Bioconvection Patterns Research Articles

Exploring stability of Jeffrey fluids in anisotropic porous media: incorporating Soret effects and microbial systems

Purpose This paper aims to present a study on stability analysis of Jeffrey fluids in the presence of emergent chemical gradients within microbial systems of anisotropic porous media. Design/methodology/approach This study uses an effective method that combines non-dimensionalization, normal mode analysis and linear stability analysis to examine the stability of Jeffrey fluids in the presence of emergent chemical gradients inside microbial systems in anisotropic porous media. The study focuses on determining critical values and understanding how temperature gradients, concentration gradients and chemical reactions influence the onset of bioconvection patterns. Mathematical transformations and analytical approaches are used to investigate the system’s complicated dynamics and the interaction of numerous characteristics that influence stability. Findings The analysis is performed using the Jeffrey-Darcy type and Boussinesq estimation. The process involves using non-dimensionalization, using the normal mode approach and conducting linear stability analysis to convert the field equations into ordinary differential equations. The conventional thermal Rayleigh Darcy number RDa,c is derived as a comprehensive function of various parameters, and it remains unaffected by the bio convection Lewis number Łe. Indeed, elevating the values of ζ and γ′ in the interval of 0 to 1 has been noted to expedite the formation of bioconvection patterns while concurrently expanding the dimensions of convective cells. The purpose of this investigation is to learn how the temperature gradient affects the concentration gradient and, in turn, the stability and initiation of bioconvection by taking the Soret effect into the equation. The results provide insightful understandings of the intricate dynamics of fluid systems affected by chemical and biological elements, providing possibilities for possible industrial and biological process applications. The findings illustrate that augmenting both microbe concentration and the bioconvection Péclet number results in an unstable system. In this study, the experimental Rayleigh number RDa,c was determined to be 4π2at the critical wave number ( δcˇ) of π. Originality/value The study’s novelty originated from its investigation of a novel and complicated system incorporating Jeffrey fluids, emergent chemical gradients and anisotropic porous media, as well as the use of mathematical and analytical approaches to explore the system’s stability and dynamics.

Multiple slip flow of nanofluid with bioconvective transport phenomenon and porosity effects

ABSTRACT With advances in the use of nanomaterials for heat transfer, various attributes have been recognized by researchers, paving the way for more dynamic and interdisciplinary applications. The assessment of heat and mass transfer facilitated by nanomaterials, especially in the context of sliding phenomena, plays a crucial role in manufacturing, industrial operations and the petroleum sector, among others. Driven by such considerations, the present study introduces bioconvective patterns into nanofluid flow, taking into account several sliding conditions. These multiple sliding conditions, related to speed, temperature, concentration and microorganisms, were examined to study the synergistic effects on heat and mass transfer. The flow examined is influenced by a porous stretchable surface, incorporating suction and injection mechanisms, while the fluid itself is electrically conductive. A firing method was designed to tackle the proposed model. The impact of various parameters is analyzed visually, revealing that the velocity profile is diminished by the velocity slip constant and the permeability of the porous medium. Furthermore, the interaction of multiple sliding effects distinctly enhances thermal processes.

Towards a low energy, stirless photobioreactor using photosynthetic motile microalgae

Microalgae have a high potential for application in different industries including renewable energy production from light and atmospheric CO2. Biotechnologies should provide high production rates while using the mass and energy resources in a restrained manner. Often, mixing and gas provision are combined to minimize the energetic requirements of photobioreactors. Recent studies suggest that bioconvective patterns naturally developed in suspensions of motile microorganisms may provide biomixing. In this paper we report a study of the phototactic response of Chlamydomonas reinhardtii to induce biomixing in a small bioreactor at macro-scale. By exploiting the phototactic mechanism, we stimulated the microalgae to swim in opposite directions repeatedly in cycles of 40 min (20′ left, 20′ right) in the photobioreactor. The resulting Phototactic Oscillatory Motion (POM) lasted 7 days and maintained an amplitude close to the width of the bioreactor. Phototaxis allowed algae to remain in suspension without the use of any mechanical stirring system. We have demonstrated that light can be controlled to effectively provide mixing and ensure access to nutrients in form of an oscillatory motion at no extra energetic costs. This strategy can replace mechanical agitation in the manner of a ‘phototactic stirring’ under certain circumstances in pursuit of energetic sobriety.

