nanofluid bioconvection in water-based suspensions containing

NANO EXPRESS Open Access

Nanofluid bioconvection in water-basedsuspensions containing nanoparticles andoxytactic microorganisms: oscillatory instabilityAndrey V Kuznetsov

Abstract

The aim of this article is to propose a novel type of a nanofluid that contains both nanoparticles and motile(oxytactic) microorganisms. The benefits of adding motile microorganisms to the suspension include enhancedmass transfer, microscale mixing, and anticipated improved stability of the nanofluid. In order to understand thebehavior of such a suspension at the fundamental level, this article investigates its stability when it occupies ashallow horizontal layer. The oscillatory mode of nanofluid bioconvection may be induced by the interaction ofthree competing agencies: oxytactic microorganisms, heating or cooling from the bottom, and top or bottom-heavy nanoparticle distribution. The model includes equations expressing conservation of total mass, momentum,thermal energy, nanoparticles, microorganisms, and oxygen. Physical mechanisms responsible for the slip velocitybetween the nanoparticles and the base fluid, such as Brownian motion and thermophoresis, are accounted for inthe model. An approximate analytical solution of the eigenvalue problem is obtained using the Galerkin method.The obtained solution provides important physical insights into the behavior of this system; it also explains whenthe oscillatory mode of instability is possible in such system.

IntroductionThe term “nanofluid” was coined by Choi in his seminalpaper presented in 1995 at the ASME Winter AnnualMeeting [1]. It refers to a liquid containing a dispersionof submicronic solid particles (nanoparticles) with typi-cal length on the order of 1-50 nm [2]. The uniqueproperties of nanofluids include the impressive enhance-ment of thermal conductivity as well as overall heattransfer [3-7]. Various mechanisms leading to heattransfer enhancement in nanofluids are discussed innumerous publications; see, for example [8-12].Wang [13-15] pioneered in developing the constructal

approach, created by Bejan [16-19], for designing nano-fluids. Nanofluids enhance the thermal performance ofthe base fluid; the utilization of the constructal theorymakes it possible to design a nanofluid with the bestmicrostructure and performance within a specified typeof microstructures.

Recent publications show significant interest in appli-cations of nanofluids in various types of microsystems.These include microchannels [20], microheat pipes [21],microchannel heat sinks [22], and microreactors [23].There is also significant potential in using nanomaterialsin different bio-microsystems, such as enzyme biosen-sors [24]. In [25], the performance of a bioseparationsystem for capturing nanoparticles was simulated. Thereis also strong interest in developing chip-size microde-vices for evaluating nanoparticle toxicity; Huh et al. [26]suggested a biomimetic microsystem that reconstitutesthe critical functional alveolar-capillary interface of thehuman lung to evaluate toxic and inflammatoryresponses of the lung to silica nanoparticles.The aim of this article is to propose a novel type of a

nanofluid that contains both nanoparticles and oxytacticmicroorganisms, such as a soil bacterium Bacillus subti-lis. These particular microorganisms are oxygen consu-mers that swim up the oxygen concentration gradient.There are important similarities and differences betweennanoparticles and motile microorganisms. In theirimpressive review of nanofluids research, Wang and Fan[27] pointed out that nanofluids involve four scales: the

Correspondence: [email protected]. of Mechanical and Aerospace Engineering, North Carolina StateUniversity, Campus Box 7910, Raleigh, NC 27695-7910, USA

Kuznetsov Nanoscale Research Letters 2011, 6:100http://www.nanoscalereslett.com/content/6/1/100

© 2011 Kuznetsov; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons AttributionLicense (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium,provided the original work is properly cited.

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molecular scale, the microscale, the macroscale, and themegascale. There is interaction between these scales.For example, by manipulating the structure and distri-bution of nanoparticles the researcher can impactmacroscopic properties of the nanofluid, such as itsthermal conductivity. Similar to nanofluids, in suspen-sions of motile microorganisms that exhibit spontaneousformation of flow patterns (this phenomenon is calledbioconvection) physical laws that govern smaller scaleslead to a phenomenon visible on a larger scale. Whilesuperfluidity and superconductivity are quantum phe-nomena visible at the macroscale, bioconvection is amesoscale phenomenon, in which the motion of motilemicroorganisms induces a macroscopic motion (convec-tion) in the fluid. This happens because motile microor-ganisms are heavier than water and they generally swimin the upward direction, causing an unstable top-heavydensity stratification which under certain conditionsleads to the development of hydrodynamic instability.Unlike motile microorganisms, nanoparticles are notself-propelled; they just move due to such phenomenaas Brownian motion and thermophoresis and are carriedby the flow of the base fluid. On the contrary, motilemicroorganisms can actively swim in the fluid inresponse to such stimuli as gravity, light, or chemicalattraction. Combining nanoparticles and motile microor-ganisms in a suspension makes it possible to use bene-fits of both of these microsystems.One possible application of bioconvection in bio-

