Hyperbolic conservation laws for continuous two-phase flow without mass exchange

It is well known that classic two-phase flow equation systems have complex characteristic roots and, therefore, constitute an ill-posed initial-value problem. Here we suggest that ill-posedness is due to working with two different material derivatives for the phases, which have varying velocities, but employing the same position vector for both operators. There follows an analysis of the conditions required for a global treatment of both phases, but using only one material derivative for both phases, now coherent with only one position vector. Consequently, new global mass- and momentum-conservation equations for a two-phase flow without mass exchange are derived by strictly following the classic Reynolds’ transport theorem. The new global mass-conservation equation proposed would only be valid if the ‘zero-net-mass-flux’ condition, another independent equation, was fulfilled. We also found that the new equation system is well-posed, i.e. its two characteristic roots are real if a new relation between velocities and densities is satisfied. Finally, we have highlighted the strong connections of new conservation laws with classic treatments, and also shown that minor modifications of the current equation system would turn it into a hyperbolic one, thus easing the computational solution of this complex problem.