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If saturation does not vary along the flow substantially then capillary effects are vanishing : so that 

LaTeX Math Inline
body\displaystyle \frac{\partial P_c}{\partial r} = \dot P_c \cdot \frac{\partial s_w}{\partial r} \approx 0
.

If flow is close to horizontal then gravity effects are vanishing too : and 

LaTeX Math Inline
body\sin \alpha \approx 0
.

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