A new numerical method for investigation of thermophoresis and Brownian motion effects on MHD nanofluid flow and heat transfer between parallel plates partially filled with a porous medium

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Results in Physics journal homepage: www.journals.elsevier.com/results-in-physics 6 7

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A new numerical method for investigation of thermophoresis and Brownian motion effects on MHD nanofluid flow and heat transfer between parallel plates partially filled with a porous medium

8

Habib-Olah Sayehvand, Amir Basiri Parsa ⇑

9

Mechanical Engineering Department, Engineering Faculty of Bu-Ali Sina University, Hamedan, Iran

3 4

10 1 2 2 5 13 14 15 16 17 18 19 20 21 22 23 24

a r t i c l e

i n f o

Article history: Received 26 November 2016 Received in revised form 22 January 2017 Accepted 6 February 2017 Available online xxxx Keywords: Brownian Nanofluid Porous medium Spectral local linearization method Thermophoresis

a b s t r a c t Numerical investigation the problem of nanofluid heat and mass transfer in a channel partially filled with a porous medium in the presence of uniform magnetic field is carried out by a new computational iterative approach known as the spectral local linearization method (SLLM). The similarity solution is used to reduce the governing system of partial differential equations to a set of nonlinear ordinary differential equations which are then solved by SLLM and validity of our solutions is verified by the numerical results (fourth-order Runge-Kutta scheme with the shooting method). In modeling the flow in the channel, the effects of flow inertia, Brinkman friction, nanoparticles concentration and thickness of the porous region are taken into account. The results are obtained for velocity, temperature, concentration, skin friction, Nusselt number and Sherwood number. Also, effects of active parameters such as viscosity parameter, Hartmann number, Darcy number, Prandtl number, Schmidt number, Eckert number, Brownian motion parameter, thermophoresis parameter and the thickness of porous region on the hydrodynamics, heat and mass transfer behaviors are investigated. Ó 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).

26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

41 42

Introduction

43

Since the energy issue is very important in today’s world, the nanofluid plays an important role in industrial phenomenon and in order to enhance heat transfer process. The heat transfers of nanofluids have received much important due to their wide range of applications in engineering and industrial processes. The nanofluids are made from a fluid containing nanometer-sized particles that called nanoparticles. In the other words, the nanofluids are engineered colloidal suspensions of nanoparticles in a base fluid [1,2]. The nanoparticles are typically made of metals, oxides, carbides, or carbon nanotubes. Common base fluids are water, ethylene glycol and oil. Nanofluids have novel properties that make them potentially useful in many applications in heat transfer including microelectronics, fuel cells, pharmaceutical processes, and hybrid-powered engines engine cooling/vehicle thermal management, domestic refrigerator, chiller, heat exchanger, in grinding, machining and in boiler flue gas temperature reduction [3]. The suspended nanoparticles exhibit enhanced thermal conductivity and the convective heat transfer coefficient compared to the

44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

pure fluid [4]. Bianco et al. [5] numerically investigated the nano fluid flow and heat transfer inside a circular tube under a laminar flow regime. They used single phase and two phase methods to model the nanofluid and reported a maximum difference of 11% between these methods. The behavior of nanofluids is found to be very critical in deciding their suitability for convective heat transfer applications [6,7]. Heris et al. [7] investigated laminar nanofluids flow of CuO/water and Al2O3/water through an annular tube, with a constant wall temperature boundary condition rather than constant heat flux condition. Comparison of experimental results of this work shows that the heat transfer coefficient enhanced with increasing volume fraction of nanoparticles as well as Peclet number while Al2O3/water showed more enhancement. In the other study, Ahmed et al. [8], numerically studied two dimensional laminar flow of different nanofluids in a triangular duct. They investigated the effect of type of the nanoparticles, particle concentration and Reynolds number on the heat transfer of the nanofluids. Nanofluids also have special acoustical properties and in ultrasonic fields display additional shear-wave reconversion of an incident compressional wave; the effect becomes more pronounced as concentration increases [9].

⇑ Corresponding author. E-mail address: [email protected] (A. Basiri Parsa). http://dx.doi.org/10.1016/j.rinp.2017.02.004 2211-3797/Ó 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

Please cite this article in press as: Sayehvand H-O, Basiri Parsa A. A new numerical method for investigation of thermophoresis and Brownian motion effects on MHD nanofluid flow and heat transfer between parallel plates partially filled with a porous medium. Results Phys (2017), http://dx.doi.org/ 10.1016/j.rinp.2017.02.004

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Nomenclature B0 Cf C cp C1 C2 D DB DT Da Ec f H K M Nb Nt Nu P Pr qw qm R Sc Sh

83 84 85 86 87 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

strength of magnetic field skin friction coefficient nanofluid concentration specific heat at constant pressure nanofluid concentration of lower plate nanofluid concentration of upper plate thickness of clear region Brownian diffusion coefficient thermophoretic diffusion coefficient Darcy number Eckert number dimensionless velocity function channel half-width permeability Hartman number Brownian motion parameter thermophoresis parameter Nusselt number pressure Prandlt number plate heat flux rate of mass flux viscosity parameter Schmidt number Sherwood number

