Approximate Solution for Two-Dimensional Free Surface Flow Using Perturbation Methods

Abstract

The two-dimensional free-surface flow of an inviscid and incompressible fluid passing over a triangular hollow at the bottom of a channel is studied in this paper. The flow is considered irrotational and steady. The effect of surface tension is taken into account, while the force of gravity is neglected. The region in the potential $W$ plane is mapped onto the auxiliary $\lambda$ plane using the Schwarz--Christoffel transformation. The primary objective of this research is to provide an approximate solution for this problem using the perturbation approach together with the Hilbert transform; the results are computed at high Weber numbers and small inclination interior angles. The solution is provided and illustrated up to first-order approximation. The shape of the free surface for different values of: the Weber number $\alpha$, the depth of the hollow $l$ and the inclination interior angles $\gamma$ and $\delta$ are plotted. The findings demonstrate the simplicity of the method used and give an approximate solution for similar problems.