Abstract
This paper presents a detailed analysis of the stability and network structure of thermal convection patterns of mixtures containing two miscible fluids. The stability of steady spatially localized solutions consisting of an even number of convection cells (even-SP) is investigated in detail because even-SPs emerge as transient states resulting from collisions between counterpropagating periodic traveling pulses, while odd-SPs do not. The even-SP branch is examined to analyze the distribution of the eigenvalues and the corresponding eigenfunctions to understand the stable/unstable direction in the vicinity of each solution and to investigate how they are connected to each other. Further, we construct a hierarchical network diagram consisting of even-SP solutions as nodes and stable and unstable manifolds connecting between them as edges, which shows a skeleton of transition processes of an arbitrary initial condition and asymptotic states.
Highlights
The Rayleigh–Benard convection, a well-known pattern formation phenomenon, shows various patterns, such as roll patterns and hexagons, depending on the specified parameters [6]
In our previous works we studied the collision of two counterpropagating periodic traveling pulse (PTP) to investigate the pattern formation process in binary fluid convection [37, 36]
The symmetric directions of the u-family steady pulses (SPs) are connected to the proximate s-family SP, and the antisymmetric directions of both families are connected to the PTP
Summary
The Rayleigh–Benard convection, a well-known pattern formation phenomenon, shows various patterns, such as roll patterns and hexagons, depending on the specified parameters [6]. This stability property is similar to those reported by Burke and Dawes [5]: drift and breathing instabilities, which are observed in the present system.
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