Weakly Nonlinear Analysis of Phototactic Bioconvection under Thermal Buoyancy
Abstract
A weakly nonlinear stability analysis is performed to investigate the influence of thermal buoyancy on phototactic bioconvection in a suspension of swimming microorganisms confined between horizontal boundaries and subjected to bottom heating or cooling. The objective is to examine how thermal forcing modifies the post-critical evolution of bioconvective instability through its influence on bifurcation behaviour and finite-amplitude convection. Linear stability theory is first employed to determine the critical conditions for the onset of convection. The corresponding direct and adjoint eigenvalue problems, together with the second-order resolvent system, are then solved numerically to derive the Stuart-Landau amplitude equation governing the nonlinear evolution of disturbances. A systematic analysis is carried out for varying thermal Rayleigh numbers. The results show that, as the thermal Rayleigh number approaches the classical Rayleigh-Benard critical value, the critical bioconvection Rayleigh number decreases to zero, indicating a gradual transition from phototaxis-dominated to thermally driven convection. The computed positive Landau coefficient predicts supercritical pitchfork and Hopf bifurcations for stationary and oscillatory instabilities, respectively. In parameter regimes with oscillatory onset, increasing the thermal Rayleigh number drives a transition from a supercritical Hopf bifurcation to a supercritical pitchfork bifurcation. The nonlinear dynamics are further characterized through amplitude evolution, phase portraits, potential functions, and finite-amplitude flow structures.
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BibTeX
@misc{kumar2026weakly,
title = {{Weakly Nonlinear Analysis of Phototactic Bioconvection under Thermal Buoyancy}},
author = {Sandeep Kumar and Suneet Singh},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04462},
eprint = {2610.04462},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04462},
}
RIS
TY - UNPB
TI - Weakly Nonlinear Analysis of Phototactic Bioconvection under Thermal Buoyancy
AU - Sandeep Kumar
AU - Suneet Singh
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2610.04462
UR - https://arxiv.org/abs/2610.04462
AB - A weakly nonlinear stability analysis is performed to investigate the influence of thermal buoyancy on phototactic bioconvection in a suspension of swimming microorganisms confined between horizontal boundaries and subjected to bottom heating or cooling. The objective is to examine how thermal forcing modifies the post-critical evolution of bioconvective instability through its influence on bifurcation behaviour and finite-amplitude convection. Linear stability theory is first employed to determine the critical conditions for the onset of convection. The corresponding direct and adjoint eigenvalue problems, together with the second-order resolvent system, are then solved numerically to derive the Stuart-Landau amplitude equation governing the nonlinear evolution of disturbances. A systematic analysis is carried out for varying thermal Rayleigh numbers. The results show that, as the thermal Rayleigh number approaches the classical Rayleigh-Benard critical value, the critical bioconvection Rayleigh number decreases to zero, indicating a gradual transition from phototaxis-dominated to thermally driven convection. The computed positive Landau coefficient predicts supercritical pitchfork and Hopf bifurcations for stationary and oscillatory instabilities, respectively. In parameter regimes with oscillatory onset, increasing the thermal Rayleigh number drives a transition from a supercritical Hopf bifurcation to a supercritical pitchfork bifurcation. The nonlinear dynamics are further characterized through amplitude evolution, phase portraits, potential functions, and finite-amplitude flow structures.
ER -