Stagnation Points Beneath Rotational Solitary Waves in Gravity-Capillary Flows

ABSTRACT

Stagnation points beneath solitary gravity-capillary waves in the weakly nonlinear weakly dispersive regime in a sheared channel with finite depth and constant vorticity are investigated. A Korteweg-de Vries equation that incorporates the surface tension and the vorticity effects is obtained asymptotically from the full Euler equations. The velocity field in the bulk fluid is approximated which allow us to compute stagnation points in the solitary wave moving frame. We show that stagnation points bellow the crest of elevation solitary waves exist for large values of the vorticity and Bond numbers less than a critical value that depends on the vorticity. Remarkably, the positions of these stagnation points do not depend on the surface tension. Besides, we show that when there are two stagnation points located at the bottom of the channel, they are pulled towards the horizontal coordinate of the solitary wave crest as the Bond number increases until its critical value.

Keywords:
gravity-capillary waves; Euler equations; KdV equation; stagnation points