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LaTeX Math Block
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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

LaTeX Math Inline
body\rho_\alpha

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body\alpha
-phase fluid density

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bodys_\alpha

volume share occupied by 

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body\alpha
-phase 

LaTeX Math Inline
body\mu_\alpha

LaTeX Math Inline
body\alpha
-phase fluid viscosity

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bodyA_\alpha

cross-sectional area occupied by 

LaTeX Math Inline
body\alpha
-phase 

LaTeX Math Inline
bodyu_\alpha

LaTeX Math Inline
body\alpha
-phase fluid velocity

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bodyA

total cross-sectional area


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titleDerivation


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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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