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

## Abstract

Interesting results on the linear bioconvective instability of a suspension of gravitactic microorganisms have been calculated. The hydrodynamic stability is characterized by dimensionless parameters such as the bioconvection Rayleigh number R, the gyrotaxis number G, the motility of microorganisms d, and the wavenumber k of the perturbation. Analytical and numerical solutions are calculated. The analytical one is an asymptotic solution for small wavenumbers (and for any motility number) which agrees very well with the numerical solutions. Two numerical methods are used for the sake of comparison. They are a shooting method and a Galerkin method. Marginal curves of R against k for fixed values of d and G are presented along with curves corresponding to the variation of the critical values of Rc and kc. Moreover, those critical values are compared with the experimental data reported in the literature, where the gyrotactic algae Chlamydomonas nivalis is the suspended microorganism. It is shown that the agreement between the present theoretical results and the experiments is very good.

### Keywords

- bioconvection
- hydrodynamic stability
- Galerkin method

## 1. Introduction

Since many years ago, efforts in the experimental and theoretical investigation of the bioconvection phenomenon have been made. These efforts, which lead to the understanding of bioconvective instability, have produced novel and interesting applications. For example, Noever and Matsos [1] proposed a biosensor for monitoring the heavy metal Cadmium based in bioconvective patterns as redundant technique for analysis, a number of researchers [2, 3, 4, 5, 6] have been working on the control of bioconvection by applying electrical fields (as in galvanotaxis) to use it as a live micromechanical system to handle small objects immersing in suspensions, Itoh et al. [7, 8] use some ideas of bioconvection in a study for the motion control of microorganism groups like *Euglena gracilis* to manipulate objects by using its phototactic orientation (as in phototaxis), and more recently possibly bioconvection seeded the investigation of Kim et al. [9, 10] for using a feedback control strategy to manipulate the motions of *Tetrahymena pyriformis* as a microbiorobot, among others. Perhaps, further applications on biomimetics [11, 12, 13] at the nano- and microscale could be driven by this contribution.

The term bioconvection was first coined by Platt [14] as the spontaneous pattern formation in suspensions of swimming microorganisms. This phenomenon has some similarity with Rayleigh-Benard convection but originates solely from diffusion and the swimming of the organisms. Reviews about this topic have been published by Pedley and Kessler [15] and Hill and Pedley [16]. Ideas and theories on cellular motility can be found in the book of Murase [17], and the effect of gravity on the behavior of microorganisms is widely explained in the book of Hader et al. [18]. In 1975, Childress et al. [19] presented a model for bioconvection of purely gravitactic microorganisms and their results of a linear theoretical study, and later Harashima et al. [20] studied the nonlinear equations of this model. According to the model of Childress et al. [19], the critical wavenumber at the onset of the instability is always zero. In ordinary particles and colloidal suspensions, the internal degrees of freedom like the internal rotation or spin are important under some geometrical and physical conditions [21, 22]. The case of a suspension of microorganisms is not an exception. For this case, Pedley et al. [23] proposed a gyrotactic model for a suspension of infinite depth. Their model includes the displacement of the gravity from the geometric center in the organisms along their axis of symmetry. Hill et al. [24] performed an analysis of the linear instability of a suspension of gyrotactic microorganisms of finite depth using the model of Pedley et al. [23]. Hill et al. [24] found finite wavenumbers at the onset of the instability, but agreement with the experiment was not good. Later, Pedley and Kessler [25] reported a model for suspensions of gyrotactic microorganisms where account was taken of randomness in the swimming direction of the cells. In a study of the linear instability of the system based on the model of Pedley and Kessler [25], Bees and Hill [26] found disagreement between their theoretical results and the experimental data reported by Bees and Hill [27]. Several experimental investigations of bioconvection have been reported by Loeffer and Mefferd [28] and Fornshell [29], by Kessler [30] and Bees and Hill [27] who take into account the gyrotaxis, by Dombrowski et al. [31] and Tuval et al. [32] who take into account the oxitaxis, and more recently by Akiyama et al. [33] who observed a pattern alteration response characterized by a rapid decrease in the bioconvective patterns. Pattern formation has been observed in cultures of different microorganisms such as *Chlamydomonas nivalis*, *Chlamydomonas reinhardtii*, *Euglena gracilis*, *Bacillus subtilis*, *Paramecium tetraurelia*, and *Tetrahymena pyriformis*.

