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{\rm Re} = \frac{ \sum_\alpha \rho_\alpha \, u_\alpha^2 \, A_\alpha}
{\sum_\alpha \mu_\alpha \, u_\alpha \, \sqrt{A_\alpha} } =
\frac{ \sum_\alpha \rho_\alpha \, q_\alpha^2 / A_\alpha}
{\sum_\alpha \mu_\alpha \, q_\alpha / \sqrt{A_\alpha} } =
\frac{1}{\sqrt{A}} \cdot \frac{ \sum_\alpha \rho_\alpha \, q_\alpha^2 / s_\alpha}
{\sum_\alpha \mu_\alpha \, q_\alpha / \sqrt{s_\alpha} } |
where
L liquid densityrho_ggas uliquid velocityL | u_g | gas velocityA_L | cross-sectional area occupied by liquid g | cross-sectional area occupied |
by gas mu_L liquid viscosity\mu_g | gas viscosity | total cross-sectional area |
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| Reynolds number represent the ration of intertial forces to viscous forces: | LaTeX Math Block |
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| {\rm Re} = \frac{\rm Intertial \ Forces}{\rm Viscocus \ Forces} |
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Homogeneous Pipe Flow
Homogeneous Pipe Flow is characterized by the same phase velocities:
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| body | u_\alpha = u_t, \, \forall \alpha \in \Gamma |
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(no slippage) and the multiphase Reynolds number takes simpler form:| LaTeX Math Block |
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{\rm Re} =\frac{ \sum_\alpha \rho_\alpha \, u_\alpha \, A_\alpha}
{\sum_\alpha \mu_\alpha \, \sqrt{A_\alpha} } =\frac{ \dot m}
{\sum_\alpha \mu_\alpha \, \sqrt{A_\alpha} } = \frac{\dot m}{\sqrt{A}} \cdot \frac{1}{ \sum_\alpha \mu_\alpha \, \sqrt{s_\alpha} } |
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