Abstract
According to mathematical theorems on initial-value problem, real-valued characteristic roots are necessary for a system of partial differential equations. However, a one-dimensional, one-pressure two-fluid model has complex-valued characteristic root, so that it is called mathematically ill-posed. Up to the present, eigenvalue analysis of a two-fluid model has been conducted only for a simplified two-fluid model without any source terms. Although these simplified analyses can yield a necessary condition for well-posedness, they cannot give a sufficient condition. A necessary and sufficient condition can be obtained if and only if the eigenvalues of a complete two-fluid model with source terms are evaluated. Furthermore, numerical stability of the two-fluid model is also affected by source terms. The examination of source terms is, hence, indispensable to clarify the mathematical and numerical nature of a two-fluid model. In this report, the method to examine the mathematical nature of a complete two-fluid model with source terms is presented. Based on this method, it is revealed that the degree of ill-posedness of the two-fluid model is greatly affected by gravity.
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