Exploration of bio-convection for slippery two-phase Maxwell nanofluid past a vertical induced magnetic stretching regime associated for biotechnology and engineering

Research in the fields of thermal and process engineering has been growing daily due to the nanoparticles' broad relevance and dynamic applications. Nanomaterials are being researched because of their potential to make remarkable contributions in a number of different areas due to their exceptional thermal and physical characteristics. The thermal efficiency of heat transfer phenomena is considerably improved when nanoparticles are combined with microbes in a solution for collective transport, a process known as mix fluids. Light, gravity, density, and electric and magnetic forces are all physical stimuli that may be used to influence the particles in a fluid mixture. Several branches of biotechnology that rely on fluids and their properties have debated the viability of the bioconvection notion. Nanofluid flow has several potential uses, including but not limited to the storage of electronic devices, the cooling and heating of industrial frameworks, and the construction of accompanying information setups for pharmaceutical governance. The flow model is constructed using relevant variables and dimensionless equations.Under the effect of remarkable Maxwell nanofluid, which is two-dimensional and electrically conducting and includes gyrotactic microorganisms, the present work aims to give some novel uses of induced magnetic field for the bio-convection pattern nanoliquid. Interference between slip effects regulates the thermal phenomena.The BVP4C numerical determination of problem is worked out through fine accuracy. The physical dynamic of flow terms is visualized graphically. It is claimed that manage of thermal transportation phenomenon is produced when the slip effects are dominant. The physical outcomes of interesting terms are visualized.The motile density profile is goes up by increasing Pe. Nano fluid velocity with Momentum slip parameter decreases near the plate and increases as one moves away from it. Motile profile behavior of microorganisms is goes down with Lb.

Bacterial bioconvection confers context-dependent growth benefits and is robust under varying metabolic and genetic conditions.

Microbes often respond to environmental cues by adopting collective behaviors-like biofilms or swarming-that benefit the population. During "bioconvection," microbes gather in dense groups and plume downward through fluid environments, driving flow and mixing on the scale of millions of cells. Though bioconvection was observed a century ago, the effects of differing physical and chemical inputs and its potential selective advantages for different species of microbes remain largely unexplored. In Bacillus subtilis, vertical oxygen gradients that originate from air-liquid interfaces create cell-density inversions that drive bioconvection. Here, we develop Escherichia coli as a complementary model for the study of bioconvection. In the context of a still fluid, we found that motile and chemotactic genotypes of both E. coli and B. subtilis bioconvect and show increased growth compared to immotile or non-chemotactic genotypes, whereas in a well-mixed fluid, there is no growth advantage to motility or chemotaxis. We found that fluid depth, cell concentration, and carbon availability affect the emergence and timing of bioconvective patterns. Also, whereas B. subtilis requires oxygen gradients to bioconvect, E. coli deficient in aerotaxis (Δaer) or energy-taxis (Δtsr) still bioconvects, as do cultures that lack an air-liquid interface. Thus, in two distantly related microbes, bioconvection may confer context-dependent growth benefits, and E. coli bioconvection is robustly elicited by multiple types of chemotaxis. These results greatly expand the set of physical and metabolic conditions in which this striking collective behavior can be expected and demonstrate its potential to be a generic force for behavioral selection across ecological contexts. IMPORTANCE Individual microorganisms frequently move in response to gradients in their fluid environment, with corresponding metabolic benefits. At a population level, such movements can create density variations in a fluid that couple to gravity and drive large-scale convection and mixing called bioconvection. In this work, we provide evidence that this collective behavior confers a selective benefit on two distantly related species of bacteria. We develop new methods for quantifying this behavior and show that bioconvection in Escherichia coli is surprisingly robust to changes in cell concentration, fluid depth, interface conditions, metabolic sensing, and carbon availability. These results greatly expand the set of conditions known to elicit this collective behavior and indicate its potential to be a selective pressure across ecological contexts.