Some of the applications of nanofluids as follow: thermal therapy for the treatment of cancer diseases, chemical process and metallurgical, power generation, micro-production, transportation, cooling, heating, air-conditioning, and ventilation. In the other words, the high quantities of thermal conductivity compared to the conventional coolants leads to enhancement of heat transfer, even when the particle concentration is very low. It is clear that conventional heat transfer is poor in various types of fluids such as water, ethylene glycol and oil that have lower thermal conductivity. Nanofluids are very helpful in handling the cooling problems in different thermal systems. Such kinds of high thermal conductivity can be useful in the automatic transmission of fluids, lubricants, coolants and engine oils. However, fluids that have lower thermal conductivity can be improved by adding solid nanoparticles [10]. Choi et al. [11] show that thermal conductivity enhancement and measured the thermal conductivity of nanofluids. The most accurate model for fluid was made by Buongiorno [12] who carefully carried out a theoretical analysis so as to estimate relative magnitudes of the terms associated with all possible slip mechanisms, namely, inertia, Brownian diffusion, thermophoresis, diffusiophoresis and Magnus effect. Sheikholeslami et al. [13] by Buongiorno model and a semi analytical method investigated the heat and mass transfer behavior of steady nanofluid flow between parallel plates in the presence of uniform magnetic field is studied. The heat transfer and nanofluids flow in porous media have received a great deal of attention in recent years, mainly due to the large number of relevant technical applications, such as fluid flow in geothermal reservoirs, catalytic reactors, separation processes and nuclear waste storage to name a few. A general literature survey concerned with this subject is given elsewhere such as Rashidi et al. [14], Kuznetsov et al. [15] and Rashidi et al. [16]. Most researches on flow in porous media have been done. The Darcy law to express the relationship between the superficial velocity with the pressure drop [10,16], though this is applicable under the assumption of slow, viscous flow conditions. Earlier studies reveal that the steady state Darcy equation is sufficient to

T T1 T2 u

v

x y

temperature temperature at the surface of lower plate temperature at the surface of upper plate velocity in x-direction velocity in y-direction distance along the plate distance normal to the plate

Greek symbols af thermal diffusivity of fluid lf dynamic viscosity of fluid h dimensionless temperature qf fluid density / dimensionless concentration rf electrical conductivity of fluid D dimensionless thickness of clear region sw skin friction along the plate g dimensionless variable Subscripts f fluid p particles

describe flow in saturated porous media, especially for low velocity values and low porosity. A lot of previous studies pertinent to heat and fluid flow in porous media with use of the Darcy flow model [17]. Darcy measured the bulk resistance to flow of an incompressible fluid through a solid matrix, as compared to the resistance at and near the surfaces confining this solid matrix. Since in his experiment the internal surface area (interstitial area) was many orders of magnitude larger than the area of the confining surfaces, the bulk shear stress resistance was dominant. The Darcy model has been examined rather extensively and is not closely followed for liquid flows at high velocities and for gas flows at very low and very high velocities. For the conditions of high velocity fluid flow, the inertia and boundary effects become important. Therefore, the Darcy model not appropriates for fast flows. On the other hand, the Darcy model does not satisfy the no-slip condition on a solid boundary: In convective heat transfer problems, heat is often transferred to the fluid saturating the porous matrix through a solid boundary. Most modern and accurate applications in porous media encompass a diversity of length and time scales, spatial in homogeneities, and local anisotropies, and the solution of either classical (Darcy) or extended (Brinkman, Forchheimer) models [18] in specific applications still represents a largely unresolved issue. The problem of convection in heated enclosures containing nanofluids are investigated through numerical and experimental published, but most of them concentrate on nanofluids in cavities and only few of them consider a applicable porous medium filled with a nanofluid. Poulikakos et al. [19] presented a series of numerical simulations which aim to document the problem of forced convection in a channel filled with a fluid-saturated porous medium. An exact solution of fully developed forced convection in a channel partially filled with a porous matrix has been done by Poulikakos et al. [20]. Bourantas et al. [21] presented the extended Darcy-Brinkman modeling of natural convection in a square porous enclosure that is saturated with a nanofluid and is heated through the bottom wall. Natural convection in a square enclosure filled with a fluid saturated porous medium using a thermal non-

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equilibrium model has been investigated for Brinkman-extended Darcy flow by Vafai et al. [22] and for Darcy flow by Poulikakos et al. [15]. Most scientific problems and phenomena such as Navier–Stokes problems occur nonlinearly. We have difficulty usually finding their exact analytical solutions. Explicit solutions to the nonlinear equations are of fundamental importance. Except a limited number of these problems that have precise analytical solution, most of them do not have analytical solution, so these nonlinear equations should be solved using other methods. In recent decades, much attempt has been done to the newly developed methods to introduce an analytic solution of these equations. The basic technique that we used is the spectral local linearization method (SLLM), which is based on Taylor series expansion. We propose a numerically and analytically to solve the present problem using a recently developed iterative method known as the spectral local linearization method (SLLM), Motsa [23]. The SLLM approach is based on transforming a system of nonlinear ordinary differential equations into an iterative scheme. The iterative scheme is then combined with the Chebyshev spectral method [24]. A similar approach to our present proposed method can be found in Motsa and Shateyi [25,26] and Motsa et al. [27]. The main aim of the present paper is to investigate the effect of nanoparticles on a fluid flow between parallel plates filled by porous media in the presence of magnetic fluid. Another basic objective of this paper is to present a new numerical and analytical method for solution of nonlinear differential equations that seeks to address some of the classical numerical difficulties. A very simple and convergent iterative algorithm for solving nonlinear systems of equations that model boundary layer flow problems is proposed. The spectral local linearization method (SLLM), is based on, separation and linearizing systems of equations using a combination of a linearization technique and a spectral collocation discretization [23]. The effects of viscosity parameter, Hartmann number, Darcy number, Prandtl number, Schmidt number, Eckert number, Brownian motion parameter and thermophoresis parameter on the hydrodynamics, heat and mass transfer behaviors are investigated.

