Non-linear stability analysis of pressure drop oscillations in a heated channel

A study of non-linear stability analysis using co-dimension two (i.e., two free parameters being varied) bifurcations related to pressure drop oscillations (PDO) in a heated channel has been carried out in this paper. In the existing literature, in the context of PDO, mostly linear stability analysis is done. A few works on non-linear stability analysis are available; however, only co-dimension one bifurcation studies of PDO have been carried out in these works. However, in practice, since the inlet temperature of the coolant is also an independent operating parameter, both inlet temperature, and inlet mass flow rate need to be considered for stability analysis. It is also noted that the existing plethora of studies on PDO is limited to the prediction of supercritical Hopf bifurcation only. However, in the current study, two types of Hopf bifurcations have been identified namely subcritical and supercritical. A subcritical Hopf bifurcation exhibits unstable limit cycles, which is a signature of the existence of unstable solutions for slightly larger perturbations even in the linearly stable region. The existence of unstable solutions indicates that linear stability analysis is not sufficient to identify the overall stability behavior. Also, a generalized Hopf point on the stability boundary has been identified which denotes a boundary between subcritical and supercritical Hopf bifurcation. Furthermore, numerical simulations are carried out in different regions (both stable and unstable) of the parameter space to understand the non-linear phenomena of the system. Moreover, the bifurcations are explained in terms of the interaction of the external and internal characteristic curves of the system. The large and small amplitude cycles are shown in the characteristic curves of the system.