Role of surface tension effect at the deformed free surface of chemotaxis coupling flow system: Weakly nonlinear study

Abstract

A comprehensive study is presented on pattern formation during the initial stages of bioconvection in chemotaxis suspensions influenced by surface tension based on the linear stability analysis done by Chakraborty and Sheu [J. Fluid Mech. 923, A14 (2021)]. We explore the intricate patterns arising when suspended oxytactic bacteria (e.g., Bacillus subtilis) are introduced into an open chamber, unveiling a compelling bioconvection phenomenon. This process stems from the bacteria's response to higher oxygen concentrations near the free surface, driving density fluctuations akin to the Rayleigh–Taylor instability. Our investigation employs a weakly nonlinear stability analysis to reveal intricate dynamics arising from multi-parameter interactions, yielding captivating transformations. A bifurcation study reveals that unstable rolls undergo sub-critical bifurcation, giving rise to hexagonal patterns and a variety of hybrid formations. Notably, altering the chamber's length accentuates this diversity, yielding pattern formation, including both rolls and hexagons, driven by surface tension. Our findings underscore the pivotal role of surface tension in shaping pattern stability. Hexagonal patterns, inherently unstable, acquire stability under heightened surface tension (decreasing Caτ). Conversely, increased Frτ values disrupt their stability. Fascinatingly, surface tension prompts the emergence of distinct hexagon subcategories—up-hexagons and down-hexagons—each exhibiting unique responses to changes in SτHτ. Numerical simulations substantiate our theoretical insight, offering tangible proof of the complex dynamics of pattern formation, as the present study elucidates the interplay between surface tension and parameter effects governing pattern stability in bioconvection onset, thereby advancing comprehension and setting the foundation for future explorations.