microsystems is for mass transport enhancement andmixing, which are important issues in many microsys-tems [28,29]. Also, the results presented in [30] suggestusing bioconvection in a toxic compound sensor due tothe ability of some toxic compounds to inhibit the fla-gella movement and thus suppress bioconvection. Also,preventing nanoparticles from agglomerating and aggre-gating remains a significant challenge. One of the rea-sons why this is challenging is because althoughinducing mixing at the macroscale is easy and can beachieved by stirring, inducing and controlling mixing atthe microscale is difficult. Bioconvection can provideboth types of mixing. Macroscale mixing is provided byinducing the unstable density stratification due tomicroorganisms’ upswimming. Mixing at the microscaleis provided by flagella (or flagella bundle) motion ofindividual microorganisms. Due to flagella rotation,microorganisms push fluid along their axis of symmetry,and suck it from the sides [31]. While the estimatesgiven in [32] show that the stresslet stress produced byindividual microorganisms have negligible effect onmacroscopic motion of the fluid (which is rather drivenby the buoyancy force induced by the top-heavy densitystratification due to microorganisms’ upswimming), theeffect produced by flagella rotation is not negligible on

the microscopic scale (on the scale of a microorganismand a nanoparticle).In order to use suspensions containing both nanoparti-

cles and motile microorganisms in microsystems, thebehavior of such suspensions must be understood at thefundamental level. Bio-thermal convection caused by thecombined effect of upswimming of oxytacic microorgan-isms and temperature variation was investigated in[33-36]. Bioconvection in nanofluids is expected to occurif the concentration of nanoparticles is small, so that nano-particles do not cause any significant increase of the visc-osity of the base fluid. The problem of bioconvection insuspensions containing small solid particles (nanoparti-cles) was first studied in [37-41] and then recently in [42].Non-oscillatory bioconvection in suspensions of oxytacticmicroorganisms was considered in Kuznetsov AV: Nano-fluid bioconvection: Interaction of microorganismsoxytactic upswimming, nanoparticle distribution andheating/cooling from below. Theor Comput Fluid Dyn2010, submitted. This article extends the theory to thecase of oscillatory convection in suspensions containingboth nanoparticles and oxytactic microorganisms.

Governing equationsThe governing equations are formulated for a water-based nanofluid containing nanoparticles and oxytacticmicroorganisms. The nanofluid occupies a horizontallayer of depth H. It is assumed that the nanoparticlesuspension is stable. According to Choi [2], there aremethods (including suspending nanoparticles usingeither surfactant or surface charge technology) that leadto stable nanofluids. It is further assumed that the pre-sence of nanoparticles has no effect on the direction ofmicroorganisms’ swimming and on their swimmingvelocity. This is a reasonable assumption if the nanopar-ticle suspension is dilute; the concentration of nanopar-ticles has to be small anyway for the bioconvection-induced flow to occur (otherwise, a large concentrationof nanoparticles would result in a large suspension visc-osity which would suppress bioconvection).In formulating the governing equations, the terms per-

taining to nanoparticles are written using the theorydeveloped in Buongiorno [43], while the terms pertain-ing to oxytactic microorganisms are written using theapproach developed by Hillesdon and Pedley [44,45].The continuity equation for the nanoparticle-microor-

ganism suspension considered in this research is

U 0 (1)

where U = (u,v,w) is the dimensionless nanofluid velo-city, defined as U*H/af; U* is the dimensional nanofluidvelocity; af is the thermal diffusivity of a nanofluid, k/(rc)f; k is the thermal conductivity of the nanofluid; and(rc)f is the volumetric heat capacity of the nanofluid.


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The dimensionless coordinates are defined as (x,y,z) =(x*, y*, z*)/H, where z is the vertically downwardcoordinate.The buoyancy force can be considered to be made up

of three separate components that result from: the tem-perature variation of the fluid, the nanoparticle distribu-tion (nanoparticles are heavier than water), and themicroorganism distribution (microorganisms are alsoheavier than water). Utilizing the Boussinesq approxima-tion (which is valid because the inertial effects of thedensity stratification are negligible, the dominant termmultiplying the inertia terms is the density of the basefluid that exceeds by far the density stratification), themomentum equation can be written as:

1 2

Pr tp Rm RaT Rn

Rb

Lbn

U

U U U k k k k ˆ ˆ ˆ ˆ (2)

where k^

is the vertically downward unit vector.

The dimensionless variables in Equation 2 are defined as:

t t H p p H

TT T

T Tn nc

h c

* *

* *

* *

* *

* **

/ , / ,

, , /

f f2 2

0

1 0

nn0 (3)

where t is the dimensionless time, p is the dimension-less pressure, � is the relative nanoparticle volume frac-tion, T is the dimensionless temperature, n is thedimensionless concentration of microorganisms, t* is thetime, p* is the pressure, μ is the viscosity of the suspen-sion (containing the base fluid, nanoparticles and micro-organisms), �* is the nanoparticle volume fraction, 0

isthe nanoparticle volume fraction at the lower wall, 1

is the nanoparticle volume fraction at the upper wall, T*is the nanofluid temperature, Tc

is the temperature atthe upper wall (also used as a reference temperature),Th is the temperature at the lower wall, n* is the con-

centration of microorganisms, and n0 is the average

concentration of microorganisms (concentration ofmicroorganisms in a well-stirred suspension).The dimensionless parameters in Equation 2, namely,

the Prandtl number, Pr; the basic-density Rayleigh num-ber, Rm; the traditional thermal Rayleigh number, Ra;the nanoparticle concentration Rayleigh number, Rn; thebioconvection Rayleigh number, Rb; and the bioconvec-tion Lewis number, Lb, are defined as follows:

Pr RmgH

Rag H T Th c

f0 f

p f0

f

f0,

( ),

* * * *0 0

3 31 f

(4)

