Numerical study of nanofluid thermo-bioconvection containing gravitactic microorganisms in porous media: Effect of vertical throughflow

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Contents lists available at ScienceDirect

Advanced Powder Technology journal homepage: www.elsevier.com/locate/apt

2

Original Research Paper

7 4 8

6

Numerical study of nanofluid thermo-bioconvection containing gravitactic microorganisms in porous media: Effect of vertical throughflow

9

Shivani Saini ⇑, Y.D. Sharma

5

10 12 11 13 1 2 5 8 16 17 18 19 20 21 22 23 24 25 26 27

Department of Mathematics, National Institute of Technology, Hamirpur, H.P. 177005, India

a r t i c l e

i n f o

Article history: Received 22 November 2017 Received in revised form 18 May 2018 Accepted 25 July 2018 Available online xxxx Keywords: Bioconvection Brownian motion Gravitactic microorganism Nanofluid Vertical throughflow

a b s t r a c t An analytical investigation of the onset of nanofluid thermo-bioconvection in a fluid saturated by porous media containing gravitactic and nanoparticles microorganisms subjected to a vertical throughflow is presented. The heat conservation equation is formulated by introducing the convective term of nanoparticle flux. The fluid is stimulated with modified Brinkman model, normal mode analysis and six-term Galerkin methods are used to solve the governing equations. The combined effects of vertical throughflow, nanoparticles, gravitactic microorganisms, and porosity have been taken into account. The effects of bioconvection Rayleigh number, bioconvection Péclet number, nanoparticle Rayleigh number, Péclet number, bioconvection Lewis number, and porosity on critical thermal Rayleigh number have been examined. The analysis leads that critical wave number is the function of bioconvection parameters, nanofluid parameters and throughflow parameters. It is also found that vertical throughflow disturbs the formation of bioconvection pattern which are necessary for the development of bioconvection. Ó 2018 Published by Elsevier B.V. on behalf of The Society of Powder Technology Japan. All rights reserved.

29 30 31 32 33 34 35 36 37 38 39 40 41 42 43

44 45

1. Introduction

46

Wooding [1] was the first who introduced the concept of throughflow. By using the concept of critical Rayleigh number, Sutton [2] evaluated the vertical temperature gradient for a hydrothermal system. The effect of throughflow in packed beds and porous media is examined by [3,4]. Quoi and Kaloni [5] performed the nonlinear stability analysis using the energy method to study the combined effects of vertical throughflow and inclined temperature. They found that destabilization starts earlier for smaller values of the Péclet number. In other investigation, Chen [6] studied the convective instability in a superposed fluid with an effect of throughflow. The impact of throughflow on bioconvection has many applications such as lithostatic pressure within the Earth’s crust, mineralization in hydrothermal systems, and convection at the ocean crust [7–9]. Avramenko and Kuznetsov [10] studied the bioconvection containing gyrotactic microorganisms in the porous layer with vertical throughflow and found that vertical throughflow stabilizes the bioconvection. Patil and Rees [11] explored the combined effects of throughflow and local thermal nonequilibrium.

47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

⇑ Corresponding author.

Bioconvection is a phenomenon that occurs when convective instability is induced by self-propelled up swimming microorganisms that are denser than cell fluid. Bioconvection has been used in water treatment plants, products like: ethanol, hydrogen gas, biofuel, fertilizers, biodiesel, and separation of vigorously swimming subpopulations and purification of cultures. Platt [13] introduced the term bioconvection and studied the moving polygonal patterns in dense cultures of Tetrahymena. Plesset and Winet [14] addressed the bioconvection in terms of Rayleigh-Taylor instability. In 1975, Childress et al. [15] were the first who proposed the extensive theory for bioconvection containing gravitactic microorganisms and also developed the mathematical model for gravitactic bioconvection. Pedley et al. [16] presented the theoretical bioconvective model for the gyrotactic microorganism. The growing volume of work devoted to experimental results, mathematical models, and mechanism of microorganisms is well documented by [17–21]. Kuznetsov and Ziang [22] found that critical value of permeability is approximately 4
107 m2 and if critical value of permeability is smaller than 4
107 m2 , then no bioconvection develops. Kuznetsov and Avramenko [23] reported that spherical shape of microorganisms produces the more unstable disturbance. Bahoul et al. [24] numerically studied the linear stability of a bioconvection in a fluid layer. For slowly swimmers, the gravitactic

E-mail address: [email protected] (S. Saini). https://doi.org/10.1016/j.apt.2018.07.021 0921-8831/Ó 2018 Published by Elsevier B.V. on behalf of The Society of Powder Technology Japan. All rights reserved.

