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Hydraulic analogy

Electricity (as well as heat) was originally understood to be a kind of fluid. This hydraulic analogy is still of some use in teaching, not only for the fact that the names of the quantities are often struck by analogy.

type hydraulic electric thermal
quantity volume V [m3] charge q [C] heat Q [J]
potential pressure p [Pa=J/m3] potential φ [V=J/C] temperature T [K=J/kB]
flux current ΦV [m3/s] current I [A=C/s] heat transfer rate \dot{Q} [J/s]
flux density velocity v [m/s] current density j [C/m2s] heat flux \dot{Q}'' [J/m2s]
linear model Poiseuille's law \Phi_{V} = \frac{\pi r^{4}}{8 \eta} \frac{\Delta p^{\star}}{\ell} Ohm's law j=-\sigma \nabla \phi Fourier's law \dot{Q}''=\kappa \nabla T
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