MHD flow of carbon in micropolar nanofluid with convective heat transfer in the rotating frame

Accepted Manuscript MHD flow of carbon in micropolar nanofluid with convective heat transfer in the rotating frame

Fouzia Rehman, Muhammad Imran Khan, Muhammad Sadiq, Assad Malook PII: DOI: Reference:

S0167-7322(16)32211-5 doi: 10.1016/j.molliq.2017.02.022 MOLLIQ 6935

To appear in:

Journal of Molecular Liquids

Received date: Revised date: Accepted date:

10 August 2016 28 December 2016 6 February 2017

Please cite this article as: Fouzia Rehman, Muhammad Imran Khan, Muhammad Sadiq, Assad Malook , MHD flow of carbon in micropolar nanofluid with convective heat transfer in the rotating frame. The address for the corresponding author was captured as affiliation for all authors. Please check if appropriate. Molliq(2017), doi: 10.1016/ j.molliq.2017.02.022

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ACCEPTED MANUSCRIPT MHD flow of carbon in micropolar nanofluid with convective heat transfer in the rotating frame Fouzia Rehmana, Muhammad Imran Khanb, Muhammad Sadiqc and Assad Malookd a

Department of Mathematics, Allama Iqbal Open University, Islamabad 44000, Pakistan

b

Department of Mechanical Engineering, University of Engineering & Technology, Peshawar, Pakistan

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Production Department, Oil and Gas Development Company Ltd, Islamabad, Pakistan

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Sarhad University of Science and Information Technology, Peshawar, Pakistan

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(*Corresponding authors: Email: [email protected], [email protected]).

Abstract:

N1 and Reynolds number R

while decreases with increase in Biot number

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parameter

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This article addresses the MHD flow of micropolar carbon-water nanofluid in the presence of rotating frame. Two types of carbon nanotubes (single and multi-walls) are homogeneously dispersed in the base fluid (water). Convective heat transfer phenomenon is also retained in this study. With the help of the similarity transformation, a mathematical model is developed by transforming the partial differential equation into ordinary differential equation. Further it is then solved analytically by using homotopy analysis method HAM. The impact of emerging parameters, skin friction coefficient and Nusselt number are also discussed in detail. It is observed that temperature profile decreases with the increase in porosity parameter and nanoparticle volume fraction while enhances for higher values of Biot number. Further Micro rotation profile increases with the increase in coupling

Keywords:

Bi,

and volume fraction

.

Introduction

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Rotating plates; Three dimensional incompressible flow; Carbon nanotubes, convective heat transfer.

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Nanofluids have many advantages over other simple fluids due to their thermal efficiency. Nanofluids are liquid suspensions in which nanoparticles including metals, polymers, metaloxides, silica etc., are added to a base fluid such as water, ethylene glycol, oil etc. Nanoparticles are basically the nanometer sized solid particles having diameter less than 100nm. Nanofluids have higher heat transfer rates, greater viscosity, higher thermal conductivity and more stability than other fluids. These fluids possess a better wetting, spreading and dispersion properties over a solid surface. The use of nanofluids spread over a wide range of fields. Nanofluids serve as a coolant in heat transfer equipment like electronic cooling system, heat exchangers and radiators etc. The efficiency of polymerase chain reaction can be improved with the use of graphene based nanofluid. Nanofluids have tunable optical properties and due to these properties, they are used in solar collectors. Nanofluids are also used in biomedical, transportation, microfluids, solid-state lighting and manufacturing. Ding et al. [1] studied heat transfer of aqueous suspensions of carbon nanotubes. Kamali and Binesh [2] examined heat transfer enhancement using carbon nanotubes in laminar flows. Khan et al. [3] discussed fluid flow and heat transfer of carbon nanotubes along a flat plate with Navier slip boundary. Imtaiz et al. [4] examined convective flow of carbon nanotubes between rotating stretchable disks with thermal radiations effects. 1