RngH

Rbg n H

DLb

D

( )( ), ,

* *

p f0

f mo

f

mo

1 03

03 (5)

where rf0 is the base-fluid density at the referencetemperature; rp is the nanoparticle mass density; g isthe gravity; b is the volumetric thermal expansion coeffi-cient of the base fluid; Δr is the density differencebetween microorganisms and a base fluid, rmo - rf0; rmo

is the microorganism mass density; θ is the averagevolume of a microorganism; and Dmo is the diffusivity ofmicroorganisms (in this model, following [44,45], allrandom motions of microorganisms are simulated by adiffusion process).The conservation equation for nanoparticles contains

two diffusion terms on the right-hand side, which repre-sent the Brownian diffusion of nanoparticles and theirtransport by thermophoresis (a detailed derivation isavailable in [43,46]):

t Ln

N

LnTAU

1 2 2 (6)

In Equation 6, the nanoparticle Lewis number, Ln, anda modified diffusivity ratio, NA (this parameter is some-what similar to the Soret parameter that arises in cross-diffusion phenomena in solutions), are defined as:

LnD

ND T T

D TA

h c

c

f

B

T

B

,( )

* *

* * *1 0

(7)

where DB is the Brownian diffusion coefficient ofnanoparticles and DT is the thermophoretic diffusioncoefficient.The right-hand side of the thermal energy equation

for a nanofluid accounts for thermal energy transport byconduction in a nanofluid as well as for the energytransport because of the mass flux of nanoparticles(again, a detailed derivation is available in [43,46]):

T

tT T

N

LnT

N N

LnT TB A BU 2 (8)

In Equation 8, NB is a modified particle-density incre-ment, defined as:

Nc

cB ( )

( )* *

p

f1 0 (9)

where (rc)p is the volumetric heat capacity of thenanoparticles.The right-hand side of the equation expressing the

conservation of microorganisms describes three modesof microorganisms transport: due to macroscopicmotion (convection) of the fluid, due to self-propelleddirectional swimming of microorganisms relative to the


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fluid, and due diffusion, which approximates all stochas-tic motions of microorganisms:

n

tn n

LbnU V

1(10)

where V is the dimensionless swimming velocity of amicroorganism, V*H/af, which is calculated as [44,45]:

V Pe

LbH C Cˆ (11)

In Equation 11 H^ is the Heaviside step function and

C is the dimensionless oxygen concentration, defined as:

CC C

C C

min

min0

(12)

where C* is the dimensional oxygen concentration,C0 is the upper-surface oxygen concentration (the

upper surface is assumed to be open to atmosphere),and Cmin

is the minimum oxygen concentration thatmicroorganisms need to be active. Equation 11 thusassumes that microorganisms swim up the oxygen con-centration gradient and that their swimming velocity isproportional to that gradient; however, in order formicroorganisms to be active the oxygen concentrationneed to be above Cmin

. Since this article deals with ashallow layer situation, it is assumed that C C minthroughout the layer thickness, and the Heaviside stepfunction, H C

^ , in Equation 11 is equal to unity.Also, the bioconvection Péclet number, Pe, in Equa-

tion 11 is defined as:

PebW

D mo

mo(13)

where b is the chemotaxis constant (which has thedimension of length) and Wmo is the maximum swim-ming speed of a microorganism (the product bWmo isassumed to be constant).Finally, the oxygen conservation equation is:

C

tC

LeC nU

1 2 ̂ (14)

The first term on the right-hand side of Equation 14represents oxygen diffusion, while the second termrepresents oxygen consumption by microorganisms.The new dimensionless parameters in Equation 14 are

LeD

H n

C CS

f

f

,min

20

0(15)

where Le is the traditional Lewis number, ^ is the

dimensionless parameter describing oxygen consumptionby the microorganisms, DS is the diffusivity of oxygen,and g is a dimensional constant describing consumptionof oxygen by the microorganisms.According to Hillesdon and Pedley [45], the layer can

be treated as shallow as long as the following conditionis satisfied:

HPe

Pe Le

C C

nPe

f

2 1

1

1 20

0

1 1 2exptan exp

/min /

1 2/

(16)

Equation 16 gives the maximum layer depth for whichthe oxygen concentration at the bottom does not drop

below Cmin .

The boundary conditions for Equations 1, 2, 6, 8, 10,and 14 are imposed as follows. It is assumed that thetemperature and the volumetric fraction of the nanopar-ticles are constant on the boundaries and the flux ofmicroorganisms through the boundaries is equal to zero.The lower boundary is always assumed rigid and theupper boundary can be either rigid or stress-free. Theboundary conditions for case of a rigid upper wall are

ww

zT

n

z

C

zz

0 1 0

1

, , ,0,

dd

0, 0 at the lower wall

(17)

ww

zT

Pe nC

z

n

zC z

0 0 1

0

, , , 0,

dd

dd

0, 1 at the upper wal

ll (18)

The fifth equation in (18) is equivalent to the state-ment that the total flux of microorganisms at the uppersurface is equal to zero: the microorganisms swim verti-cally upward at the top surface but (because their con-centration gradient at the top surface is directedvertically upward) they are simultaneously pusheddownward by diffusion; the two fluxes are equal butopposite in direction).If the upper surface is stress-free, the second equation

in (18) is replaced with the following equation:

2

2

w

z 0 (19)

Basic stateThe solution for the basic state corresponds to a time-independent quiescent situation. The solution is of thefollowing form:


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Ub b b

b b b

( ),

( ), ( ),

0, ( ),p p z T T z

z n n z C C z (20)