More recently, investigations have been reported for a semi-dilute suspension of swimming microorganisms where cell–cell interactions are considered [34, 35, 36, 37, 38]. On the other hand, Kitsunezaki et al. [39] investigated the effect of oxygen and depth on bioconvective patterns in suspensions with high concentrations of *Paramecium tetraurelia*. Bioconvection is also studied from other points of view in gravitational biology. Interesting results are also available in Refs. [40, 41, 42] about the pattern formation in suspensions of *Tetrahymena* and *Chlamydomonas* subject to different gravity conditions. Further results are due to Sawai et al. [43] who investigate the proliferation of *Paramecium* under simulated microgravity, to Mogami et al. [44] who report an investigation of the formed patterns by *Tetrahymena* and *Chlamydomonas* as well as a physiological comparison, to Takeda et al. [45] who give an explanation of the gravitactic behavior of single cells of *Paramecium* in terms of the swimming velocity and swimming direction, to Mogami et al. [46] who present theory and experiments of two mechanisms of gravitactic behavior for microorganisms, and to Itoh et al. [47] who investigate the modification of bioconvective patterns under strong gravitational fields.

This chapter presents interesting results about the bioconvective linear stability of a suspension of swimming microorganisms. Use is made of the equations presented by Ghorai and Hill [48, 49] some years ago. In their approach, Ghorai and Hill [48, 49] used a different dimensionalization scale for the concentration microorganisms which gives distinct meaning to the basic state for the concentration of microorganisms and a bioconvective Rayleigh number defined in terms of the mean cell concentration. To the authors best knowledge, those equations along with the change in the basic state and Rayleigh number definitions have not been used to determine the linear bioconvective instability in an infinite horizontal fluid layer and to compare the results with experiment. These results were obtained by means of both numerical and analytical techniques. The critical values of the Rayleigh number *G* and the motility of microorganisms *d*, that characterize the hydrodynamic stability of the system are compared with the experimental data presented in Table I of Bees and Hill [27] and Table II of Bees and Hill [26] where the gyrotactic biflagellate alga *Chlamydomonas nivalis* is used as suspended microorganism. Below, it is shown for the first time that the numerical results have a very good agreement with the experimental data.

The chapter is organized as follows. The governing equations and boundary conditions [48, 49] as well as the basic state can be found in Section 2. Nondimensionalization and linearization of the system of equations is outlined in Section 3. In Section 4, use is made of an asymptotic expansion [50, 51, 52, 53] method and a Galerkin method [54] to find limiting cases and predict critical values of *R* and *k* for the instability. The numerical calculations done by means of the shooting method along with the graphics corresponding to the marginal curves are given in Section 5. In Section 6, the experimental data [27] are compared with the numerical results. A discussion is given in the final section.

## 2. Equations of motion

We consider an infinite horizontal layer of a suspension of gyrotactic microorganisms. The fluid layer is bounded at

where **u****k** is the acceleration due to gravity, **k** is the vertical unit vector, **J**

where **p**

In the basic state, the fluid velocity is zero and **p****k**. Thus for

where

## 3. Linear stability

We make the governing Eqs. (1–3) nondimensional by scaling all lengths with *H*, the time with

where the nondimensional quantity *d* is called the motility of the microorganisms. In order to investigate the linear stability of the system, small perturbations have to be considered. They are

where **u**^{1} are

where

are the Schmidt and bioconvection Rayleigh numbers, respectively. Pedley and Kessler [55] give a definition of the vector **p****p****u**

where the subscript **p**

with boundary conditions

where the superscripts have been deleted. Notice that the adimensionalization of the equations is different from that of Hill et al. [24]. Here, an application of a more general asymptotic analysis for any magnitude of *d* is used. An analytic Galerkin method and a shooting numerical method for the solution of the proper value problem allowed us to have an interesting perspective of the stability of the present problem under research. The results are used here to compare with the experimental data of the flagellated alga *Chlamydomonas nivalis*.