Bioconvection pattern of Euglena under periodical illumination.

Microorganisms respond to environmental conditions and often spontaneously form highly ordered convection patterns. This mechanism has been well-studied from the viewpoint of self-organization. However, environmental conditions in nature are usually dynamic. Naturally, biological systems respond to temporal changes in environmental condition. To elucidate the response mechanisms in such a dynamic environment, we observed the bioconvection pattern of Euglena under periodical changes in illumination. It is known that Euglena shows localized bioconvection patterns under constant homogeneous illumination from the bottom. Periodical changes in light intensity induced two different types of spatiotemporal patterns: alternation of formation and decomposition over a long period and complicated transition of pattern over a short period. Our observations suggest that pattern formation in a periodically changing environment is of fundamental importance to the behavior of biological systems.

Stability analysis of thermo-bioconvection flow of Jeffrey fluid containing gravitactic microorganism into an anisotropic porous medium

This paper presents a novel study on the stability of thermo-bioconvection due to gravitactic microorganisms into an anisotropic porous fluid layer saturated with Jeffrey liquid. A Jeffrey-Darcy model along with Boussinesq approximations is utilized. The field equations are treated with non-dimensionalization, linear stability analysis, and the normal mode technique to formulate a set of ordinary differential equations. These equations along with Robin boundary conditions are then analytically solved by employing the weighted residual Galerkin method utilizing the trigonometric trial functions. The traditional thermal Rayleigh-Darcy number Ra,c is obtained as a well-compiled function of the mechanical anisotropy parameter ξ, Jeffrey parameter γ, the thermal anisotropy parameter η, bioconvection Rayleigh-Darcy number Rb, and Péclet number Q, while it is independent from bioconvection Lewis number Lb. It is observed that mounting ξ and γ in between 0 to 1 hasten the formulation of bioconvection patterns and also enlarges the size of convective cells. The results demonstrate that increasing bioconvection Péclet number and microorganism concentration constitute an unstable system. η ranged between 0 to 1 has shown dual effect which is dominated by the concentration of gravitactic microorganism. For small microorganism concentration, augmenting thermal anisotropy strength stabilizes the system and increases the size of the convective cells. Mathematically, the stabilizing nature of η is bounded by the feasibility of the inequality (π2+δc2ξ)(1+γ)ξ>2Rbδc2(eQ−1)π2Q2(π2+δc2)(π2+Q2)(4π2+Q2). This study may find relevance in applications related to pharmaceutical, bio-mechanics, and in microbial enhanced oil recovery (MEOR). The experimental Ra,c value of measure 4π2 at critical wave number value δc=π is also regained as a limiting case from this study.

Pattern wavelengths and transport characteristics in three-dimensional bioconvection generated by chemotactic bacteria

