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For homogeneous pipe flow: 

LaTeX Math Inline
bodyu_\alpha = u_m, \, \forall \alpha
 and volumetric shares are going to be:

LaTeX Math Block
anchorLTRUSTFZOO
alignmentleft
s_\alpha =  \frac{\dot m_\alpha}{\rho_\alpha} \cdot \left( \sum_\beta \frac{\dot m_\beta}{\rho_\beta} \right)^{-1}

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