193

Flow analysis and mathematical formulation

194

Heat and mass transfer analysis in the steady fully developed hydromagnetic flow of an incompressible viscous nanofluid between two parallel plates, partially filled with a porous medium, is considered. One of the plates is situated at y ¼ 0 and the other is situated at y ¼ H and the thickness of the clear fluid region, attached to lower plate is D. A uniform magnetic field of strength B0 is applied perpendicular to the plates along the y-axis. The

155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191

195 196 197 198 199 200

plates and the fluid rotate in unison about the y-axis with an angular velocity (Fig. 1). The presence of magnetic field is more important in the clear medium but in the porous medium because of the presence of the Darcy drag, the magnetic field is not strong enough for the magnetic drag to be significant in comparison with the Darcy drag. Servati et al. [28] numerically showed that different magnitudes of magnetic field have no noticeable effect on the heat transfer rate in the porous zone. Also, this prediction will be seen in the results section. Under these assumptions, the problem becomes two dimensional and interest lies in the investigation of flow characteristics at any cross section. The governing equations of continuity, momentum, energy and concentration are as follows: Continuity:

@u @ v þ ¼ 0; @x @y

ð1Þ

Momentum conservation: Region-I ð0 < y < DÞ: clear medium

! rf B20 u @u @u 1 @p lf @ u @ u  þv ¼ þ  u þ ; @x @y @x2 @y2 qf @x qf qf 2

202 203 204 205 206 207 208 209 210 211 212 213 214

215 217 218 219

220

2

ð2Þ 222 223

!

@v 1 @p lf @2v @2v ; u þ þ  ¼ @x qf @x qf @x2 @y2

ð3Þ 225

Region- II ðD < y < HÞ: porous medium

! rf B20 u lf @u @u 1 @p lf @2u @2u  þv ¼ þ  u þ  u; @x @y @x2 @y2 qf @x qf qf K qf @v 1 @p lf @2v @2v þ  u þ ¼ @x qf @x qf @x2 @y2 Energy:

201

!

226

227

ð4Þ 229 230

lf  v; K qf

ð5Þ 232 233

234

!

@T @T @2T @2T u þ v ¼ af  þ @x @y @x2 @y2 (    2  2 !) ðqcp Þp @C @T @C @T DT @T @T þ þ DB þ þ @x @x @y @y @x @y ðqcp Þf T1  2 lf @u ; þ ðqcp Þf @y ð6Þ Nanofluid concentration: 2

2

@C @C @ C @ C þv ¼ DB  u þ @x @y @x2 @y2

!

( ) DT @ 2 T @ 2 T þ þ ; T 1 @x2 @y2

237

238

ð7Þ 240

All of the parameters in the above equations are defined in nomenclature. The viscous dissipation effect has been taken into account to study the temperature distribution. The temperature of the lower plate and upper plates are considered at the constant temperature T 1 and T 2 , respectively. The boundary conditions necessary to complete the problem formulation are



u ¼ ax; v ¼ 0; T ¼ T 1 ; C ¼ C 1 at y ¼ 0; u ¼ 0; v ¼ 0; T ¼ T 2 ; C ¼ C 2 at y ¼ H:

236

ð8Þ

Since the clear fluid region is attached to the lower plate, the stretching sheet condition can be assumed for this boundary. In order to obtain the velocity, temperature and concentration fields, the Eqs. (1)–(7) subject to the appropriate boundary conditions (8) should be solved. For this we use the following non-dimensional similarity transformation Fig. 1. The schematic of the problem and coordinate system.

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g ¼ Hy ; D ¼ HD ; 0 u ¼ axf ðgÞ; v ¼ aHf ðgÞ; 2 2 hðgÞ ¼ TTT ; /ðgÞ ¼ CCC : 1 T 2 1 C 2

258

Substituting Eqs. (9) in Eqs. (2)–(7), we get

259

260

000

262

f 

aqf H2

lf

02

f þ

aqf H2

lf

00

ff 

263

H2

lf



  1 @P 0 ; rf B20 þ d f ¼ ax @x K



lf 

ð10Þ



qf aH 02 H2 1 @P ; f  xf þ d f ¼ lf K lf @ g 00

265

ð9Þ

ð11Þ

266

h00 þ

aH2

af

f h0 þ

ðqcp Þp DB ðC 1  C 2 Þ 0 0 ðqcp Þp DT ðT 1  T 2 Þ 02 /h þ h ðqcp Þf af ðqcp Þf af T 1

lf Hax 002 þ f ¼ 0; ðqcp Þf af ðT 1  T 2 Þ

268

ð12Þ

269 271 272

273

/00 þ 

d ¼ 0; Region  I : ð0 < g < DÞ

ð14Þ

From (10) and (11), elimination of P yields

"

