Relation between resistance to the flow of fluid, pressure and radius of the vessel
I just need a clarification, that may seem absurdly easy to some, fair enough, but I am not a physicist and would be glad to hear some answers.
Regarding blood floow, or any flow through a tube for that matter, it is intuitive that decreasing the diameter or radius of the vessel increases the pressure (since pressure is the force of the particles exterted upon a certain surface area of the vessel wall, F/S). However, we then have resistance that is of course, using analogy with Ohm's law, inversely proportional to flow; Q (flow) = Pressure gradient /R. Bigger the resistance, smaller the flow. However from that same equation Pressure difference is directly proportional to resistance, meaning bigger the resistance bigger the pressure difference - bigger the decline in pressure is. The pressure difference is driving pressure, though, and that is a "force" that drives fluid from one end of the vessel to the other. So am I right if I say that, when resistance is increased, the flow decreases but the velocity of the flow increases because of bigger pressure difference (also, conservation of mass and smaller radius since usually resistance increases with smaller radius)? I just need a confirmation of that one, however my real question is as follows.
Resistance according to <PERSON> is inversely proportional to the fourth power radius. 439
So smaller the radius much bigger the resistance, much more decline in pressure. But I also said in the beginning that the pressure increases with decreased radius because of more collisions between particles and vessel walls. This seems like a bit of a paradox to me so I would need a bit explanation for it. I see the same "contradiction" in <PERSON>'s equation, where because of total energy conservation, pressure is decreased when velocity is increased (when there is a decrease in radius of the vessel). From a conservation of energy's standpoint that is completely logical however, I cannot imagine pressure decreasing with smaller radius from a "molecular" point of view. Thank you in advance. 439