Conformal maps of viscous electron flow in the Gurzhi crossover

We investigate the impact of geometric constriction on the viscous flow of electron liquid through quantum point contacts. We provide analysis on the electric potential distribution given the setup of a slit configuration and use the method of conformal mapping to obtain analytical results. The potential profile can be tested and contrasted experimentally with the scanning tunneling potentiometry technique. We discuss intricate physics that underlies the Gurzhi effect, i.e. the enhancement of conductivity in the viscous flow, and calculate the temperature dependence of the momentum relaxation time as a result of impurity assisted quasi-ballistic interference effects. We caution that spatially inhomogeneous profiles of current in the Gurzhi crossover between Ohmic and Stokes flows might also appear in the non-hydrodynamic regime where non-locality plays an important role.