Please cite this article in press as: S. Saini, Y.D. Sharma, Numerical study of nanofluid thermo-bioconvection containing gravitactic microorganisms in porous media: Effect of vertical throughflow, Advanced Powder Technology (2018), https://doi.org/10.1016/j.apt.2018.07.021

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Nomenclature DB Dm DT D a Da g H j ^ k km K

Lb Le n  n NA NB p Qv Qb Ra Rb Rm Rn t T Tc V W0

88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119

Brownian diffusion coefficient diffusivity of microorganisms thermophoresis diffusion coefficient Darcy number modified Darcy number gravity vector dimensional layer depth flux of microorganisms vertically upward unit vector effective thermal conductivity of porous media permeability of the porous media bioconvection Lewis number Lewis number microorganism concentration average concentration of microorganisms modified diffusivity ratio particle density increment number pressure Péclet number bioconvection Péclet number thermal Rayleigh number bioconvection Rayleigh number basic density Rayleigh number nanoparticle Rayleigh number time temperature reference temperature dimensionless Darcy velocity vertical upward velocity

bioconvection is similar to Bénard convection, while for faster swimmers, this phenomenon is quantitatively and qualitatively different from Bénard convection [25]. Kuznetsov [26] found the correlation between two Rayleigh number, bioconvection Rayleigh number, and traditional Rayleigh number. Due to a vast range of applications, nanofluids are widely used in cooling, micro heat pipes, microchannel heat sinks, microreactors, cancer therapy, sterilization of medical suspensions, process industries, polymer coatings, aerospace tribology, microfluid delivery devices etc [27,28]. Buongiorno [29] was perhaps the first who proposed a model, which predicts the behavior of nanoparticles. Using the Buongiorno model, Tzou [30] and Nield and Kuznetsov [31,32] investigated the thermal Rayleigh instability of nanofluid and found that nanoparticles enhance the thermal conductivity of the fluid. Nield and Kuznetsov [33] also examined the effect of throughflow on instability in nanofluid saturated by porous medium. In 2013, Baehr and Stephan [34] gave the concept of physically realistic boundary conditions and proposed zero nanoparticle flux on the boundaries. Incorporating the suggestions made by [34], Nield and Kuznetsov [35,36] revised their work [31,33] by considering the more realistic boundary conditions. Double diffusive mixed convection in a porous cavity is analytically studied by Sheremet et al. [37]. Sheremet et al. [38] also examined the natural convection of a nanofluid in a wavy-walled porous cavity and they found that local heat source has an efficient influence of the heat transfer rate. Recently, Saini and Sharma [39] studied the thermal instability in Rivlin-Erickson Elastico-Viscous nanofluid with the effect of throughflow and found that throughflow stabilizes the system. Kuznetsov [40,41] extended the work of [31,32] for the suspension containing both gyrotactic microorganisms and nanoparticles. He observed that adding the microorganisms to a nanofluid

Wc microorganisms velocity ðu; v ; wÞ Darcy velocity components ðx; y; zÞ space co-ordinates Greek symbol Wave number critical wave number volumetric thermal expansion coefficient bT Dq ¼ qcell  qf difference between cell density and a fluid density l viscosity

a ac



l

effective viscosity

qp qf ðqcÞp ðqcÞf ðqcÞm

average volume of microorganisms nanoparticles volume fraction density of nanoparticles density of the base fluid heat capacity for the nanoparticles heat capacity for the fluid effective heat capacity for the porous media

h /

Subscript b Basic state c Upper boundary h Lower boundary Superscript * dimensional variable 0 perturbed state

increases the stability of a suspension. Later, Sheremet and Pop [42] extended the work of [26] to the case of bioconvection in a square porous cavity filled by microorganisms. Nanofluid with bioconvection may find useful applications in different biomicrosystems, such as inflammatory responses, chip-size microdevices for assessing nanoparticle toxicity, toxic of the lung to silica nanoparticles, enzyme biosensors, mass transport enhancement, and mixing [43,44]. In the present paper, we study the effect of vertical throughflow on nanofluid thermo-bioconvection using the modified mass flux condition. Our attention is mainly focused on the dependence of various parameters such as nanofluid parameters, bioconvection parameters, and throughflow parameters on thermal Rayleigh number and wave number. Also, this work has some relevance to highly efficient microbial fuel cells utilizing Bacillus licheniformis, bioconvection nanotechnological devices, and bioconvection in motile thermophilic microorganisms.

