APT 2033
No. of Pages 8, Model 5G
6 August 2018 Advanced Powder Technology xxx (2018) xxx–xxx 1
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
65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87
APT 2033
No. of Pages 8, Model 5G
6 August 2018 2
S. Saini, Y.D. Sharma / Advanced Powder Technology xxx (2018) xxx–xxx
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
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
121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136
139 140 141 142 143 144 145 146 147 148
APT 2033
No. of Pages 8, Model 5G
6 August 2018 3
S. Saini, Y.D. Sharma / Advanced Powder Technology xxx (2018) xxx–xxx
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
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
223
APT 2033
No. of Pages 8, Model 5G
6 August 2018 4
S. Saini, Y.D. Sharma / Advanced Powder Technology xxx (2018) xxx–xxx
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
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
333
APT 2033
No. of Pages 8, Model 5G
6 August 2018 5
S. Saini, Y.D. Sharma / Advanced Powder Technology xxx (2018) xxx–xxx
334
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 .
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
362 363 364 365 366 367 368
APT 2033
No. of Pages 8, Model 5G
6 August 2018 6
S. Saini, Y.D. Sharma / Advanced Powder Technology xxx (2018) xxx–xxx
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
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
372 373 374 375
APT 2033
No. of Pages 8, Model 5G
6 August 2018 S. Saini, Y.D. Sharma / Advanced Powder Technology xxx (2018) xxx–xxx 376 377 378
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;
426
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.
427
7. Conclusions
428
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
379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425
429 430 431 432 433 434 435 436 437 438
7
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.
439
8. Uncited reference
446
440 441 442 443 444 445
[12].
447
References
448
[1] R.A. Wooding, Rayleigh instability of a thermal boundary layer in flow through a porous medium, J. Fluid Mech. 9 (1960) 183–192. [2] F.M. Sutton, Onset of convection in a porous channel with net through flow, Phys. Fluids 13 (1970) 1931–1934. [3] G.M. Homsy, A.E. Sherwood, Convective instabilities in porous media with through flow, AIChE J. 22 (1976) 168–174. [4] M.C. Jones, J.M. Persichetti, Convective instability in packed beds with throughflow, AIChE J. 32 (1986) 1555–1557. [5] P.N. Kaloni, Z. Qiao, Non-linear convection in a porous medium with inclined temperature gradient and variable gravity effects, Int. J. Heat Mass Transf. 44 (2001) 1585–1591. [6] F. Chen, Throughflow effects on convective instability in superposed fluid and porous layers, J. Fluid Mech. 231 (1991) 113–133. [7] C. Zhao, B.E. Hobbs, H.B. Mühlhaus, A. Ord, G. Lin, Analysis of steady-state heat transfer through mid-crustal vertical cracks with upward throughflow in hydrothermal systems, Int. J. Numer. Anal. Methods Geomech. 26 (2002) 1477–1491. [8] A. Khalili, I.S. Shivakumara, S.P. Suma, Convective instability in superposed fluid and porous layers with vertical throughflow, Transp. Porous Media 51 (2003) 1–18. [9] G. Lin, C. Zhao, B.E. Hobbs, A. Ord, H.B. Mühlhaus, Theoretical and numerical analyses of convective instability in porous media with temperaturedependent viscosity, Int. J. Numer. Methods Biomed. Eng. 19 (2003) 787–799. [10] A.A. Avramenko, A.V. Kuznetsov, The onset of convection in a suspension of gyrotactic microorganisms in superimposed fluid and porous layers: effect of vertical throughflow, Transp. Porous Media 65 (2006) 159–176. [11] P.M. Patil, D.A.S. Rees, Linear instability of a horizontal thermal boundary layer formed by vertical throughflow in a porous medium: the effect of local thermal nonequilibrium, Transp. Porous Media 99 (2013) 207–227. [12] J.R. Platt, Bioconvection patterns in cultures of free-swimming organisms, Science 133 (80) (1961) 1766–1767. [13] M.S. Plesset, H. Wine, Bioconvection patterns in swimming microorganism cultures as an example of Rayleigh-Taylor instability, Nature 248 (1974) 441– 443. [14] S. Childress, M. Levandowsky, E.A. Spiegel, Pattern formation in a suspension of swimming micro-organisms: equations and stability theory, J. Fluid Mech. 69 (1975) 591–613. [15] T.J. Pedley, N.A. Hill, J.O. Kessler, The growth of bioconvection patterns in a uniform suspension of gyrotactic micro-organisms, J. Fluid Mech. 195 (1988) 223. [16] N.A. Hill, T.J. Pedley, J.O. Kessler, Growth of bioconvection patterns in a suspension of gyrotactic micro-organisms in a layer of finite depth, J. Fluid Mech. 208 (1989) 509–543. [17] N.A. Hill, D.-P. Häder, A biased random walk model for the trajectories of swimming micro-organisms, J. Theor. Biol. 186 (1997) 503–526. [18] M.A. Bees, N.A. Hill, Wavelengths of bioconvection patterns, J. Exp. Biol. 200 (1997) 1515–1526. [19] S. Ghorai, N.A. Hill, Development and stability of gyrotactic plumes in bioconvection, J. Fluid Mech. 400 (1999) 1–31. [20] Y.D. Sharma, V. Kumar, Overstability analysis of thermo-bioconvection saturating a porous medium in a suspension of gyrotactic microorganisms, Transp. Porous Media 90 (2011) 673. [21] Y.D. Sharma, V. Kumar, The effect of high-frequency vertical vibration in a suspension of gyrotactic micro-organisms, Mech. Res. Commun. 44 (2012) 40– 46. [22] A.V. Kuznetsov, A.A. Avramenko, A 2D analysis of stability of bioconvection in a fluid saturated porous medium – estimation of the critical permeability value, Int. Commun. Heat Mass Transf. 29 (2002) 175–184. [23] A. Bahloul, T. Nguyen-Quang, T.H. Nguyen, Bioconvection of gravitactic microorganisms in a fluid layer, Int. Commun. Heat Mass Transf. 32 (2005) 64–71. [24] Z. Alloui, T.H. Nguyen, E. Bilgen, Stability analysis of thermo-bioconvection in suspensions of gravitactic microorganisms in a fluid layer, Int. Commun. Heat Mass Transf. 33 (2006) 1198–1206. [25] Z. Alloui, T.H. Nguyen, E. Bilgen, Bioconvection of gravitactic microorganisms in a vertical cylinder, Int. Commun. Heat Mass Transf. 32 (2005) 739–747.
