This work studies nonlinear spatiotemporal stability of two-dimensional electroconvection between two flat plates subjected to a through-flow, using numerical simulations and weakly nonlinear analyses. We found that the traveling speeds of the leading and trailing edges of the wave packet in the nonlinear regime are consistent with the linear ones. We derived for the first time the Ginzburg-Landau equation (GLE) using an amplitude expansion method extending earlier work of Pham and Suslov. This GLE can predict the absolute growth rate even when the parameters are away from the linear critical conditions, outperforming the GLE derived using a multiple-scale expansion method.
This article presents a combined approach to enhancing the effectiveness of the Jacobianfree Newton–Krylov (JFNK) method by deep reinforcement learning (DRL) in identifying fixed points within the two-dimensional Kuramoto– Stravinsky equation (KSE). The JFNK approach entails a good initial guess for improved convergence when searching for fixed points. With a properly defined reward function, we utilize DRL as a preliminary step to enhance the initial guess in the converging process. The main advantage brought about by the reward function in DRL is to identify potential initial guess candidates with similar spectral structures over time, which facilitates the search of fixed points. We report new results of fixed points in the two-dimensional KSE that have not been reported in the literature. Additionally, we explored control optimization for the two-dimensional KSE to navigate the system trajectories between known fixed points, based on parallel reinforcement learning techniques. This combined method underscores the improved JFNK approach to finding new fixed-point solutions within the context of two-dimensional KSE, which may be instructive for other high-dimensional dynamical systems.
This study presents analytical and numerical investigations of Marangoni interfacial instability in a two-liquid-layer system with constant solute transfer across the liquid interface. Previous research has demonstrated that both viscosity ratio and diffusivity ratio can influence the system's hydrodynamic stability via the Marangoni effect, but the distinctions in process and mechanisms are not yet fully understood. To gain insights, we developed a numerical model based on the phase-field method, rigorously validated against linear stability analysis. The parameter space explored encompasses Schmidt number ($Sc \in [0.2, 200]$), Marangoni number ($Ma_c \in [10, 2000]$), Capillary number ($Ca \in [0.01, 1]$), viscosity ratio ($\zeta_\mu \in [0.1, 10]$), and diffusivity ratio ($\zeta_D \in [0.1, 10]$). We identified two key characteristics of Marangoni instability: the self-amplification of triggered Marangoni flows in flow-intensity-unstable case and the oscillation of flow patterns during flow decay in flow-intensity-stable case. Direct numerical simulations incorporating interfacial deformation and nonlinearity confirm that specific conditions, including solute transfer out of a higher viscous or less diffusive layer, amplify and sustain Marangoni interfacial flows. Furthermore, the study highlights the role of interfacial deformation and unequal solute variation in system instability, proposing corresponding instability mechanisms. Notably, traveling waves are observed in the flow-intensity-stable case, correlating with the morphological evolution of convection rolls. These insights contribute to a comprehensive understanding of Marangoni interfacial instability in multicomponent fluids systems, elucidating the mechanisms underlying variations in viscosity and diffusivity ratios across liquid layers.
Axisymmetric viscoelastic pipe flow of Oldroyd-B fluids has been recently found to be linearly unstable by Garg et al. ( Phys. Rev. Lett. , vol. 121, 2018, 024502). From a nonlinear point of view, this means that the flow can transition to turbulence supercritically, in contrast to the subcritical Newtonian pipe flows. Experimental evidence of subcritical and supercritical bifurcations of viscoelastic pipe flows have been reported, but these nonlinear phenomena have not been examined theoretically. In this work, we study the weakly nonlinear stability of this flow by performing a multiple-scale expansion of the disturbance around linear critical conditions. The perturbed parameter is the Reynolds number with the others being unperturbed. A third-order Ginzburg–Landau equation is derived with its coefficient indicating the bifurcation type of the flow. After exploring a large parameter space, we found that polymer concentration plays an important role: at high polymer concentrations (or small solvent-to-solution viscosity ratio $\beta \lessapprox 0.785$ ), the nonlinearity stabilizes the flow, indicating that the flow will bifurcate supercritically, while at low polymer concentrations ( $\beta \gtrapprox 0.785$ ), the flow bifurcation is subcritical. The results agree qualitatively with experimental observations where critical $\beta \approx 0.855$ . The pipe flow of upper convected Maxwell fluids can be linearly unstable and its bifurcation type is also supercritical. At a fixed value of $\beta$ , the Landau coefficient scales with the inverse of the Weissenberg number ( $Wi$ ) when $Wi$ is sufficiently large. The present analysis provides a theoretical understanding of the recent studies on the supercritical and subcritical routes to the elasto-inertial turbulence in viscoelastic pipe flows.
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