On the propagation of a two-dimensional viscous density current under surface waves

This study aims to develop an asymptotic theory for the slow spreading of a thin layer of viscous immiscible dense liquid on the bottom of a waterway under the combined effects of surface waves and density current. By virtue of the sharply different length and time scales (wave periodic excitation being effective at fast scales, while gravity and streaming currents at slow scales), a multiple-scale perturbation analysis is conducted. Evolution equations are deduced for the local and global profile distributions of the dense liquid layer as functions of the slow-time variables. When reflected waves are present, the balance between gravity and streaming will result, on a time scale one order of magnitude longer than the wave period, in an undulating water/liquid interface whose displacement amplitude is much smaller than the thickness of the dense liquid layer. On the global scale, the streaming current can predominate and drive the dense liquid to propagate with a distinct pattern in the direction of the surface waves.