\dot m = \dot m_1 + \dot m_2

A = A_1 + A_2

s_1 = A_1/A

s_2 = A_2/A

s_1 + s_2 = 1

u_m = s_1 \cdot \dot u_1 + s_2 \cdot \dot u_2

q_1 = \dot m_1 / \rho_1 = A_1 \, u_1 \Rightarrow \dot m_1 = \rho_1 \, A_1 \, u_1

q_2 = \dot m_2 / \rho_2 = A_2 \, u_2 \Rightarrow \dot m_2 = \rho_2 \, A_2 \, u_2

s_1 =  \frac{\dot m_1 \, \rho_2 \, u_2}{\dot m_1 \, \rho_2 \, u_2 + \dot m_2 \, \rho_1 \, u_1}

s_2 = \frac{1}{1+\omega_{12}} \cdot A = \frac{\dot m_2 \, \rho_1 \, u_1}{\dot m_1 \, \rho_2 \, u_2 + \dot m_2 \, \rho_1 \, u_1}

For homogeneous 2-phase pipe flow: and volumetric shares are going to be:

s_1 =  \frac{\dot m_1 \, \rho_2 }{\dot m_1 \, \rho_2  + \dot m_2 \, \rho_1 }

s_2 =  \frac{\dot m_2 \, \rho_1}{\dot m_1 \, \rho_2 + \dot m_2 \, \rho_1}

See also