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Alsaedi et al. [5] studied the effect of convective boundary conditions and heat generation/absorption in the stagnation point flow of nanofluid past a permeable stretching surface. Bhattacharyya and Layek [6] examined steady boundary layer flow of nanofluid over an exponentially stretching permeable sheet in the presence of magnetic field. Sheikholeslami et al. [7] analyzed the influence of natural convection on the heat transfer of Cu-water nanofluid inside an enclosure with hot elliptic cylinder. The flow is also subjected to magnetic field. The analysis of entropy generation for the MHD flow of nanofluid due to a rotating porous disk is performed by Rashidi et al. [8]. Turkyilmazoglu [9] discussed the heat and mass transfer effects in the flow of nanofluid over a vertically infinite flat plate with thermal radiation. The researchers at present have much interest in the investigation of non-Newtonian liquids. It is due to their exceedingly significance in numerous organic, mechanical and designing procedures, for example, glass arrangement, fiber sheet fabricating, wire drawing, sustenance items, paper creation, precious stone development and so on. Analysis of boundary layer flow has special significance in the situations when fluid is passing over the surface. The researchers in recent times are looking for increasing the efficiency of various machines through reduction of drag/friction forces. Different endeavors therefore have been made about lessening of drag powers/forces for flow over the surface of a wing, tail plane and wind turbine rotor and so forth. Hence heat transfer and boundary layer flow by a moving surface has wide coverage in the industrial manufacturing procedures. Such procedures may incorporate glass fiber creation, hot moving, paper era, wire drawing, relentless tossing, metal turning, metal and polymer ejection, drawing of plastic motion pictures etc. The last thing in toughening and lessening of copper wires hugely depends on heat transfer rate at the stretching surface. Such flow thought in region of magnetic field has significant part in the metallurgical procedure. Specific inspiring stream issues containing non-Newtonian liquid can be found in the studies [10-20]. Researcher and scientists are still interested to disclose the characteristics of magnetohydrodynamic behavior due to its promising applications in various engineering and industrial processes. Such flow characteristics are inter connected in the design of cooling system with liquid metals, accelerators, MHD generators, nuclear reactor, pumps and flow meters and blood flow measurement. Having such feature in mind, numerous researchers explore the behavior of magnetohydrodynamic flow for different flow configurations. Makinde [21] investigated the MHD flow due moving surface in the existence of convective boundary condition. Heat transfer and MHD flow due to exponentially stretching sheet embedded in a thermally stratified medium is explored by Mukhopadhyay [22]. Analysis of electrically conducting viscoelastic fluid considering nonlinear radiation and heat generation/absorption is reported by Cortell [23]. Abdel-wahed et al. [24] disclose the hydromagnetic flow of nanofluid over a moving surface with variable thickness and non-linear velocity. The characteristics of MHD and heterogeneous and homogeneous reactions due stretched flow of viscoelastic fluid with melting heating is presented by Hayat et al. [25]. Recently the researchers and scientists are interested to reduce the skin friction coefficient and enhance the rate of heating or cooling in the advanced technological processes. Thus various attempts have been made about the reduction of skin friction or drag forces for flows over the surface of a wing, tail plane and wind turbine rotor, etc. However these forces can be reduced by keeping the boundary layer away from separation and to delay the transition of laminar to turbulent flow. This task can be performed through different physical aspects such as moving the surface, through fluid suction and injection and the presence of body forces. Similarly most of the researchers have been tried to enhance the rate of cooling/heating by using different types of 2

ACCEPTED MANUSCRIPT boundary conditions over a flat plate. Thus here our main objective is to analyze convective heat transfer aspect of micropolar carbonwater nanofluid over a rotating frame. In addition, magnetohydrodynamic (MHD) effect is also considered. Homotopy analysis method [26-40] is used to obtain the convergence of the series solutions. A detailed study is carried out to discuss the plots for various parameters of interest. Nusselt number and skin friction coefficient are computed and analyzed numerically. Comparisons with the existing literature in limiting sense is found in an excellent agreement.

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Formulation

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Consider steady, incompressible 3D flow of a micropolar fluid rotating between the two parallel plates with angular velocity  in the presence of carbon nanotubes. The plates are kept at y  0 and y  h . Moreover, the lower plate is stretched with velocity u  ax where a  0. The hot base fluid is also placed at lower surface of the lower plate such that the temperature at lower plate To is greater than the temperature at upper plate Th . A uniform magnetic field Bo is applied along the y-axis. The base fluid and the carbon nanotubes are considered to be in thermal equilibrium. Convective heating conditions is considered. Flow assumption is: 1. Three dimensional incompressible flow 2. Rotating plates 3. Carbon nanotubes 4. Convective heat transfer.