120

2. Problem formulation

137

We consider a plane horizontal porous layer with thickness H, saturated by nanofluid with gravitactic microorganisms confined between the planes Z  ¼ 0 and Z  ¼ H (see Fig. 1). It is assumed that the nanoparticles suspended in the base fluid are stable [45], and the concentration of nanoparticles is than 1% (since the larger concentration of nanoparticles would suppress bioconvection instability [2]). The base fluid is water so that microorganisms can stay alive in it. Nanoparticles do not affect the velocity and direction of gravitactic microorganisms. It is assumed that motion of microorganism’s can be split into random and directional components [15]. Nanofluid is assumed to be Newtonian, laminar, and

138

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Z Gravitactic Microorganisms

Microscopic view of nanoparticles

T * = Tc*

Z=H

g = (0, 0, -g)

H

Z

Z=0

T * = Th* Y

X

W0 Fig. 1. Physical model and coordinate system.

149 150 151 152 153 154 155 156 157 158

159 161 162

164

incompressible. Each boundary wall is assumed to be thermal conducting and permeable to throughflow. Local thermal equilibrium and homogeneity are assumed. To neglect the effect of thermal transport attribute to the small size of nanoparticles, thermophoresis and Brownian diffusion coefficient are taken to be time independent [46]. We use the modified Brinkman model and the Oberbeck–Boussinesq approximations are employed. The thermal energy equation is based on Nield and Kuznetsov [36]. The conservation equations for the momentum, nanoparticles and gravitactic microorganisms are based on [15,16,31,32].

r  V  ¼ 0

qf @V  2 l ¼ r p þ l r V  V e @t K h i
 þ / qp þ ð1  / Þqf 1  bT ðT   T c Þ þ n h Dq g 

165

167 168 170 171 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187

ð1Þ

ð2Þ

@T ðqcÞm  þ ðqcÞf V  r T  @t   2 DT V  r T  ð3Þ ¼ km r T  þ eðqcÞp DB r / þ  r T   ð/  /0 Þ Tc e 

h i @n        ^  ¼ r  n V þ n W c k  Dm r n @t



@w w ¼ W0; ¼ 0; T  ¼ T h  ; @z @/ DT @T  W 0  ^ ¼ 0; DB  þ   ð/  /0 Þ ¼ 0 ; j  k @z T c @z e 

@w w ¼ W0; ¼ 0 ; T  ¼ Tc ; @z @/ DT @T  W o  ^ ¼ 0; DB  þ   ð/  /0 Þ ¼ 0 ; j  k @z T c @z e

at z ¼ 0

ð6aÞ

188 189 190 191 192 193 194 195

196

198 199





 @ 1  DT  / ¼ DB r2 / þ  r2 T  þ V  r @t  e Tc

We are studying infinite layer of nanofluid bounded by two horizontal planes at z = 0 and z = H. The lower and upper boundary layer is assumed to be rigid (since the upper surface of the layer is not fully stress free [16]). On the boundaries, temperatures at the upper and lower wall are taken to be T c and T h . Throughflow velocity has a constant value (W 0 ). Total nanoparticle flux is assumed to be zero on the boundaries [16,36,40]. The boundary conditions are

ð4Þ

^  Dm r n . To non-dimensionalise the govwhere j ¼ n V þ n W c k erning equations, we define the dimensionless variables as follows: 



rH2 ; am

h

c

In Eqs. (1)–(5), l is the viscosity, V is the Darcy velocity, t is time, g is gravity vector, p is pressure, T  is temperature of nanofluid, T c is reference temperature, / is the nanoparticles volume fraction, /0 is reference volume fraction, h is the average volume of microorganism, n is the microorganism concentration, qp is density of nanoparticles, ðqcÞf is the heat capacity for the base

fluid, km is the thermal conductivity of nanofluid, qf is the density of the nanofluid, ðqcÞm is the effective heat capacity for the porous media, Dm is the diffusivity of microorganism, bT is the volumetric thermal expansion coefficient, ðqcÞp is the heat capacity for the nanoparticles, DT is the thermophoresis diffusion coefficient, DB is the Brownian diffusion coefficient, W c is the microorganism swimming velocity, e is porosity, and K is the permeability of the porous medium.

