## Wednesday, July 7, 2021

### Python Library for Calculation of Balloon Properties

I have now put up a python library with numerical implementations of the fundamental relationships for pressure, volume, mass, etc. of a gravity balloon.

https://github.com/AlanCoding/gravitational-balloon-mathematics

The old work on this blog was done largely in an Excel spreadsheet with macros. In terms of the library of methods, I feel that the python version is now better and more correct. Some library methods haven't yet been converted, but the important ones will, and that should go quite fast.

Doing this allows me to put work in Juypter notebooks, which is much better for sharing my work down to the numerical implementation. For a demo, I have re-done the graphs for the PR and PV curve.

These are showing the same thing as the prior post did:

That had some errors in it, and I'm now confident those are corrected.

The python library is much much easier to understand and modify, and any further improvements will become considerably faster.

## Friday, July 2, 2021

### A More Detailed Run-through of the Pressure-Volume Relationship

I've enabled MathJax on this blog and started repairing numerous equations, after a few years of the equations not rendering. Some parts have still not been recovered, but I've also realized that some parts are unclear.

So there will be value in running through the basic equations of the gravity balloon with fresh eyes.

## Objective

The equations here will relate multiple quantities. At different times we are interested in different quantities. The most common is to seek pressure (P) and volume (V), given values for all the others. Given that a gravity balloon is to be built out of some asteroid, then available mass and density of rock is fixed.

## Mass

We will temporarily introduce variables for dimensions. The big R is used for the radius of the inner volume of air - the inner surface of the rock. The little t is used for the thickness of the rock. Thus, the outer surface of the inflated gravity balloon would be (R+t). Less commonly, the original radius of the asteroid (assuming spherical) will be denoted $R_0$. Using those, we will use the general formula for volume of a sphere both before turning into a gravity balloon and after.

$$M = \rho V = \rho \frac{4}{3} \pi R_0^3 = \rho \frac{4}{3} \pi \left( (R + t)^3 - R^3 \right)$$

## Gravitational Field

Consider any homogeneous spherical planet, we have a simple $1/r^2$ formula for gravity on the surface. I'm keeping M, the mass of the planet, in there as an independent variable of the function, as this will be important.

Spherical Planet Field Surface and Beyond

$$g_{space}(r, M) = G M \frac{1}{r^2}$$

As you consider the interior, Gauss' law dictates that we can use that same formula, if we include all the mass below the radius of consideration.

Spherical Planet Field Interior, not for Gravity Balloon

$$g_{interior}(r) = G M_{inside} \frac{1}{r^2} = G \rho \frac{4}{3} \pi r^3 \frac{1}{r^2} = G \rho \pi \frac{4}{3} r$$

These are universal expressions for field inside and outside a solid sphere. I don't want to make them out to be anymore than that, because the important thing is how they get reused in for the gravity balloon.

### Gravity Balloon Gravitational Field

The gravity balloon has 3 distinct regions:

• livable air on the inside
• rock walls
• space outside

I will write the gravitational field for the gravity balloon as a piecewise function covering all 3 regions here.

$$g_{gb}(r) = \begin{cases} 0 & r > R \\ g_{rock}(r) & R < r < R + t \\ G M \frac{1}{r^2} & r > R + t \\ \end{cases}$$

As you can imagine, the hard part is that $g_{rock}(r)$, and that is what the rest of the work here is for. The right way is to use $g_{interior}(r)$ to find the field contributions from the rock, but this isn't valid by itself. Instead, we pretend that this is a solid planet of radius $(R+t)$, and write $g_{interior}(r)$ for that, but then subtract the field you would get from the air volume if it were made of rock. This subtraction ("superposition" if you will) is valid for Newtonian field calculations.

$$g_{rock}(r) = g_{interior}(r) - g_{space}(r, M_{\text{air as rock}}) \\ = g_{interior}(r) - g_{space}(r, \rho \pi \frac{4}{3} R^3)$$

(ASIDE: this is the critical step, and it is easy to get it wrong. I have gotten it wrong before, that, and lack of clarity in prior posts, is why I revive this now. Although I did eventually correct my actual numbers, I also had a habit of skipping a ton of steps, so I'm publicly taking it slowly here.)

