## 1 Introduction
Moreover, the parallel supercritical heated channel systems could be stabilized by the increases in inlet flow resistance or channel diameter, while the unstable influences on the system stability could be attributed to the increases in the outlet flow resistance or channel length [14]. The seismic issues are very crucial for the safe operations of various power generation systems. However, the relevant studies of seismic impacts on the flow stabilities were very limited in the literatures, which majorly conducted in nuclear boiling systems [17, 18, 19, 20, 21]. Among these studies, two major approaches, i.e., external force method [17, 18, 19, 20, 21] and moving grid method [21], were employed to evaluate the seismic impact on two phase flows. The external force method is usually considered in the rigid systems, where no deformation occurs in the system structure during the seismic motions; otherwise, it would require the moving grid method instead. The previous study indicated that the impact of seismic vibration on two-phase boiling flows may depend on its intensity and the inherent stability natures of operating states. The resonance oscillation can be triggered by the seismic wave if its oscillation frequency is equivalent to the natural frequency of working flow [17]. In a parallel nuclear-coupled boiling channel system, the external vibration acceleration could result in the resonance phenomena with either regional or core-wide instabilities. The impact of vertical acceleration would generate a much stronger effect on the core power oscillations with respect to those in the other directions [18]. Our recent studies also illustrated that complicated seismic-induced oscillations could exist in a nuclear-coupled boiling channel system [19]. For a natural circulation loop involving parallel nuclear-coupled boiling channels, the resonance effects in different natural circulation stable states would dominate seismic-induced flow oscillations. Under the impact of vertical seismic motion, the parallel channels tended to exhibit in-phase mode of oscillations. A more significant resonance effect could be triggered by the natural circulation state with a higher core inlet subcooling [20]. To the authors' best knowledge, at present, the research regarding the influence of external vibration, such as seismic acceleration, on either single channel or parallel channels of the supercritical heated system is very sparse. The unique nature of parallel supercritical heated channels exhibits the complex interactions distributing among them, in which the unstable phenomenon is commonly the compound type of density-wave instability merged with parallel channel instability. The seismic impact interacting with the distinctive stability characteristics of such systems should need to be explored. Thus, the major objective of the present study is aimed at developing a non-linear dynamic model to deal with the effects of external vertical acceleration and seismic acceleration on the transient behaviors of parallel uniformly-heated channels with supercritical water. The non-linear model can also be applied to examine the seismic effects on the single supercritical heated channel after the proper simplifications.