Stability analysis and solitary waves on Plateau border flows

We investigate the linear stability of a Plateau border and the existence of solitary waves. Firstly, we formulate a new non-orthogonal coordinate system that describes the specific geometry of a Plateau border. Within the framework of this coordinate system, the equations of motion for the fluid potential and free surface are derived. By performing a linear stability analysis we find that the Plateau border is stable under small perturbations. Next, a weakly nonlinear theory is developed, leading to a Korteweg–de Vries equation for the free surface profile. Our weakly nonlinear evolution equation predicts depression solitary waves, such as those observed by Bouret et al. ( Phys. Rev. Fluids. , 2016, vol. 1 no. 4, p. 043902).