An asymptotic solution to bio-porousconvection in a suspension of microscopic swimming phototactic algae

This paper deals with a detailed study of asymptotic solution to bio-porous convection in a suspension of microscopic swimming phototactic algae. Experimental observations indicate that bio convection patterns are modified by illumination. This phenomenon will be observed in a large number of different algal species where the cells are denser than water and tend to swim upward on average. The continuum model for phototaxis and suspension shading was formulated in a porous medium. Here, the length scale of the bulk motions and the concentration distribution are large when compared to cell diameters and cell spacing. Pure phototaxis is a valid limiting case to consider in order to understand the complexities of the effects of photo taxis on BPC (bio-porous-convection) before moving in to a higher dimension. Further the diffusion tensor is a constant orthotropic tensor. The linear stability problem is discussed in detail and solvability conditions are derived up to the third order. Analyses for the two cases namely a) upper rigid and b) upper stress free boundary are discussed. Extensive graphs are drawn for the various computed results and the permeability number has strong influence on the inhibition and enhancement of bio convection.