  We obtain a rigorous upper bound on the resistivity $\rho$ of an electron
fluid whose electronic mean free path is short compared to the scale of spatial
inhomogeneities. When such a hydrodynamic electron fluid supports a non-thermal
diffusion process -- such as an imbalance mode between different bands -- we
show that the resistivity bound becomes $\rho \lesssim A \, \Gamma$. The
coefficient $A$ is independent of temperature and inhomogeneity lengthscale,
and $\Gamma$ is a microscopic momentum-preserving scattering rate. In this way
we obtain a unified and novel mechanism -- without umklapp -- for $\rho \sim
T^2$ in a Fermi liquid and the crossover to $\rho \sim T$ in quantum critical
regimes. This behavior is widely observed in transition metal oxides, organic
metals, pnictides and heavy fermion compounds and has presented a longstanding
challenge to transport theory. Our hydrodynamic bound allows phonon
contributions to diffusion constants, including thermal diffusion, to directly
affect the electrical resistivity.
