 Complex characteristic speeds of a pressure-coupled two-fluid model

Take incompressible phase mass equations for fractions $\alpha ,1-\alpha$ and momentum equations ${\rho }_i[(u_i{)}_t+u_i(u_i{)}_x]+p_x=0$ with a common pressure. At a uniform state with $0<\alpha <1$, elimination of the mass and momentum perturbation amplitudes givesIts roots are the displayed speeds, whose imaginary parts are $\pm s(u_1-u_2)/(1+s^2)$. A nonzero slip velocity therefore gives a Fourier branch growing as $e^{|k||\mathrm{Im}c|t}$. No finite-order Sobolev estimate can control all high wavenumbers, showing ill-posedness of this unregularized closure. At zero slip the roots coincide and this nonreal-root argument no longer settles the degenerate case. Phase-fraction endpoints also lie outside the root formula's derivation.