= Complex characteristic speeds of a pressure-coupled two-fluid model
{title2=$c_\pm=\frac{u_2\pm is u_1}{1\pm is},\quad s^2=\frac{\rho_1(1-\alpha)}{\rho_2\alpha}$}
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 gives
$$
\rho_1(1-\alpha)(u_1-c)^2+\rho_2\alpha(u_2-c)^2=0.
$$
Its 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 $e^{|k|\,|\operatorname{Im}c|t}$. 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.
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