2

# lf  0 2 f ¼ 0; rf B 0 þ d K

2

2

d 000 aqf H 02 aqf H 00 H f  f þ ff  lf lf lf dg

ð15Þ

So that

280

281 000

f 

aqf H2

lf

02

f þ

aqf H2

lf

00

ff 

H

2

lf



rf B20 þ d

lf  K

0

f ¼ C;

ð16Þ

285

where C is a constant to be evaluated at g ¼ 0. From Eqs. (15) and (16) we get

288

f

284

286

iv

 0 00 1 00 000   R f f  ff  ðM þ d Þf ¼ 0: Da

ð17Þ

290

The final form of dimensionless energy and concentration equations are as below:

293

h00 þ R:Prf h0 þ Nb/0 h0 þ Nth02 þ Ec:Prf

289

291

002

¼ 0;

ð18Þ

294 296 297 298

299

301 302

Nt 00 /00 þ R:Scf /0 þ h ¼ 0: Nb

ð19Þ

The dimensionless boundary conditions corresponding to Eqs. (14)–(16) reduce as

(

0

f ¼ 1; f ¼ 0; h ¼ 1; / ¼ 1 at 0

f ¼ 0; f ¼ 0; h ¼ 0; / ¼ 0 at

g ¼ 0; g ¼ 1:

ð20Þ

where R ¼ aH2 =mf is the viscosity parameter, M ¼ B20 H2 rf =lf is 2

304

Hartmann number, Da ¼ K=H is Darcy number, Pr ¼ mf =af is the Prandtl number, Sc ¼ lf =DB is Schmidt number, Ec ¼ aHx=

305

ðT 1  T 2 Þcpf is the Eckert number, Nb ¼ ðqcp Þp DB ðC 1  C 2 Þ=ðqcp Þf af

306

is the Brownian motion parameter and Nt ¼ ðqcp Þp DT ðT 1  T 2 Þ=

307

ðqcp Þf af T 1 is the thermophoresis parameter.

303

308 309 310

312 313 315

ð23Þ

The important Parameters such as the skin coefficient friction ðC f Þ, Nusselt number ðNuÞ and Sherwood number ðShÞ are derived from substituting the dimensionless variables in Eqs. (18)–(20) for the lower plate:

322

ð25Þ

326 328

Sh ¼ /0 ð0Þ:

ð26Þ

329 331

Spectral local linearization method (SLLM)

332

Basic idea of the spectral local linearization method (SLLM)

333

Consider a system of differential equations such as Z ¼ ½z1 ðgÞ; z2 ðgÞ; . . . ; zm ðgÞ that satisfies the system (see Motsa [23,25])

334

ð27Þ

that m is the number of differential equations. Also and each Hi is a function of g 2 ½a; b. Also, Li and Ni are linear and nonlinear components of differential equations, respectively. Generally, the spectral local linearization method is an iterative method for solving differential equations similar to (24) which begins with an initial approximation Z 0 Successive application of the SLLM generates approximations Z 1 ; Z 2 ; . . . where Z r ¼ ½Z 1;r ; Z 2;r ; . . . ; Z m;r  for each r ¼ 0; 1; 2; . . .. After the linearization of component N i , the differential Eq. (27) can be solved numerically using the Chebyshev spectral collocation method [24]. For this porous the i th differential Eq. (27) after the first r þ 1 iterations of the SLLM can be written as bellow

Li jrþ1 þ Ni jrþ1 ¼ Hi :

ð28Þ

The nonlinear component N i can be linearized using the Taylor series

Ni jrþ1 ¼ Ni jr þ rNi jr ðwrþ1  wr Þ

ð29Þ

when wr is an n tuple of Z i;r and its derivatives. So that (25) can be approximated as

Li jrþ1 þ rNi jr :wrþ1 ¼ Hi þ rNi jr :wr  Ni jr

ð30Þ

335 336

337 339 340 341 342 343 344 345 346 347 348 349 350 351

352 354 355 356

357 359 360 361

362 364

The current equation can be solved using the Chebyshev spectral collocation method, now. The more details of SLLM can be studied in Motsa [23].

365

Solution by means the SLLM

368 0

In order to reduce order of Eq. (17) let us f ¼ q; q0 ¼ p; then Eqs. (17)–(19) become

  1 00 0 p ¼ 0; f  Rðqp  fp Þ  M þ a Da

ð31Þ

h00 þ R:Prf h0 þ Nb/0 h0 þ Nth02 þ Ec:Prp2 ¼ 0;

ð32Þ

366 367

369 370

371 373 374 376 377

Eqs. (31)–(33) with the mentioned change of variables in the form of Eq. (27) may be written as

ð22Þ

321

Nu ¼ hð0Þ;

sw ¼ lf

 @T  qw ¼ kf  @y y¼0

320

323 325

Nt 00 h ¼ 0: / þ R:Scf / þ Nb

ð21Þ

319

ð24Þ

The shear stress, the rate of heat transfer and the rate of mass flux on the lower plate can be written as

 @u @yz¼0

318

C f ¼ f ð0Þ;

00

Li þ Ni ¼ Hi ! i ¼ 1; 2; . . . ; m

d ¼ 1; Region  II : ðD < g < 1Þ

276 277

283

ð13Þ

where

275

279

aH2 0 DT ðT 1  T 2 Þ 00 f/ þ h ¼ 0: DB T 1 DB ðC 1  C 2 Þ

316

 @T  qm ¼ DB  @y y¼0

00

0

L1 þ N1 ¼ H1 ;

