= Phase momentum profile coefficient
{title2=$D_1=\langle\chi_1v_x^2\rangle/(\alpha u_1^2)$}
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 $D_1\geq1$ 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 $\partial_t(\alpha u_1)+\partial_x(\alpha D_1u_1^2)$ 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.
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