7. Conclusions

The governing equations of bioconvection were used to investigate the problem of an infinite horizontal microorganism suspension fluid layer. The theoretical predictions of the critical wavenumber kc and Rayleigh number Rc were compared with their experimental counterparts [27]. Very good, good, and fair agreements were found. But in general, we may say that our numerical results improve by far those obtained by Bees and Hill [26].

With the asymptotic analysis for k < < 1, it was possible to calculate a Rayleigh number not reported before without any restrictions on the magnitudes of d and G. This result is important because it was also possible to calculate a critical value of the gyrotaxis parameter Gc which indicates the boundary between the possibility of a marginal curve with kc ¼ 0 (G < Gc) and another one with kc >0(G > Gc).

However, it is clear from the experimental results that the critical wavenumbers are finite and large and that the former case is not physical. Therefore, this Gc also defines the limit of validity of the theory. Note that it agrees very well with numerical analysis.

An analytic Galerkin method was also used to obtain a general expression of R without any restriction on the magnitudes of d, G, and k � Oð Þ1 . This gave us an explicit expression of R not reported before which proved to be very useful when checking with the numerical computations.

Numerical results have shown that the system becomes more unstable when the layers are shallow. The physical interpretation of such situation is that the accumulation of microorganisms near the top of the layer in the shallow case is faster than in the deeper case, due to the smaller depth of suspension H.A consequence of this is that the critical wavenumber is smaller for shallower layers. This can be explained by means of the boundary conditions of the microorganism concentration. If the parameter d tends to zero, the boundary conditions tend to those similar to the "fixed heat flux" boundary conditions of the problem of natural convection heated from below [50–53, 56]. Moreover, it has been shown above that by a change of variable, it is possible to transform the boundary conditions of the concentration into those similar to the "fixed heat flux" boundary conditions. In that problem it has been shown that the critical wavenumber tends to zero. However, due to the gyrotaxis, the critical wavenumber is not zero in the present problem if G > Gc, which, from the experimental results, is the case here. But notice in Figures 1–3 that in fact, also in this case, the critical wavenumber decreases with a decrease of d. The change of the critical wavenumber with respect to G is also clear in the figures. The critical wavenumber decreases with a decrease of G.

Finally, we would like to point out that it is our hope that the results presented in this chapter may stimulate researchers to make more new and precise experiments on bioconvection in the near future.
