# Q: Fluid drainage along the outside of an inclined cylinder

• Andy Resnick
Andy Resnick
TL;DR Summary
Looking for references analyzing fluid drainage along the outside of an inclined cylinder
Asking on behalf of a colleague who is studying rainwater transport/drainage in the context of plants (trees, mostly):

There's a lot of published work analyzing the flow of fluid within a tube, or along an interior corner. I can't seem to find much of anything about the flow of water along the outside of a cylinder- think rainwater on a cable. Clearly interfacial energy has a lot to say about how much water is transported along the cable and how much drips off onto the ground. I guess a pendant drop analysis is a reasonable starting point, but the contact line isn't pinned as the usual case.

Has anyone worked out the fluid dynamics?

Just hoping someone can point us in the right direction for a literature search. Thanks in advance!

Andy Resnick said:
TL;DR Summary: Looking for references analyzing fluid drainage along the outside of an inclined cylinder

Asking on behalf of a colleague who is studying rainwater transport/drainage in the context of plants (trees, mostly)
Interesting question.

Can you provide a bit more context? What is the flow rate of the rainwater, and what are the dimensions and incline angle of the trees? Are you wanting to figure out how much water falls off of the inclined tree? Are you trying to optimize how much of the runoff you can collect in a cistern near the base of the tree?

berkeman said:
Interesting question.

Can you provide a bit more context? What is the flow rate of the rainwater, and what are the dimensions and incline angle of the trees? Are you wanting to figure out how much water falls off of the inclined tree? Are you trying to optimize how much of the runoff you can collect in a cistern near the base of the tree?
I can try and answer these...

The flow rate is (most likely) driven by the rate of rainfall, and my colleague, and environmental scientist, is interested in how solutes from the tree (leaves, bark) dissolve into the water and are transported elsewhere- to other parts of the tree, to runoff, etc. So there's a range of inclination angles and 'cylinder' radii (abaxial surface of leaf vasculature/stems/branches/trunk).

At this point, he's just looking for some 'reference frame' to build a model, any quantitative results would be helpful.

Others will have better replies, but I wonder if it might be good to start with some experiments to gain some insights. Certainly the type and surface roughness of the bark will make a difference, in addition to the branch diameter and inclination angle. At least with the experimental data, they could start to get some ideas of the flow rates for all the different parts of the tree. Wow, complicated problem.

Andy Resnick said:
The flow rate is (most likely) driven by the rate of rainfall, and my colleague, and environmental scientist, is interested in how solutes from the tree (leaves, bark) dissolve into the water and are transported elsewhere- to other parts of the tree, to runoff, etc.
Drops that fall to the side, will flow to that side, but with an elliptical slope due to the section of a sloping cylinder.

Drops that fall on the upper central axis will flow down the axis, but as flow volume increases, that flow will radiate out to the sides, then flow down the elliptical side.

The water that wraps around under the cylinder will [converge and] flow down the lower axis until film thickness, or surface roughness, [detaches it, and] drops it to a cylinder below, or to the ground.

[edited]

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