With pretty good confidence in this established, let's expand it because it will be integrated.

$$g_{rock}(r) = G \rho \pi \frac{4}{3} r - G \rho \pi \frac{4}{3} R^3 \frac{1}{r^2} = G \rho \pi \frac{4}{3} \left( r - \frac{R^3}{r^2} \right)$$

This wraps up the gravitational field. The field is fully described by the expressions for $g_{gb}$ with the supporting $g_{rock}$. There are other tangents I can go on, like contributions from the gravity of the air itself, but this should not be interesting until numerical solutions come into play. I have little very interest in air gravitational contributions as a calculus problem.

## Pressure

To get air pressure we integrate the gravitational field, and then multiply by density. This is justified by our intuition that pressure is $\rho g h$, which is (density)x(gravity)x(height). This is the integral form, specific to the gravity balloon.

$$P = \rho \int_R^{R+t} g_{rock}(r) dr = G \rho^2 \pi \frac{4}{3} \int_R^{R+t} \left( r - \frac{R^3}{r^2} \right) dr$$

Now we perform the integral.

$$P = G \rho^2 \pi \frac{4}{3} \left( \frac{1}{2} r^2 + \frac{R^3}{r} \right) \Big|_R^{R+t} \\ = G \rho^2 \pi \frac{4}{3} \left( \frac{1}{2} (R + t)^2 + \frac{R^3}{(R+t)} - \left( \frac{1}{2} R^2 + \frac{R^3}{R} \right) \right)$$

I feel like it is important to write everything out here so that people can follow. That will simplify a great deal, which I do here.

$$P = G \rho^2 \pi \frac{4}{3} \left( \frac{1}{2} (R + t)^2 + \frac{R^3}{R+t} - \frac{3}{2} R^2 \right)$$

Prepare to combine the fraction.

$$P = G \rho^2 \pi \frac{4}{3} \left( \frac{(R + t)^3}{2 (R+t)} + \frac{2 R^3}{2 (R+t)} - \frac{3 R^2 (R + t)}{2 (R+t)} \right) \\ = G \rho^2 \pi \frac{4}{3} \left( \frac{(R + t)^3 + 2 R^3 - 3 R^3 - 3 R^2 t}{2 (R+t)} \right)$$

The cubic expands into a lot of terms, but the $R^3$ power from it cancels out with other terms.

$$P = G \rho^2 \pi \frac{4}{3} \left( \frac{ t ( 3 R^2 + t (3 R + t) ) - 3 R^2 t}{2 (R+t)} \right) \\ = G \rho^2 \pi \frac{4}{3} \left( t \frac{ 3 R^2 + t (3 R + t) - 3 R^2 }{2 (R+t)} \right) \\ = G \rho^2 \pi \frac{2}{3} \left( t^2 \frac{ 3 R + t }{R+t} \right)$$

Years ago, I know that I did this many times on paper. Now, I realize that I just don't want any of it to get lost. The pressure equation is slightly non-trivial, but it's still within the realm of basic college physics / calculus.

How you use the equations is the one other slightly non-trivial part.

## Usage

The equations above should be thought about in terms of their independent variables. We have functions to give one variable in terms of other.

$$P(R, t) \\ M(R, t)$$

Usually, we prefer to put things in terms of volume, and this can be swapped one-for-one with R in the relationships above, because the relationship between M and R is trivial and only involves those 2.

$$V = \frac{4}{3} \pi R^3$$

This can be done as a preprocessing step in a method numerically. So we can always freely swap V and R as independent variables. I have more commonly written:

$$P(V, t) \\ M(V, t)$$

Referring back to their definitions, neither of these methods are trivial to invert symbolically, as they carry cubic terms. It can be done, but it's not the best for technical communication.

At this point, I stop with math.