We conducted a three-dimensional numerical simulation of bioconvection generated by oxygen-reactive chemotactic bacteria. This study investigated the bioconvection patterns, interference between plumes, and the wavelength of bioconvection patterns. In addition, we clarified the transport characteristics of cells and oxygen in the bioconvection. Multiple plumes occur in the suspension, and three-dimensional bioconvection is formed around the plumes by the cells with vortex rings arising around the plumes. Even if bioconvection at a high Rayleigh number is disturbed, the bioconvection is strongly stable with respect to disturbances, and the pattern does not change due to disturbances. Bioconvection changes depending on the physical properties of bacteria and oxygen, and in particular, the rate of oxygen consumption by bacteria significantly affects the strength of bioconvection. Bioconvection patterns with different plume arrangements and shapes are formed for different Rayleigh numbers or initial disturbances of the cell concentration. As a result, the wavelengths of the patterns also vary. As the Rayleigh number increases, interference between plumes is strengthened by the shortening of the pattern wavelength, so the velocities of both upward and downward flows increase. Many cells are located under the plumes, and a strong shear flow occurs in these regions. As the pattern wavelength decreases, the cells are affected by high shear stress. Then the convective transport of the entire suspension strengthens and the transport characteristics of cells and oxygen improve. When the chamber boundary is changed to side walls, bacteria adhere to the wall surface, and plumes are arranged regularly along the side walls.

Numerical Simulation of Reiner–Rivlin Nanofluid Flow under the Influence of Thermal Radiation and Activation Energy over a Rotating Disk

In current study, the numerical computations of Reiner–Rivlin nanofluid flow through a rotational disk under the influence of thermal radiation and Arrhenius activation energy is considered. For innovative physical situations, the motile microorganisms are incorporated too. The multiple slip effects are considered in the boundary conditions. The bioconvection of motile microorganism is utilized alongside nanofluids to provide stability to enhanced thermal transportation. The Bioconvection pattern in various nanoparticles accredits novel applications of biotechnology like the synthesis of biological polymers, biosensors, fuel cells, petroleum engineering, and the natural environment. By deploying some suitable similarity transformation functions, the governing partial differential equations (PDEs) of the flow problem are rehabilitated into dimensionless forms. The accomplished ordinary differential equations (ODEs) are solved numerically through the bvp4c scheme via a built-in function in computational MATLAB software. The upshots of some prominent physical and bioconvection parameters including wall slip parameters, thermophoresis parameter, Brownian motion parameter, Reiner–Revlin nanofluid parameter, Prandtl number, Peclet number, Lewis number, bioconvection Lewis number, and the mixed convection parameter against velocity, temperature, nanoparticles concentration, and density of motile microorganism profiles are dichotomized and pondered through graphs and tables. The presented computations show that the velocity profiles are de-escalated by the wall slip parameters while the thermal and solutal fields are upgraded with augmentation in thermophoresis number and wall slip parameters. The presence of thermal radiation enhances the temperature profile of nanofluid. The concentration profile of nanoparticles is boosted by intensification in activation energy. Furthermore, the increasing values of bioconvection Lewis number and Peclet number decay the motile microorganisms’ field.

Effectiveness of induced magnetic force and non-uniform heat source/sink features for enhancing the thermal efficiency of third grade nanofluid containing microorganisms

Owing to the multidisciplinary significance and dynamic applications of nanoparticles, the research in areas of thermal and process engineering has been expanding every day. The nanomaterials with extraordinary thermal and physical properties are being studied to pose some astonishing contributions in various fields. The nanomaterials when taken with microorganisms in a solution for collective transport, referred as mix fluids, greatly improve the thermal efficiency during heat transfer phenomena. Usually, the mixed fluids particles have greater advantage of being controlled by some physical stimuli of light, gravity and density along with electrical and magnetic forces. The current study offers some innovative applications of induced magnetic field for the bio-convection pattern third grade nanofluid under the influence of extraordinary activation energy and non-uniform hear source/sink factors. The assumptions of stagnation points are considered under the influence of stretched. The nano-materials have been mixed with microorganism to prepare a mixed fluid in order to obtain more stability and directed transportation. The flow constraints for the thermal transport phenomenon have been explained by using convective boundary approach. The couple and nonlinear equations have been put forward to present the model and the solution has been derived using shooting algorithm technique. Moreover, the numerical and graphical outcomes from the study are presented using tables and figures. The improved profile of velocity is predicted against higher values of velocity ratio, Reynolds number and third grade fluid parameter. The induced magnetic field profile enhanced for reciprocal magnetic Prandtl number and magnetic parameter. The consideration of non-uniform heat source/sink is more effective to improve and control and thermal transportation process.

