the onset of rayleigh-bénard convection in a suspension of ... · microorganisms, a type of...

International Journal of Applied Engineering Research ISSN 0973-4562 Volume 14, Number 4 (2019) pp. 1031-1040 © Research India Publications. http://www.ripublication.com

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The Onset of Rayleigh-Bénard Convection in a Suspension of Gyrotactic

Microorganisms with ITBT

Rasika V. Pote1

, Azalea C. K. Marak2

, Smita S. Nagouda3

1,2,3 Department of Mathematics, CHRIST (Deemed to be University), Bangalore-560029, India.

Abstract

In this paper, the onset of Rayleigh-Bénard convection of an incompressible fluid suspended with motile gyrotactic microorganisms which is imposed to time-periodic boundary temperature is investigated. Linear stability analysis along with Venezian approach is used to obtain the correction Rayleigh number. The shift of the critical Rayleigh number is obtained as a function of frequency of modulation. Three different cases of oscillating temperature fields are considered namely, Symmetric, Asymmetric and bottom wall modulated. Graphs are plotted and analysed accordingly for these three cases.

Keywords: Gyrotactic microorganisms, Bioconvection, ITBT, Thermophilic.

1. INTRODUCTION

Convection is a phenomenon in fluids where transfer of heat occurs due to an imbalance between the buoyancy and viscous forces. Depending on the nature of the boundaries and the type of imbalance that occurs between these two forces, convection problems can be classified into several types. Rayleigh-Bénard convection is one such in which a fluid confined between two infinite horizontal plates, separated by a finite depth, is heated from below. The temperature difference between the plates at which convection occurs is called the Rayleigh number. If an inactive layer of fluid is subjected to a sudden temperature gradient, there is a transition from conduction to convection at the critical Rayleigh number and thereafter, the motion of the fluid becomes superficial leading to turbulence as the Rayleigh number becomes large.

There are numerous mechanisms that can be utilized to control the onset of convection. Maintaining a non-uniform temperature gradient across the fluid layer by sudden heating or cooling at the boundaries is one of the effective ways to control convection. Along with a fixed temperature difference between the boundaries, an additional perturbation, which varies sinusoidally in time, is applied to the wall temperatures. This is achieved by properly tuning the amplitude and the frequency of modulation applied to the boundaries. This was initially investigated by Venezian [1] after being motivated by Donnelly after he investigated the onset of instability of viscous flow between rotating cylinders under the effect of modulation of the rate of rotation of the inner cylinder [2] . Later, many other researchers like Rosenblatt and Tanaka [3] ,

Bhadauria [4] - [5] , Pranesh and Sangeetha [6] , Pranesh and Riya [7] , Siddheshwar and Pranesh [8] have all investigated the same under different physical models.

When a layer of fluid is suspended with self propelled motile microorganisms, a type of convection called bioconvection sets in even in the absence of any heat. This type of convection is induced by a density gradient that occurs when a large number of microorganisms (heavier than water) gather in a certain region of the fluid layer and not due to the swimming of microorganisms. When the upper surface of the suspension become too dense due to the accumulation of micro-organisms, it becomes unstable and micro-organisms fall down to cause this phenomenon called bioconvection. This bioconvection pattern is maintained by the return upswimming microorganisms.

The initial observations of bioconvection had been carried out by Wager and was described in his article [9] but the matter was not mentioned again until Platt [10] found cells resembling “Benard cells” in dense cultures of free-swimming microorganisms which are however not due to thermal convection. In 1988, a continuum model for suspensions of gyrotactic microorganisms was proposed by Pedley et al. [11] . The following year, Hill et al. [12] made a thorough study based on the continuum model of Pedley et al. of the growth of bioconvection patterns in a suspension of gyrotactic microorganisms in a finite depth fluid layer. This continuum model had been used since then by many researchers like Ghorai and Hill in 1999 [13] , and in 2000 [14] , and Kuznetsov (2005) [15] who first studied the stability of a suspension of gyrotactic microorganisms in a fluid layer of finite depth when heated from below. He was the one who established that heating from below makes the system more unstable helping the development of bioconvection. Thereafter, researchers like Srimani and Roopa in 2011, investigated the result of rotation on the onset of bioconvection in a fluid layer suspended with motile gyrotactic microorganisms [16] . In 2012, Srimani and Sujatha conducted a study of the cumulative effect of uniform rotation and the heating or cooling from below on the stability of a suspension of microorganisms,which are motile and gravitactic, in a porous shallow horizontal layer [17] .

