Alternating currents and shear waves in viscous electronics

Strong interaction among charge carriers can make them move like viscous fluid. Here we explore alternating current (ac) effects in viscous electronics. In the Ohmic case, incompressible current distribution in a sample adjusts fast to a time-dependent voltage on the electrodes, while in the viscous case, momentum diffusion makes for retardation and for the possibility of propagating slow shear waves. We focus on specific geometries that showcase interesting aspects of such waves: current parallel to a one-dimensional defect and current applied across a long strip. We find that the phase velocity of the wave propagating along the strip respectively increases/decreases with the frequency for no-slip/no-stress boundary conditions. This is so because when the frequency or strip width goes to zero (alternatively, viscosity go to infinity), the wavelength of the current pattern tends to infinity in the no-stress case and to a finite value in a general case. We also show that for dc current across a strip with a no-stress boundary, there are only one pair of vortices, while there is an infinite vortex chain for all other types of boundary conditions.