Global classical solutions for the 2D four-component chemotaxis-Navier-Stokes equations

In this paper, we consider a mathematical model describing the emergence of bioconvective patterns. Then we have a chemotaxis system with logistic source terms coupling with a Navier-Stokes equation. Using the Fourier localization technique and the regularity criterion, we establish the global classical solutions for the incompressible four-component chemotaxis-Navier-Stokes equations in R2.

Slip flow of micropolar nanofluid over a porous rotating disk with motile microorganisms, nonlinear thermal radiation and activation energy

A three-dimensional (3-D) investigation of micropolar nanofluid has been presented under the novel impact of nonlinear thermal radiation, activation energy and magnetohydrodynamic features. The induced flow is reflected by a rotating disk which rotates with uniform velocity along z-directions. The fundamental features of Brownian diffusion and thermophoretic assessment is determined by following Buongiorno's nanofluid model. The nanofluid contains gyrotactic microorganisms for which bioconvection pattern is examined for stability assessment. The model flow problem under the certain flow assumptions are retained into the set of dimensionless eqs. A numerical procedure is employed to achieve the solution of such equations. The prime features of parameters against velocity, temperature, concentration and microorganism profiles are graphically examined with justified physical consequences. The numerical computations for wall shear stress, effective local Nusselt number, local Sherwood number and local motile density number are elaborated in terms of numerical data. The simulated theoretical results yield dynamic applications in era of thermal engineering and bio-technology area

Significance of bioconvection in flow of Williamson nano‐material confined by a porous radioactive Riga surface with convective Nield constrains

AbstractThe improved thermal assessment of nano‐particles in presence of magnetic force, thermal radiation and activation energy involve dynamic applications in thermal engineering, industrial processes, and modern technologies. The bioconvection pattern in various nanoparticles attributes novel bio‐technology applications like bio‐fuels petroleum engineering, enzymes, bio‐sensors, and many more. On this end, present theoretical analysis endeavors to examine the thermal characteristics of time dependent Williamson nanofluid confined by an unsteady stretched radioactive Riga plate. Here, nanofluid bioconvection is developed by the combined aspects of buoyancy forces and magnetic field with collaboration of motile microorganisms and nanoparticles. Present analysis is further carried out in presence of activation energy, nonlinear thermal radiation and chemical reaction. The whole exploration is exposed to convective Nield constraints on boundary. The dimensionless form of formulated nonlinear system is achieved with utilization of apposite transformations and then approximate solution is elucidated by shooting technique. The graphical interpretation for numerous relevant parameters on velocity, concentration, temperature and motile microorganism profiles are displayed. The results reveal that an increasing variation in velocity is noticed with modified Hartmann. An increasing nanofluid temperature is associated with Williamson parameter and temperature ratio constant. Moreover, the motile microorganism distribution enhances with bioconvected Lewis and mixed convection constant.

Effects of anisotropic scattering on the onset of phototactic bioconvection with diffuse and collimated irradiation

The ability of many photosynthetic micro-organisms (algae) to detect a localized light source in their environment and move toward (away from) it is known as positive (negative) phototaxis. The phototactic algae absorb the incident light and scatter it isotropically or anisotropically thereafter across the suspension, which is illuminated by both diffuse and collimated irradiation. In this paper, we use the generic model of Panda et al. [“Effects of both diffuse and collimated incident radiation on phototactic bioconvection,” Phys. Fluids 28, 124104 (2016)] for phototaxis and investigate the effects of anisotropic (forward) scattering at the onset of phototactic bioconvection. In the motionless basic state, the up and down swimming caused by positive and negative phototaxis is balanced by diffusion due to randomness in the swimming behavior of algae. The forward scattering improves the underwater light distribution (radiation field) by propagating the light deeper into the algal suspension. At the equilibrium state, the bimodal (respectively, unimodal) base concentration profiles are shifted into the unimodal (respectively, bimodal) ones due to forward scattering for some fixed governing parameters when self-shading (absorption) is insignificant. As a result of forward scattering, for some parameter values, the most unstable solution shifts from mode 1 to mode 2 and a single oscillatory branch bifurcates from the stationary branch or disappears at bioconvective instability when self-shading (absorption) is significant. The dominant bioconvection pattern wavelength at instability is also qualitatively in good agreement with the experimental observations as the forward scattering coefficient is varied.