We have 2 equations and 4 variables. That means that, somehow, a user has to specify 2 variables, and the other 2 can be calculated. The system is fully defined for any correctly-phrased question. In my younger days, I was inclined to go further into problem solving through math. Now, I am more than happy to let code solve the rest. The only challenge I have not addressed is selecting workable initial values.

I have posted one demo method here:

https://github.com/AlanCoding/gravitational-balloon-mathematics/blob/master/gb/inflation.py

This is the rebooted form. I used Excel Visual Basic macros before. I have successfully converted those to python, which I will use to bootstrap some initial testing.

## Tuesday, January 31, 2017

### Very Preliminary Thoughts on a Hybrid Taper Solution

Material strength is the enemy in the design philosophy of the gravity balloon, as I have laid it out. Honestly, I think that it is always the enemy. It is one more thing that can go wrong operationally, it's not something one would like to rely on for safety. It also lacks a sort of elegance needed for the massive scales and volume of industrial replicability that is needed for massive zero-gravity industry of the future.

Looking at both the approaches for the the task of tapering - bringing the friction buffer sheets into a smaller radius without defeating their economic point - I feel like I'm still missing something. Neither of the designs I wrote about does a good job of eliminating the need for tensile strength due to the different radial acceleration of the air between sheets. On that point, I think I might have already conceded defeat. In gross terms, the extra tensile strength isn't actually a problem. It would pale in comparison to what the rotating tube's hull itself would require, and that would, again, be vastly less than a comparable fully 1-bar pressurized hull. No, I think the problem is requiring both strength and numbers at the same time. The failure modes are not good in that case. There is a level of material quality assurance that must be very robust. Additionally, it has intense tie-ins with the stability questions still outstanding for the many-layer hydrodynamic stability. Oscillation is also a problem if the friction buffers have good mass to them, and the idea of them as something more ephemeral has always been something much more appealing from an engineer's perspective.

I have briefly mentioned a 3rd possibility in addition to the taper-nested and taper-zero options that I have been playing around with. You could basically combine the two. In particular, on the large scale I think that taper-zero is hard to avoid. It keeps the outer motion in the right kind of reference frame, the air ingress patterns give some very needed design flexibility, and it doesn't have those darned problems of undesirable connection points. It's the connection points that I think will sink the other design.

So it's worth while following this rabbit hole for some time. Trash the taper-nested option. Start with taper-zero, and use something apparently exotic - a hierarchical type of design to them. I've used that word before, but I think that "group" might be better, because there would only reasonably be 1 layer to the hierarchy.

The idea is that flow divider sheets are divided into 2 categories. One consists of infrequent "heavy" (although not all that heavy in absolute terms) dividers that hold a pressure boundary across its surface. These do substantial radial tapering toward the ends. The other type consists of a sheet that really has no material strength, and merely rests on air pressure. The connection points for the "flimsy" sheets are almost directly to the side, on the inner surface of their neighborhood "heavy" sheet. For the "flimsy" sheet, this avoids the problems of pressure ride-up in the taper altogether by outsourcing the task to the other type. For the "heavy" sheets, it consolidates the manufacturing and operations complexity by reducing the numbers to around maybe 4, when it might have otherwise required 20. This can also be a go-to solution for centralization of the pressure management of the "flimsy" sheets.

I'm still not sure how contact might be maintained between the heavy sheets and the flimsy sheets. There might still need to be some tapering... but maybe not.

I hope to return to this, and I'm sure I'll make a detailed post sometime after I can get around to sketching some of the basics and doing brief mathematics on it. I just want to throw it out there for now. I'm definitely thinking about it, and it absolutely has tie-ins to the most major problems of the gravity balloon design.

I also find it interesting that this makes vastly more sense as you talk about larger and larger artificial gravity tubes. I've kept assuming a 500 meter diameter, but realize that might be too little. Once you talk about > 1 km, then the non-hybrid approaches start to look more and more unrealistic.