For steady and incompressible fluid, electrically conducting fluid we have continuity, 3

ACCEPTED MANUSCRIPT momentum, energy and micro rotation equations are as (1)

(2)

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(3)

(4) (5) (6)

here u, v and w are the components of the velocity in x, y and z direction respectively. nf ,

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nf , and nf are density, viscosity and thermal diffusivity of the nanofluid respectively. The convective condition along with micro rotation takes the form,

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T u  Af (To  Th ), N  n , at y  0, y y u u , v  vo , w  0, T  Th , N  n , at y  h , y

(7) (8)

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u  ax, v  0, w  0,  k f

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where a is stretching constant, the thermal conductivity of the hot fluid and convective heat transfer coefficient are represented with k f and Af respectively. Appling the similarity transformation

u  axf (), v  ahf (), w  axg(), T  Th axG() y N ,  ()  ,  . h To  Th h

(9)

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

(10)

Eqs. (2) to (6) become

   1 1    N1  ) f iv  N1G  Mf    R( f f   f f )  2Krg  0, CNT 2.5  (1)  (  f )   (1)   

   1 1    2Krf   0,    N1  )g  Mg   R  gf  f g CNT 2.5  (1)  (  f )   (1   )    4

(11)

(12)

ACCEPTED MANUSCRIPT

(13)

  1    N G  N1 ( f   2G)  N3R(Gf   fG)  0.  (1)  ( CNTf )  2  

(14)

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 (C )CNT  knf    Peh f   (1) ( p )   0,   kf (  ) C f p  

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The non dimensional boundary conditions take the form,

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f (0)  0, f (0)  1, g(0)  0,  (0)  Bi(1 (0)), G(0)  nf (0), f (1)  , f (1)  0, g(1)  0,  (1)  0, G(1)  nf (1),

(15) (16)

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where N1 shows coupling parameter, N2 represents spin-gradient viscosity parameter, R shows Reynolds number, M indicates the magnetic parameter, Kr indicates rotation parameter, Bi is Biot number and Peclet number is represented by Peh which are defined as

h2 f

f

( s ) f  h2 Bo2 j ah2 , N  , R  , M  ,  f h2 3 h2  f

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f

, N2 

(17)

hA v L , Peh  R Pr,    o ,   , Bi  f . h h kf

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Kr 

k

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N1 

Cf 

w

(18)

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, 1 nf uw2 2 hqw Nu  , knf (To Th )

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The skin friction coefficient Cf and Nusselt number Nu are described as

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where

(19)

 w represents shear wall stress and qw is heat flux.

 w  ((nf  k ) qw  knf (

u   k N) y0 , y

(20)

T ) . y y0

(21)

After applying similarity transformations, skin friction coefficient and Nusselt number take the form,

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ACCEPTED MANUSCRIPT

Cf 



1 (1 )2.5



 N1 f (0)  N1G(0)

1 ( ) CNT f

(22)

,

Nu   (0).

(23)

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Solution methodology

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Here homotopy analysis method is used for the computations of nonlinear systems (11-14) subject to boundary conditions (15 and 16). Initial guesses ( fo , go ,o , Go ) and linear operators (Lf , Lg , L , LG ) are required for homotopy solution. The initial guesses and linear operators are defined as

(24)

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(25)

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Lf {Co  C1  C2  C3}  0, Lg{C4  C5}  0, L {C6  C7}  0, LG{C8  C9}  0,

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fo ()  (1 2)3  (2  3) 2 , go ()  0, Bi(1) o ()  , 1 Bi Go ()  n[6  4  2], Lf  f iv , Lg  g, L   , LG  G,

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where all Ci (i  0,1,..9) represent arbitrary constants.

