# Bioconvection in Microalgae: Review of Mathematical Models
## 2 Models
Though the models described so far deal with the effect of phototaxis on fluid flows within non-porous media, bioconvection in porous media also deserves attention since it presents unique and important considerations for both theoretical understanding and practical applications [51; 52; 53]. This fascinating phenomenon combines biological behavior with fluid dynamics, where the collective motion of microorganisms creates convection patterns within fluid-filled porous materials characterized by a solid matrix containing distributed void spaces or pores, and it is ubiquitous in both natural and engineered contexts, ranging from soil and rock formations to biological tissues, and from foam materials to concrete and ceramics. Also in this case, the basic concept is bioconvection, which occurs when phototactic microorganisms swim with a directional bias. In porous media, these organisms must navigate through interconnected void spaces, and their movement creates density instabilities that lead to convection patterns. The behavior of bioconvection in porous media is fundamentally shaped by several key characteristics of the medium itself. The porosity \(\varphi\), defined as the ratio of the void volume to the total volume, determines the space available for fluid flow and organism movement. Permeability K measures how easily fluid can flow through the medium, directly affecting the formation and maintenance of convection patterns. Additionally, tortuosity--the ratio of the actual fluid path length to the straight-line distance--influences both organism movement patterns and the formation of chemical gradients within the system. The mathematical description of bioconvection in porous media provided by [PERSON] and [PERSON] [51] requires a comprehensive framework incorporating several key equations. The system is governed by conservation equations for mass, and the momentum is determined as follows:  
\[\frac{\rho}{\varphi}\ (\frac{\partial u}{\partial t}+(\frac{1}{\varphi})u \
abla u)=-\
abla P_{e}+\mu\
abla 2u-(\frac{u}{K})u-n\theta g\Delta\rho z \tag{8}\]  
In this equation \(\rho\) is the fluid density, \(\varphi\) is the porosity of the medium, \(t\) is the time, \(P_{e}\) is the pressure above the hydrostatic, \(\mu\) is the fluid viscosity, \(K\) is the permeability of the medium, \(u\) is the fluid velocity vector, \(n\) is the concentration of microorganisms, \(\theta\) is their average volume, \(g\) is the gravitational acceleration, \(\Delta\)\(\rho\) is the density difference between microorganisms and the base fluid, and z is the unit vector in vertical direction. Swimming orientation and light distribution are intimately coupled through two key equations (swimming orientation and Lambert-Beer law) that work together to describe how microorganisms respond to and affect light conditions in their environment. The swimming orientation is described by  
\[S_{o}=Ph(I)\ z \tag{9}\]  
where \(S_{o}\) is the mean swimming orientation vector, \(Ph(I)\) is the phototaxis function, \(I\) is the local light intensity, and \(z\) is the unit vector in vertical direction. The phototaxis function determines the vertical orientation of the microorganisms; its values range between \(-1\) and \(+1\), indicating downward swimming for negative values and upward swimming for positive values. The swimming orientation affects the distribution of organisms, which in turn affects light absorption and distribution. Areas with a higher organism concentration will absorb more light, creating shadows for organisms below them, where the light intensity distribution follows the Lambert-Beer law. The implications of bioconvection in porous media span multiple fields and applications. In environmental science, this phenomenon plays a crucial role in understanding nutrient cycling in soils, groundwater contamination, and microbial distribution patterns. These processes are fundamental to bioremediation efforts and the overall health of soil and aquatic ecosystems.