By R. T. Lahey Jr., D. A. Drew (auth.), Jeffery Lewins, Martin Becker (eds.)

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In particular, let us examine the mathematical classification of the one dimensional two-fluid model given in Equations (145) - (152), and the appropriate jump and closure relations given in the one dimensional version of Equations (110) - (143). This system of equations can be written in matrix form as (153) 52 R. T. LAHEY AND D. A. DREW where (154) u and ~ and are 7 x 7 square matrices. ~ To classify the mathematical system given in Equation (153), we must evaluate the eigenvalues (v,). This is accomplished by calculating (15) 1.

CLOSURE AND CONSTITUTIVE EQUATIONS The equations of motion, Equations (107)-(109) for each phase, and the jump conditions in Equations (110) represent fifteen scalar equations. The equations of motion can be used to predict the pressures, velocities, temperatures and volume fraction of each phase. e. the primary unknowns) and/or the four independent variables (x j and t). Table III lists the 337 scalar parameters which must be specified to obtain closure. Let us now consider a set of constitutive laws which allow closure of the problem.

Hence, if our problem is properly formulated, we should expect only real eigenvalues. The fact that many two-fluid models do not yield real eigenvalues is apparently because some essential physical phenomenon has been left out of the conservation or constitutive equations, such as the higher-order spatial derivatives which are normally neglected in two-fluid models. It should be noted in Equation (155) that the only phenomena which can change the eigenvalues are those which contain space and time derivatives.