(1)	$\begin{array}{l}\displaystyle \dot m = \dot m_1 + \dot m_2\end{array}$

(2)	$\begin{array}{l}\displaystyle A = A_1 + A_2\end{array}$

(3)	$\begin{cases} s_1 = A_1/A \\ s_2 = A_2/A \end{cases}$

(4)	$\begin{array}{l}\displaystyle s_1 + s_2 = 1\end{array}$

(5)	$\begin{array}{l}\displaystyle u_m = s_1 \cdot \dot u_1 + s_2 \cdot \dot u_2\end{array}$

(6)	$\begin{cases} q_1 = \dot m_1 / \rho_1 = A_1 \, u_1 \Rightarrow \dot m_1 = \rho_1 \, A_1 \, u_1 \\ q_2 = \dot m_2 / \rho_2 = A_2 \, u_2 \Rightarrow \dot m_2 = \rho_2 \, A_2 \, u_2 \end{cases}$

The areas ratio:

(7)	$\begin{array}{l}\displaystyle \omega_{12} = \frac{A_1}{A_2} = \frac{\dot m_1 \, \rho_2 \, u_2}{\dot m_2 \, \rho_1 \, u_1}\end{array}$

(8)	$\begin{cases} A_1 = \frac{\omega_{12}}{1+\omega_{12}} \cdot A \\ A_2 = \frac{1}{1+\omega_{12}} \cdot A \end{cases}$

For homogeneous 2-phase pipe flow: $\begin{array}{l}u_1 = u_2 = u_m\end{array}$ and areas ratio are going to be:

(9)	$\begin{array}{l}\displaystyle \omega_{12} = \frac{\dot m_1 \, \rho_2 }{\dot m_2 \, \rho_1}\end{array}$

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See also