Intrinsic phase averages in a two-fluid pipe model ($\alpha =⟨{\chi }_1⟩,{\rho }_1=⟨\rho {\chi }_1⟩/\alpha ,u_1=⟨\rho v_x{\chi }_1⟩/(\alpha {\rho }_1)$)

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.