Multiphase flow

A flow containing two or more distinguishable material phases, such as two immiscible liquids or a gas-liquid mixture. Each phase can have its own volume fraction, density and velocity. Cross-sectional or ensemble averaging produces coupled phase balance equations whose mass, momentum and energy exchange terms must be specified consistently. A common pressure alone does not automatically make an averaged model well posed.

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.

Here $⟨\cdot ⟩$ means integration over a fixed pipe cross-section divided by its area, and ${\chi }_1$ is the indicator of one immiscible fluid. The volume fraction is $\alpha$; intrinsic density averages over the phase-occupied area rather than over the whole section. The velocity must be density-weighted for its product with mean phase density to represent mass flux exactly.

If there is no interphase mass transfer, integrating ${\partial }_t(\rho {\chi }_1)+\nabla \cdot (\rho v{\chi }_1)=0$ over a section with impermeable outer wall gives ${\partial }_t({\rho }_1\alpha )+{\partial }_x({\rho }_1\alpha u_1)=0$. The second phase uses $1-\alpha$ and its own intrinsic averages. For constant phase density the velocity reduces to the ordinary phase-area average. These definitions apply where the corresponding phase fraction is positive; zero-fraction intrinsic quantities need a separate convention.

For an incompressible phase, this coefficient compares its actual axial momentum flux with the uniform-profile flux at the same intrinsic mean velocity. Cauchy-Schwarz gives $D_1\ge 1$ where the mean is nonzero. A spatially varying profile can have a larger second moment even when its mean speed is unchanged.

With constant $D_1$, expansion of ${\partial }_t(\alpha u_1)+{\partial }_x(\alpha D_1{u}_1^2)$ and use of ${\alpha }_t+(\alpha u_1{)}_x=0$ gives $\alpha (u_1{)}_t+(2D_1-1)\alpha u_1(u_1{)}_x+(D_1-1)u_1^2{\alpha }_x$. Thus a profile correction changes both convection and phase-fraction-gradient coupling in the averaged momentum equation. If $D_1$ varies, an extra $\alpha u_1^2(D_1{)}_x$ remains; a zero mean should be treated through the undivided second moment rather than an undefined coefficient.

The source ${\Gamma }_i$ is the mass gained by phase $i$ per mixture volume and time, through processes such as evaporation or condensation. Internal exchange conserves total mass, so its phase sources sum to zero. In an immiscible material-interface model without phase change, every ${\Gamma }_i$ is zero. The interface may still move: absence of transfer means that mass does not cross it relative to its own motion.