Bio-thermal convection induced by two different species of microorganisms☆
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
Basic state
In the basic state, the solution is of the form
Utilizing Eq. (15), Eqs. (1), (2), (3), (4), (5), (6) are simplified as
Using boundary conditions (3), (4), Eq. (17) is integrated twice to give
The solution for ng, b(z) is obtained by integrating Eq. (18) and accounting for the fact that ng(z) is normalized such that
Linear instability analysis
For the linear instability analysis, perturbations are superimposed on the basic solution, according to the following equation:
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:
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.