Characteristics and Stability Analyses of Transient One-Dimensional Two-Phase Flow Equations and Their Finite Difference Approximations

AbstractEquation systems describing one-dimensional, transient, two-phase flow with separate continuity, momentum, and energy equations for each phase are classified by use of the method of characteristics. The main purpose of this paper is to study the mathematical nature of these equations, not their physics, although it is realized that these two problems are not entirely independent. Many of the equation systems possess complex-valued characteristics and hence, according to well-known mathematical theorems, are not well-posed as initial-value problems (IVPs). Real-valued characteristics are necessary but not sufficient to ensure well-posedness. In the absence of lower order source or sink terms, that can affect the well-posedness of IVPs, the complex characteristics associated with these two-phase flow equations imply unbounded exponential growth for disturbances of all wavelengths.Analytical and numerical examples show that the ill posedness of IVPs for the two-phase flow partial differential equatio...

[1]  L. Payne,et al.  ON SOME IMPROPERLY POSED PROBLEMS FOR QUASILINEAR EQUATIONS OF MIXED TYPE( , 1967 .

[2]  Ole Rathmann,et al.  Numerical Experience with the Two-Fluid Model Risque , 1977 .

[3]  T. B. Anderson,et al.  Fluid Mechanical Description of Fluidized Beds. Equations of Motion , 1967 .

[4]  Numerical calculation of two-phase flows , 1976 .

[5]  L. Hamilton,et al.  THE EFFECTS OF NONEQUILIBRIUM HEAT, MASS AND MOMENTUM TRANSFER ON TWO-PHASE SOUND SPEED , 1972 .

[6]  L. M. Milne-Thomson,et al.  Theoretical hydrodynamics / by L.M. Milne-Thomson , 1955 .

[7]  Richard Courant,et al.  On nonlinear partial differential equations with two independent variables , 1949 .

[8]  R. D. Richtmyer,et al.  Survey of the stability of linear finite difference equations , 1956 .

[9]  James D. Murray,et al.  On the mathematics of fluidization Part 1. Fundamental equations and wave propagation , 1965, Journal of Fluid Mechanics.

[10]  R. Harrop.  An Introduction to the Method of Characteristics , 1953 .

[11]  A. A. Amsden,et al.  A numerical fluid dynamics calculation method for all flow speeds , 1971 .

[12]  W. Chester,et al.  Mathematical Theory of Compressible Fluid Flow , 1958 .

[13]  P. Schaefer.  On uniqueness, stability and pointwise estimates in the Cauchy problem for coupled elliptic equations , 1973 .

[14]  M. Fischer.  The dynamics of waves including shocks in two-phase flow , 1970 .

[15]  N. Zuber.  ON THE DISPERSED TWO-PHASE FLOW IN THE LAMINAR FLOW REGIME. , 1964 .

[16]  Percy Williams Bridgman,et al.  The thermodynamics of electrical phenomena in metals, and A condensed collection of thermodynamic formulas. , 1961 .

[17]  Paul Garabedian,et al.  Partial Differential Equations , 1964 .

[18]  G. Wallis.  One Dimensional Two-Phase Flow , 1969 .

[19]  A. A. Amsden,et al.  Numerical calculation of multiphase fluid flow , 1975 .

[20]  L. Payne,et al.  On some improperly posed problems for the Chaplygin equation , 1967 .

[21]  Fritz John,et al.  A note on “improper” problems in partial differential equations , 1955 .

[22]  A. Jefrey,et al.  Non-Linear Wave Propagation , 1964 .

[23]  J. Hinze,et al.  Turbulence: An Introduction to Its Mechanism and Theory , 1959 .

[24]  Pointwise Bounds in the Cauchy Problem for an Elliptic System , 1967 .

[25]  Ronald L. Panton,et al.  Flow properties for the continuum viewpoint of a non-equilibrium gas-particle mixture , 1968, Journal of Fluid Mechanics.

[26]  G. Rudinger,et al.  ANALYSIS OF NONSTEADY TWO-PHASE FLOW. , 1964 .

[27]  A. E. Bergles,et al.  Review of two-phase flow instability , 1973 .

[28]  L. Payne,et al.  Bounds in the Cauchy problem for the Laplace equation , 1960 .

[29]  H. S. Isbin,et al.  The "calculated" loss-of-coolant accident : a review , 1972 .

[30]  C. Tchen,et al.  Mean Value and Correlation Problems connected with the Motion of Small Particles suspended in a turbulent fluid , 1947 .