I'm not really able to discuss fluid mechanics with you since I'm an economist and I've never studied this matter before. But an overview shows that maybe you didn't consider the rate of the sebum flowing (10micron/second? 10micron/minute?) Also micron is an unidimensional measure, I believe it would have to be measured in square milimeters? Also its volume could increase because of the fluid build up, since there are thousands of follicles, it is not an unique flow.
But anyway, that's just a layman's view. I do believe in these because there are anedoctal evidence (mine also) of having a more oily face when having more hair. Also, mainly, (off-topic) there's an evolutionary theory that hair (in caucasians) is made for cold protection. Being exposed to excessive heat would turn the hair purposeless, even working against the body cooling. Sebum could be the carrier of that message. Makes me think if there is a relation with sebaceous glands hyperplasia.
Edit: (maybe off-topic) A sebaceous glands hyperplasia study:
https://www.ncbi.nlm.nih.gov/pubmed/26147300
CONCLUSIONS:
The overgrowth (multilobulation) of the sebaceous gland and relative preservation of the follicular stem cells suggest that the changes in the sebaceous gland could be an important factor in the pathology of Androgenetic Alopecia.
There is a disconnect between what I am saying and what you are suggesting:
But an overview shows that maybe you didn't consider the rate of the sebum flowing (10micron/second? 10micron/minute?)
The rate is irrelevant in this matter. What causes a fluid to flow? The forces acting on it. What kind of forces can act on it? Viscous forces - pertaining to viscosity and fluid shearing, capillary forces - forces due to surface tension, intertial forces - forces due to a push or pull from an external object, or body forces - field forces such as an electromagnetic field or gravitational field.
How do we know which forces are causing the fluid to flow in a given scenario? On large scales such as in an ocean or on the sun, capillary forces are unimportant and body forces dominate. What this means is that the contribution to the dynamics of the fluid due to capillary forces are small. You can neglect capillary forces when you look at solar flares, but you cannot neglect electromagnetic forces.
What I am suggesting, and what I have shown through the computation of the Bond number, is a comparison of which forces dominate fluid motion at that scale. Capillary and viscous forces overwhelmingly contribution to any fluid flow that occurs, and the fluid essentially does not "feel" a gravitational force because the contribution of gravity is negligible. I cannot stress this further. This is the correct answer.
If a grain of sand is thrown down a well, it is unscathed. If a pebble is tossed down a well, it may scuff. If a boulder is thrown down a well, it will be obliterated. Assume all test materials mentioned above are composed of the same material. The material properties - hardness, elastic modulus, toughness, etc. are the same. But the length and time scales are not. In the case of the pebble, inertial force is not important. In the case of the boulder, it is. All relative of course, to the material properties of the object - which remain constant.
Also micron is an unidimensional measure, I believe it would have to be measured in square milimeters? Also its volume could increase because of the fluid build up, since there are thousands of follicles, it is not an unique flow.
Yes this particular nuance is a bit harder to explain without familiarity of non-dimensional equations. The Bond number we computed earlier is not itself a universal law or governing equation but rather a scaling parameter. Basically, I'm not saying that gravity doesn't exist, but if it only contributes 0.0001% of the force acting on the fluid, then we can solve the equation by setting any contribution by gravity to 0 - this would be a safe assumption. Of course if one wanted to be rigorous, one would input all of the effects, all of the time, but this will make the problem more computationally rigorous all the while changing the end result by a negligible amount. The best example I can in relation to your field (I do apologize because I am not an economist) is that if you are computing the cash flows in relation to a particular company or organization, neglecting the smallest 0.001% of the cash flows for a quick estimate of the financials would be safe to do. In other words, you do not need to count every nickel and dime on every receipt in order to get an accurate idea of where the money is. The same is true with my calculation.
Now, given that the Bond number is a simple scaling parameter, L in this case represents the "characteristic" length scale. It doesn't have to be an exact number, but rather the order of magnitude. The width of the channel through which the sebum is flowing is the relevant length scale in this problem, as is the diameter of a pipe in a pipe flow problem. This is why 10 micron was chosen instead of an area. I apologize if this is still mildly confusing but it is somewhat difficult to justify without the ground work. But the simple explanation is that you give the Bond number calculation the approximate sizes and strengths of the various physical parameters of the problem and we can find out which terms contribute and which terms can be neglected. In this case, gravity's contributed force to the fluid will be around 1/10000th ish what surface tension forces will contribute. That's why we can assume that gravity is unimportant.
You are free to continue to believe it if you want. But it is false, and I don't mean that in an arrogant way; physically speaking, gravity does not contribute on that scale. This however bears no insight into the role of sebum in hair loss by virtue of its overproduction or chemical composition. It is just incorrect to suggest that the flow of sebum is significantly impacted by gravity.
Here is a final sanity check if you remain suspicious of the argument outlined about. A great portion of the population, including myself, spend ~8 hours per day, a non-negligible portion of time, sleeping on their side or back. Yet there is no noticeable effect of uneven sebum distribution before versus after sleeping, or in the case of sleeping on one's side, we do not observe greatly asymmetric rates of balding on one side versus another. We also do not normally see the sides and back thin, despite your suggestion that sebum pool in a particular way depending on its orientation. Sleeping, according to your suggestion, should also enable the sebum on the hair on top of the head to "drain" overnight. However, this is not the case because gravity is unimportant to determining the dynamics of fluid flow at that scale.