Thermo-bioconvection is pertinent in applications where hemophilic microorganisms such as Bacillus licheniformis and Bacillus thermoglucosidasius are utilized, as in


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fuel-efficient cells, and to motile thermophilic microorganisms thriving in hot springs. Microorganisms are needed abundantly for pharmaceutical purposes and in the food and chemical industries for the production of everyday foods and insecticides respectively. Moreover, they play a vital role in our intestines, not only in digestion and absorption of ingested food, but also helps our immune system in fighting off infections.

In this paper, a fluid layer of finite depth suspended with motile upswimming gyrotactic microorgansims have been subjected to thermal modulation. This fluid layer exhibiting bioconvection is investigated for the onset of Rayleigh- Bènard convection using Venezian approach and accordingly, graphs have been plotted for the same for three cases: symmetric, asymmetric and bottom wall modulation.

Nomenclature

* * * *= ( , , )q u v w fluid convection velocity vector

w the density of water at 0=T T

*t time

*p excess pressure

Dynamic viscosity of the suspension *n Number density of motile

microorganisms

Average volume of microorganisms

Density difference

g Gravitational acceleration

Volume expansion coefficient of water at constant pressure

*T Temperature

pc Specific heat of water

Thermal conductivity of water *j Flux of microorganisms due to

macroscopic convection of the fluid, self-propelled swimming of microorganisms and diffusion of microorganisms

p̂ Unit vector indicating the direction of swimming of microorganisms

ˆcW p Vector of average swimming velocity relative to the fluid

cW Constant swimming speed of the microorganisms

D Diffusivity of microorganisms

k̂ Unit vector which is positively oriented downward

A small amplitude of temperature modulation

Temperature modulation frequency

Phase angle

Sc Schmidt number

Rb Bioconvection Rayleigh Number

Ra Thermal Rayleigh Number

Lb Bioconvection Lewis Number

Thermal diffusivity of the fluid

cV Swimming speed of microorganisms

a , b Semi-major and semi-minor axes of the spheroidal cells

0 Measure of the cell eccentricity

B Gyrotactic orientation parameter

Dimensionless constant relating viscous torque to the relative angular velocity of the cell

h Displacement of the centre of mass of the cell from the centre of buoyancy

* Denotes dimensional variable

2. MATHEMATICAL FORMULATION

Figure 1: Physical Configuration

Following are the governing equations:

Equation of Continuity :

* = 0q (1)

Conservation of Linear Momentum:


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** * * * * *2 *

*

* * *

[ ( ) ] =

( )

w

w o

q q q p qt

n g g T T

(2)

Conservation of Energy:

** * * *2 *

*[ ( ) ] =p wTc q T Tt

(3)

Conservation of Cell Number:

** *

* =n jt

(4)

where, * * * * * *ˆ= cj n q n W p D n

The boundary walls are imposed with the following temperature conditions:

0

0

[1 cos ] = 02=[1 cos( )] =

2

TT t zT

TT t z d

(5)

Thermal Modulations:

1. Symmetric (in-phase, =0)

2. Asymmetric (out-of-phase, = )

3. Upper wall is at constant temperature and temperature of lower wall is modulated. ( = i )

We use the following dimensionless variables to get the non-dimentionalized form of the equations (1)-(4) given by equations (6)-(9):

* * * *

02 *

* * *2

[ , , ]= ; [ , , ] = ;

= ; = ; = ( ) ; = ;

n x y zn x y zn d

d D d Tq q t t p p TD d D T

= 0q (6)

2

0

1 ( ) =

q q q p qSc t

TRbn RaLb TT

(7)

2( ) =T q T Lb Tt

(8)

2ˆ= ( )cn q n nV p nt

(9)

where the dimensionless constants are given below,

=w

ScD

,

30= n gdRbD

,

3 2

= w pg Td cRa

,

=LbD

, =p wc

, = cc

W dVD

2.1 Basic State

The quantities in the basic state are given by:

= (0,0,0); = ( );= ( ); = ( , ); = ( )

b b

b b b

q q p p zz T T z t n n z

Using the above quantities, equations (6)-(9) can be rewritten as given below:

= 0bq (10)

0=bb b

p TRaLb T Rbnz t

(11)

2

2=b bT TLbt z

(12)

Equation (12) is solved for ( )bT z subject to the boundary conditions.