## Wednesday, January 4, 2017

### Global Heat Transfer Without Alternating Layers

Here, I hope to offer an alternative to the previous method I discussed for exchanging hot air from the rotating tubes to the edges of the gravity balloon (from where they would need to go through another heat transport mechanism into the space radiator on the surface).

Because global heat transport is a relatively loosely constrained problem for gravity balloons of most conceivable sizes and densities, we are able to spend some of that margin in order to achieve an interior that may be more desirable to the inhabitants. The previous method is described here:

http://gravitationalballoon.blogspot.com/2014/12/global-air-heat-transport-in-gravity.html

The basic ideas of the first version are:
• The tubes are arranged in a regular lattice where each tube has a neighbor counter-rotating tube
• They are further arranged into cross-sectional layers where all tubes rotate in the same direction
• Sheets are placed spanning the space between similar-rotating tubes to block airflow going from layer to layer
• The flow makes a U-turn once it gets to the heat exchanger on the inner surface of the global gravity balloon wall and enters the neighboring sheet to continue a criss-crossing pattern
The potentially undesirable aspect of this is that the inner space has a great deal of clutter in the form of flow-dividing sheets. You could still have some holes in the sheets for travel of people and goods to go through, but the sheets would still be a nuisance. It would also be mostly mutually exclusive with large open spaces.

The alternative proposal is only marginally modified from that idea. My overall sketch is this:

Here, I am illustrating a pair of counter-rotating tubes. In the previous version of this idea, the air in the above diagram would be moving downward (by the directions in the sketch) within the space between the two tubes. The modification here is that we add a scaffolding around the surface of the tubes in this space. This achieves the goal of bulk flow in one single direction over a large volume within the gravity balloon.

QUALIFIER: Someone unfamiliar with the broader gravity balloon concept in this blog might find it easy to mistake the motion depicted here as directly corresponding to the outer surface of the artificial gravity tube itself. Instead, this is only the outermost friction buffer layer, traveling at a few m/s, instead of on the order of 100 m/s, which would be the outer hull of the artificial gravity tube itself.

What about the global circulation patterns? The air can't only move in one direction, it has to have a net loop in some sense. The best solution to couple with this design is to make the flow regions as large as possible, resulting in a "apple core" sort of circulation pattern. In this sense, there is one major river of flow going straight through the center, which fans out and goes along the outer regions to return to the other side and back again.

The main challenges, as I see them, are the management of the integrity of the shape of the outer layer friction buffer. It is true that I expect some natural wedge effect to help maintain stability, but there will be more deformation caused by the significant asymmetrical drag on the inner segment. Intentionally putting asymmetrical forces on the outer layer will also push them closer to the limit of the stability criteria (whatever that specific criteria may be) and may also increase the drag forces.

I started thinking about this topic more after seeing some artwork in the Accelerando blog. In future posts I do hope to provide more specific commentary and back-links those works. My intent here was just to put this concept out there.

## Monday, May 16, 2016

### Massive Topic List with Half-Finished Writings on Gravity Balloons

NOTE: extremely meta post follows.

Several past posts have been a dump of a large set of topics all balled into one. Now, I would like to do something different. I have used a text file as a staging area where I battled ideas against each other (in terms of where they rank in priority) and also just spitball ideas to see what sticks. While I still like to do these things, I'm depreciating the text file as a tool. I would rather keep the extremely premature ideas on the back of a napkin (or email draft), and use the blog itself as a better format for medium-low effort writings.

Even so, I still have that massive backlog of stuff that I never got to finishing. In my personal philosophy, I want to follow the open source model, and just let information free whenever there's no compelling reason not to. So here it is, my newly retired backlog:

http://pastebin.com/p1qSQkdu

If I'm too wordy in my general writings, then there won't be much of interest in there. Nevertheless, not everything is terrible, and I'm always somewhat taken aback when I read my old writings. It might be nice to polish it up and put it in a nicely displayed format (at least with working links). But let me just go through and dig a few things out of the dredges...

