Correction for non-horizontal pipes

If the pipe considered in the previous analysis were not horizontal, one should include gravity effects in calculating velocity and volumetric flow rates. Referring to Fig.7.8. we see that if the flow is inclined at an angle $\theta$ to the horizontal, the pressure difference term need be modified. Accordingly, Eqn. 7.3 becomes

$\displaystyle {{\Delta p - \gamma l \sin \theta} \over l} = {{2 \tau} \over r}$ (7.16)

We need to replace the pressure difference term in other equations as well. Accordingly,

$\displaystyle Q = { {\pi D^4 (\Delta p - \gamma l \sin \theta)} \over {128 \mu l}}$ (7.17)

and

$\displaystyle V = {{(\Delta p - \gamma l \sin \theta) D^2} \over {32 \mu l}}$ (7.18)