5. Numerical computations by a shooting method

Here, the shooting method [61] is used to solve the eigenvalue problem posed by the system of Eqs. (20) and (21) subjected to the boundary condition Eq. (22). Curves of marginal stability in the plane ðk; RÞ were calculated for fixed values of the parameters d, G, and α0. Notice that very good agreement was always found among the values of the R of the asymptotic analysis (in the limit k ! 0), of the Galerkin method, and of the numerical computations. Calculations were made in two ways. First, the parameters d, G, and α0 were varied in order to obtain a representative set of marginal curves for the problem of bioconvection. Second, experimental data were also used to fix the values of d, G, and α0 and used to find theoretical values of kc and Rc that could be compared with their corresponding experimental values. Here, in particular, a selection is made of α0 ¼ 0:4, which corresponds to the flagellated alga Chlamydomonas nivalis. Figures 1–3 show marginal curves for different values of the gyrotaxis parameter G, while d remains fixed with magnitudes 0.1, 1, and 5, respectively. These figures clearly show the effect the gyrotaxis parameter G has on the critical wavenumber. When the magnitude of G is large enough, the critical wavenumber changes from zero to a finite value which increases with G, as shown by the squares located at the minimum value of R. Notice that it is found that the critical value Gc, which represents the magnitude at which the properties of the marginal curves change, from having kc ¼ 0 to kc > 0, is very well approximated by Eq. (46). This critical value is important because it represents the magnitude of G below which the present theory ceases to predict the experimental results which always show critical wavenumbers kc > 0.

In the curves shown in Figure 1a–b, the critical values of the gyrotaxis parameter are Gc ¼ 0:0306; 0:0266; 0:0060, respectively. As mentioned above for G > Gc, the critical wavenumber is finite, and for G < Gc the critical wavenumber is always zero. The combined effects of the velocity of the swimming of microorganisms, d,

#### Figure 1.

(a) Graphs of log R vs. k for fixed d ¼ 0:1. (b) Graphs of log R vs. k for fixed d ¼ 1. The black square markers indicate the position of the critical wavenumber and Rayleigh number.

Figure 2.

(a) Graphs of log R vs. k for fixed d ¼ 5. (b) Graphs of log R vs. k for experiments 35, 2, 4, and 9 with d increasing from the curve below to above. The black square markers indicate the position of the critical wavenumber and Rayleigh number.

#### Figure 3.

(a) Graphs of log R vs. k for experiments 26–29, 24, 31, and 16 with d increasing from the curve below to above. (b) Graphs of log R vs. k for experiments 13, 10, and 20 with d increasing from the curve below to above. The black square markers indicate the position of the critical wavenumber and Rayleigh number.

and that of gyrotaxis, G, change the location of the critical wavenumber. Note also that for fixed d, when G increases, the system becomes more unstable. From the Figure 1a and b, it can be seen that the most unstable case corresponds to that for d ¼ 0:1 and G ¼ 10 where kc ¼ 4:45 and Rc ¼ 9:0618. This may be understood by the fact that the accumulation of microorganisms near to the top of a shallow layer is faster than in a deeper one. This is due to the important role that the mass diffusion of microorganism Dc and the depth of suspension H play on the instability of the system. The value of Gc in the limit of d, k ! 0 can also be calculated from Eq. (46) by means of an asymptotic analysis. That is,

$$\mathcal{G}\_c = \frac{17}{132(5 - 2\alpha\_0)} + O\left(d^2\right) \tag{54}$$

Here, some theoretical curves are presented of which some have a very good agreement and others a reasonable agreement with the experiments 2, 4, 9, 10, 13, 16, 20, 24, 26, 27, 28, 29, 31, and 35, performed by Bees and Hill [27].

The values for the motility d and the gyrotaxis parameter G used in Figures 2 and 3 were calculated based on experimental data by Bees and Hill [26, 27], which


#### Table 1.

Experimental measurements of Bees and Hill [27] and their theoretical prediction [26].

in here are presented in Tables 2 and 3 of the following section. In order to observe in detail the position of the critical point ðkc; RcÞ in Figures 2 and 3, a local magnification is included.

Here, a comparison is done of our theoretical results of ðkc; RcÞ with the theoretical ones presented by Bees and Hill [26] in their Table VI. According to Bees and Hill [26], experiments 2 and 23 in their Table V have ðkc; RcÞ of comparable order with those in their Table VI. In our Table 1, we reproduce the comparison made by Bees and Hill [26] of their own theoretical and experimental results of their Table V, and we added the corresponding error in percent of the wavenumbers and Rayleigh numbers, respectively. Note that the value α0 ¼ 0:4 corresponds to flagellated microorganisms such as Chlamydomonas nivalis, while α0 ¼ 0:2 corresponds to nonflagellated. Notice that their experimental and theoretical values of d are not exactly the same.

