Momentum interactions in dispersed flow: an averaging and a variational approach

Abstract It is shown that spatial averaging of the local instantaneous conservation laws for inviscid incompressible dispersed flow yields identical results to equations obtained from a variational principle, in the dilute limit. The equations deduced from the variational approach extend easily to the nondilute case. The drift flux concept reduces the 4 × 4 system of equations to a 2 × 2 system which is cast into the framework of continuous Hamiltonian systems. The hyperbolicity of the equations is examined and it is concluded that the system is ill-posed for isotropic dispersions. It is then demonstrated that the model is Lyapunov stable, provided it is hyperbolic.

[1]  G. Wallis.  On Geurst's equations for inertial coupling in two phase flow , 1990 .

[2]  G. Wallis.  INERTIAL COUPLING IN TWO-PHASE FLOW: MACROSCOPIC PROPERTIES OF SUSPENSIONS IN AN INVISCID FLUID , 1990 .

[3]  G. Yadigaroglu,et al.  On the various forms of the conservation equations in two-phase flow , 1976 .

[4]  Darryl D. Holm,et al.  A multipressure regularization for multiphase flow , 1986 .

[5]  P. Lax.  Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves , 1987 .

[6]  A.M.C. Chan,et al.  Separated flow models—I. Analysis of the averaged and local instantaneous formulations , 1980 .

[7]  Donald A. Drew,et al.  The virtual mass and lift force on a sphere in rotating and straining inviscid flow , 1987 .

[8]  R. L. Seliger,et al.  Variational principles in continuum mechanics , 1968, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[9]  P. Smereka,et al.  Bubbly flow and its relation to conduction in composites , 1991, Journal of Fluid Mechanics.

[10]  J. A. Geurst.  Virtual mass in two-phase bubbly flow , 1985 .

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

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

[13]  G. Whitham,et al.  Linear and Nonlinear Waves , 1976 .

[14]  Jerrold E. Marsden,et al.  Nonlinear stability of fluid and plasma equilibria , 1985 .

[15]  A. B. Basset,et al.  Treatise on Hydrodynamics , 1889, Nature.

[16]  D. Drew.  Mathematical Modeling of Two-Phase Flow , 1983 .

[17]  L. van Wijngaarden,et al.  Two-phase flow equations for a dilute dispersion of gas bubbles in liquid , 1984, Journal of Fluid Mechanics.

[18]  Robert Lipton,et al.  Darcy's law for slow viscous flow past a stationary array of bubbles , 1990 .

[19]  R. Isaacs,et al.  Applied Mathematics , 1901, Nature.

[20]  Hamiltonian structure and Lyapunov stability of a hyperbolic system of two‐phase flow equations including surface tension , 1986 .

[21]  Morton E. Gurtin,et al.  Multiphase thermomechanics with interfacial structure , 1990 .

[22]  J. A. Geurst.  Variational principles and two-fluid hydrodynamics of bubbly liquid/gas mixtures , 1986 .

[23]  Victor H. Ransom,et al.  RELAP5 two-phase fluid model and numerical scheme for economic LWR system simulation , 1981 .

[24]  J. W. Humberston.  Classical mechanics , 1980, Nature.

[25]  D. Drew,et al.  THE ANALYSIS OF VIRTUAL MASS EFFECTS IN TWO-PHASE FLOW , 1979 .

[26]  Philip J. Morrison,et al.  Poisson brackets for fluids and plasmas , 1982 .

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

[28]  G. Wallis.  The averaged Bernoulli equation and macroscopic equations of motion for the potential flow of a two-phase dispersion , 1991 .

[29]  A. Clebsch,et al.  Ueber die Integration der hydrodynamischen Gleichungen. , 1859 .

[30]  Sanjoy Banerjee,et al.  INTERPHASE MOMENTUM INTERACTION EFFECTS IN THE AVERAGED MULTIFIELD MODEL , 1986 .

[31]  D. Lhuillier.  Phenomenology of inertia effects in a dispersed solid-fluid mixture , 1985 .

[32]  Robert I. Nigmatulin,et al.  Spatial averaging in the mechanics of heterogeneous and dispersed systems , 1979 .

[33]  A. Bedford,et al.  A variational theory of immiscible mixtures , 1978 .