In this case, the solution of Equations 6, 8, 10, and 14subjects to boundary conditions (17) and (18) is (theparticular form of hydrodynamic boundary conditions atthe upper surface is not important because the solutionin the basic state is quiescent):

b z N

N N

Lnz

N N

Ln

NA

A B

A BA

exp

exp

(

11

11

1 ))z 1 (21)

T z

N N

Lnz

N N

Ln

A B

A Bb

exp

exp

11

11

(22)

n zA

Pe Le

A zb

2

12

2 1 1

2ˆ sec

(23)

C zPe

A z

Ab

12 1 2

21

1

lncos /

cos /(24)

where A1 is the smallest positive root of the transcen-dental equation

tan^

APe

Le

A1

12

(25)

The solutions given by Equations 23 and 24 were firstreported in [44].The pressure distribution in the basic state, pb (z), can

then be obtained by integrating the following form of themomentum equation (which follows from Equation 2):

dd

bb b b

p

zRm Ra T Rn

Rb

Lbn 0 (26)

Equations 21 and 22 can be simplified if characteristicparameter values for a typical nanofluid are considered.Based on the data presented in Buongiorno [43] for analumina/water nanofluid, the following dimensional para-

meter values are utilized: 0 0 01* . , af = 2 × 10-7m2/s,

DB = 4 × 10-11m2/s, μ = 10-3 Pas, and rf0 = 103 kg/m3.The thermophoretic diffusion coefficient, DT, is esti-

mated as 0 , where, according to Buongiorno [43], τ

is estimated as 0.006. This results in DT = 6 × 10-11m2/s.The nanoparticle Lewis number is then estimated as

Ln = 5.0 × 103. The modified diffusivity ratio, NA, andthe modified particle-density increment, NB, depend onthe temperature difference between the lower and theupper plates and on the nanoparticle fraction decrement.Assuming that T Th c

* * 1 K , 1 0 0 001* * . , and

Tc* 300 K , gives the following estimates: NA = 5 and

NB = 7.5 × 10-4. This suggests that the exponents inEquations 21 and 22 are small and that these equationscan be simplified as:

b z z 1 (27)

T z zb (28)

Linear instability analysisPerturbations are superimposed on the basic solution, asfollows:

U

U

, , , , , , , , , ,

, ,

T n C p T z z n z C z p z

t x y

0 b b b b b

,, , , , , , , , , ,

, , , , , , , ,

z T t x y z t x y z

n t x y z C t x y z p

tt x y z, , , (29)

Equation 29 is then substituted into Equations 1, 2, 6,8, 10, and 14, the resulting equations are linearized andthe use is made of Equations 27 and 28. This procedureresults in the following equations for the perturbationquantities:

U 0 (30)

1 2

Pr tp RaT Rn

Rb

Lbn

UU k k kˆ ˆ ˆ (31)

T

tw T

N

Ln z

T

z

N N

Ln

T

zB A B2 2

(32)

tw

Ln

N

LnTA1 2 2 (33)

n

tw

dn

dz

Pe

Lb

C

z

dn

dz

dC

dz

n

zn

d C

dzn Cb b b b

b

2

22

1 2

Lbn (34)

C

tw

C

z LeC n

dd

1b 2 ̂ (35)

Equations 30 to 35 are independent of Rm since thisparameter is just a measure of the basic static pressure gra-dient. In order to eliminate the pressure and horizontalcomponents of velocity from Equations 30 and 31, Equa-

tion 31 (see [46]) is operated with k^ curl curl and the use

is made of the identity curl curl ≡ grad div - ∇2 togetherwith Equation 30. This results in the reduction of Equations30 and 31 to the following scalar equation which involvesonly one component of the perturbation velocity, w’:


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1 2 4 2 2 2

Pr tw w Ra T Rn

Rb

Lbn

H H H (36)

where H2 is the two-dimensional Laplacian operator

in the horizontal plane and ∇4w’ is the Laplacian of theLaplacian of w’.Equations 17 and 18 then lead to the following

boundary conditions for the perturbation quantities forthe case when both the lower and upper walls are rigid:

ww

zT

n

z

C

zz

0 0 0

1

, , ,0,

dd

0,dd

0 at the lower wall

(37)

ww

zT

Pe nC

z

C

zn

n

zC

0 0 0, , ,0,

dd

dd

dd

0,bb

00 at the upper wallz 0(38)

If the upper boundary is stress-free, the second equa-tion in Equation 38 is replaced by

2

2 0w

zz0 at (39)

The method of normal modes is used to solve a linearboundary-value problem composed of differential Equa-tions 32 to 36 and boundary conditions (37), (38) (or(39)). A normal mode expansion is introduced as:

w T n C W z z z N z z f x y st, , , , ( ), ( ), ( ), , , exp( ) ,, (40)

where the function f(x,y) satisfies the following equa-tion:

2

2

2

22f

x

f

ym f (41)

and m is the dimensionless horizontal wavenumber.Substituting Equation 40 into Equations 36 and 32 to

35, utilizing Equation 41, and letting ^ (so that

the resulting equation for amplitudes would depend onthe product Pe

^ rather than on Pe and ^ indivi-

dually), the following equations for the amplitudes, W,Θ, F, N, and , are obtained:

d

d

d

d

d

d

4

42

2

24

2

22

2 2 2

2W

zm

W

zm W

s

Pr

W

zm

sW

Ram RnmRb

Lbm N

Pr

00

(42)

