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 Poiseulle is inversely proportional to the fourth power radius. 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 Bernoulli'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.

• Please use paragraphs - this very hard to read – user140434 Jan 7 '17 at 13:46
• I thought I did, but they somehow got lost. – Whiterabbit Jan 7 '17 at 14:05

No, you are mixing up concepts at random. First of all, there's no reason for pressure to increase just because the surface area of the tube decreases. The definition of pressure $p=F/A$ does not allow you to draw this conclusion, since you have no way (without additional information) to know what happens to the numerator of this expression.

Second, you need to keep your boundary conditions straight: Are you performing an experiment where you keep the flow rate constant? If that is the case then, yes, your pressure gradient will increase with decreasing tube radius, and the velocity will increase, of course. Often, however, we keep the pressure gradient constant (e.g. in a gravity-driven flow), in which case your flow rate decreases with the tube radius. Or, you may have a pump driving your system that has constant available power, in which case the product of pressure drop times flow rate is constant, and reducing the tube diameter may both increase the pressure gradient and decrease the flow rate, depending on the type of pump you use. In this case the flow velocities may or may not increase. Long story short, no, you're wrong on this one, too.

Finally, no, your ideas around how and why the pressure changes with reduced tube diameter and the connection with particle collisions with the wall are wrong. The fact that the Bernoulli equation assumes frictionless flow has been mentioned already.

Let's talk about blood flow. It's a closed system with a pump and resistance, exactly analogous to an electrical circuit with resistors (and capacitors and inductors, but let's ignore those.)

Blood is approximately water, and it is not flowing very fast, and it is flowing through a lot of long narrow tubes (capillaries). In such tubes, there is resistance due to viscosity, so the longer and narrower the tubes, the greater the resistance.

Bernoulli's principle does not deal with viscosity. It simply says that a fluid cannot gain speed without a pressure difference pushing it forward. Similarly it cannot lose speed without a pressure difference pushing against it. This is both conservation of kinetic energy and conservation of momentum.

When fluid moves through a long narrow tube it is not conserving kinetic energy, because the viscous resistance causes the fluid to convert kinetic energy to heat.