On the existence of solutions of flows due to pressure distribution

Abstract

Steady two-dimensional flows due to an applied pressure distribution in water of finite depth are considered. Gravity is included in the dynamic boundary condition. We use perturbation technique to derive the forced KdV equation. Existence theorems for different types of solution to this equation are given and proved. Numerical solutions are provided as the confirmation to these findings. There exist up to two solutions that correspond to the same value of froude number greater than unity for positive pressure distributions, and a unique solution for negative pressure distribution. When the Froude number is less than unity, solutions are characterized by a train of waves behind the pressure distribution. These waves diminish when the Froude number approaches some critical values.