Bacterial chemotaxis in thin fluid layers with free surface

Thin fluid layers are common natural habitats for various species of aerobic bacteria. Collective behaviors in bacterial colonies caused by chemotaxis can form complex bioconvection patterns, which often work in favor of the colony’s survival and growth. The connection between the biology of bacterial aerotaxis and the physics of buoyancy effects caused by non-uniform suspension density is numerically investigated for a suspension of oxytactic bacteria placed in the Petri dish. The upper surface is free and open to the atmosphere, and through it oxygen diffuses into the suspension. Surface tension and dynamic contact line are incorporated into the mathematical and numerical models. A comparison has been made between dynamic free surface and fixed free surface models, and differences have been revealed. The parametric study in the case of dynamic free surface has been performed, and the non-linear dynamics of the phenomenon has been investigated. Resulting from upward aerotaxis and downward gravitational force, Rayleigh–Taylor-like instabilities develop between layers of different densities in the suspension. Bacterial plume patterns and their dynamics, such as sinking, merging, and birth of new plumes, characterize the phenomenon for particular intervals of dimensionless parameters. Accordingly, categorization of the phenomenon based on bacterial plume evolution has been made, and significant intervals of dimensionless parameters have been extracted.

Well‐posedness and finite element approximation of time dependent generalized bioconvective flow

AbstractWe consider the finite element approximation of a time dependent generalized bioconvective flow. The underlying system of partial differential equations consists of incompressible Navier–Stokes type convection equations coupled with an equation describing the transport of micro‐organisms. The viscosity of the fluid is assumed to be a function of the concentration of the micro‐organisms. We show the existence and uniqueness of the weak solution of the system in two dimensions and construct numerical approximations based on the finite element method, for which we obtain error estimates. In addition, we conduct several numerical experiments to demonstrate the accuracy of the numerical method and perform simulations of the bioconvection pattern formations based on realistic model parameters to demonstrate the validity of the proposed numerical algorithm.

The free surface effect on a chemotaxis–diffusion–convection coupling system

Suspension of an oxytactic bacteria (e.g. the species Bacillus subtilis) placed in a container with its upper surface open to the atmosphere results in the formation of complex bioconvection patterns. The bacteria consume the oxygen diluted in the water, thereby causing the decrease of oxygen concentration everywhere except on the free surface. Through the free surface, which is in direct contact with the air, oxygen diffuses into the water. Slightly denser than water, the oxytactic bacteria are able to swim towards the higher concentration of oxygen (i.e. upwards) and they concentrate in a thin layer below the free surface. This causes the change of the suspension density and Rayleigh–Taylor type instabilities to occur. The chemotaxis phenomenon has been successfully modeled within continuum mechanics approach under certain simplifications. The set of (non-linearly) coupled equations describing the process involves the Boussinesq approximation of the Navier–Stokes equations governing the fluid motion and two convection–diffusion type equations governing the bacteria and oxygen concentrations. One of the simplifications that might significantly influence numerical simulations is the boundary condition for fluid equation on the free surface. This condition ensures that the vertical component of the velocity is zero, thus keeping the position of free surface fixed. This assumption significantly simplifies numerical procedure since the non-linearly coupled system can then be solved on stationary grid. However, allowing the motion of the free surface and completing the system with appropriate boundary conditions on contact line (liquid–solid–gas interface), a more realistic model is derived and new insights on nonlinear dynamics of the chemotaxis phenomenon are obtained. Our aims in this paper are to upgrade the currently available model into a more realistic one in both two and three dimensions, to propose a numerical procedure to deal with the new system (now posed on time-dependent domain) and, finally, to show the difference between this new model and the previous simplified one.

