As I thought, this is an extremely well-studied & modelled area. The following is for incompressible fluid (blood) flow. For example, Murray's Law for a bifurcation: r^alpha = r1^alpha + r2^alpha where r is the radius of the parent node, and r1,r2 are the radii of the child nodes. alpha is a constant >=2. alpha=2 => sum of cross sectional areas of the children = cross sectional area of the parent. alpha=3 => minimizes work for laminar flow. alpha=7/3 => minimizes work for turbulent flow. https://en.wikipedia.org/wiki/Murray%27s_law Also, Adam, John A. "Blood Vessel Branching: Beyond the Standard Calculus Problem". Math. Mag. 84 (2011), 196-207. Adam calculates the height of the binary tree as around k ~ 30, so we don't need a 64-bit machine! Adam claims (sum area of children)/(area of parent) ~ 1.26 ~ 2^(1/3). https://en.wikipedia.org/wiki/Hemodynamics Cross-sectional area of the aorta (root) ~ 3-5 cm^2 velocity ~ 40 cm/s. Total area of the capillaries ~ 4500-6000 cm^2 velocity ~ 0.03 cm/s. i.e., total area ratio ~ 1000 = 10^3. --- So if rc is the radius of each child (exactly 2 children), and if rp is the radius of the parent, then (2*pi*rc^2)/(pi*rp^2) = 2^(1/3) So, rc/rp ~ 2^(-4/9) ~ 0.735 But Murray's Law gives rp^alpha = 2*rc^alpha, or 1/2 = (rc/rp)^alpha, or rc/rp = (1/2)^(1/alpha) Putting these equations together yields a Murray-type law with alpha = 9/4 = 2.25, i.e., rp^(9/4) = 2*rc^(9/4). The next thing to do is to estimate the flow resistance for each level 1-30, which are all in *series*. At 04:21 PM 10/9/2018, Henry Baker wrote:

I'm interested in a balanced binary tree of air flow; the flow comes in at the root, and goes through k levels of binary branching, yielding 2^k outputs.

I'm assuming that air flow resistance varies superlinearly with the flow -- i.e., trying to push 2x amount of air should face a resistance which is >2x -- perhaps 4x or 8x. Let's assume that whatever exponent is used, perhaps 2<alpha<3, the same alpha occurs at each level of the tree, so that the scaling is simple fractal.

For the moment, let's assume that these pipes are cylindrical, and that the joint where one pipe becomes two is designed so that the primary resistance isn't the joint itself, but merely the sum of the resistances of the branches. There is also a slight "enlargement factor" at each branch, whereby the area of the branches at level k+1 is slightly larger than the area of the branches at level k.

My intuition is that as the flow increases due to increased pressure at the root, the maximum resistance occurs at smaller and smaller k's -- i.e., closer to the root. Equivalently, as the pressure at the root decreases, the maximum resistance occurs at larger & larger k's -- i.e. further from the root.

Is this intuition correct?

Any references or links?

Next question: what happens if the fluid is *incompressible* -- e.g., water?