Phototaxis denotes swimming towards (positive) or away (negative) from light. The suspension containing phototactic algae is illuminated by both the diffuse and collimated solar radiation. The algae absorb the incident light and scatter it. We use the phototaxis model of Ghorai et al. [“Bioconvection in a suspension of isotropically scattering phototactic algae,” Phys. Fluids 22, 071901 (2010)] and investigate the onset of bioconvection with particular emphasis on the effects of diffuse irradiation. The basic equilibrium state of the bioconvective governing system is defined by assuming that the bulk velocity of the fluid to be zero and the up and down swimming, caused by the positive and negative phototaxis, is balanced with the diffusion. For some values of the parameters, the bimodal steady-state profile transits to a unimodal equilibrium state as the diffuse irradiation is increased. For a small scattering albedo, at the onset of bioconvective instability, this model differs significantly from the up-swimming model of Vincent and Hill [“Bioconvection in a suspension of phototactic algae,” J. Fluid Mech. 327, 343 (1996)], even for small wavelengths. Furthermore, the solutions show a transition of the most unstable mode from the stationary to oscillatory state, and then back to the stationary state again, as the governing parameters are varied. A significant stabilizing effect on suspension has also been observed due to the effects of diffuse irradiation. The effect of the diffuse irradiation on a dominant bioconvection pattern wavelength at instability is also qualitatively in good agreement with the bioconvection experiments.

Light gradient can allow many motile photosynthetic microorganisms to bias their motion towards moderate light (positive phototaxis) or away from intense light (negative phototaxis). The proposed work presents the penetrative phototactic bioconvection in a non-scattering algal suspension. The suspension is confined by a stress-free top boundary, and rigid bottom and periodic lateral boundaries. The resulting bioconvective patterns of the problem strongly resemble to that of a spatially extended domain in the same vicinity. The bioconvection solution appears in the form of a two-rolls pattern (or any even number of rolls) due to the periodic lateral boundaries.

Bioconvection occurs as the result of the collective behavior of many micro-organisms swimming in a fluid and is realized as patterns similar to those of thermal convection, which occur when a layer of fluid is heated from below. We consider the phenomenon of pattern formation due to gyrotaxis, an orientation mechanism which results from the balance of gravitational and viscous torques acting on bottom-heavy micro-organisms. Using the continuum model of Pedley et al. ͓"The growth of bioconvection patterns in a uniform suspension of gyrotactic micro-organisms," J. Fluid Mech. 195, 223 ͑1988͔͒, the linear stability of a gyrotactic plume ͑descending line of concentrated micro-organisms͒ is investigated. Linear stability analysis predicts that a plume is always unstable to both the varicose and meandering modes. The growth rates of these instability modes and their dependence on parameter values are investigated. Comparisons are made with the experimental and numerical results.

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