### Small Gravity Balloons

I attacked this concept from several different angles. One specific scale I was interested in was 1 km size asteroids, since these get closer to the NEA territory. Air pressure might still be used to resist against self-gravitation or some other uses, but exactly what is an open question.

### Stability Topics

Nothing I've posted has come close to an exhaustive accounting of the stability topics which span from the gravity balloon part itself, to the internals, to the friction buffers. Then, multiply some of these by all the different size classes. Every problem you look at, you can probably think of at least five different mathematically simplified cases. Each one of those problems could have an enormous amount of analysis, and even radically different formulations.

### Phobos Reference Values

The moon of Mars is just too interesting to give up on. But fairly small, and also close to the Roche Limit, this moon also presents a host of different mathematics that can complicate its scenario. In spite of that, there is still a strong motivation to give definite values for some kind of maximally populated colony.

### Space Development Plan

Where does the gravity balloon exist within some kind of space roadmap? There are some efforts to make a detailed space development roadmap (I have one in mind that was on Kickstarter). These can offer an interesting starting place, but may be too near-term to have much connection to gravity balloons. The NSS also has some interesting roadmaps that involve extremely far-reaching development goals. I am interested in making an ordering. What is likely to happen before a gravity balloon is built? What is strictly necessary?

### Cross-Structures in Artificial Gravity Tubes

Unlike most artist illustrations and typical guidance from 1970s era concepts, the artificial gravity tubes inside a gravity balloon would be extremely friendly for developers who want to build skyscrapers straight across the tube. In fact, it would be necessary if the mass was not evenly distributed (this is more important for gravity balloons, since the shielding mass isn't integral to the hull, making the livable structures relatively heavier compared to the walls).

### Lower-Gravity Tubes and Microgravity Industry

Where would sewage processing happen? Probably in tubes that don't have the burden of friction buffers (accomplished by spinning at very low rates). This seems perfectly fine for algae and other smaller and easy to manage husbandry of animals, etc. What other industry (like shipyards) would happen in the microgravity (but not airless) space? Probably lots.

### Procurement and Management of Air

I address "air" in the basic physical sense a lot, but the chemical realities of production of air is more challenging.

### Other Agenda Items

These topics aren't in that list, because I've only recently been kicking them around:
• Looking into the work by Dr. Forward in his 1990s (and hard to obtain) books
• Of course, actually conducting the scaled experiment
• Inner solar system asteroid Delta V versus mass map (not perfect math, but good-enough with some help from JPL pre-calculated data)

## Sunday, May 15, 2016

### Tether Superstructure for Large Space Cities

Given a large number of artificial gravity tubes located near to each other, they all need to fly in formation. A common picture of how to approach this problem is to have many orbital habitats in synchronized orbits. Since their mutual gravity is small, they all essentially trace their own ballistic orbit, and for many orbital locations (like some L4/L5 point) lots of stable non-crossing orbits exist. This does require some station keeping because we can not arrange the orbits with perfect accuracy to begin with.

Another approach is to tie together several cylinders together to create a larger space station. Like many of these deep-future scenarios I write about on this blog, massive structures would be much better off without needing manufactured steel beams with high compression strength. At a glance, working around the constraints to achieve this ideal might sound impractical, but it's actually entirely reasonable. I will address both scenarios of 1.) the superstructure inside of large gravity balloons and 2.) a generic case for orbital habitats in typical vacuum conditions. These are quite different, but some of the underlying principles are the same.

## Purposes of Tethers

Let's get the first things straight - what specifically the engineering role for a superstructure is.

• Rotation - spinning up and stopping the rotation of the tubes that creates gravity
• Contingencies - if a tube breaks, there is a space traffic accident, or a host of other scenarios, the tethers must be in place to arrest the motion and avoid a local conflagration from destroying the entire space city in a domino effect
• Position - must be maintained against tube-to-tube self-gravitation, external gravitational fields, momentum exchanges from transport, and so on
While this blog might have hit on one or two of these roles, I have never faced them all in any capacity. Elsewhere, I have not seen much of any analysis or reference to the superstructure notion to begin with.