Zeroth order problem For system of coupled equations we define the zeroth order problem as follow

(1 q)Lf [ fˆ (, q)  fo ()]  q1 hf N f [ fˆ (, q), gˆ (, q),ˆ(, q), Gˆ (, q)], (1 q)Lg [gˆ (, q)  fo ()]  q2 hgNg [ fˆ (, q), gˆ (, q),ˆ(, q), Gˆ (, q)], (1 q)L [ˆ(, q)  fo ()]  q3 h N [ fˆ (, q), gˆ (, q),ˆ(, q), Gˆ (, q)], (1 q)LG[Gˆ (, q)  fo ()]  q4 hG NG[ fˆ (, q), gˆ (, q),ˆ(, q), Gˆ (, q)],

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(26)

ACCEPTED MANUSCRIPT fˆ (0, q) fˆ (0, q)  0,  1, gˆ (0, q)  0, ˆ(0, q)  Bi(1 ()),   0,  (27) ˆ (1, q)  f fˆ (1, q)  ,  0, gˆ (1, q)  0, ˆ(1, q)  0,  1,  where h f , hg , h and hG are auxiliary parameters which are non-zero and the nonlinear operators Lf , Lg , L and NG are defined as

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1 1 4 fˆ (, q) 2Gˆ (, q) ][(  N1)  N1 4  2 (1)  ( CNTf ) (1)2.5

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N f [ fˆ (, q), gˆ (, q),ˆ(, q), Gˆ (, q)]  [

(28)

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2 fˆ (, q) fˆ (, q) 2 fˆ (, q) ˆ 3 fˆ (, q) M ]  R (  f (  , q ) )  2   2 3 gˆ (, q) 2Kr , 

(29)

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  1   2 gˆ (, q) 1  Ng [ fˆ (, q), gˆ (, q),ˆ(, q), Gˆ (, q)]    N  Mg   1  CNT 2.5 2  (1)  (  f )    (1)       gˆ (, q) ˆ fˆ (, q)  fˆ R  f  gˆ (, q)   2Kr (, q),     

(30)

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 k 2ˆ(, q) ˆ  1 ˆ , N [ fˆ (, q), gˆ (, q),ˆ(, q), Gˆ (, q)]  nf  Pe f h  k f 2   (1) ( ( ( CCp p)CNT ) )f  

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  2Gˆ (, q)  ˆ (, q)   N  N (2 G   2 1 2       1   NG[ fˆ (, q), gˆ (, q),ˆ(, q), Gˆ (, q)]     (1)  ( CNTf )   2 fˆ (, q)     2       fˆ (, q) ˆ  Gˆ (, q) ˆ N3R  G(, q)  f (, q) .    

mth-order deformation problem mth order deformation is defined as

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(31)

ACCEPTED MANUSCRIPT Lf [ fm (, q)  X m fm1 ()]  hf Rf , m (), Lg [ gm (, q)  X m gm1 ()]  hg Rg , m (), L [m (, q)  X mm1 ()]  h R , m (), LG[Gm (, q)  X mGm1 ()]  hG RG, m (), fm (0)  0, fm (1)  , fm (0)  1, fm (1)  0, gm (0)  gm (1)  0, m (0)  Bi(1 (0)), m (1)  0, Gm (0)  nfm (0), Gm (1)  nfm (0),

(32)

(34)

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(35)

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  1  iv 1     Rmf ()    N f  N G  Mf  1 m  1 1 m  1 m 1     (1)  ( CNTf )    (1)2.5       m1   m1  R   fm 1n fn    fm1n fn    2Krgm 1,   n0    n0

(36)

 knf   m1  1 m1  Peh  fm1nn   kf  n0   (1)  

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Rm () 

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  1   1  Rmg ()    N1  gm 1  Mgm1  CNT 2.5  (1)  (  f )    (1)       m1   m1  R   fm1n gn    gm1n fn    2Krfm 1,   n0    n0



( Cp )CNT ( Cp ) f

  1  N2Gm 1  N1  2Gm1  fm1  RmG ()   CNT  (1)  (  f )      m1    m1  N3R   fm1nGn    Gm 1n fn   ,   n0    n0 X m  0, m  1 . X m  1, m  1



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 

 ,  

(37)

(38)

(39)

ACCEPTED MANUSCRIPT When q  0 and q  1,

fˆ (;0)  fo (), fˆ (;1)  f (), gˆ (;0)  go (), gˆ (;1)  g(), ˆ(;0)  o (), ˆ(;1)   (),

(40)

Gˆ (;0)  Go (), Gˆ (;1)  G().

