A topic that has repeatedly given me confusion is the notion of fluid flow through animal vasculature. I find that many of the 'physics 101' basic notions of fluid dynamics are not well-suited to deal with the more complicated context of animal vasculature (or, more likely, I am misapplying them). I suppose the easiest model to play with (that aims to approximate organism vasculature) is something like a closed loop, mixed in-series / in-parallel system with varying radii of 'pipes' along the length of the system and a temporally varying cross-sectional area (capturing the capacity of blood vessels to expand and contract). Consider, for example, the following picture:
One instance of a confusing claim associated with such systems is the following:
Increasing the diameter of a blood vessel will increase blood flow.
One particular application of this statement is: "Small vessels (capillaries) embedded within muscle tissue can send a signal upstream to larger vessels (arterioles) that increases the diameter of the larger vessel. Consequently, greater blood flow will be observed in the small vessels (for the duration of of the above-normal cross sectional area of the larger vessel)"
It is completely unclear to me why this is the case. Consider the following graphic:
Why should the blood flow at the red arrow increase? What equations are being used to justify this? The only thing that physics 101 stuff tells me is that the segment of the vessel (dashed green lines) that experiences an increase in cross sectional area will contain particles moving at a commensurately reduced velocity (by conservation of mass). Moreover, given that there is a conservation of mass along that segment, why should the amount of exiting fluid (from the most downstream dashed green line) be any greater than it was previously? Any clarifications would be greatly appreciated.
The only thing I can think to consider is that there must also be simultaneous vasoconstriction of nearby vessels.
EDIT
After thinking about this question a little bit more, I felt that the following illustration would be useful to reference (which can be viewed as a segmental subset of the original illustration I provided).
Here is some relevant information that may facilitate more biologically-tailored responses:
We can assume that the pressure generating device (i.e. the heart) is outputting the same force at time one ($T_1$) and time two ($T_2$) $\quad (\dagger_1)$.
Also, it is my belief that the core concept under investigation is unrelated to the pulsatile nature of the heart (so let us just assume that there is no temporal variation to the pressure generated by the heart).
Further, suppose I am uninterested in the 'temporal dynamics' that follow the geometric changes to the $r_2$ vessel. Instead, let us assume that the flow behavior we will discuss is specific to the eventual arrival of steady state conditions...i.e. $T_1$ and $T_2$ represent points in time where flow behavior has reached steady state.
With these stipulations in place, my questions are as follows:
Does the flow rate in the $r_1$ or $r_3$ pipe change between the $T_1$ and $T_2$ geometries? If so, why? How is it that the pressure at the inlet of the $r_1$ or $r_3$ pipe changes between $T_1$ and $T_2$ despite the fact that no changes in geometry take place?
Is the pressure at the $r_2$ inlet different than the pressure at the $r_2^*$ inlet?
I think most of the subsequent understanding can be handled if I can get some insight into these two questions. For example, if the pressure changes at the inlet of a pipe, then flow rate through that pipe changes, which may explain why my original understanding is incorrect! Specifically, it is incorrect because the mass conservation argument is based on the premise that the flow rate is the same between the two different geometries.
