There are many flexible vessels in human body, such as artery, air way and urethra. When these vessels subject to a negative transmural pressure (internal minus external) while conveying flow, self-excited oscillations can occur. We explore the mechanism of self-excited oscillations in a two-dimensional rigid channel conveying flow. One segment of the upper rigid wall is replaced by a flexible wall under external pressure. In particular, we construct a general framework for analyzing the energy budget of self-excited oscillations about a non-uniform basic state. We then apply this general framework to consider two particular models for the flexible wall, namely a fluid-beam model [1] and simple fluid-membrane model [2] with an external pressure gradient. For fluid-beam model, a modified constitutive law is used to ensure the elastic beam is energetically conservative. The steady system shown multiple steady configurations: an (inflated) upper branch and a (collapsed) lower branch, connected by a pair of limit point bifurcations to an unstable intermediate branch. Both upper and lower steady branches can each become unstable to self-excited oscillations. In addition, detailed energy budget over a period of oscillation was calculated, where we shown that both upper and lower branch instabilities require an increase in the work done by the upstream pressure to overcome the increased dissipation. For fluid-membrane model, numerical simulations indicate that the baseline state two unstable normal modes: the Tollmien–Schlichting (TS) mode and a surface-based mode which manifests as one of two flow-induced surface instabilities (FISI), known as travelling wave flutter (TWF) and static divergence (SD), respectively. We find that both FISI are primarily driven by the working of normal stress on the flexible wall, lower-branch SD has negative activation energy, while upper-branch SD approaches zero activation energy in the limit of large wall damping. [1] Danyang Wang, Xiaoyu Luo, Zishun Liu and Peter S. Stewart. Instabilities in collapsible channel flow with a pre-tensioned elastic beam. Journal of Fluid Mechanics, Vol. 1024, pp. A58, 2025. [2] Danyang Wang, Xiaoyu Luo, Zishun Liu and Peter S. Stewart. Flow-induced surface instabilities in a flexible-walled channel with a heavy wall. Journal of Fluid Mechanics, Vol. 956, pp. A1, 2023.