Bio-thermal convection induced by two different species of microorganisms

https://doi.org/10.1016/j.icheatmasstransfer.2011.02.006Get rights and content

Abstract

This paper develops a theory of bio-thermal convection in a suspension that contains two species of microorganisms exhibiting different taxes, gyrotactic and oxytactic microorganisms. The developed theory is applied to investigating the onset of bio-thermal convection in such a suspension occupying a horizontal layer of finite depth. A linear stability analysis is utilized to derive the equations for the amplitudes of disturbances. The obtained eigenvalue problem is solved by the Galerkin method. The case of non-oscillatory instability in a layer with a rigid lower boundary and a stress-free upper boundary is investigated. The resulting eigenvalue equation relates three Rayleigh numbers, the traditional Rayleigh number (Ra) and two bioconvection Rayleigh numbers, one for gyrotactic (Rbg) and one for oxytactic (Rbo) microorganisms. The neutral stability boundary is presented in the form of a diagram showing that boundary in the (Ra, Rag) plane for different values of Rao.

Introduction

Recently there has been increased interest in research addressing biononvection, a macroscopic convective fluid motion induced by upswimming of motile microorganisms [1], [2], [3], [4], [5]. In [6] the effect of bioconvection on the dynamics of plankton population was analyzed. Recently, Kuznetsov [7] suggested using motile microorganisms to induce mixing and prevent nanoparticle agglomeration in nanofluids.

The purpose of this paper is to introduce the theory of bio-thermal convection induced by the presence of two species of microorganisms exhibiting different taxes, gyrotactic and oxytactic microorganisms. This system is fundamentally interesting and important because the presence of two species exhibiting different taxes would give an experimentalist an additional control of the system's behavior. This paper investigates the onset of instability in such a suspension occupying a horizontal layer of finite depth.

Section snippets

Governing equations

A water-based dilute suspension containing both gyrotactic and oxytactic microorganisms is considered. The suspension occupies a horizontal layer of depth H. The governing equations are based on the theory of bioconvection in suspensions of gyrotactic microorganisms developed in [8], [9], [10], the theory of bioconvection in suspensions of oxytactic microorganisms developed in [11], [12], and the theory of bio-thermal convection developed in [13]. The dimensionless governing equations are·U=01

Basic state

In the basic state, the solution is of the formU=0,p=pb(z), T=Tb(z),ng=ng,b(z),no=no,b(z),C=Cbz

Utilizing Eq. (15), Eqs. (1), (2), (3), (4), (5), (6) are simplified asdpbdzRm+RaTbRbgLbgng,bRboLbono,b=0d2Tbdz2=0ng,bPegdng,bdz=0PeodCdzdno,bdz+ Peono,bd2Cdz2d2no,bdz2=0d2Cdz2βˆLeono,b=0

Using boundary conditions (3), (4), Eq. (17) is integrated twice to giveTbz=1z

The solution for ng, b(z) is obtained by integrating Eq. (18) and accounting for the fact that ng(z) is normalized such that 01ngzdz

Linear instability analysis

For the linear instability analysis, perturbations are superimposed on the basic solution, according to the following equation:U,pˆ,T,ng,no,C,p=0,kˆ,Tbz,ng,bz,no,bz,Cbz,pbz+εUt,x,y,z,pˆt,x,y,z,Tt,x,y,z,ngt,x,y,z,not,x,y,z,Ct,x,y,z,pt,x,y,z

Upon the substitution of Eq. (26) into Eqs. (1), (2), (3), (4), (5), (6), the linearization of the results, and the utilization of Eq. (21), the following equations for the perturbation quantities are obtained:·U=01PrUt=p+2U+RaTkˆRbgLbgngkˆ

Conclusions

The developed theory shows the interaction of three agencies influencing the stability of a horizontal layer: unstable density stratification induced by heating from the bottom and upswimming of gyrotactic and oxytactic microorganisms. Each effect is characterized by its own Rayleigh number. The gyrotactic and oxytactic Rayleigh numbers can be changed by changing the concentration of corresponding microorganisms in the suspension. Increase in any of the three Rayleigh numbers decreases the

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Communicated by W.J. Minkowycz.

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