For the sake of comparison of our theoretical results with those of the experiments, Table 1 shows the percent of error calculated by taking the difference of the experimental and theoretical values and then dividing by the smallest one. In Table 1, the more realistic value α0 ¼ 0:4 for Chlamydomonas nivalis is included, which corresponds to the second line of experiment 2 of Bees and Hill [26] predictions. It is clear from Table 3 that our theoretical results show a very important improvement in the reduction of the percent error with respect to experiment 2.

#### 6. Comparison with experiments

In this section a comparison is done of our theoretical results of Rc and kc with the corresponding experimental values obtained by Bees and Hill [27]. Here use is made of the results of the 39 experiments shown in Table I of Bees and Hill [27]. Besides, the more realistic value of the parameter α0 ¼ 0:4, corresponding to the flagellated algae Chlamydomonas nivalis, is also used to calculate d, G, and R.

In Table 3, the values of d, G, and R resulting from the experimental data are presented. Note in Table 2 that the experimental results of the cell swimming speed Vs and of the cell diffusivity Dc are given inside a range of values. In this case, a particular value inside the range has to be selected. The swimming speed used here is 63�10�<sup>4</sup>cm=s. The decision is based on the results obtained by Hill and Hader [62], Pedley and Kessler [25], and Bees and Hill [26]. The value of the cell diffusivity was decided to be that corresponding to an average over the range given in <sup>2</sup> Table <sup>2</sup>, that is, Dc <sup>¼</sup> <sup>27</sup>:5 � <sup>10</sup>�<sup>5</sup> cm =s.

Very recent experimental measurements on the diffusivity for different microorganisms like the biflagellated alga Chlamydomonas reinhardtii have been reported by Polin et al. [63]. Bees and Hill [27] state that there is some evidence to suggest that cells of Chlamydomonas nivalis are not gyrotactic during the first week of subculturing; then if it is not the case for the cells of Chlamydomonas reinhardtii, more measurements for the parameters α0, B, Vs, H, n, and kc would be needed in order to perform comparison between theoretical and experimental results. The definitions of d, G, and R are related with those of Bees and Hill [26] dBH, η, and RBH, respectively, as follows:

$$d = \frac{V\_s^2 \pi K\_2}{D\_c K\_1} d\_{BH\_1} \quad G = \frac{D\_c}{V\_s^2 \pi} \eta\_b \quad R = \frac{K\_2^2 \pi^3 V\_s^5}{D\_c^2 H K\_1^2} \left(1 - \exp\left[-\frac{K\_1 H}{K\_2 V\_s \pi}\right]\right) R\_{BH} \tag{55}$$

where the constants K2 ¼ 0:15 and K1 ¼ 0:57 (see [26] for more details). τ is a direction correlation time which equals 1:3s in the nonflagellated case and 5s in the flagellated case. The data corresponding to the suspension depth H and the average cell concentration of microorganisms n of each experiment (see [27] for more details) have not been reported in Table 3. Only the parameters d, G, k, and R are presented in that table. It is also found that the value of the G of each experiment is greater (but sometimes near) than their corresponding critical value Gc of Eq. (46). Under these conditions, all the critical wavenumbers have to be kc > 0.

By using the data of our Table 2 and Table I of Bees and Hill [27], the experimental values for d, G, and RE were calculated and listed in Table 3. The experimental value of the wavenumber kE was also obtained from Table I of Bees and Hill [27] and was calculated as follows: the wavelength λ0 ð Þ cm is nondimensionalized with the corresponding suspension depth Hð Þ cm to get λE, and then the critical wavenumbers were calculated from kE ¼ 2π=λE. RT and kT are our theoretical wavenumber and Rayleigh number obtained by the shooting method. The curves of marginal stability corresponding to experimental results with good and very good agreement with theory are shown in Figures 2 and 3. As explained above, we have a substantial improvement in the agreement of the critical wavenumbers and Rayleigh numbers with respect to the experimental results (see Table 3). A great number of experimental data have been compared with the present theory in Table 3.


#### Table 2.

Estimates and measurements of typical parameters for a suspension of the alga Chlamydomonas nivalis [24, 25, 64, 65].


#### Bioconvective Linear Stability of Gravitactic Microorganisms DOI: http://dx.doi.org/10.5772/intechopen.83724


Heat and Mass Transfer - Advances in Science and Technology Applications

Table

3. Experimental measurements of wavenumbers [27] and present theoretical predictions.

#### Bioconvective Linear Stability of Gravitactic Microorganisms DOI: http://dx.doi.org/10.5772/intechopen.83724

Some numerical results agree very well with experiments, as can be seen in the experiments 4, 10, 12, 13, 20, and 35 of Table 3. Others are good, such as the results of experiments 9, 16, 24, 26, 27, 28, 29, and 31. With respect to the other data in Table 3, it might be possible that if the experimental measurements are improved, the agreement with theory will be better. The results given here show that the approximate and numerical solutions of the system of governing equations presented in this paper may bring a light to the solution of many other problems of bioconvection.