By elimination of the pressure from Eqs. (13–15), it is possible to obtain a coupled system of two equations, for *w* and *n*, to describe the instability of the system. The perturbations of the variables will be analyzed in terms of normal modes of the form

where

subject to the boundary conditions

where

Then, Eqs. (17) and (18) and the boundary conditions Eq. (19) become

subject to the new boundary conditions

In this form, the equations are very similar to those of the well-known problem of thermal convection in an infinite horizontal fluid layer between nonconducting boundaries [50, 51, 52, 53, 56]. The familiar fixed heat flux boundary condition is the main characteristic of those thermal convection problems and is analogous to that presented in Eq. (22). The equations derived by Childress et al. [19] can also be analyzed from the present view point of this change of variable. In the theory of thermal convection as in that of Childress et al. [19], a zero critical wavenumber is found as a result of the fixed flux boundary condition. In more recent models, which include the effects of gyrotaxis, the similarity with the thermal convection problem is not valid unless

## 4. Asymptotic analysis

In this section, the eigenvalue problem stated in the system of Eqs. (13–15) with boundary condition Eq. (16) is investigated by means of two analytic methods. The magnitude of the marginal value of *R* is a function of all the other parameters. The way in which the solution of the stability problem is to be carried out is as follows. For a given value of *d* and *G*, we must determine the lowest value for *R* with respect to the wavenumber *k*. The values obtained are the critical Rayleigh numbers *Rc* at which instability will first occur.

### 4.1 Asymptotic analysis

We conducted a general asymptotic analysis in comparison with those used before [19, 24, 26] which included the restrictions of the limits *G*. In a similar way, as in other problems of convection, we follow the steps of Chapman and Proctor [51], Dávalos-Orozco [52], and Dávalos-Orozco and Manero [53]. Under the above conditions, the analysis is very complex, the reason why use has been made of the Maple algebra package. Thus, we look for a solution to Eqs. (20) and (21) using the following expansions:

where

At order

subject to

At order

subject to

At order

subject to

The systems of equations at order *R* as an eigenvalue in terms of the other parameters of the problem. Solvability conditions are found as usual [57]: each inhomogeneous system is multiplied by the solution to the adjoint of the homogeneous system and integrated over the range of the independent variable. The resulting integral must vanish.

Thus, the solvability conditions at

The solutions of the system of equations at leading order are

where the function

The constant

After substitution of

The growth rate can now be obtained by substitution of

Now, the transition from stability to instability via a stationary state is investigated by setting

where some simplifications have been made with the use of

From the expression for the Rayleigh number given in Eq. (42), it is possible to calculate the limit for

Here we point out that in the present chapter, our definition of the Rayleigh number differs from that defined by Hill et al. [24]. If our approximation given in Eq. (44) is multiplied by

then

From this equation, two admissible cases are possible when Eq. (45) is satisfied. First, for fixed values of

### 4.2 Analytic Galerkin method

Here use is made of the analytical Galerkin method to study the eigenvalue problem of Eqs. (17)–(18) with the boundary condition Eq. (19). This method has been used before by Pellew and Soutwell [58], Chandrasekhar [54], and Gershuni and Zhukovitskii [59]. Even though this is an approximate method, it has a very high precision. The advantage of the method is that it is possible to obtain an explicit expression of the Rayleigh number *R*. Here, it is supposed that

Briefly, the method consists in assuming a trial function which satisfies the boundary conditions for each of the dependent variables. Let that variable be *W*. Both trial functions are now substituted into the other coupled equation. Then, use is made of the orthogonality properties of the solutions in this equation to obtain the proper value of the Rayleigh number as a function of the other parameters [60].

In this way, the proposed expansions of

then, after substitution of

which is subjected to the conditions

The solution is

where

Next, Eq. (18) is multiplied by

After substitution of

This determinant, calculated with the help of the software Maple, is the solvability condition from which the eigenvalue

where *k* and *d* and can be obtained from authors upon request. This result is new because it includes, for the first time, all the parameters of the problem without any approximation. In the limit of *R* reduces to the well-known value of 720. Higher-order estimates of *R* can be obtained from Eq. (52), which provides a useful check on numerical calculations. The comparison of

## 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 *Chlamydomonas nivalis*. Figures 1–3 show marginal curves for different values of the gyrotaxis parameter