Wz

N

Ln z

N N

Ln zm s

N

Ln zB A B Bd

d

dd

dd

dd

2

222

0 (43)

WN

Lnm

Lnm s

N

Ln z Ln zA A2 2

2

2

2

2

1 10 d

d

d

d(44)

212

112

1

12

1 1 13 2

1A Le A zN

zA A z

A

tan sec

tan

dd

112

2

2

12

1 2

z LbWz

Le m NN

z

A

dd

d

d

secc 21

22

2

12

1 2 0A z Le N mz

Lb Le s N

d

d

(45)

N A A z Wm

sLe z

1 1

2 2

2

12

1 0tan

Led

d(46)

where Equation 25 for A1 is reduced to

tanA Le

A1

12

(47)

In Equations 42 to 46 s is a dimensionless growth fac-tor; for neutral stability the real part of s is zero, so it iswritten s = iω, where ω is a dimensionless frequency (itis a real number).For the case of rigid-rigid walls, the boundary condi-

tions for the amplitudes are

WW

zN

z zz

0 0 0

1

, , ,dd

0,

dd

0,dd

0 at the lower wall

(48)

WW

zn

zPe

C

z

NN

zz

zz

0 0 0

0

00

, , ,dd

0,dd

dd

dd

0, 0 at t

bb

hhe upper wall (49)

If the upper surface is stress-free, the second equationin (49) is replaced by

d

d0 at

2

2 0W

zz (50)

Equations 42 to 46 are solved by a single-term Galer-kin method. For the case of the rigid-rigid boundaries,the trial functions, which satisfy the boundary condi-tions given by Equations 48 and 49, are

W z z z z z z

N z z z

12 2

1 1

12

1

1 1 1

112

12

( ) , ( ), ( ),

,

zz 2 (51)

where

A A Le A

Le A1 1 1

11

sin

cos(52)


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and A1 is given by Equation 47.If the upper boundary is stress-free, W1 is replaced by

W z z z13 43 2 (53)

and the rest of the trial functions are still given byEquation 51. W1 given by Equation 53 satisfies theboundary condition given by Equation 50.

Results and discussionRigid-rigid boundariesFor the case of the rigid-rigid boundaries the utilizationof a standard Galerkin procedure (see, for example[47]), which involves substituting the trial functionsgiven by Equation 51 into Equations 42 to 46, calculat-ing the residuals, and making the residuals orthogonalto the relevant trial functions, results in the followingeigenvalue equation relating three Rayleigh numbers,Ra, Rn, and Rb:

F Ra F Rn F Rb F1 2 3 4 0 (54)

where functions F1, F2, F3, and F4 are given in theappendix [see Equations A1 to A4], they depend on Lb,Le, Ln, Pr, NA, ϖ, ω, and m. It is interesting that Equa-tion 54 is independent of NB at this order (one-termGalerkin) of approximation.In order to evaluate the accuracy of the one-term

Galerkin approximation used in obtaining Equation 54the accuracy of this equation is estimated for the case ofnon-oscillatory instability (which corresponds to ω = 0)for the situation when the suspension contains no micro-organisms (this corresponds to n0 0 , which leads toRb = 0) and no nanoparticles (this leads to Rn = 0).In this limiting case Equation 54 collapses to

Ram m m

m

28 10 504 24

27

2 2 4

2(55)

The right-hand side of Equation 55 takes the mini-mum value of 1750 at mc = 3.116; the obtained criticalvalue of Ra is 2.5% greater than the exact value(1707.762) for this problem reported in [48]. The corre-sponding critical value of the wavenumber is 0.03%smaller than the exact value (3.117) reported in [48].Based on the data presented in [44,45] for soil bacter-

ium Bacillus subtilis, the following parameter values forthese microorganisms are used: Dm = 1.3 × 10-10 m2/s,Ds = 2.12 × 10-9 m2/s, Δr = 100 kg/m3,n0

1510* cells/m3 , θ = 10-18 m3, and H = 2.5 × 10-3 m(or 2.5 mm, this is a typical depth of a shallow layer;this size is also typical for a microdevice). Also, accord-ing to Hillesdon et al. [45], for Bacillus subtilis dimen-sionless parameters can be estimated as follows: Pe =15H,

^/ 7 2H Le , where the layer depth, H, must be

given in mm. Based on [43], the following parametervalues for a typical alumina/water nanofluid are utilized:0 0 01* . , rf0 = 103 kg/m3, rp = 4 × 103 kg/m3, (rc)p =3.1 × 106 J/m3, af = 2 × 10-7 m2/s, DB = 4 × 10-11 m2/s,DT = 6 × 10-11 m2/s, and μ = 10-3 Pas. It is alsoassumed that 1 0 0 001* * . , b = 3.4 × 10-31/K, (rC)f= 4 × 106J/m3, T Th c

* * 1K , and Tc* 300 K .