We get the solution for ( )bT z as given below:

( ) =2

( ) ( )

b o

z z i td d

T TT z T zd

Re a e a e e

(13)

where 2

2 = i dLb

and ( ) = ( )2

iT e eae e

2ˆ( ) =b c bn V p n (14)

Solving for bn we get,

= V zcbn e (15)


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where is integration constant. The basic number density at bottom of the layer n is given by

0

1= ( )d

bn n z dzd

Therefore

=1

cV dc

nV de

(16)

3. LINEAR STABILITY

The perturbations are introduced as follows: ' ' '

' '

= ; = ; = ;ˆˆ ˆ= ; =

b b b

b

q q q p p p T T T

n n n p k p

' = 0q (17)

'' 2 ' ' '1 =q p q n Rb RaLbT

Sc t

(18)

'' 2 '0 =dTT w Lb T

t dz

(19)

where, 0dTdz

is the non-dimensional form of bdTdz

0, = 1 ( )dTwhere f zdz

(20)

( ) = ( ) ( )z z i tf z Re A e A e e

, ( ) =2

ie ewhere Ae e

'' ' ' 'ˆˆ= ( )c c

n div n q V p nV k nt

(21)

Where = Vcn e

According to Pedley et al.[11] , Since gyrotactic behaviour of the microorganism is not affected by the variation in temperature within the fluid.

'ˆ = ( , ,0)p B (22)

where ' '

= (1 ) (1 )o ow vy z

' '

= (1 ) (1 )o ow ux z

2 2

2 2=oa ba b

=2 o

Bh g

Therefore equation (21) becomes,

' ''

2 '2 ' 2 '1 2

=

(1 ) (1 )

c

c o o

n n nw Vt z z

wV Bn w nz

(23)

Curling twice and eliminating the pressure term from (18), we get

2 '4 ' 2 ' 2 '

1 11 ( ) =w w Rb n RaLb TSc t

(24)

The above equations (after dropping ' for simplicity) are solved subject to the free-free boundary conditions given by

2

2= = = 0

= = 0 = 1

c

WW Tz

dnand V n at z and zdz

Elimination of T and n from (19), (23), (24) we get the equation for W as follows

2 2 2 2 4

22 2 21 1 2

2 2 2 01

1

(1 ) (1 )

c

c c o o

c

V Lbt t Sc t

WRb Lb V V B n

t z

TRa V Wt z

(25)

Boundary conditions for solving (25) are in the following form

2 4 6

2 4 6= = = = 0 = 0 =1W W WW at z and zz z z

4. PERTURBATION PROCEDURE

The value of W and R are determined as the eigen-function and eigenvalue of the (25) from the linear profile for the distribution of basic temperature 0 = 1T

z

by


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quantities of order . Thus, there is a difference in the value of the eigen-values of ordinary Bénard convection and the present problem by order of . The solution of (25) is of the form:

20 0 1 1 2 2( , ) = ( , ) ( , ) ( , ) ..........R W R W R W R W (26)

Here, =R Ra

Substituting into (25) and equating different coefficient of we get the resulting system of equations.

1 = 0oLW (27)

2 2 21 1 0 1= c oLW R V fW

t

(28)

2 2 21 2 1 1 2= c o oLW V fR W R W

t

(29)

Where,

2 2 2 2 4 2 21 0 1

1= cL Lb V Rt t Sc t

2 2

22 2 21 1 2(1 ) (1 )

c

C c o o

Vt

Rb Lb V V B nt z

(30)

4.1 Solution to the Zeroth Order Problem

The zeroth order problem is equivalent to the Rayleigh-Bénard problem without temperature modulation. We introduce a vertical velocity perturbation to investigate system’s stability in absence of temperature modulation corresponding to lowest mode of convection as:

= ( ) [ ( )]oW Sin z exp i lx my (31)

where, l and m are horizontal wave numbers in x and y direction.