/ /0 . /0

201 202 203

H; aHm ;

204

and lKam to scale the distance componentsðx ; y ; z Þ, Darcy

205

c T ¼ TT  T ; n ¼ n h; / ¼ T 

We

employ

the

entities

velocity components V ðu ; v  ; w Þ, time ðt  Þ, and pressure ðp Þ. f



ð6bÞ

 



m Where, am ¼ ðqkcÞ and

ð5Þ

at z ¼ H

r ¼ ððqqcÞcÞmf . Resulting dimensionless form of

Eqs. (1)–(5) are as follows

r V ¼0

 @ NB N N þ V  r T ¼ r2 T þ r/  rT þ A B rT  rT @t Le Le NB /V  rT 

1 @/ 1 1 NA 2 þ V  r/ ¼ r2 / þ r T Le r @t e Le   1 @n Q ^ 1  rn ¼ r  nV þ n b k Lb r @t Lb

209 211 212

ð8Þ

214 215



e

207 208

ð7Þ

 Da @V ^ þ Ra T k ^  Rn / k ^  Rb n k ^ ¼ rp þ D a r2 V  V  Rm k ePr @t Lb v

206

ð9Þ

217 218

ð10Þ

220 221

ð11Þ

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In Eq. (19), r2H is the 2-D Laplacian operator in the horizontal plane. The boundary conditions are

With the non-dimensional boundary conditions as

224

225

@w ¼ 0; T ¼ 1; @z @/ @T Q v Le dn þ NA  / ¼ 0 ; ðQ b þ Q v Lb Þn ¼ @z @z dz e

w ¼ Qv; 227 228

@w ¼ 0; T ¼ 0; @z @/ @T Q v Le dn þ NA  / ¼ 0 ; ðQ b þ Q v Lb Þn ¼ @z @z dz e

at z ¼ 0

ð12aÞ

w ¼ Qv; 230

at z ¼ 1

ð12bÞ

231

The non-dimensional parameters are in Eqs. (8)–(11) namely,

232

modified Darcy number D a ¼ l K=lH2 , the Prandtl number



233 234 235 236 237 238 239 240 241 242 243

244 246



Pr ¼ l=qf am , Darcy number Da ¼ K=rH2 , thermal Rayleigh number Ra ¼ qbT KHðT h  T c Þg=lam , basic density Rayleigh number Rm ¼ ½qP /0 þ qf ð1  /0 ÞgKH=lam , bioconvection Rayleigh number Rb ¼ Dqgv KH=lDm , nanoparticle Rayleigh number Rn ¼ n o  ðqp  qf Þ/o gKH=lam , Lewis number Le ¼ am =DB , bioconvection

Lewis number Lb ¼ am =Dm , bioconvection Péclet number Q b ¼ W c H=Dm , Péclet number Q v ¼ W 0 H=am , particle density increment number N B ¼ eðqcÞp ð/0 Þ=ðqcÞf , and modified diffusivity ratio N A ¼ DT ðT h  T c Þ=DB T c /0 . The basic state of nanofluid is assumed to be time-independent and is described by as follows

p ¼ pb ðzÞ; V ¼ ð0; 0; Q v Þ; n ¼ nb ðzÞ; / ¼ /b ðzÞ; T ¼ T b ðzÞ

ð13Þ

249

The Eqs. (8)–(11) together with boundary conditions give the basic solutions for the nanoparticle volume fraction, temperature, and microorganisms concentration as:

250

eN A Þðe 1Þ NA ðe 1Þ /b ¼ ðeLðeeL þ ðeeL , Q Q L =e e Þðe v 1Þ e Þðe v e 1Þ

247 248

251

252

Q v Le z=e

v v

Qv z

v T b ¼ e evQe v 1 , Q

Q z

nb ðzÞ ¼

and

expððQ v Lb þ Q b ÞzÞ.Where m is the integration constant given by  R1  nðQ v Lb þQ b Þ ¼ expðQ and n ¼ nb ðzÞdz is the average dimensionless v L þQ Þ1 b

b

0

253

concentration of microorganisms.