### Scale-Dependence

For a small gravity balloon, it might be completely reasonable to pull against a tether anchored into asteroid rock to maintain or start up rotation. On any larger scales, however, you would just use force-exchange with the neighbor cylinders for this task.

Also, let me modify and expand on a statement I previously made regarding wall-connected tethers:
These must be anchored deep into the asteroid rock
Larger gravity balloons will see different physics in this respect. The tethers can be (and likely would be) held in place by the air pressure itself. This will demand some curvature of the wall, as I illustrate in the sketch below.

Going further, there are many many more interesting things that need to be taken into account on larger scales. As scale increases, the self-gravitation effect becomes larger and larger and for something like Virga, you might very need to imagine something like carbon nanotubes in order to keep all the habitats from collapsing in on the center.

### Tensioners and Enabling Technology

For the superstructure concept to work, we need some form of control input. I have made several references to the generic type of technology whereby a tether can have slack pulled in (thus making it tighter) on command. This is the control machinery side of the equation - we also need good control signals. Simply put, on a large enough scale, the system must have more give than what simple material properties would allow you.

As a counterpoint alternative, think about what the superstructure would be like if it were made of springs. This could accomplish most of what a control system could otherwise do.

Momentum will be fully looked at as a commodity in this system. This implies it can be stored and transferred from one place to another as needed. There are large-scale balance equations that will eventually be satisfied, and a gravity balloon would presumably have an agency that followed these numbers (like the various energy agencies are to us on Earth today).

## A Reference Case

Illustrating what I have in mind - tethers extend from the pressure liner into the habitat. These fork into different directions to connect with the different connection points (and have a curve so they can reach closer to the wall) and also with the rotating cylinders themselves. All throughout the middle space you must imagine a vast number of tubes. Each one of these tubes are connected into the tether network - and the tether network will do the work to hold them safely in place, depending on what the particular needs at the time and location are.

I have drawn an empty space between the tethers and the liner. This isn't strictly necessary because you could just have more tethers connected to the liner. I did not go this route because I assume that additional connections would have an additional cost, but this is not strictly true. So you are free to imagine an innumerable number of tethers extending out from the wall and no "dead space" around the edges. Also, I am burdened by the need to illustrate something.

You must also use your imagination to picture a network of ever-thinner tethers that reaches out to each individual habitat in the entire volume, and connects (eventually) to the superstructure.

## Comparison to Vacuum Orbital Superstructures

In the cases I've described, air pressure is the "ultimate" tension force. That allows you to pull something away from the center. In an arbitrary orbital location, tidal forces might be one of the easily available options. Even a space elevator could be an extreme version of this.

Going in another direction, an orbital ring can be used to pull in another direction. Combining tidal and the orbital ring tug, you could arrange habitats in a 2D plane which is essentially a disk around a planet. This structure could hold all the constituent colonies in place. But let me stress that in the radial directions, "anchors" are needed in order to get a good tug on the tether, while operating on habitats that have relatively small micro-g accelerations acting on them.

These could also be very useful for other mega-structure projects, like solar power arrays, or electromagnetic launchers and catchers. These projects might need precise alignment, and you need something with which to act on in order for them to be viable. The disc-type megastructure might be the ideal option for that.

Another option also presents itself (which I have sometimes briefly hit upon), you could have a large bag of low-pressure air that envelops a large space of orbital habitats just for the purpose of holding the tethers in place and keeping tension on them. This would enable massive 3D habitats, but would not allow free-air access to the different habitats. Since the latter (a full gravity balloon) requires access to a large mass source (a large asteroid), many locations would not have that option and may possibly find a space city of this type the best available option.

## Thursday, April 28, 2016

### Scaled Experiment Metrics and Development Pathway

Stability remains an issue - it is one of the core weaknesses of the proposition for the gravity balloon concept. The issue isn't whether the fundamental principle of laminarization will work, or even necessarily that the channel flow physics are suitable for this problem, but the behavior of a complex and fairly high-energy system.