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From Taylor series theorem, we get 

fˆ (; q)  fo ()  fm ()qm ,

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n0



gˆ (; q)  go ()  gm ()qm , 1  g( ; p) , m! qm q0

1 m ( ; p) , m! qm q0

(44)

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m () 

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

n0

(43)

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m

ˆ(; q)  o ()  m ()qm ,

(42)

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n0

gm () 

(41)

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1 m f ( ; p) fm ()  , m! qm q0



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Gˆ (; q)  fo ()  Gm ()qm , n0

(45)

1  Gˆ ( ; p) Gm ()  . m! qm q0

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m

The values of h f ,

hg ,

convergent at q  1.

h and hG are chosen in such a way that the series become



f ()  fo ()  fm (), n0 

(46)

g()  go ()  gm (), n0

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ACCEPTED MANUSCRIPT 

 ()  o ()  m (), n0 

G()  Go ()  Gm ().

(47)

n0

The general solutions are

fm ()  fm ()  Co  C1  C2 2  C33 , gm ()  gm ()  C4  C5, m ()  m ()  C6  C7, Gm ()  Gm ()  C8  C9,

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where fm (),

(48)

gm (), m () and Gm () represent particular solutions.

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Convergence of HAM solution

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h and hG play a major role to get the convergence of the Auxiliary parameters hf , hg , series solution. With the help of these auxiliary parameters, convergence region can easily be controlled. Therefore the h -curves are plotted for velocity, temperature and micro rotation profiles for both SWCNT as well as MWCNT.

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Table 1 : For SWCNT, the convergence of the HAM solution for several order of approximations when N 1  N 2  1, N 3  0. 1, R  0. 5, M  0. 1, Kr  0. 05, Pe  0. 31,   0. 1,

M

h  0. 7, Bi  0. 7,   0. 03 for n  0.

f  0

1

-0.67812979 0.0039041887 -0.42584822 0.19714012

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G  0

-0.67806618 0.0038595577 -0.42585114 0.19744288

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  0

-0.68108269 0.0053666667 -0.42291892 0.14691366

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g  0

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Order of approximation

-0.67807017 0.0038624586 -0.42585140 0.19745839 -0.67807011 0.0038624248 -0.42585140 0.19745837 -0.67807011 0.0038624246 -0.42585140 0.19745837

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-0.67807011 0.0038624246 -0.42585140 0.19745837

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Order of approximation

g  0

  0

G  0

-0.68078256 0.0038333333 -0.41408066 0.10803968

5

-0.67816122 0.0037503469 -0.41602858 0.19309304

8

-0.67806646 0.0037517252 -0.41612106 0.19698964

13

-0.67805656 0.0037518548 -0.41613167 0.19745861

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-0.67805634 0.0037518573 -0.41613191 0.19747169

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-0.67805634 0.0037518573 -0.41613192 0.19747186

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-0.67805634 0.0037518573 -0.41613192 0.19747186

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Table 3 : For SWCNT, the convergence of the HAM solution for several order of

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approximations when N 1  N 2  1, N 3  0. 1, R  0. 5, M  0. 1, Kr  0. 05, Pe  0. 31,   0. 1, h  0. 7, Bi  0. 7,   0. 03 for n  0. 5.

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-0.68108269 0.0053666667 -0.42291892 -0.33357654

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  0

-0.68100968 0.0038775451 -0.42573757 -0.33622549

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Order of approximation

-0.68100396 0.0038377856 -0.42574135 -0.33622360 -0.68100436 0.0038401366 -0.42574139 -0.33622409 -0.68100436 0.0038401138 -0.42574139 -0.33622409

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-0.68100436 0.0038401137 -0.42574139 -0.33622409

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-0.68100436 0.0038401137 -0.42574139 -0.33622409

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-0.68100436 0.0038401137 -0.42574139 -0.33622409

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Order of approximation

g  0

  0

G  0

-0.68078256 0.0038333333 -0.41408066 -0.32991387

5

-0.68098517 0.0037303009 -0.41600195 -0.33596775

8

-0.68099023 0.0037301908 -0.41608755 -0.33611928

13

-0.68099054 0.0037301840 -0.41609700 -0.33613020

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-0.68099055 0.0037301839 -0.41609721 -0.33613035

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-0.68099055 0.0037301839 -0.41609721 -0.33613035

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-0.68099055 0.0037301839 -0.41609721 -0.33613035

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-0.68099055 0.0037301839 -0.41609721 -0.33613035