254

3. Linear instability analysis

255

Perturbations are superimposed on the basic solutions in the form V ¼ Vb þ V0 ; p ¼ pb þ p0 ; T ¼ T b þ T 0 / ¼ /b þ /0 ; and n ¼ nb þ n0 . Substituting these values in Eqs. (8)–(11) and neglecting the product of prime quantities, we get

256 257 258 259

260 262 263 265

r  V0 ¼ 0

266 @T 0 @t

 0 0 0 b @T þ dTdzb w0 þ Q v @T ¼ r2 T 0 þ NLeB d/ þ dTdzb @/ þ 2NLeA NB dTdzb @z dz @z @z  0 0  NeB Q v dTdzb /0 þ /b dT w0 þ Q v /b @T dz @z

269 271 272 274 275 276

277

1 @/0 Q v @/0 1 d/b 0 1 NA 2 0 r T þ þ w ¼ r2 /0 þ Le r @t e @z e dz Le

ð21Þ

Substituting Eq. (21) in Eqs. (16)–(19), we get the following equations 



1 þ þ





ð22Þ

ð23Þ  dnb 1 ðQ Lb þ Q b Þ s DN þ D2  a2  N ¼ 0 W  v Lb Lb dz r



½D aðD  a 2

285 286

287 289 290 291

292



ð25Þ

With boundary conditions

297

300

303 304

305

W ¼ 0; DW ¼ 0; H ¼ 0; Q Le DU þ NA DH  v U ¼ 0;ðQ v Lb þ Q b ÞN ¼ DN

ð26Þ

307

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 and kx þ ky ¼ a is dimensionless horizontal wave

308

e

d where D ¼ dz

294

301

 sDa Rb 2 Þ  1þ a N¼0 ðD2  a2 ÞW  Ra a2 H þ Rn a2 U þ ePr Lb v

2 2

at z ¼ 0;z ¼ 1

number.

309

4. Method of solution

310

Oscillatory convection turns oblivious due to the absence of two opposing forces. Thus, we set s ¼ 0 in Eqs. (22)–(25). Six-term Galerkin weighted residual method is used to obtain approximate solutions to the system of Eqs. (22)–(25). Trial functions which satisfy the boundary conditions exactly, are taken as

311

6 P

Pi W i ¼ P1 W 1 þP2 W 2 þP 3 W 3 þP4 W 4 þP5 W 5 þP 6 W 6

ð27aÞ

i¼1

312 313 314 315

316 318 319

6 X

Q i Hi ¼ Q 1 H1 þQ 2 H2 þQ 3 H3 þQ 4 H4 þQ 5 H5 þQ 6 H6

ð27bÞ

i¼1

321 322

U¼

6 X

Ri Ui ; ¼ R1 U1 þ R2 U2 þ R3 U3 þ R4 U4 þ R5 U5 þ R6 U6

ð27cÞ

i¼1

324 325

ð17Þ ð18Þ

^ Operating the Eq. (15) with k:curl curl and consider the solenoidal velocity field, Eqs. (14) and (15) are reduced as

Da @ 2 0 R r w  D ar4 w0 þ r2 w ¼ Ra r2H T 0  Rn r2H u0  b r2H n0 ePr @t Lb v

283

298

ð24Þ

N¼

6 X

Si N i ¼ S1 N 1 þ S 2 N 2 þ S 3 N 3 þ S 4 N 4 þ S 5 N 5 þ S 6 N 6

ð27dÞ

i¼1

1 @n0 1 @nb ðQ b þ Lb Q v Þ @n0 ¼ r2 n0  w0  Lb r @t Lb @z @z

282

295

  1 d/b NA 2 1 2 Q s W ðD  a2 ÞH  ðD  a2 Þ  v D  U¼0 e dz Le e r Le



281

N B Q v dT b N B d/b 2N A N B dT b NB Q v W þ D2 þ Dþ D / D  Q v D  a2  s H e dz Le dz Le dz e b

  N B dT b N B Q v dT b D U¼0 Le dz e dz

H¼

ð16Þ



279

½w0 ; T 0 ; /0 ; n0  ¼ ½WðzÞ; HðzÞ; UðzÞ; NðzÞeðstþikx xþiky yÞ

ð15Þ

268

ð20Þ

Analyzing the disturbances into the normal modes as follows

W¼

@T 0 @z

0

@w @/ @T Q Le w ¼ 0; ¼ 0; T 0 ¼ 0; þ NA  v /0 ¼ 0; @z @z @z e dn0 0 n ðQ v Lb þ Q b Þ ¼ at z ¼ 0; z ¼ 1 dz 0