Evaporation-driven convective flows in suspensions of nonmotile bacteria

We report a novel form of convection in suspensions of the bioluminiscent marine bacterium $Photobacterium~phosphoreum$. Suspensions of these bacteria placed in a chamber open to the air create persistent luminiscent plumes most easily visible when observed in the dark. These flows are strikingly similar to the classical bioconvection pattern of aerotactic swimming bacteria, which create an unstable stratification by swimming upwards to an air-water interface, but they are a puzzle since the strain of $P.~phosphoreum$ used does not express flagella and therefore cannot swim. Systematic experimentation with suspensions of microspheres reveals that these flow patterns are driven not by the bacteria but by the accumulation of salt at the air-water interface due to evaporation of the culture medium; even at room temperature and humidity, and physiologically relevant salt concentrations, the rate of water evaporation is sufficient to drive convection patterns. A mathematical model is developed to understand the mechanism of plume formation, and linear stability analysis as well as numerical simulations were carried out to support the conclusions. While evaporation-driven convection has not been discussed extensively in the context of biological systems, these results suggest that the phenomenon may be relevant in other systems, particularly those using microorganisms of limited motility.

Numerical study of nanofluid thermo-bioconvection containing gravitactic microorganisms in porous media: Effect of vertical throughflow

An analytical investigation of the onset of nanofluid thermo-bioconvection in a fluid saturated by porous media containing gravitactic and nanoparticles microorganisms subjected to a vertical throughflow is presented. The heat conservation equation is formulated by introducing the convective term of nanoparticle flux. The fluid is stimulated with modified Brinkman model, normal mode analysis and six-term Galerkin methods are used to solve the governing equations. The combined effects of vertical throughflow, nanoparticles, gravitactic microorganisms, and porosity have been taken into account. The effects of bioconvection Rayleigh number, bioconvection Péclet number, nanoparticle Rayleigh number, Péclet number, bioconvection Lewis number, and porosity on critical thermal Rayleigh number have been examined. The analysis leads that critical wave number is the function of bioconvection parameters, nanofluid parameters and throughflow parameters. It is also found that vertical throughflow disturbs the formation of bioconvection pattern which are necessary for the development of bioconvection.

Stability and dynamics of a chemotaxis system with deformed free-surface in a shallow chamber

Complex bioconvection patterns have been studied analytically, experimentally, and numerically previously only for a flat free-surface of a suspension of chemotaxis bacteria in a shallow/deep chamber. In this paper, we have considered a two-dimensional chemotaxis-diffusion-convection system with a deformed free-surface. The influence of aggregation of chemotactic cells on the deformed free-surface of a shallow chamber is studied analytically. The aim of this paper is to investigate the nature of the instability in the chemotaxis-diffusion-convection system. We performed a detailed linear stability analysis of a steady-state cell and oxygen concentration distribution. The system becomes dominated by nonlinear convection terms beyond a critical Rayleigh number Raτ, which also depends on the critical wavenumber k as well as the other parameters. We have investigated that how the critical Rayleigh number in this system varies with three different sets of parameters. A weakly nonlinear analysis is carried out as well to determine the relative stability of the pattern formation at the onset of instability. A reactance between rolls, squares, hexagons, and mixed mode pattern is investigated in detail. Further research should link the weakly nonlinear analysis with the bifurcation analysis. Some important direct numerical simulation results have been presented in the support of linear stability analysis. Comparison of the analytical steady-state solution shows good agreement with the numerical result.