449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514
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
APT 2033
No. of Pages 8, Model 5G
6 August 2018 8 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546
S. Saini, Y.D. Sharma / Advanced Powder Technology xxx (2018) xxx–xxx
[26] A.V. Kuznetsov, The onset of thermo-bioconvection in a shallow fluid saturated porous layer heated from below in a suspension of oxytactic microorganisms, Eur. J. Mech. 25 (2006) 223–233. [27] S. Ebrahimi, J. Sabbaghzadeh, M. Lajevardi, I. Hadi, Cooling performance of a microchannel heat sink with nanofluids containing cylindrical nanoparticles (carbon nanotubes), Heat Mass Transf. Und Stoffuebertragung. 46 (2010) 549– 553. [28] X. Fang, Y. Xuan, Q. Li, Experimental investigation on enhanced mass transfer in nanofluids, Appl. Phys. Lett. 95 (2009) 203108. [29] J. Buongiorno, Convective transport in nanofluids, J. Heat Transf. 128 (2006) 240–250. [30] D.Y. Tzou, Thermal instability of nanofluids in natural convection, Int. J. Heat Mass Transf. 51 (2008) 2967–2979. [31] D.A. Nield, A.V. Kuznetsov, Thermal instability in a porous medium layer saturated by a nanofluid, Int. J. Heat Mass Transf. 52 (2009) 5796–5801. [32] A.V. Kuznetsov, D.A. Nield, Thermal instability in a porous medium layer saturated by a nanofluid: Brinkman model, Transp. Porous Media 81 (2010) 409–422. [33] D.A. Nield, A.V. Kuznetsov, The effect of vertical throughflow on thermal instability in a porous medium layer saturated by a nanofluid, Transp. Porous Media 87 (2011) 765–775. [34] H.D. Baehr, K. Stephan, Heat conduction and mass diffusion, in: Heat Mass Transf., Springer, 2013, pp. 107–273. [35] D.A. Nield, A.V. Kuznetsov, Thermal instability in a porous medium layer saturated by a nanofluid: A revised model, Int. J. Heat Mass Transf. 68 (2014) 211–214. [36] D.A. Nield, A.V. Kuznetsov, The effect of vertical throughflow on thermal instability in a porous medium layer saturated by a nanofluid: a revised model, J. Heat Transf. 137 (2015) 52601. [37] M.A. Sheremet, I. Pop, A. Ishak, Double-diffusive mixed convection in a porous open cavity filled with a nanofluid using Buongiorno’s model, Transp. Porous Media 109 (2015) 131–145.
[38] M.A. Sheremet, D.S. Cimpean, I. Pop, Free convection in a partially heated wavy porous cavity filled with a nanofluid under the effects of Brownian diffusion and thermophoresis, Appl. Therm. Eng. 113 (2017) 413–418. [39] S. Saini, Y.D. Sharma, The effect of vertical throughflow in Rivlin-Ericksen elastico-viscous nanofluid in a non-Darcy porous medium, Nanosyst. Phys. Chem. Math. 8 (2017) 606–612. [40] A.V. Kuznetsov, The onset of nanofluid bioconvection in a suspension containing both nanoparticles and gyrotactic microorganisms, Int. Commun. Heat Mass Transf. 37 (2010) 1421–1425. [41] A.V. Kuznetsov, Non-oscillatory and oscillatory nanofluid bio-thermal convection in a horizontal layer of finite depth, Eur. J. Mech. 30 (2011) 156– 165. [42] M.A. Sheremet, I. Pop, Thermo-bioconvection in a square porous cavity filled by oxytactic microorganisms, Transp. Porous Media 103 (2014) 191–205. [43] T.H. Tsai, D.S. Liou, L.S. Kuo, P.H. Chen, Rapid mixing between ferro-nanofluid and water in a semi-active Y-type micromixer, Sensors Actuators, A Phys. 153 (2009) 267–273. [44] D. Huh, B.D. Matthews, A. Mammoto, M. Montoya-Zavala, H.Y. Hsin, D.E. Ingber, Reconstituting organ-level lung functions on a chip, Science 328 (80) (2010) 1662–1668. [45] K.B. Anoop, T. Sundararajan, S.K. Das, Effect of particle size on the convective heat transfer in nanofluid in the developing region, Int. J. Heat Mass Transf. 52 (2009) 2189–2195. [46] Keblinski, D.G. Cahil, Comments on model for heat conduction in nanofluids, Phys. Rev Lett. 95 (2005) 209401. [47] J. Guo, P.N. Kaloni, Double-diffusive convection in a porous medium, nonlinear stability, and the Brinkman effect, Stud. Appl. Math. 94 (1995) 341–35846. [48] B.A. Finlayson, The Method of Weighted Residuals and Variational Principles, Academic Press, New York, 1972, chapter 6.
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
547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576