ð14Þ

 Da @V0 ^  Rn /0 k ^  Rb n0 k ^ ¼ rp0 þ D ar2 V0  V0 þ Ra T 0 k ePr @t Lb v

0

0

280

ð19Þ

where Pi, Qi, Ri, and Si are constants. The base functions are taken as h

½W i ; Hi ; Ui ; Ni  ¼ z ð1  zÞ ; z ð1  zÞ; N A z ð1  zÞ; ði þ 2  ðQ b þ Q v Lb ÞÞz i

2

i

i

 þððQ b þ Q v Lb Þ  i  1Þziþ2 Þ ; i ¼ 1;2;:::; 6

327 328

329

iþ1

ð28Þ

331

Using the Eqs.(27a)–(28) into Eqs. (22)–(25) and following the Galerkin procedure [48], we get the following Eigenvalue equation

332

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336

Aji D ji Gji 0

337

where

338

Bji Eji

C ji F ji

Hji

Iji

0

K ji

0 0 ¼0 Jji L

Here, hfgi =

ð30Þ

ji

NB d/b 2NA NB dT b DHj DHi þ Hj DHi þ Hj DHi Le dz Le dz  NB Q v /b Hj DHi  Q v Hj DHi  a2 Hj Hi ; 

Aji ¼

e

340

344

346 347 349 350 352

In order to validate the accuracy of numerical method carried out in this paper, the results obtained using six-term Galerkin weighted residual method for the regular fluid (in the absence of throughflow, nanoparticles, and gravitactic microorganisms) are compared with earlier reported work [47] (see Table 1). It is observed that maximum error committed for Rac and ac is less than 0.17%, which may be taken as reasonably very good estimate.

361

1645

  Gji ¼ Ra a2 W j Hi ;

 Rb 2 a W j Ni ; Lb v

J ji ¼

356

1644

1643





Hji ¼ Rn a2 W j Ui ;

1642

K ji ¼

Lji ¼

1641



dnb Nj W i ; dz

 1 DNj DN i  ðQ b þ Q v Lb ÞNj DNi  a2 Nj Ni Lb

1640 0.20



0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

Qv Fig. 3. Variation of Rac with Q v .

Table 1  Comparison of Rac and ac for different values of D a with [47]. 

Guo and Kaloni [47]

Da 0.01 0.1 1 1

Present Paper

Error (%)

Rac

ac

Rac

ac

Rac

ac

60.36 215.06 1752.20 1707.70

3.2357 3.1501 3.1199 3.1161

60.38 215.08 1752.23 1707.76

3.2309 3.1496 3.1145 3.1159

0.033 0.009 0.002 0.003

0.148 0.016 0.173 0.006

1700

2000

1600

Qb

0

Ra c

358

360

D E Iji ¼ D aðD2 W j D2 W i  2a2 DW j DW i þ a4 W j W i Þ  ðDW j DW i  a2 W j W i Þ ;

353 355

5. Validation

 NA Dji ¼  ðDUj DHi  a2 Uj Hi Þ ; Le

  1 Q ; ðDUj DUi  a2 Uj Ui Þ  v Uj DUi Ejs ¼  Le e

 1 d/b F ji ¼ Uj W i ; e dz

359

Rac

343

 NB dT b N Q dT Bji ¼ Hj DUi  B v b Hj Ui ; Le dz e dz 

  N B Q v dT b Hj W ; C ji ¼  1 þ e dz

fgdz; j; i ¼ 1; 2; :::; 6

0

2

Qb

1 Qb

2000 Qb Qb

4000 Qb 6000

20

3 Lb

4 Lb

1400

5

1300

4

1200

6 1100

8 10

Lb

1500

0.1

Ra c

341

R1

30

40

50

1000

10

20

30

Rb

Rb

(a)

(b)

40

50

Fig. 2. Variation of Rac with Rb for different values of (a)Q b ; (b) Lb .

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20

20 Rb

Fig. 4. Variation of Rac with Rb for different values of (a)Rn ;(b) e.

Fig. 5. Variation of ac with Q b for different values of (a)Rn ;(b)Le , (c)Q v , and (d) Rb .