ð33Þ

ð34Þ

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385 387

L2 þ N2 ¼ H2 ;

ð35Þ

Dfrþ1 ¼ qrþ1 ;

ð56Þ

461

388 390

L3 þ N3 ¼ H3 ;

ð36Þ

Dqrþ1 ¼ prþ1 ;

ð57Þ

462 464

391 393

L4 þ N4 ¼ H4 ;

ð37Þ

A3 Prþ1 ð0Þ ¼ B3 ;

ð58Þ

465 467

394 396

L5 þ N5 ¼ H5 :

ð38Þ



397

where

A45

A44

A54

A55



hrþ1 /rþ1



 ¼

B4 B5

468

:

ð59Þ

470

398 400

L1 ¼ f  q; N1 ¼ 0; H1 ¼ 0;

ð39Þ

401 403

L2 ¼ q0  p; N2 ¼ 0; H2 ¼ 0;

ð40Þ

f rþ1 ðsN Þ ¼ 0;

ð60Þ

473 475

  1 0 P; N3 ¼ Rðqp  fp Þ; H3 ¼ 0; L3 ¼ p00  M þ d Da

ð41Þ

qrþ1 ðsN Þ ¼ 1;

ð61Þ

476 478

ð62Þ

479 481

404 406

0

407

Prþ1 ðsN Þ ¼ a;

0

L4 ¼ h00 ; N4 ¼ R:Pr:f h0 þ Nb/0 h0 þ Nth02 þ Ec:Pr:P2  fp Þ; H4 ¼ 0;

409

when O is zero matrices of order ðN þ 1Þ  ðN þ 1Þ. Subject to the boundary conditions

Prþ1 ðs0 Þ ¼ b;

ð42Þ

410 412 413

414

L5 ¼ /00 þ

Nt 00 h ; N5 ¼ R:Sc:f /0 ; H5 ¼ 0: Nb

For example, derivation of Eq. (27) for i ¼ 3 is as bellow:      @N 3  @N 3  @N 3  @N 3  0 L3 jrþ1 þ p þ p ¼ H þ p þ p0  N3 r 3 rþ1 r rþ1 @p rþ1 @p0 rþ1 @p rþ1 @p0 rþ1 r

ð44Þ

416 417

418 420

Substituting Eq. (37) in Eq. (30) and after simplifying:

  1 p00rþ1  M þ d Prþ1  R:qr prþ1 þ R:f r p0rþ1 ¼ 0: Da

ð45Þ

The similar procedure for each i ¼ 1; 2; 4; 5 yields

421

422

ð43Þ

424

0 f rþ1

¼ qrþ1 ;

ð46Þ

425 427

q0rþ1 ¼ prþ1 ;

ð47Þ

h00rþ1 þ R:Pr:f r h0rþ1 þ Nb:/0rþ1 h0r þ Nt:h0rþ1 h0r ¼ Ec:Pr:P2r ;

ð48Þ

428 430 431 433 434 435 436

437 439

/00rþ1 þ

Nt 00 h þ R:Sc:f r /0rþ1 ¼ 0: Nb rþ1

ð49Þ Fig. 2. Comparison between the SLLM and Runge-Kutta solutions.

Since the transformed boundary condition (20) for Eq. (45) is incompatible, thus we must consider the transformed boundary conditions as follows

Prþ1 ð0Þ ¼ a; Prþ1 ð1Þ ¼ b;

ð50Þ

440 442

f rþ1 ð0Þ ¼ 0;

ð51Þ

443 445

qrþ1 ð0Þ ¼ 1;

ð52Þ

446 448

hrþ1 ð0Þ ¼ 1; hrþ1 ð1Þ ¼ 0;

ð53Þ

449 451

/rþ1 ð0Þ ¼ 1; /rþ1 ð1Þ ¼ 0;

ð54Þ

452 453 454

455

457 458

459

Therefore, all of the boundary conditions are according to Eqs. (45)–(49), now. Chebyshev differentiation [24] replaces equations Eqs. (45)–(49) with the discrete form

2

I D

D 6 6O 6 6O 6 6 4O

O I

O O

O

A3

O

O

O

A44

O

O

O

A54

32 3 2 3 f rþ1 0 O 76 7 6 7 O 76 qrþ1 7 6 0 7 76 7 6 7 7 6 7 6 O 7 76 prþ1 7 ¼ 6 B3 7: 76 7 6 7 A45 54 hrþ1 5 4 B4 5 A55 /rþ1 B5

In the other words

ð55Þ

Fig. 3. Comparison of the results between the present work and Sheikholeslami et al. [13].

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Fig. 4. Effect of viscosity parameter on vertical velocity profile. Fig. 6. Effect of viscosity parameter on temperature profile.

Fig. 5. Effect of viscosity parameter on horizontal velocity profile. Fig. 7. Effect of viscosity parameter on concentration profile.