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Results and discussion

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In this part from Figs. (9  27) the influence of several emerging parameters on velocity, temperature and micro-rotation profiles are discussed. In order to find the impact of different parameters i.e. coupling parameter N1, spin gradient viscosity parameter N2 , rotation

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parameter Kr, magnetic parameter M , Reynolds number R, porosity parameter , and volume fraction  on velocity profile f '( ), , graphs (9 15) have been plotted. It can be noticed that coupling parameter N1, spin gradient viscosity N2 , Reynolds number R and magnetic parameter M have same impact on velocity profile. Physically for higher estimation of magnetic parameter Lorentz forces enhances, which is a resistive forces. Therefore velocity field decreases. Similarly velocity field decay for larger Reynold number. Physically Reynold number is the ratio of inertial forces to viscous forces. Therefore higher estimation of Reynold number increase the inertial forces as a result velocity profile decreases. Figs. (13 14) show the effect of rotation parameter Kr, and volume fraction  on velocity profile. It can be seen that initially, at lower plate the velocity increases and then decreases towards the upper plates with the increase in rotation parameter, and volume fraction. Because viscosity decreases with the increase in Kr which tends to increase in velocity profile. The impact of porosity parameter on velocity can be observed in Fig. 15 . Velocity increases with the increase in porosity parameter . To observe the effect of different emerging parameters on transverse velocity profile g(), , graphs (16  21) have been plotted. In Figs. 16 and 17 it is observed that transverse velocity decreases with the increase in coupling parameter N1 and volume fraction  . Figs. (18 19) describe the impact of rotation parameter Kr and Hartman number M on g(). It can be noticed that both parameters have similar behavior on transverse velocity profile i.e. initially it increases at lower plate and then start decreasing towards upper plate. Figs. 20 16

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and 21 show the influence of Reynolds number R and porosity parameter  on g . It can be noted that both parameters have increasing behavior on g ( ) . The effect of different appearing parameters on micro rotation profile G() can be observed in Figs. (22  25). Fig. (22) is plotted to analyze the behavior of coupling parameter N1 on micro rotation profile. It can easily be observed that micro rotation profile increases with the increase in coupling parameter N1. Spin gradient viscosity parameter N2 , Reynolds number R, and volume fraction  have same impact on micro rotation profile i.e. G() decreases with the increase in these parameters and this can be observed in Figs. (23  25) . Figs. (26  27) have been plotted to discuss the behavior of different emerging parameters on temperature profile  (). Fig. (26) represents the influence of porosity parameter  on temperature profile. It can be seen that temperature decreases with the increase in . To observe the effect of Biot number Bi on temperature profile, Fig. 27 has been plotted. Temperature increase with the increase in Biot number. Because the higher values of volume fraction enhance the motion of molecules which tend to increase in thermal conductivity. To analyze the effect of different parameters i.e. N1, N2 , M , Kr, , and  on skin friction coefficient, tables 5 and table 6 are prepared for both single walled carbon nanotubes as well as multiple walled carbon nanotubes respectively. From Tables 5 and 6, we can see that skin friction coefficient enhances for N1, N2 , M , Kr, ,  and n. Also these effects are more prominent in strong concentration

M

n  0.0 when compared with weak concentration n  0.5. The impact of N1, N2 , N3, M , Kr, , and  on Nusselt number -  () are described in tables 7 and 8. From tables it can be seen that the coupling parameter N1 has positive effect on Nusselt number but Nusselt number

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decreases with the increase in rest of the parameters. Moreover the results are more clear for strong concentration n  0.0 as compare to the weak concentration n  0.5 .

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n  0. 0

n  0. 5

M Kr

0.0

0.1 0.1 0.1 0.1 -1.47208 -1.47208

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0.1

-1.58457 -1.52844

0.5

-2.03132 -1.75511 -1.52866

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0.1 0.5 0.1 0.1 0.1 0.1 -1.5844

-1.58453 -1.52849

1.0

-1.58457 -1.52844 0

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0.1 0.1 0.1 -1.58265 -1.52668

0.5

-1.59132 -1.53503

1.0

-1.59977 -1.54318

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

N1 N2

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0.1 1.0 0.1 0.5 0.1 0.1 -1.58462 -1.52848 1.0