The parameter values given above result in the follow-ing representative values of dimensionless parameters: Lb= 1.5 × 103, Le = 94, Ln = 5.0 × 103, Pr = 5.0, NA = 5, NB

= 7.5 × 10-4, Pe = 37, ^

. 0 46 , ϖ = 17, Ra = 2.7 × 103,Rb = 1.2 × 105, Rm = 8.0 × 105, and Rn = 2.3 × 103. Thevalues of Ra and Rb can be controlled by changing thetemperature difference between the plates and the micro-organism concentration, respectively, and Rn depends onnanoparticle concentrations at the boundaries.For Figure 1a,b,c, the following values of dimension-

less parameters are utilized: Lb = 1500, Le = 94, Ln =5000, Pr = 5, NA = 5, ϖ = 17, and Rb = 0 (which corre-sponds to the situation with zero concentration ofmicroorganisms). Rn is changing in the range between-1.2 and 1.2. In Figure 1a, the boundary for non-oscilla-tory instability (shown by a solid line) is obtained by set-ting ω to zero in Equation 54, solving this equation forRa and then finding the minimum with respect to m ofthe right-hand side of the obtained equation. Theboundary for oscillatory instability (shown by a dottedline) is obtained by the following procedure. Twocoupled equations are produced by taking the real andimaginary parts of Equation 54. One of these equationsis used to eliminate ω, and the resulting equation isthen solved for Ra; the critical value of Ra is againobtained by calculating the minimum value that theexpression for Ra takes with respect to m.Figure 1a shows that for Rb = 0 the curve representing

the instability boundary for non-oscillatory convection(solid line) is a straight line in the (Rac, Rn) plane. Rn isdefined in Equation 5 in such a way that positive Rncorresponds to a top-heavy nanoparticle distribution.Therefore, the increase of Rn produces the destabilizingeffect and reduces the critical value of Ra. A comparisonbetween instability boundaries for non-oscillatory (solidline) and oscillatory (dotted line) cases indicates that inorder for the oscillatory instability to occur, Rn generallymust be negative, which corresponds to a bottom-heavy(stabilizing) nanoparticle distribution. In this case thedestabilizing effect of the temperature gradient (positiveRa corresponds to heating from the bottom) and desta-bilizing effect from upswimming of oxytactic microor-ganisms compete with the stabilizing effect of thenanoparticle distribution.Figure 1b shows that the critical value of the wave-

number, mc, is independent of Rn and for the case dis-played in Figure 1a (Rb = 0) is equal to 3.116; also, it is


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almost independent of the mode of instability (non-oscillatory versus oscillatory).Figure 1c shows the square of the oscillation fre-

quency, ω2, versus the nanoparticle concentration Ray-leigh number, Rn. The value of ω2 for the oscillatoryinstability boundary is obtained by eliminating Ra fromthe two coupled equations resulting from taking the realand imaginary parts of Equation 54 and solving theresulting equation for ω2. The solution is presented interms of ω2 rather than ω because the resulting equa-tion is bi-quadratic in ω. For oscillatory instability tooccur, ω2 must be positive so that ω is real. Figure 1cshows that for Rb = 0 ω is real when Rn is negative.Figure 2a,b,c is computed for the same parameter

values as Figure 1a,b,c, but now with Rb = 120000. Figure2a,b,c thus shows the effect of microorganisms. By com-paring Figure 2a with 1a, it is evident that the presence ofmicroorganisms produces the destabilizing effect andreduces the critical value of Ra. For example, at (NA +Ln) Rn = -5000 in Figure 1a the value of Rac correspond-ing to the non-oscillatory instability boundary is 6750and in Figure 2a the corresponding value of Rac is 6437.At (NA + Ln) Rn = 5000 in Figure 1a the value of Rac cor-responding to the non-oscillatory instability boundary is-3250 and in Figure 2a the corresponding value of Rac is-3563. The destabilizing effect of oxytactic microorgan-isms is explained as follows. These microorganisms areheavier than water and on average they swim in theupward direction. Therefore, the presence of microorgan-isms produces a top-heavy density stratification and con-tributes to destabilizing the suspension.The comparison of Figure 2b with 1b shows that the

presence of microorganisms increases the critical wave-number (in Figure 1b it was 3.116 and in Figure 2b it is3.441).Figure 2c brings an interesting insight. Apparently, if

the concentration of microorganisms is above a certainvalue, the oscillatory mode of instability is not possible.Indeed, ω2 in Figure 2c is negative for the whole rangeof Rn (-1.2 ≤ Rn ≤ 1.2) used for computing this figure.This means that ω is imaginary and oscillatory instabil-ity does not occur for the value of Rb used in comput-ing Figure 2.

Rigid-free boundariesFor the case when the upper boundary is stress-free, theeigenvalue equation is

F Ra F Rn F Rb F5 6 7 8 0 (56)

where functions F5, F6, F7, and F8 are given in theappendix [see Equations A10 to A13].Again, to evaluate of the accuracy of the one-term

Galerkin approximation in this case, the accuracy of

(NA+Ln)Rn

2

-4000 -2000 0 2000 4000-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Rb=0 oscil=0

(c)

(NA+Ln)Rn

mc

-4000 -2000 0 2000 40002

2.5

3

3.5

4

4.5

5

Rb=0 non-oscilRb=0 oscil

(b)

(NA+Ln)Rn

Ra c

-4000 -2000 0 2000 4000

-4000

-2000

0

2000

4000

6000

8000


(a)

Figure 1 The case of rigid upper and lower walls, Rb = 0 (nomicroorganisms): (a) Oscillatory and non-oscillatory instabilityboundaries in the (Rac, Rn) plane. (b) Critical wavenumber in the(Rac, Rn) plane. (c) Square of the oscillation frequency, ω2, versusthe nanoparticle concentration Rayleigh number (for oscillatoryinstability to occur, ω2 must be positive so that ω remains real).