482 484

hrþ1 ðsN Þ ¼ 1;

hrþ1 ðs0 Þ ¼ 0;

ð63Þ

485 487

/rþ1 ðsN Þ ¼ 1;

/rþ1 ðs0 Þ ¼ 0:

ð64Þ

488

where

489

1 A3 ¼ D2 þ R:diagff r gD  R:diagff r g  ðM þ d Da ÞI;

A44 ¼ D2 þ ðR:Pr:diagff r g þ Nt:diagfDhr gÞD; A45 ¼ Nb:diagfDhr g; 491 492 493 494 495 496

Nt 2 D ; A54 ¼ Nb

B3 ¼ 0; 498

ð65Þ

B4 ¼ Ec:Pr:P2r ;

A55 ¼ D2 þ R:Sc:diagff r gD;

B5 ¼ 0:

when 0 is zero matrices of order ðN þ 1Þ  1. Also, diag is the vector elements that are placed on the main diagonal of a matrix whose entries everywhere else are zero. For solution of linear system (42), we need to apply boundary conditions, accordingly. The Eqs. (43)–(46) with the boundary conditions are: Please cite this article in press as: Sayehvand H-O, Basiri Parsa A. A new numerical method for investigation of thermophoresis and Brownian motion effects on MHD nanofluid flow and heat transfer between parallel plates partially filled with a porous medium. Results Phys (2017), http://dx.doi.org/ 10.1016/j.rinp.2017.02.004

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Fig. 8. Effect of Hartman number on vertical velocity profile.

Fig. 10. Effect of Darcy number on vertical velocity profile.

Fig. 9. Effect of Hartman number on horizontal velocity profile.

Fig. 11. Effect of Darcy number on horizontal velocity profile.

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The initial approximations are selected as follow

509

510

f 0 ðgÞ ¼ b6 a g3 þ a2 g2 þ g; q0 ðgÞ ¼ b2 a g2 505 506

þag þ 1; p0 ðgÞ ¼ ðb  aÞg þ a; h0 ðgÞ ¼ 1  g; /0 ðgÞ ¼ 1  g:

ð70Þ 512

These initial approximations satisfy boundary conditions (50)(54). So that the SLLM generates subsequent approximations f r ; qr ; pr ; hr ; /r for each r ¼ 1; 2; . . .. Ultimately, using original boundary condition (20) ,we can obtain a and b.

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Results and discussion

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The simulations are performed of heat transfer and fluid flow of an incompressible viscous flow of nanofluid between two horizon-

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Fig. 12. Effect of Prandtl number on temperature profile.

Fig. 14. Effect of Brownian parameter on temperature profile.

Fig. 13. Effect of Prandtl number on concentration profile.

Fig. 15. Effect of Brownian parameter on concentration profile.

tal parallel plates partially filled with a porous medium under the effect of uniform magnetic field, when fluid and the plates rotate together around the y-axis. The dimensionless velocity components, temperature and concentration distributions are analyzed for different emerging physical parameters. The main results of the numerical results iteratively generated by the SLLM are reported and discussed in this section. The effects of the viscosity parameter, Hartmann number, Darcy number, Prandtl number, Schmidt number, Eckert number, Brownian motion parameter, thermophoresis parameter and thickness of the porous region are analyzed and discussed in this section. All the SLLM results presented in this work were obtained using N ¼ 40 collocation points, although relative convergence was achieved in as few as five iterations. The present numerical solution code (SLLM approximate) is validated by comparing the achieved numerical results using shooting technique with Runge-Kutta method, for selected default values of

the active parameters. As shown in Fig. 2, they are in a very good agreement. Also, in order to verify the correctness of the present SLLM code, we have compared the results for the velocities, temperature and nanoparticles concentration profiles with those reported by Sheikholeslami [13] when D ¼ 0; Ec ¼ 0 (clear medium and ignoring the viscous dissipation). This comparison shows a good agreement (Fig. 3). Figs. 4–7 depict the profiles of velocities, temperature and concentration obtained by the SLLM for various values of viscosity parameter ðRÞ when other physical parameters are fixed. Figs. 4 and 5 show that the vertical and horizontal velocities decreases with increasing viscosity parameter especially at middle point. This reduction in velocity is due to inertia effect that is increased by viscosity parameter. According to Figs. 6 and 7 the thickness of thermal boundary layer decreases with an increase of viscosity parameter but an opposite trend is observed for concentration

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Fig. 16. Effect of thermophoresis parameter on temperature profile.

Fig. 18. Effect of Eckert number on temperature profile.

Fig. 17. Effect of thermophoresis parameter on concentration profile.

Fig. 19. Effect of Eckert number on concentration profile.

boundary layer thickness. For high qualities of viscosity parameter, the concentration of nanoparticles becomes more even from the boundary value. It is clear that temperature reduction is occurred due to reduction of viscose dissipation. Considering the temperature of plates, this temperature reduction lead to a local concentrations amplification. Figs. 8–11 show the variations of dimensionless velocities for various values of Hartman number ðMÞ and Darcy number ðDaÞ. It is interesting to note that the effects of Hartman number on velocity profiles are similar to that of viscosity parameter. In the other words, the Lorentz force acts as more retarding force on the velocity field, so that the vertical and horizontal velocities decreases with increasing Hartman number. Also, as Darcy number increases velocity profiles decreases while opposite trend is observed for Hartman number. A reason for the sudden rise of horizontal velocity in the near of upper plate, is the effects of porous medium. As we know, in the clear medium the presence of mag-

netic field is more important but in the porous medium because of the presence of the Darcy drag, the magnetic field is not strong enough for the magnetic drag to be significant in comparison with the Darcy drag. This fact is well visible form Figs 8–11. Fig. 12 illustrates the effect of Prandtl number ðPrÞ on the temperature distribution. The temperature increases with the increasing Prandtl number, extremely. This is in agreement with the physical fact that the thermal boundary-layer thickness increases with the increasing Pr. Also, For high qualities of Prandtl number, the temperature of nanoparticles becomes more even from the boundary value, interestingly. It can be occurred because of the sharp rise in viscose dissipation. An increase in Prandtl number leads to irregularities and overall reduction in the particles distribution that can be seen as well in Fig. 13. Due to the temperature of plates, the temperature increase of the flow field is the most important reason for reduction of nanoparticles concentration.