-1.58477 -1.52863 -1.58502 -1.52863

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1.5

0.1 1.0 0.1 0.1 0.1 0.1 -1.58457 -1.52844 -1.30696 -1.26074

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-1.02844 -0.992122

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0.1 1.0 0.1 0.1 0.1 0.1 -4.16259 -4.01729 -2.95695 -2.85298 -1.37247 -1.32379

0.1 1.0 0.1 0.1 0.1 0.0 -1.44457 -1.37935

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Kr 



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n  0. 5

0.1

-1.73404 -1.67259

0.5

-2.22304 -1.92069

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0.1 0.5 0.1 0.1 0.1 0.1 -1.73386 -1.67283 -1.73399 -1.67265

1.0

-1.73404 -1.67259

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0.1 0.5 0.0 0.1 0.1 0.1 -1.73212 -1.67075 -1.74161 -1.67989

1.0

-1.75087 -1.68882

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0.1 1.0 0.1 0.0 0.1 0.1 -1.73403 -1.67259 1.0

-1.73422 -1.67277 -1.73476 -1.67329

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2.0

0.1 1.0 0.1 0.1 0.1 0.1 -1.73404 -1.67259

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-1.43009 -1.37949 -1.12523 -1.08546

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0.1 -1.73404 -1.67259 0.2 -2.16052 -2.10245

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Kr 

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n0.5

0.5

0.428328 0.428259

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0.428217 0.428230

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0.0 1.0 1.0 0.1 0.1 0.1 0.428266 0.428266

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0.428274 0.428259

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0.1 1.0 0.1 0.1 0.1 0.1 0.428274 0.428270 0.428274 0.428269

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0.1 1.0 1.0 0.0 0.1 0.1 0.428296 0.428290 0.428187 0.428181

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0.428080 0.428075

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0.1 1.0 1.0 0.1 0.1 0.1 0.428274 0.428268 0.428264 0.428259

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0.428253 0.428247

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0.1 1.0 1.0 0.1 0.1 0.1 0.433995 0.433987 0.427375 0.427370 0.425025 0.425022

0.1 1.0 1.0 0.1 0.1 0.0 0.429519 0.429512

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0.1 0.428274 0.428268 0.2 0.427021 0.427017

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n  0. 5

0.5

0.429069 0.428998

1.0

0.429061 0.428966

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0.0 1.0 1.0 0.1 0.1 0.1 0.429005 0.429005

0.429015 0.429005

1.0

0.429013 0.429008

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0.1 0.2 1.0 0.1 0.1 0.1 0.429019 0.428996

0.1 1.0 0.1 0.1 0.1 0.1 0.429013 0.429010 0.429013 0.429010

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0.429013 0.429010

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0.1 1.0 1.0 0.0 0.1 0.1 0.429036 0.429031 0.428922 0.428917

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0.428811 0.428806

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0.1 1.0 1.0 0.1 0.1 0.1 0.429013 0.429008 0.429005 0.428999

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0.428995 0.428989

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Concluding remarks: MHD flow of a micropolar fluid between the horizontal plates in the presence of carbon-water nanotubes in rotating system has been discussed. Some important results of the present study are as follow:  Coupling parameter N1, viscosity parameter N2 , Reynolds number R, and Hartman number M have similar influence on velocity profile f () while rotation parameter Kr and volume fraction  have similar effect. Porosity parameter have positive effect on velocity profile f ( ). 29

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profile g while rotation parameter Kr, Reynolds number R, Magnetic parameter M and porosity parameter  have similar impact on g(). g ( ) increases with the increase in Kr, R, M and . Temperature decreases with the increase in porosity parameter  and volume fraction  while increases with the increase in Biot number Bi. Micro rotation profile increases with the increase in coupling parameter N1 and Reynolds number R while decreases with increase in Biot number Bi, and volume fraction  . For both strong concentration and weak concentration as well, skin friction coefficient increases with the increase in coupling parameter N1, , viscosity parameter N2 , Magnetic parameter M , rotation parameter Kr, porosity parameter  and volume fraction  in both SWCNT as well as MWCNT. Nusselt number decreases with the increase in viscosity parameter N2 , Hartman number M , Reynold number R, volume fraction  , and rotation parameter Kr where as increases with the increase in coupling parameter N1.