Substituting (31) in (27) 8 2 2 2 2

0 00 2 2 2 2

( ) ( 1)(1 {(1 ) (1 ) })=( )

c c c

c

k V k RbV V B aRk a V k

(32)

Where 2 2 2=k a , 2 2 2=a l m

4.2 Solution to the First Order Problem

Now, (28) becomes 2

1 1 1= o oLW R Aa fW (33)

where, 2 2 2

1 = ( )cA k V k

To obtain solution for (33), the R.H.S must be orthogonal to null space of 1L , which implies that the time independent part of R.H.S must be orthogonal to sin z . 2 2

1 ( )R a k sin z is the only term in (33) which is steady since we know that f varies with time sinusoidally, so that R1 = 0. Therefore in (26) all the odd coefficients i.e. R3 = 0, R5 = 0.

Expanding right-hand side of (33) using Fourier series

expansion and inverting 1L term by term we obtain 1W

21 0 1

=1 1

( )= ( ) ( )( , )

i tr

r

BW R a A Re e Sin r zL r

(34)

1 1( ) = ( ) ( ) ( ) ( )r r rB A g A g

2 2

2 2 2 2 2 2

2 [ ( 1) ( )( ) =( )[ ( 1) ][ ( 1) ]

r i i

rr e e e eB

e e r r

1

1 0= 2 ( ) ( )z

rg e Sin r z Sin z

2 2

1 2 2 2 2 2 2

4 [1 ( 1) ]=[ ( 1) ][ ( 1) ]

r

rr eg

r r

1 1 2( , ) =L r Y iY ,

Where, 2 2 2 2 2

2 2 2 2 4 41

2 2 2 2 2 4 2 2 20 0 0

( )= ( ( ) )

( ) (1 {(1 ) (1 ) })

c r rr c r r r

Vcr c r r c

V k k LbY k Lb V k k kSc Sc

R a k Lb V k RbLba k V e B a r

2 2 232 2 2 4 4 2 2

2 0

4 2 2 2 20 0

( )= ( ( ) )

(1 {(1 ) (1 ) })

r c rr c r r r r

Vcr c

k Lb V kY k V k Lbk k R a k LbSc Sc

Rbk Lba V e B a r

Here, 2 2 2 2=rk r a

The equation of 2W is

2 21 2 0 2 1 2 1= oLW R a A fW R a AW (35)

Where 2 2 22 = ( )r c rA k V k i


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Now we solve (35) to find the value of R2. For the solution it is essential that the steady part of R.H.S should be orthogonal to sinn z . This gives

1

2 2 101

2= ( )oRR A fW Sin z dzA

(36)

here, over-bar denotes the time average.

From (33),

2 1 1 12 1 2

1

=o

A W LWA fWR a A

(37)

simplifying we get, 2 2

212 1 2 =1

1

| ( ) |( ) = ( ) ( )2 ( , )

o rr

R a A BA fW Sin z A Re Sin zL r

Finally, 2 2

02 2

=1 2

( ) | ( ) |=2 | ( , ) |

r

r

R a BR Re AL r

(38)

2 2 2 *0 2 2 2

2=1 2

| ( ) | | | [ ( , ) ( , )]=2 2 | ( , ) |

r

r

R a B A L r L rRL r

(39)

Where 2 2 1( , ) = ( , )L r A L r , *2L is conjugate of 1( , )L r

5. MINIMUM VALUE FOR R FOR CONVECTION

Rayleigh number is obtained by finding the eigen value of the eigen function W , which is bounded with respect to time. Since Rayleigh number is a function of amplitude of modulation and wave number,

20 0( , ) = ( ) ( ) ......R a R a R a (40)

After evaluating value of 0R and 2R at 0=a a , we can

evaluate 4R at 2=a a , which minimizes 2R . To evaluate

critical Rayleigh number 2cR we substitute the critical

0=a a in 2R , where 0a is the value for which 0R is

minimum. 2cR for three different cases is discussed below.

Case(1): The wall temperature are modulated with = 0 because the oscillating field is symmetric. Therefore, we have,

( ) =0r

r

b r is evenB

r is odd

Case(2): Here the walls are modulated out-of-phase with = 0 because temperature field is anti-symmetric.