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Fig. 20. Effect of Schmidt number on concentration profile.

Fig. 21. Effect of thickness of the porous region on vertical velocity profile.

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Effects of Brownian parameter on the temperature and concentration profiles and are depicted in Figs. 14 and 15, respectively. It is obvious that the temperature increases with augment of Brownian parameter. This temperature rise is happen due to effect of nanoparticles’s heat capacity. Also, As Brownian motion increases concentration of nanofluid decreases. But for the small amounts of Brownian parameter, concentration of nanofluid has a oscillatory behavior, according to Fig. 15. This unusual behavior is happen due to very high quantity of thermal diffusivity of base fluid. Figs. 16 and 17 demonstrate the effect of thermophoresis parameter on the temperature and concentration distributions. This parameter have a effect similar to Brownian parameter on the temperature profile. It means that temperature increases with increasing of thermophoresis parameters. Again, This is happen due to effect of nanoparticles’s heat capacity. It is clear that the concentration of nanofluid decreases with increasing thermophoresis parameters that can be justified according to the tem-

Fig. 22. Effect of Schmidt number on horizontal velocity profile.

perature of boundaries and the temperature increasing of the flow field. Figs 18 and 19 illustrate the effects of viscous dissipation for which the Eckert number ðEcÞ on the temperature and concentration profiles. On observing the temperature graph (Fig. 18), the wall temperature gradient of the plate increases as the values of Ec increase. Moreover, when the values of Ec increase, the thermal boundary layer thickness increases. This is due to the fact that heat energy is stored in the fluid due to the frictional heating. Fig. 19 illustrates the effect of the Eckert number (viscous dissipation effect) on the concentration profiles. Increases in the value of Eckert number have the tendency to increase decrease the solute concentration. For high qualities of Eckert number, the concentration of nanoparticles becomes fewer even from the boundary value. This is due to the fact that the heat transfer rate at the surface decrease as Ec increases. The effects of Schmidt number on the concentration distributions has been studied through Fig. 20. The Schmidt number is an important parameter in heat and mass transfer processes as it characterizes the ratio of thicknesses of the viscous and concentration boundary layers. Its effect on the species concentration has similarities to the Prandtl number effect on the temperature. That is, increases in the values of Sc cause the concentration of nanoparticles and its boundary-layer thickness to decrease, significantly. Since the increasing Schmidt number is occurred as a result of increasing the viscosity of the base fluid, so lead to a local concentrations reduction. Figs. 21 and 22 display the effects of porous medium thickness on the velocity profiles. As it is seen, by decreasing porous medium thickness (increasing clear medium thickness) more flow is pushed towards clear region with less resistance, which leads to increase of vertical velocity peak (Fig. 21). In the case of horizontal velocity, in the near of lower plate (clear region) by increasing clear medium thickness, less resistance and more flow inertia lead to increase of horizontal velocity, but in the near of upper plate (porous region) by increasing porous medium thickness (decreasing of D) because of simultaneous effects of pressure gradient and flow inertia, the horizontal velocity is decreased in opposite direction. A comparison with Runge-Kutta method results has been done in order to check the accuracy of the present method (SLLM). As it is shown in Tables 1–3 the numerical values of the skin friction

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Table 1 00 Comparison of the SLLM results of f ð0Þ, h0 ð0Þ and /0 ð0Þ with those obtained by Runge-Kutta for different values of the viscosity parameter when M ¼ 1, Da ¼ 0:1, Ec ¼ 0:03, Pr ¼ 6:7, Sc ¼ 0:1, Nb ¼ 0:1, Nt ¼ 0:1, D ¼ 0:5. 00

h0 ð0Þ

/0 ð0Þ

R

f ð0Þ SLLM results

Runge-Kutta results

SLLM results

Runge-Kutta results

SLLM results

Runge-Kutta results

1 2 3

4.315144967 4.412521320 4.523652122

4.315144962 4.412521325 4.523652113

0.652315652 1.002156897 1.496235874

0.652315663 1.002156885 1.496235885

1.563212578 1.007852031 0.598521367

1.563212563 1.007852025 0.598521356

Table 2 00 Comparison of the SLLM results of f ð0Þ, h0 ð0Þ and /0 ð0Þ with those obtained by Runge-Kutta for different values of the Hartman number when R ¼ 1, Da ¼ 0:1, Ec ¼ 0:03, Pr ¼ 6:7, Sc ¼ 0:1, Nb ¼ 0:1, Nt ¼ 0:1, D ¼ 0:5. 00