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References

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presence of Brownian motion, Appl. Math. Comput. 254 (2015) 49  62. 25. T. Hayat, Z. Hussain, M. Farooq and A. Alsaedi, Effects of homogeneous and heterogeneous reactions and melting heat in the viscoelastic fluid flow, J. Mol. Liq. 215 (2016) 749  755. 26. S. J. Liao, Homotopy analysis method in nonlinear differential equations, Springer & Higher Education Press, 2012. 27. T. Hayat, M. Imtiaz and A. Alsaedi, Impact of magnetohydrodynamics in bidirectional flow of nanofluid subject to second order slip velocity and homogeneous-heterogeneous reactions, J. Magn. Magn. Mater. 395 (2015) 294-302. 28. T. Hayat, M. Waqas, S. A Shehzad and A. Alsaedi, Mixed convection radiative flow of Maxwell fluid near a stagnation point with convective condition, J. Mech. 29 (2013) 403409. 29. Y. Lin and L. Zheng, Marangoni boundary layer flow and heat transfer of copper-water nanofluid over a porous medium disk, AIP Adv. 5 (2015) 107225. 30. Sui, L. Zheng, X. Zhang and G. Chen, Mixed convection heat transfer in power law fluids over a moving conveyor along an inclined plate, Int. J. Heat Mass Transf. 85 (2015) 1023 1033. 31. T. Hayat, M. Tamoor, M.I. Khan and A Alsaedi, Numerical simulation for nonlinear radiative flow by convective cylinder, Results Phy. 6 (2016) 1031–1035. 32. T. Hayat, M. I. Khan, M. Farooq, T. Yasmeen and A. Alsaedi, Stagnation point flow with Cattaneo-Christov heat flux and homogeneous-heterogeneous reactions, J. Mol. Liq. 220 (2016) 49-55. 33. T. Hayat, M. I. Khan, M. Farooq, A. Alsaedi, M. Waqas and T. Yasmeen, Impact of Cattaneo-Christov heat flux model in flow of variable thermal conductivity fluid over a variable thicked surface, Int. J Heat Mass Transfer 99 (2016) 702--710. 34. T. Hayat, M. I. Khan, A. Alsaedi and M. I. Khan, Homogeneous-heterogeneous reactions and melting heat transfer effects in the MHD flow by a stretching surface with variable thickness, J. Mol. Liq. 223 (2016) 960-968. 35. T. Hayat, M. I. Khan, M. Farooq, N. Gull and A. alsaedi, Unsteady threedimensional mixed convection flow with variable viscosity and thermal conductivity, J. Mol. Liq. 223 (2016) 297-1310. 36. T. Hayat, M. I. Khan, M. Farooq, A. Alsaedi and M. I Khan, Thermally stratified stretching flow with Cattaneo-Christov heat flux, Int. J. Heat Mass Transfer 106 (2017) 289-294. 37. T. Hayat, M. I. Khan, M. Farooq, T. Yasmeen and A. Alsaedi, Water-carbon nanofluid flow with variable heat flux by a thin needle, J. Mol. Liq. 224 (2016) 786-791. 38. T. Hayat, M. I. Khan, M. Waqas, A. Alsaedi and M. Farooq, Numerical simulation for melting heat transfer and radiation effects in stagnation point flow of carbon-water nanofluid, Computer Methods Appl. Mech. Eng. (2017) DOI: 10.1016/j.cma.2016.11.033. 39. T. Hayat, M. I. Khan, M. Farooq, A. Alsaedi and T. Yasmeen, Impact of Marangoni convection in the flow of Carbon-water nanofluid with thermal radiation, Int. J. Heat Mass transfer (2017) DOI: 10.1016/j.ijheatmasstransfer.2016.08.115.

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40. T. Hayat, M. I. Khan, M. Waqas and A. Alsaedi, Magnetohydrodynamic stagnation point flow of third grade liquid towards variable sheet thickness, Neural Comput. Appl. DOI: 10.1007/s00521-016-2827-1.

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ACCEPTED MANUSCRIPT Highlights • • • •

Characteristics of heat transfer are explored in the presence of convective heat transfer. Three dimensional incompressible flow of micropolar carbon-water nanofluid with rotating plate is considered. Skin friction coefficient and Nusselt number are computed numerically. Homotopy analysis method is used to obtain the analytical solution of ordinary differential equations.

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