369

6. Results and discussion

370

Using the data given by [15,29,36,41], the value of N B is taken in the order of 103–102, Rn in the order of 100–101, Q v in the order

371

of 102–101, and Le in the order of 100–103. The value of N A is not more than 10. The value of porosity is lies between of 0 to 1. The used values of Q b are taken in the order of 101–101, and the values of Rb is taken in the order of 1–50. We have fixed the parameters

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values for alumina/water nanofluid with alga Chlamydomonas nivails are as follows: Le ¼ 500; Lb ¼ 4; Q b ¼ 3:0; Q v ¼ 0:05; Rb ¼ 3; Rn ¼ 0:1; N A ¼ 5; 

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N b ¼ 0:01; e ¼ 0:7 and D a ¼ 0:8: Fig. 2(a) displays the influence of bioconvection Péclet number ðQ b Þ and bioconvection Rayleigh number ðRb Þ on critical thermal Rayleigh number ðRac Þ. From figure, it is evident that with an increase in bioconvection Rayleigh number (concentration of microorganisms), thermal Rayleigh number decreases, showing thereby Rb promotes the nanofluid thermo-bioconvection. This means that increasing the density difference between cell and fluid destabilizes the system, which helps to construct the bioconvection pattern. From Fig. 2(a), it is also observed that bioconvection Péclet number (swimming speed of microorganisms) destabilizes the nanofluid thermo-bioconvection and this effect becomes more significant for higher values of the bioconvection Rayleigh numberðRb Þ. The effect of bioconvection Lewis number ðLb Þ is shown in Fig. 2(b). From figure, it is clear that Lb accelerates the bioconvection. Since by definition, Lb is inversely proportional to the microorganism diffusivity and directly proportional to the thermal conductivity of nanofluid. Therefore, an increase in microorganism’s diffusivity stabilizes the system. Fig. 3 illustrates the effect of vertical upward flow ðQ v Þ. It is observed that increasing the Péclet number stabilizes the system, which hinders the development of nanofluid thermobioconvection. This may attribute to the fact that an additional flow disturbs the formation of bioconvection pattern which are necessary for the development of bioconvection. Influence of porosity and nanoparticle Rayleigh number are illustrated in Fig. 4(a) and (b) respectively. It is observed that as nanoparticle Rayleigh number increases, the value of Rac decreases. By definition, Rn is directly proportional to volumetric fraction of nanoparticles, therefore it is concluded that both Brownian motion and thermophosis destabilise the suspension. From Fig. 4(b), it is observed that porosity delays the nanofluid thermo-bioconvection. In Fig. 5, the variation of critical wave number ðac Þ against bioconvection Péclet number ðQ b Þ is analyzed graphically with respect to (a) Lewis number, (b) nanoparticle Rayleigh number, (c) Péclet number, and (d) bioconvection Rayleigh number. From Fig. 5, it is noted that as swimming speed of microorganisms increases, cell size become narrower. The effect of low bioconvection Péclet numbers (slowly and intermediate swimmers, Q b < 4) on critical wave number is slight as compared to high bioconvection Péclet numbers (faster swimmers, Q b > 4). From Fig. 5(a) and (b), it is evident that with an increase in nanofluid parameters (Rn and Le ), critical wave number decreases, showing thereby nanofluid parameters increase the size of cell. From Fig. 5(c) and (d), it is observed that with an increase in Péclet number and bioconvection Rayleigh number, critical wave number increases. Thus, both the parameters ðRb and Q v Þ reduce the size of cells.

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The effect of vertical throughflow on the onset of nanofluid thermo-bioconvection in a porous media containing gravitactic microorganisms is examined. Boundary conditions in terms of zero nanoparticle flux (sum of Brownian diffusion term, convective term, and thermophoretic diffusion term) are considered. The present study reveals that nanoparticle Rayleigh number, bioconvection Péclet number, and bioconvection Lewis number destabilize the nanofluid thermo-bioconvection while porosity delays the onset of nanofluid thermo-bioconvection. Péclet number (vertical throughflow) disturbs the formation of bioconvection pattern which are necessary for the development of bioconvection. In this

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work, it is also observed that due to the convective term of nanoparticle flux in the thermal energy equation, critical wave number is the function of bioconvection parameters, nanofluid parameters, and throughflow parameters. Lewis number and nanoparticle Rayleigh number increase the size of a cell, while the bioconvection Rayleigh number and Péclet number reduce the size of a cell.

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[12].

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