h0 ð0Þ

/0 ð0Þ

M

f ð0Þ SLLM results

Runge-Kutta results

SLLM results

Runge-Kutta results

SLLM results

Runge-Kutta results

0 6 20

4.152365478 4.963215879 6.325689788

4.152365462 4.963215889 6.325689796

0.663254888 0.586321587 0.432588789

0.663254863 0.586321589 0.432588785

1.385213697 1.453221201 1.598663231

1.385213695 1.453221208 1.598663237

Table 3 00 Comparison of the SLLM results of f ð0Þ, h0 ð0Þ and /0 ð0Þ with those obtained by Runge-Kutta for different values of the Darcy number when R ¼ 1, M ¼ 1, Ec ¼ 0:03, Pr ¼ 6:7, Sc ¼ 0:1, Nb ¼ 0:1, Nt ¼ 0:1, D ¼ 0:5. 00

h0 ð0Þ

f ð0Þ

Da

0.001 0.01 1

/0 ð0Þ

SLLM results

Runge-Kutta results

SLLM results

Runge-Kutta results

SLLM results

Runge-Kutta results

4.653220148 4.478522301 4.246530021

4.653220149 4.478522309 4.246530036

0.586230145 0.621032587 0.645203258

0.586230152 0.621032575 0.645203263

1.462301258 1.452123687 1.402569875

1.462301263 1.452123675 1.381300978

Fig. 23. Effect of active parameters on skin coefficient friction.

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00

coefficient f ð0Þ; local Nusselt number h0 ð0Þ and local Sherwood number /0 ð0Þ in this paper for different values of R; M and Da are in excellent agreement with the result obtained by Runge-Kutta method. Therefore, we are confident that our results are highly accurate to analyze this problem. Seeing these Tables, the skin friction and Nusselt number are observed to increase while the Sherwood number decreases with the viscosity parameter. We also noticed that as Hartman number increases, both the skin friction and Sherwood number increase, while the Nusselt number decreases. Finally, a slight decreasing for skin friction and Sherwood number is observed when Darcy number is increased. Effects of active parameters on the skin friction coefficient of lower plate are depicted in Fig. 23. As Hartmann number is augmented, the gradient of horizontal velocity near the lower wall enhances. So, skin friction coefficient increases with augment of

these parameters. According to the direction of a magnetic field (Fig. 1), this result is expected. More velocity gradient due to increasing of porous medium thickness (decreasing of D) increases skin friction coefficient. As permeability is grown due to increasing of Darcy number, fluid Flow become more freely and the skin friction coefficient is decreased. For very low quantities of Darcy number an unusual behavior can be seen due to blockage of fluid flow. Also as expected, the general trend of variation of skin friction coefficient with the viscosity parameter is additive due to effects of viscosity, except for very small quantities of Darcy number. In Fig. 24, the effects of the active parameters on the Nusselt number are portrayed. Increasing the viscosity parameter leads to decrease in the thermal boundary layer thickness. Hence the temperature gradient and Nusselt number on the lower plate increases as the viscosity parameter increases. As it can be seen,

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Fig. 24. Effect of active parameters on Nusselt number.

Fig. 25. Effect of active parameters on Sherwood number.

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variation of Darcy number has no significant effect on Nusselt number, except for small quantities of Darcy number. As seen in Fig. 18, temperature increases with an increase of viscous dissipation. So Nusselt number decreases with an increase of Eckert number. Also Nusselt number is increased with Prandtl number, as expected. This is in agreement with the physical fact that the thermal boundary-layer thickness increases with the increasing Pr. Effects of thermophoretic and Brownian parameter on Nusselt number are remarkable. These parameters have a similar effect on temperature profile. It means that temperature gradient

increases with augment of these parameters due to effect of nanoparticles’s heat capacity and in turn Nusselt number has reverse relationship with thermophoretic and Brownian parameter. Ultimately, Fig. 25 shows the influence of thermophoretic and Brownian parameter on Sherwood number. The dimensionless mass transfer rates is decreased with the increase in Nb and it is decreased with Nt. These observations can be justified according to the temperature of plates and the temperature profiles of flow field.

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Conclusions

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Heat and mass transfer of a MHD nanofluid flow between two horizontal parallel plates partially filled with a porous medium has been numerically investigated to count in the effects of Brownian motion and thermophoresis in the nanofluid model. The governing partial differential equations by using suitable similarity variables are transformed into a system of nonlinear ordinary differential equations. These equations are solved numerically by a recently developed technique known as the spectral local linearization method (SLLM). The accuracy of the SLLM is validated against the Runge-Kutta method. Also a good agreement with results of previously published paper available in the literature was observed. The following conclusions were drawn in our investigation. The effects of physical parameters including: viscosity parameter, Hartmann number, Darcy number, Prandtl number, Schmidt number, Eckert number, Brownian motion parameter, thermophoresis parameter and thickness of the porous region on the profiles of velocities, temperature and concentration are examined. The reported results show that skin friction increases with an

697

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increase of viscosity and Magnetic parameters while it decreases with an increase of Darcy number and clear medium thickness. However, an unusual behavior can be seen due to blockage of fluid flow, for very low quantities of Darcy number. Also it can be found that Nusselt number increases with increase of viscosity parameter and Prandtl number while it decreases with an increase of Eckert number, thermophoretic and Brownian parameter. The Nusselt number does not change with Darcy number, significantly. An increasing in Brownian parameter leads to decrease in the Sherwood number while it is increased with increase of thermophoretic parameter.

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