Therefore, we have,

( ) =0r

r

b r is oddB

r is even

Case(3): The upper wall is considered to be held at constant temperature and lower wall is modulated with = i . Therefore,

( ) = ,2r

rbB for all integer value of r

where,

2

2 2 2 2 2 2

4=[ ( 1) ][ ( 1) ]r

rbr r

(41)

is already defined, but in terms of the dimensionless frequency, it reduces to

= i

and therefore equation (41) becomes 2 4 2

22 2 2 2 2 2 2 2

16| | =[ {( 1) } ][ {( 1) } ]r

rbr r

(42)

Hence the expression for 2cR becomes,

2 2 2 *0 2 2 2

2=1 2

| ( ) | | | [ ( , ) ( , )]=2 2 | ( , ) |

rc

r

R a B A L r L rRL r

(43)

6. RESULTS AND DISCUSSIONS

The onset of Rayleigh Bénard convection in a suspension of gyrotactic microorganisms with imposed time periodic boundary temperature is comprehended with relevant parameters. The parameters that affects the onset of convection of this system are Rb , Sc , Lb , Vc , Ra , .

In-phase:

Figure (1) and Figure (2) shows frequency of modulation versus correction Rayleigh number when modulation is in-phase. For various values of Bioconvection Rayleigh number(Rb), Bioconvection Peclet number(Vc), Schmidth number(Sc), and Bioconvection Lewis number(Lb), we observe that, for low frequency, 2cR becomes more negative indicating that has a destabilizing effect for low values of but has a stabilizing effect for moderate values of the same.

In other words, if c is the frequency where the correction Rayleigh number changes from destabilizing to stabilizing, then we can infer that, the system destabilizes for < c ,

and the system stabilizes for <c .


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(a) Vs 2cR for different values of Rb

(b) Vs 2cR for different values of cV

Figure 1

Figure 1(a) shows the graph of versus 2cR for different

values of bioconvection-Rayleigh number Rb . As Rb increases correction Rayleigh number becomes less negative indicating that it stabilizes the system. This implies that the presence of gyrotactic microorganisms decreases the effect of modulation.

Figure 1(b) shows the graph of versus 2cR for different

values of bioconvection Peclet number cV . As observed in

[11] , for increasing values of cV , bR decreases. From

Figure 1(a), it is evident that with decrease in bR , the system stabilizes. This is because, as the swimming speed of the microorganisms increases, the effect of temperature modulation decreases.

(a) Vs 2cR for different values of Sc

(b) Vs 2cR for different values of Lb

Figure 2


values of Schmidth number Sc . As Sc increases, the viscosity of the fluid increases which leads to more consumption of heat in overcoming the viscous drag. This leads to an increase in the Rayleigh number thereby stabilizes the system.


values of bioconvection Lewis number Lb . As Lb correction Rayleigh number increases and hence Lb stabilizes the system.

Out-of-phase:

Figure (3) and (4) shows frequency of modulation versus correction Rayleigh number when modulation is out-of-phase for different values of Bioconvection Rayleigh number(Rb), Bioconvection Peclet number(Vc), Schmidth number(Sc), and Bioconvection Lewis number(Lb). In this case, we find that the values that 2cR take are positive. This is because the


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temperature gradient is linear with respect to time, therefore R is supercritical and subcritical for first half and second half of the cycle.


values of bioconvection Rayleigh number Rb . We see that as Rb increases correction Rayleigh number becomes more positive. Hence, as in the case of in-phase modulation, the presence of gyroctactic microorganisms decreases the effect of modulation.


values of bioconvection Peclet number cV . As cV increases, correction Rayleigh number becomes more positive and hence, stabilizes the system.



Figure 3


values of Schmidth number Sc . As Sc values increase, the fluid becomes more viscous and hence, the correction Rayleigh number increases.

Figure 4(b) shows the graph of Vs 2cR for different

values of bioconvection Lewis number Lb . We see that as Lb increases correction Rayleigh number increases.



Figure 4

Bottom wall Modulated:

The effect of modulation on correction Rayleigh number in case only lower wall boundary is modulated is qualitatively same as in case of out-of-phase. When modulation is lower wall boundary or out-of-phase the convection wave inhibits instability of system. So the convection takes place at higher vales of Rayleigh number.


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Figure 5



Figure 6

7. CONCLUSION

From the study of thermal modulation of a fluid containing gyrotactic microorganisms, the following conclusions can be drawn:

1. In the case of in-phase modulation, the maximum stability is achieved at c =9. When < c

modulation effect is increases, for <c effect of modulation increases.