If you took a simplistic picture of the wedge effect, the prediction would be a slight restorative force for each layer of the friction buffers. The exact directionality of this force gets complicated. It gets even more complicated by the non-rigid nature of the buffers. What we should really raise our eyebrows at, however, is what happens when you consider all the numerous sheets in tandem. Loosely connected dynamic systems can be prone to failure, and this situation seems like a candidate for that. But then again, maybe not. The solution becomes fairly non-trivial.

Due to the complexity of the problem, our only best option left is to perform scaled experiments. Ideally, we would like to use a fluid that is more convenient, in that the same flow patterns can be produced at a smaller scale. To do this, we will look at the Reynold's number.

Re = rho v d / mu

We would like something with a high density and a low viscosity (in particular, relative to air). It becomes fairly obvious that water is our best option for this. Also, you can see that we have 2 dimensions over which to flex our abilities - that is the velocity and the distance metric. Now it seems pretty clear that we want the smallest structure that we can spin at workable speeds.

These 2 degrees of freedom would still dictate a massive scale of experiment, if taken to apply to the entire system all at once. By that, you can say that d is the diameter of the outer-most friction buffer (for example), and that the system will be a scaled model in terms of all geometric dimensions. The enormous cost of this nudges us to seek cheaper solutions that might go 80% of the way with 20% of the effort. In a more accurate accounting, I'm looking for something more like 10% of the ultimate discoveries with 0.001% of the effort.

Obviously, we might instead take d to measure the distance between sheets, while also relaxing the requirement for strict geometric similarity. This might be nice in order to make something testable by reducing the number of sheets compared to the gravity balloon reference design. Thus, the overall scale and velocities will be dramatically less, while still demonstrating channel-to-channel interactions with the same flow patterns.

This still won't be sufficient to make an immediately tenable experiment. We'll need to relax something related to the channel flow pattern itself. The obvious candidate is to change the channel width relative to the overall tube diameter. However, I will not count this as an independent variable, because I think that (for most cases) it will fall out of the selection of the number of sheets. In any format you choose, it's likely that the ratio of the overall thickness of the friction buffer region will be about 50% of the tube radius. So greater channel width will follow with fewer sheets.

Our hypothetical experiment has been cut-and-slashed a lot by this point, but we're not finished yet! What is truly the really important point? What would we want to learn from this? I would argue that it is the interaction of multiple friction buffers in a (generally sufficiently) turbulent flow regime. Even if you cut that out, there's still some value because it answers some questions about this broader notion of friction buffers (which can even have other applications). However, we do want to answer questions about the friction buffers used in a gravity balloon regarding their stability. We basically know that the answer will be different for laminar and turbulent (or at least somewhat independent). Let me illustrate my thinking in a sketch.

(let me volunteer that I know I illustrated the transition region poorly)

Basically, we want to probe on the minimum edge of turbulent flow regimes with a multi-sheet friction buffer system. Just opt for an outright change of Reynolds number according to the abilities available for experimentation. This would be the 10% of ultimate knowledge I'm interested in. This would set the stage for everything that may (or may not) come in the future.

The good news - all this slashing of the metrics gets our experiment size way down (the scale is highly sensitive to Reynolds number). And that gives us some wiggle room. With lower Reynolds number, we can play around with more sheets (even up to 10 or 16 as I'm dreamed about), while staying at least in the turbulent regime. With the same general equipment, dial the numbers the other direction, and see about higher Reynolds number channels with fewer sheets.

Once you start chewing on this, something new starts to take form - a general format of the development path. Because as these numbers are dialed back up to the full scale (with more resources), its possible to speculate when many different components of the design will be proven in principle. After that, you can image at what point those components will mature into a representative suite of technologies. For instance, at certain numbers, the intra-sheet flow management will become testable. At another point later on, active controls for maintaining the seals could be strapped on.

So now we've covered 4 (mostly) independent factors. I think this is probably the right way to look at scaling of real, physical, experiments. These can start on a household scale.