Interdisciplinary Note (3 of 16)
Analogy between Ohm's law and Poiseuille's Equation.
Resistance to flow is analogous to electrical resistance.

There are a useful set of analogies to draw between Ohm's law and Poiseuille's law from fluid mechanics. Ohm's law governs the relationship between voltage, current, and resistance in a DC circuit. Poiseuille's law describes the relationship between the pressure gradient, volume flux, and various fluid and flow vessel parameters determining the flow of a real fluid, ie. a fluid with viscosity.

This analogy is productive towards intuitive conceptual understanding. (Medical school cardiovascular physiology will draw this analogy out into a whole analytical framework.) You can think of the voltage across a resistor like a kind of pressure. The voltage gradient impels the flow of current through the resistor, and you can see the voltage drop that occurs across the resistor like you see the pressure falling off down the flow-line. In both cases, energy is dissipating. The voltage drop tells you the loss of energy in joules per coulomb (volts). In the flow of a real fluid, the pressure drop tells you the loss of energy through viscous dissipation in joules per m3.

It's especially useful to put some attention on the bundle of terms occupying the place of "resistance" in Poiseuille's law when you compare it to Ohm's law. The more viscous the fluid and the longer the vessel segment, the greater the resistance to flow through that segment. The greater the radius, the less the resistance. Decreasing the vessel radius would increase resistance fast. Half the radius equals sixteen times greater resistance, all else being equal.

You can actually find the equivalent resistance of vessel segments in series in the cardiovascular system by adding them, just like you do in DC circuits, and the equivalent resistance of parallel vessel segments is also computed the same way as parallel resistors in a DC circuit. If the resistance to flow through a segment is great, the pressure gradient will be steep there. There will be a pressure drop just like a voltage drop. This gives you insight into why most of the pressure drop in the cardiovascular system is in the arterioles. You might think most of the pressure drop should be in the capillaries, because the capillaries are so narrow, but because the number of capillaries is so huge, the actual pressure drop there is not as great as though the arterioles. Like an enormous set of parallel resistors, the equivalent resistance through the capillary beds is not as great as through the arterioles.

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