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For the electrical resistance of a conductor, we have $$R = \rho \frac{l}{A}$$

Noting the structural similarity between the Hagen-Poiseuille law and Ohm's law, we can define a similar quantity for laminar flow through a long cylindrical pipe: $$R_V = 8\eta \frac{l}{Ar^2}$$

So there's a structural difference of a factor of $r^2$ between the two. What's the intuition behind this?

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  • $\begingroup$ For a round wire we have $R\propto r^{-2}$ while Hagen-Poiseuille will give you $R_V\propto r^{-4}$ and we never have to use $A$ as flow in non-round pipes is not characterized by any such law with area dependence, to begin with. Also keep in mind that for very slow flow Hagen-Poiseuille wouldn't apply, at all, while there is no such low-current limit for Ohmic resistors. $\endgroup$ – CuriousOne Mar 22 '16 at 0:49
  • $\begingroup$ The length divided by cross section area is common due to the similar geometry, and how the parameters are defined. Nothing more to it. $\endgroup$ – Peter Diehr Mar 22 '16 at 2:18
  • $\begingroup$ Thanks for your comments. Maybe it doesn't make much sense if I write $\pi r^4$ as $Ar^2$ because doubling the area doesn't halve the resistance. But my question still remains: what are the physical principles which lead to the different dependencies in the two cases? $\endgroup$ – Marc Mar 22 '16 at 11:21
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Viscosity is the difference. Hagen-Poisseuille thinks of the fluid as concentric cylinders dragging on one another with drag being proportional to the contact area and the relative speed.

Motion of elecrons is not usually like this. Electrons move as diffracting waves scattering from imperfections of the material lattice, their kinetic energy being partly absorbed by the lattice. This absorption is what gives rise to ohmic resistance. Electrons don't normally drag on one another like the fluid, so the conductance of an ohmic conductor is proportional to the cross sectional area. If you double this area, you double the possible paths.

Electrons in some materials do indeed behave like viscous fluids. There have to be quite unusual correlations and conditions for this to happen, but it can nonetheless. Electron viscosity is a fairly active, recent research topic.

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Not much more than perhaps an analogy for circular cross sections. And the analogy can fall apart for different cross sections (consider a Venturi nozzle).

But for limited situations one could use such analogies and linear electrical circuit laws (Kirchoff's, Norton, etc.) to solve for fluid system behaviors. But beware... the analogy is limited since fluid relations can go nonlinear. The Pouiselle relation assumes steady state axi-symmetric (one -dimensional) fluid flow. And flow doesn't necessarily seem to behave the way you want it to - like creating vortices and eddies. You just don't see that happening with electrons in a copper wire!

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