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\frac{\partial (\rho_A \, \phi) }{\partial t} + \nabla (\rho_A \, {\bf u}_A) = 0, \quad A = \{ O, G, W \}

\int_V \, \frac{\partial (\rho_A \, \phi) }{\partial t} \, dV = - \int_V \, \nabla (\rho_A \, {\bf u}_A) \, dV = - \int_{\partial V} \, \rho \, {\bf u}_A \, d {\bf A}

V \cdot \delta (\rho_A \, \phi) = \delta \, m_A 

V \cdot \delta \left( \phi \, \sum_\alpha \rho_{A,\alpha} \, s_\alpha \right) = \mathring{\rho}_A \cdot \delta \, qQ_A, \quad \alpha = \{ o, g, w \}

\delta \left( \phi \, \sum_\alpha \frac{\rho_{A,\alpha}}{\mathring{\rho}_A} \, s_\alpha \right) =  V^{-1} \cdot \delta \, qQ_A  

\delta \left( \phi \, \sum_\alpha \frac{\mathring{V}_{A,\alpha}}{V_\alpha} \, s_\alpha \right) =  V^{-1} \cdot \delta \, qQ_A

 \delta \left( \phi \, \sum_\alpha \xi_{A,\alpha}\, s_\alpha \right) =  V^{-1} \cdot \delta \, qQ_A

  \xi_{A,\alpha} =  \frac{\mathring{V}_{A,\alpha}}{V_\alpha}