\dot m = \sum_\alpha \dot m_\alpha

A = \sum_\alpha A_\alpha

\sum_\alpha s_\alpha = 1

u_m = \sum_\alpha s_\alpha \cdot \dot u_\alpha

q_\alpha = \dot m_\alpha / \rho_\alpha = A_\alpha \, u_\alpha \Rightarrow \dot m_\alpha = \rho_alpha \, A_\alpha \, u_\alpha

s_\alpha =  \frac{\dot m_\alpha}{\rho_\alpha \, u_\alpha} \cdot \left( \sum_\beta \frac{\dot m_\beta}{\rho_\beta \, u_\beta} \right)^{-1}

For homogeneous pipe flow: and volumetric shares are going to be:

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

See also