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Equation 56 is estimated for the case of non-oscillatoryinstability (which corresponds to ω = 0) for the situationwhen the suspension contains no microorganisms (Rb =0) and no nanoparticles (Rn 0). In this limiting caseEquation 56 collapses to

Ram m m

m

28 10 4536 432 19

507

2 2 4

2(57)

The right-hand side of Equation 57 takes the mini-mum value of 1139 at mc=2.670; the obtained value ofRac is 3.48% greater than the exact value (1100.65) forthis problem reported in [48]. The corresponding criticalvalue of the wavenumber is 0.45% smaller than the exactvalue (2.682) reported in [48].For Figures 3a,b,c and 4a,b,c, which show the results

for the rigid-free boundaries, the same parameter valuesas for Figures 1 and 2 are utilized. Figure 3a, which iscomputed for Rb = 0 (no microorganisms), showsboundaries of non-oscillatory and oscillatory instabilities.This figure is similar to Figure 1a, but since now thecase of the rigid-free boundaries is considered, thevalues of the critical Rayleigh number in Figure 3a aresmaller than those in Figure 1a. Again, the comparisonbetween the non-oscillatory and oscillatory instabilityboundaries indicates that in order for oscillatoryinstability to occur Rn must be negative; in this case atthe instability boundary the effect of the nanoparticledistribution is stabilizing and the effect of the tempera-ture gradient is destabilizing; the presence of these twocompeting agencies makes the oscillatory instabilitypossible.The critical wavenumber shown in Figure 3b (mc =

2.670) is smaller than the corresponding critical wave-number for the rigid-rigid boundaries shown in Figure1b. Again, it is independent of Rn and almost indepen-dent of the mode of instability (non-oscillatory versusoscillatory).Figure 3c, similar to Figure 1c, shows that ω is real

when Rn is negative, which means that for negativevalues of Rn oscillatory instability is indeed possible.Figure 4a,b,c shows the results for rigid-free bound-

aries computed with Rb = 120000, meaning that the dif-ference with Figure 3a,b,c is the presence ofmicroorganisms. As in the case with rigid-rigid bound-aries, the presence of microorganisms produces a desta-bilizing effect and reduces the critical value of theRayleigh number (compare Figures 4a and 3a).Also, the presence of microorganisms increases the

critical value of the wavenumber (compare Figures 4band 3b).Figure 4c again shows that for the range of Rn used

for this figure the presence of microorganisms makesthe oscillatory mode of instability impossible (corre-sponding values of ω are imaginary).

ConclusionsThe possibility of oscillatory mode of instability in a nano-fluid suspension that contains oxytactic microorganisms is

(NA+Ln)Rn

2

-4000 -2000 0 2000 4000-1

-0.8

-0.6

-0.4

-0.2

0

Rb=120000 oscil=0

(c)

(NA+Ln)Rn

Ra c

-4000 -2000 0 2000 4000

-4000

-2000

0

2000

4000

6000

8000


(a)

(NA+Ln)Rn

mc

-4000 -2000 0 2000 40002

2.5

3

3.5

4

4.5

5


(b)

Figure 2 Similar to Figure 1, but now with Rb = 120000.


of 13

investigated. Since these microorganisms swim up the oxy-gen concentration gradient, toward the free surface (whichis open to the air), and they are heavier than water, theyalways produce the destabilising effect on the suspension.The destabilizing effect of microorganisms is larger if their

(NA+Ln)Rn

2

-4000 -2000 0 2000 4000-5

-4

-3

-2

-1

0

Rb=120000 oscil=120000

(c)

(NA+Ln)Rn

Ra c

-4000 -2000 0 2000 4000

-4000

-2000

0

2000

4000

6000

8000


(a)

(NA+Ln)Rn

mc

-4000 -2000 0 2000 40002

2.5

3

3.5

4

4.5

5


(b)

Figure 4 Similar to Figure 3, but now with Rb = 120000.(NA+Ln)Rn

2

-4000 -2000 0 2000 4000-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Rb=0 oscil=0

(c)

(NA+Ln)Rn

Ra c

-4000 -2000 0 2000 4000

-4000

-2000

0

2000

4000

6000

8000


(a)

(NA+Ln)Rn

mc

-4000 -2000 0 2000 40002

2.5

3

3.5

4

4.5

5


(b)

Figure 3 The case of a rigid lower wall and a stress-free upperwall, Rb = 0 (no microorganisms): (a) Oscillatory and non-oscillatory instability boundaries in the (Rac, Rn) plane. (b) Criticalwavenumber in the (Rac, Rn) plane. (c) Square of the oscillationfrequency, ω2, versus the nanoparticle concentration Rayleighnumber (for oscillatory instability to occur, ω2 must be positive sothat ω remains real).


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concentration in the suspension is larger. The concentra-tion of microorganisms is measured by the bioconvectionRayleigh number, Rb, which by definition is always non-negative (the zero value of Rb corresponds to a suspensionwith no microorganisms). The increase of Rb thus destabi-lizes the suspension. It is also shown that the presence ofmicroorganisms increases the critical wavenumber.The effect of the temperature distribution can be

either stabilizing (heating from the top, negative thermalRayleigh number Ra) or destabilizing (heating from thebottom, positive Ra). The effect of nanoparticles canalso be stabilizing (bottom-heavy nanoparticle distribu-tion, negative nanoparticle concentration Rayleigh num-ber Rn) or destabilizing (top-heavy nanoparticledistribution, positive Rn).The results obtained in this article indicate that in

order for the oscillatory instability to occur, Rn generallymust be negative, which corresponds to a bottom-heavy(stabilizing) nanoparticle distribution. In this case thedestabilizing effect of the temperature gradient (positiveRa) and destabilizing effect from upswimming of oxytac-tic microorganisms compete with the stabilizing effect ofthe nanoparticle distribution.In order for the oscillatory mode of instability to

occur, the dimensionless oscillation frequency, ω, mustbe real. Since increasing Rb pushes ω2 to negativevalues, oscillatory instability is possible only if the con-centration of microorganisms is below a certain value.The results for the rigid-rigid and rigid-free bound-

aries are similar, but the critical Rayleigh number forthe rigid-free boundaries is smaller. The critical wave-number for the rigid-free boundaries can be either smal-ler or larger, depending on the concentration ofmicroorganisms. For Rb = 0 the critical wavenumber issmaller for the rigid-free boundaries but for Rb =120000 it is larger than for the rigid-rigid boundaries.

