是被一族相互独立的直线所覆盖,沿着每个特征平面u,v,w从而p,ρ,c均为常数.     

Existence of global solutions to isentropic gas dynamics equations with a source term

In this paper we prove existence of isentropic gas dynamic equations with a source term (1.2). To this end we construct a sequence of regular hyperbolic systems (1.1) to approximate the inhomogeneous system of isentropic gas dynamics (1.2). First,for each fixed approximation parameter δ and very general condition on P (ρ),we establish the existence of entropy solutions for the Cauchy problem (1.1) with bounded initial date (1.4). Second,letting=o(δ),we obtain a complete proof of the H-1loc compactness of weak entropy pairs of system (1.2) in the form η(ρ,u) =ρH(ρ,u) given in Chen-LeFloch (2003). Finally,for the conditions of P(ρ) given in Chen-LeFloch (2003),applied to the results in Theorems 1 and 2,we obtain the global existence of entropy solutions for the Cauchy problem (1.2) with bounded initial date (1.4).     

Lie group analysis and propagation of weak discontinuity in one-dimensional ideal isentropic magnetogasdynamics

The aim of this paper is to carry out symmetry group analysis to obtain important classes of exact solutions from the given system of nonlinear partial differential equations (PDEs). Lie group analysis is employed to derive some exact solutions of one dimensional unsteady flow of an ideal isentropic, inviscid and perfectly conducting compressible fluid, subject to a transverse magnetic field for the magnetogasdynamics system. By using Lie group theory, the full one-parameter infinitesimal transformations group leaving the equations of motion invariant is derived. The symmetry generators are used for constructing similarity variables which leads the system of PDEs to a reduced system of ordinary differential equations; in some cases, it is possible to solve these equations exactly. Further, using the exact solution, we discuss the evolutionary behavior of weak discontinuity.     

Speeds of sound and isothermal compressibility of ternary liquid systems: Application of Flory’s statistical theory and hard sphere models

Speeds of sound and densities of three ternary liquid systems namely, toluene + n-heptane + n-hexane (I), cyclohexane + n-heptane + n-hexane (II) and n-hexane + n-heptane + n-decane (III) have been measured as a function of the composition at 298.15 K at atmospheric pressure. The experimental isothermal compressibility has been evaluated from measured values of speeds of sound and density. The isothermal compressibility of these mixtures has also been computed theoretically using different models for hard sphere equations of state and Flory’s statistical theory. Computed values of isothermal compressibility have been compared with experimental findings. A satisfactory agreement has been observed. The superiority of Flory’s statistical theory has been established quite reasonably over hard sphere models.      

Realizing Quasiordered Sets by Subspaces of ‘Continuum-Like’ Spaces

Given an ordered set E and a topological space X, we say that E can be realized within X if there is an injection j from E into the class of (homeomorphism classes of) subspaces of X such that, for x, y in E, x y if and only if j(x) is homeomorphically embeddable into j(y). It is known, for instance, that transfinite induction demonstrates that every partially-ordered set of cardinality c (and some larger ones) can be realized within the real line. We explore aspects of the realizability problem, indicating, in particular, how to weaken the hypothesis on E from partial- to quasi-order, and seeking to isolate the characteristics of the real line that are relevant here.