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
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 RayleighBenard 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
More recently, investigations have been reported for a semidilute 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
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
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
2. Equations of motion
We consider an infinite horizontal layer of a suspension of gyrotactic microorganisms. The fluid layer is bounded at
where
where
In the basic state, the fluid velocity is zero and
where
3. Linear stability
We make the governing Eqs. (1–3) nondimensional by scaling all lengths with
where the nondimensional quantity
where
where
are the Schmidt and bioconvection Rayleigh numbers, respectively. Pedley and Kessler [55] give a definition of the vector
where the subscript
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
By elimination of the pressure from Eqs. (13–15), it is possible to obtain a coupled system of two equations, for
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 wellknown 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
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
where
At order
subject to
At order
subject to
At order
subject to
The systems of equations at order
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
Briefly, the method consists in assuming a trial function which satisfies the boundary conditions for each of the dependent variables. Let that variable be
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
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
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
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
6. Comparison with experiments
In this section a comparison is done of our theoretical results of
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
where the constants
NE 

Error 
Error 


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
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
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
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
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
dimensional gyrotactic parameter, s
wavenumber
cell diffusivity, cm^{2}s^{−1}
motility of microorganisms
dimensionless gyrotactic parameter
acceleration due to gravity, cms^{−2}
layer depth, cm
flux density of organisms
average cell concentration
concentration of microorganisms
pressure
Rayleigh number
Schmidt number
time
cell swimming speed, cms^{−1}
fluid velocity
Cartesian coordinates
cell eccentricity
viscosity, gcm^{−1}s^{−1}
kinematic viscosity, cm^{2}s^{−1}
water density, gcm^{−3}
cell volume, cm^{−3}
result obtained by Bees and Hill [26]
critical value
experimental result
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).
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