In the curves shown in Figure 1a–b, the critical values of the gyrotaxis parameter are

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

Here, a comparison is done of our theoretical results of *Chlamydomonas nivalis*, while

Experimental results | Theoretical predictions | Error (%) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

EN | ||||||||||

2 | 44.7 | 5.11 | 3.25 | 40 | 16 | 0.2 | 51 | 5.0 | 898 | 53 |

40 | 16 | 0.4 | 51 | 9.0 | 898 | 176 | ||||

23 | 204 | 7.84 | 863 | 200 | 32 | 0.2 | 270 | 1700 | 3343 | 96 |

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 *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 *Chlamydomonas nivalis*, is also used to calculate

In Table 3, the values of

Name | Description | Value |
---|---|---|

Cell volume | 5 × 10^{−10} cm^{3} | |

Acceleration due to gravity | 10^{3} cms^{−2} | |

Cell diffusivity | 5 × 10^{−5}–5 × 10^{−4} cm^{2}s^{−1} | |

Fluid density | 1 gcm^{−3} | |

Cell density | 1.05 gcm^{−3} | |

Kinematic viscosity | 10^{−2} cms^{−2} | |

Cell swimming speed | 0–2 × 10^{−2} cms^{−1} | |

Dimensional gyrotaxis parameter | 3.4 s | |

Including flagella | 6.3 s | |

Cell eccentricity | 0.20–0.31 | |

Including flagella | 0.4 |

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

where the constants

NE | d | RE | RT | kE | kT | Error k (%) | Error R (%) | |
---|---|---|---|---|---|---|---|---|

