ABSTRACT
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A.V. Kuznetsov
CONTENTS 16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646 16.2 Stability of a Suspension of Gyrotactic Microorganisms
in a Fluid Saturated Porous Medium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648 16.2.1 Stability Analysis under the Assumption that a Porous
Matrix does not Absorb Microorganisms . . . . . . . . . . . . . . . . . . . . . . . 648 16.2.1.1 General linear stability analysis . . . . . . . . . . . . . . . . . . . . . . . . 648 16.2.1.2 Investigation of stability under the assumption
that the principle of exchange of stabilities is satisfied . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655
16.2.2 Stability Analysis Accounting for the Absorption of Microorganisms by the Porous Matrix and the Reduction of its Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656
16.2.3 Stability Analysis of Bioconvection of Gyrotactic Microorganisms in a Layer of Final Depth . . . . . . . . . . . . . . . . . . . . . . 659
16.3 Stability of a Suspension of Oxytactic Microorganisms in a Fluid Saturated Porous Medium and Analysis of a Bioconvection Plume Caused by these Microorganisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664 16.3.1 Stability of a Shallow Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664 16.3.2 Self-Similarity Solution for a Falling Plume in
Bioconvection of Oxytactic Bacteria in a Deep Fluid Saturated Porous Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672
Acknowledgment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685
Bioconvection is a new area of fluid mechanics, which has been developed during the last few decades. The term “bioconvection” refers to macroscopic convection induced inwater by the collectivemotion of a large number of selfpropelledmotilemicroorganisms. This convection is usually characterized by
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regularfluid circulationpatterns. Bioconvection is inducednotbymomentum generated as a result of the swimming of individual microorganisms, but rather by a density gradient, which occurs when a large number of these microorganisms (which are heavier thanwater) accumulate in a certain region of the fluid. This chapter concentrates on the theory of bioconvection in porous media.
By using porous media with different permeabilities, it is possible to have either a stable suspension (this happens if permeability is small) or unstable suspension (this happens if permeability is large), in which case bioconvection plumes will develop. Utilizing porous plugs in composite porous/fluid domainsmakes it possible to control and, if necessary, suppress bioconvection. In this chapter, the stability criteria for bioconvection of gyrotactic microor-
ganisms in porous media are derived. It is established that there is a critical permeability of a porous medium. If permeability is larger than critical, bioconvection develops; if it is smaller than critical, it is suppressed. Critical permeability is determined as a function of parameters of upswimming microorganisms through a linear stability analysis of governing equations. The effect of cell deposition and resuspension as well as the effect of fouling of porous media on the critical permeability is investigated. The chapter also presents a theory of bioconvection plume in a suspension
of oxytactic bacteria in a deep chamber filled with a fluid-saturated porous medium. The plume transports oxygen from the upper boundary layer, which is rich in cells and oxygen, to the lower part of the chamber, which is depleted of both cells and oxygen. A similarity solution of full governing equations (without utilizing the boundary layer approximation) that describe fluid flow as well as oxygen and cell transport in the plume is obtained. The resulting ordinary differential equations are singular when the similarity variable approaches zero; therefore, a series solution of these ordinary differential equations, which is valid for small values of the similarity variable, is obtained. This series solution is used as a starting point for a numerical solution that makes it possible to investigate the plume for the whole range of values of the similarity variable.