2. The system is the most stable when the modulation is out-of-phase.

3. The study of temperature modulation enables us to advance or delay the convection.

4. The suspended gyrotactic microorganisms decreases the effect of modulation.

5. As , 2 0cR

REFERENCES

[1] G. Venezian,1969 “Effect of modulation on the onset of thermal convection,” Journal of Fluid Mechanics, vol. 35, no. 2, pp. 243–254.

[2] R. J. Donnelly, 1964 “Experiments on the stability of viscous flow between rotating cylinders iii. enhancement of stability by modulation,” Proc. R. Soc. Lond. A , vol. 281, no. 1384, pp. 130–139.

[3] S. Rosenblat and G. Tanaka,1971 “Modulation of thermal convection instability,” The Physics of Fluids, vol. 14, no. 7, pp. 1319–1322.

[4] B. Bhadauria, P. Siddheshwar, and O. P. Suthar,2012 “Nonlinear thermal instability in a rotating viscous fluid layer under temperature/gravity modulation,” Journal of Heat Transfer , vol. 134, no. 10, p. 102502.

[5] B. Bhadauria, P. Siddheshwar, J. Kumar, and O. P. Suthar,2012 “Weakly nonlinear stability analysis of temperature/gravity-modulated stationary rayleigh–bénard convection in a rotating porous medium,” Transport in porous media , vol. 92, no. 3, pp. 633–647.

[6] S. Pranesh and S. George,2010 “Effect of magnetic field on the onset of rayleigh-bénard convection in boussinesq- stokes suspensions with time periodic boundary temperatures,” International Journal of Applied Mathe- matics and Mechanics , vol. 6, no. 16, pp. 38–55.

[7] S. Pranesh and R. Baby,2012 “Effect of non-uniform temperature gradient on the onset of rayleigh-bénard electro convection in a micropolar fluid,” Applied Mathematics , vol. 3, no. 05, p. 442


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[8] P. Siddheshwar and S. Pranesh,2000 “Effect of temperature/gravity modulation on the onset of magneto- convection in electrically conducting fluids with internal angular momentum,” Journal of magnetism and magnetic materials , vol. 219, no. 2, pp. 153–162.

[9] H. Wager,1911 “On the effect of gravity upon the movements and aggregation of euglena viridis, ehrb, and other micro-organisms,” Philosophical Transactions Of The Royal Society Of London Series B-Containing Papers Of A Biological Character , no. 201.

[10] J. R. Platt,1961 “Bioconvection patterns in cultures of free-swimming organisms,” Science , vol. 133, no. 3466, pp. 1766–1767.

[11] T. Pedley, N. Hill, and J. Kessler,1988 “The growth of bioconvection patterns in a uniform suspension of gyrotactic micro-organisms,” Journal of Fluid Mechanics , vol. 195, pp. 223–237.

[12] N. Hill, T. Pedley, and J. Kessler,1989 “Growth of bioconvection patterns in a suspension of gyrotactic microorganisms in a layer of finite depth,” Journal of Fluid Mechanics , vol. 208, pp. 509–543.

[13] S. Ghorai and N. A. Hill,1999 “Development and stability of gyrotactic plumes in bioconvection,” Journal of Fluid Mechanics , vol. 400, pp. 1–31.

[14] S. Ghorai and N. A. Hill,2000 “Wavelengths of gyrotactic plumes in bioconvection,” Bulleting of Mathematical Biology , vol. 62, pp. 429–450.

[15]A. Kuznetsov,2005 “The onset of bioconvection in a suspension of gyrotactic microorganisms in a fluid layer of finite depth heated from below,” International Communications in Heat and Mass Transfer , vol. 32, no. 5, pp. 574–582.

[16] P. K. Srimani and M. C. Roopa,2011 “The cumulative effect of rotation on the onset of bio-porousconvection in a suspension of gyrotactic microorganisms in a layer of finite depth under adverse temperature gradient,” International Journal of Current Research , vol. 3, no. 7, pp. 204–210.

[17] P. K. Srirami and Sujatha D.,2012 “The effect of rotational constraint on thermo-bioconvection in a suspension gravitactic microorganisms- a numerical study”, International Journal of Current Research , vol. 4, no. 3, pp. 126–131.

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