AppendixThe functions F1, F2, F3, and F4 defining the eigenvalueequation for the layer with the rigid-rigid boundaries[given by Equation 54] are

F Lbm Pr m i Ln

Le I A I m I

12 2

3 12

42

2

1265

10

15 5 2 15 5

22 2

4 15 2 5 5 2 2

2

2 2

m i Le

Le m i Lb m i Le

(A1)

F Lbm Pr m Ln N i Ln

Le I A I m

A22 2

3 12

42

1265

10

15 5 2

15 5 2 2

4 15 2 5 5 2 2

22

2 2

I m i Le

m i Lb m i Le

Le

(A2)

F A m Pr m i

m i Ln I Le I A I m

3 12 2

25 3 1

24

2

294 14 5 10

10 15

32 5 2 212

12A I Lb m i Le

(A3)

F Lb m i m i Ln

m m Pr i m

42 2

2 4 2

39215

10 10

504 24 12

15 5 2 15 5 2 2

4 15 2 5

3 12

42

22Le I A I m I m i Le

Le

m i Lb m i Le2 25 2 2

(A4)

The integrals I1 to I5 in Equations A1 to A4 are func-tions of Le and ϖ. The expressions for these integralsfor the rigid upper boundary case are given below:

I z z z z

A z A

12 2

0

1

31

41

1 112

2

112

1

csc sin zz z

d

(A5)

I z z Le A z z2

0

1

12

2

112

2 2 2

1

4 Le

sec22

1 4 112

11 1 1A z A Le z A z

tan

(A6)

I z z A A z

A

3 12 2

1

0

1

1

112

2 212

1

2

sec

33 21 11

12

112

1

z A z A z z sec tan d

(A7)

I z z z z A z z42

0

1

12 112

212

1

sec d (A8)

I z z z A z z52 3

1

0

1

2 112

1

tan d (A9)

The functions F5, F6, F7, and F8 defining the eigenva-lue equation for the layer with the rigid-free boundaries[given by Equation 56] are

F Lbm Pr m i Ln I m i Le

Le I

52 2

22

3

23665

10 30 5 2 2

30 5

ˆ

ˆ

2 15 5 2

4 15 2 5 5 2 2

12

42

2 2

A I m

m i Lb m i Le

ˆ

(A10)

F Lbm Pr m Ln N i Ln

I m i Le Le

A62 2

22

23665

10

30 5 2 2

ˆ

30 5 2 15 5 2

4 15 2 5 5

3 12

42

2

ˆ ˆI A I m

m i Lb

22 22m i Le

(A11)


of 13

F A m Pr m i m i Ln

I I Le A

7 12 2 2

3 5 12

147 126 41 10 10

30

ˆ ˆ 115 4 5 2 24 52

12ˆ ˆ ˆI I Lem I Lb m i Le

(A12)

F Lb m i m i Ln

m m Pr i

82 2

2 4

39215

10 10

4536 432 19 216

119

30 5 2 2 30 5 2

15

2

22

3

12

4

m

I m i Le Le I

A I

ˆ ˆ

ˆ mm

m i Lb m i Le

2

2 2

5 2 4 15 2 5

5 2 2

(A13)

The integrals I^1to I

^5in Equations A10 to A13 are

functions of Le and ϖ. The expressions for these inte-grals for the stress-free upper boundary case are givenbelow:

ˆ

sec

I z z z z z z

A z

1

0

12

21

1 1 2 112

2

12

1

tan12

11A z zd

(A14)

ˆ

se

I z z Le A z z2

0

1

121

12

2 212

2 2

Le

cc sec

tan

21 1

21

12

1 112

1

12

A z A Le z A z

A

11 1 1

21

1 1 1

12

1

z A Le z A z

A z

cos

sec

tan12

11A z dz

(A15)

ˆ secI z z A A z

A

3

0

1

12 2

1

13

112

212

1

1

zz A z A z z

sec tan21 1

12

112

1 d

(A16)

ˆ secI z z z z A z z4

0

12

12 112

212

1

d (A17)

ˆ tanI z z z z A z z5

0

12 2

12 1 1 212

1

d (A18)

Authors’ contributionsAVK carried out all the work regarding the development of the model,performing simulations, writing and revising the paper and approving thefinal manuscript.

Competing interestsThe author declares that he has no competing interests.

Received: 20 September 2010 Accepted: 25 January 2011Published: 25 January 2011

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doi:10.1186/1556-276X-6-100Cite this article as: Kuznetsov: Nanofluid bioconvection in water-basedsuspensions containing nanoparticles and oxytactic microorganisms:oscillatory instability. Nanoscale Research Letters 2011 6:100.

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