1 | 7.63 | 1.56 | 7043.64 | 10384.23 | 5.67 | 5.65 | 0.353 | 47.6 |

2 | 9.07 | 1.10 | 10599.14 | 17075.28 | 5.12 | 6.22 | 21.5 | 61.1 |

3 | 8.36 | 1.30 | 23319.62 | 13531.06 | 8.59 | 5.95 | 44.4 | 72.3 |

4 | 10.2 | 0.879 | 24835.78 | 23783.25 | 5.96 | 6.71 | 12.6 | 4.42 |

5 | 11.9 | 0.636 | 12433.03 | 37984.11 | 6.81 | 7.69 | 12.9 | 205 |

6 | 16.7 | 0.326 | 59993.82 | 105787.08 | 6.61 | 10.9 | 64.9 | 76.3 |

7 | 9.14 | 1.09 | 4676.34 | 17544.33 | 6.01 | 6.31 | 4.99 | 275 |

8 | 8.73 | 1.19 | 7198.47 | 15309.02 | 7.25 | 6.09 | 19.0 | 112 |

9 | 10.4 | 0.833 | 20618.78 | 25437.60 | 8.33 | 6.84 | 21.8 | 23.4 |

10 | 15.8 | 0.364 | 88709.37 | 89546.43 | 8.34 | 10.3 | 23.5 | 0.943 |

11 | 6.46 | 2.18 | 3700.78 | 6613.53 | 5.24 | 5.31 | 1.33 | 78.7 |

12 | 12.1 | 0.621 | 39993.68 | 40261.72 | 5.66 | 7.94 | 40.3 | 0.670 |

13 | 14.8 | 0.416 | 77622.32 | 73779.00 | 7.85 | 9.71 | 23.7 | 5.20 |

14 | 8.80 | 1.17 | 8561.41 | 15655.83 | 6.93 | 6.11 | 13.4 | 82.9 |

15 | 7.28 | 1.71 | 4022.37 | 9146.74 | 5.46 | 5.54 | 1.46 | 127 |

16 | 7.10 | 1.80 | 6960.80 | 8540.98 | 6.43 | 5.49 | 17.1 | 22.7 |

17 | 10.7 | 0.788 | 18965.81 | 27708.92 | 4.16 | 7.02 | 68.7 | 46.1 |

18 | 10.7 | 0.788 | 18965.81 | 27708.92 | 8.32 | 7.02 | 18.1 | 46.1 |

19 | 10.7 | 0.788 | 18965.81 | 27708.92 | 4.89 | 7.02 | 43.5 | 46.1 |

20 | 16.6 | 0.331 | 107793.38 | 104192.94 | 8.7 | 10.9 | 25.3 | 3.45 |

21 | 8.80 | 1.17 | 8561.41 | 15655.83 | 7.01 | 6.11 | 14.7 | 82.9 |

22 | 8.13 | 1.37 | 6910.81 | 12466.94 | 6.11 | 5.84 | 4.62 | 80.4 |

23 | 10.7 | 0.791 | 41777.52 | 27799.28 | 7.84 | 7.06 | 11.0 | 50.3 |

24 | 6.67 | 2.04 | 6260.75 | 7202.62 | 6.07 | 5.36 | 13.2 | 15.0 |

25 | 4.26 | 5.01 | 1044.17 | 2301.46 | 6.22 | 4.82 | 29.0 | 120 |

26 | 6.46 | 2.18 | 5663.12 | 6613.53 | 6.71 | 5.31 | 26.4 | 16.8 |

27 | 6.46 | 2.18 | 5663.12 | 6613.53 | 6.05 | 5.31 | 13.9 | 16.8 |

28 | 6.46 | 2.18 | 5663.12 | 6613.53 | 5.37 | 5.31 | 1.13 | 16.8 |

29 | 6.46 | 2.18 | 5663.12 | 6613.53 | 5.94 | 5.31 | 11.9 | 16.8 |

30 | 7.83 | 1.48 | 33599.24 | 11220.81 | 7.46 | 5.74 | 30.0 | 199 |

31 | 6.80 | 1.96 | 6478.07 | 7585.38 | 6.00 | 5.39 | 11.3 | 17.0 |

32 | 4.47 | 4.56 | 4519.70 | 2583.99 | 6.66 | 4.86 | 37.0 | 74.9 |

33 | 2.70 | 12.4 | 475.88 | 2180.73 | 4.97 | 6.52 | 31.2 | 358 |

34 | 3.85 | 6.14 | 1958.29 | 4879.03 | 6.21 | 6.46 | 4.02 | 149 |

35 | 7.42 | 1.65 | 8259.40 | 9655.91 | 6.15 | 5.60 | 9.82 | 16.9 |

36 | 7.83 | 1.48 | 33599.24 | 11220.81 | 6.49 | 5.74 | 13.1 | 199 |

37 | 5.22 | 3.33 | 2418.53 | 3794.61 | 6.37 | 4.99 | 27.6 | 56.9 |

38 | 6.87 | 1.92 | 20572.53 | 22011.51 | 10.4 | 7.13 | 46.0 | 6.99 |

39 | 6.87 | 1.92 | 20572.53 | 22011.51 | 10.8 | 7.13 | 51.9 | 6.99 |

By using the data of our Table 2 and Table I of Bees and Hill [27], the experimental values for *d*, *G*, and

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.

## 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

With the asymptotic analysis for *d* and *G*. This result is important because it was also possible to calculate a critical value of the gyrotaxis parameter

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

An analytic Galerkin method was also used to obtain a general expression of *R* without any restriction on the magnitudes of *d*, *G*, and *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, 51, 52, 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.

## Nomenclature

*B*

dimensional gyrotactic parameter, s

*k*

wavenumber

cell diffusivity, cm^{2}s^{−1}

motility of microorganisms

*G*

dimensionless gyrotactic parameter

*g*

acceleration due to gravity, cms^{−2}

*H*

layer depth, cm

flux density of organisms

average cell concentration

concentration of microorganisms

*p*

pressure

*R*

Rayleigh number

Schmidt number

time

cell swimming speed, cms^{−1}

fluid velocity

Cartesian coordinates

*Greek symbols*

cell eccentricity

viscosity, gcm^{−1}s^{−1}

kinematic viscosity, cm^{2}s^{−1}

water density, gcm^{−3}

cell volume, cm^{−3}

*Subscripts*

*BH*

result obtained by Bees and Hill [26]

*c*

critical value

*E*

experimental result

*T*

theoretical result

## Acknowledgments

The authors would like to thank Alberto López, Alejandro Pompa, Cain González, Raúl Reyes, Ma. Teresa Vázquez, and Oralia Jiménez for technical support. I. Pérez Reyes would like to thank the Programa de Mejoramiento del Profesorado (PROMEP).