Are there basic rules to properly design vein channels?

Question

I am trying to model a prototype resembling a section of vein channels. I am wondering if there are some basic rules for such fluid channels, for instance when a vein splits in two capillaries, is there some quantity conserved (area of the section or else)? I observed that with my current design (which is done by hand just imitating pictures I saw) the fluid I pump in tends to flow very fast in some channels and very slow in others. I'd want some sort of common front to advance. I am a mathematician and have ZERO education on fluid dynamics or else :) any help is much appreciated!

EDIT: I saw the comment answer and I apologise because I am not sure I will be able to formulate my question in a decent way for engineers :). I am printing my model in rigid plastic and the application is scanning its section in the X-ray CT lab. Here's a couple of pics that I hope will clarify my position:

The idea is to have a viscous fluid (honey+iodine) put in the top "funnel" and start measuring in the CT lab, observing it flowing into the bottom channels. Observe that the channels have a rectangular section.

Answer 1

If I understand your question correctly, you are designing and building a small network of pipes or the like that involves different diameters. You have little experience with this kind of design work. You are looking for rough heuristics to help you get started. I assume pipe flow, not open channels.

... when a vein splits in two capillaries, is there some quantity conserved (area of the section or else)?

This is actually a good idea. In a first approximation, if you conserve area flow velocity will be equal in all branches.

I'd advise you however to look at and understand the Darcy-Weisbach equation. Looks daunting, but its probably less work to understand than to build endless prototypes. The upshot is that specific pressure loss (pressure / length) depends strongly on flow velocity, but not only - so our first approximation is wrong but maybe useful.

The next thing you need to understand is pipe network analysis. Then you can build a simulated model of your system.

Give that you say "ZERO education on fluid dynamics" I'll throw in a link on how to read pump and system curves.