# relationship between electrical resistance and Hagen Poiseuille's law

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?

• 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. – CuriousOne Mar 22 '16 at 0:49
• The length divided by cross section area is common due to the similar geometry, and how the parameters are defined. Nothing more to it. – Peter Diehr Mar 22 '16 at 2:18
• 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? – Marc Mar 22 '16 at 11:21