I was listening lectures from Mark Saltzman via Open Yale Courses. Now, if I did not understand him wrong, he said that the amount of blood that travels through every vein and artery in our bodies per minute is a constant. Five liters of blood go through the aorta every minute, and since the blood circulation is a closed system, also every other vein/artery must pass that amount of blood per minute. Is this true? How could I (mathematically) show that it is true?
I think the question is not whether the blood flux is constant in time, but rather in space: if 5L/min is the flux through aorta (in given physiological conditions), then at the same moment 5L/min will cross the capillaries. This is volume conservation, for a closed system filled with an incompressible fluid, see this sketch:
If 5L are pushed through the main pipe, the same volume needs to be pushed through the system downstream of it.
Then comes the fact that the system is not perfeclty closed, and some water (and solutes) will "leak" in and out, as @inf3rno points, so this is only approximatively true. Another limitation is that vessel walls are actually elastic, so a change in vessel radius does occur during blood flow. However I believe it is a fair approximation.
It is ofc. not true, because it is not a closed system. The kidneys filter blood and excrete water and other compounds (which are reabsorbed or go to urine), the sweat glands remove water and salts. By drinking daily 2 liters (the total blood volume is about 5 liters) of fresh water and salts are absorbed from the intestines to the circulation, by eating just the opposite happens. Every cell has aquaporins which are responsible for water transport, so they can dilute the blood for a while, if necessary. The spleen can withhold red blood cells (before injury) which reduces the blood volume. The white blood cells can leave the circulation. By inflammation water leaves the vessels because of increased permeability, and so on...
Nope. All blood flows in the body can change at any time(more or less).
If the heart rate and blood pressure, etc, is constant the amount of blood that travels through the aorta is constant, yes. But there is far more capacity in the arterial and venous systems for the actual amount of blood flow there normally is. Veins and arteries contract and expand to route blood flow to different parts of the body, so the amount of blood flowing through any particular artery can change without changing the aortic flow.
A good example of this: Vasomotion. It happens all the time in your hands (sometimes it's even visible as patches of red/white skin. On my hands they're a little smaller than the head of a thumbtack) on the order of minutes. fMRI is a neat technology based around watching brain blood vessel changes in flow as a proxy for activity.
All the flows through all your arteries don't even have to sum to the aortic flow, due to compliance(a fluid dynamics term. essentially, the aorta or other arteries/veins can 'inflate' so more blood is flowing in than is flowing out. Then later they 'deflate' and more blood flows out than in. Arteries tend to be more compliant because they have to be springier to deal with heartbeat). There are also osmotic effects where fluid leaves the arteries and becomes lymph, which then later drains back into veins. It's not a huge effect but it's there. If you measure total blood flow through the aorta and then total blood flow through all the capillaries, there will be less total blood flowing through the capillaries. Not much less( 10% at the most) depending on blood salt and blood vessel position and whether you count the kidneys, etcetera etcetera.
So: the amount of blood that passes through a given artery given the aortic blood flow is not fixed. The amount of blood that passes through all arteries given the aortic blood flow is much closer to fixed, but even that can be different(even if you're not bleeding). And if you were asking if the amount of blood that flows through the aorta is constant, that's absolutely not true. That can vary through the heart rate, the contractility of the heart, the venous return pressure, the overall blood pressure, and a bunch of other things that very slightly affect flow rates.
The amount of blood flow is not constant in general. As you exert yourself, your heart rate and therefore blood flow increases. Furthermore, individual vessels dilate or constrict to increase or decrease flow by modulating their resistance to flow.
Your circulatory system is closed however, so all of the blood leaving the heart must eventually come back to it.