Effects of surface roughness on mixed convective nanofluid flow past an exponentially stretching permeable surface

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Effects of surface roughness on mixed convective nanofluid flow past an exponentially stretching permeable surface P.M. Patil , Madhavarao Kulkarni , P.S. Hiremath PII: DOI: Reference:

S0577-9073(19)31004-4 https://doi.org/10.1016/j.cjph.2019.12.006 CJPH 1026

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Chinese Journal of Physics

Received date: Revised date: Accepted date:

29 October 2018 26 September 2019 4 December 2019

Please cite this article as: P.M. Patil , Madhavarao Kulkarni , P.S. Hiremath , Effects of surface roughness on mixed convective nanofluid flow past an exponentially stretching permeable surface, Chinese Journal of Physics (2019), doi: https://doi.org/10.1016/j.cjph.2019.12.006

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Effects of surface roughness on mixed convective nanofluid flow past an exponentially stretching permeable surface P. M. Patil* and Madhavarao Kulkarni Department of Mathematics, Karnatak University, Pavate Nagar, Dharwad – 580003, India P. S. Hiremath Department of Computer Science (MCA), KLE Technological University, BVB Campus, Hubli – 580031, India

Highlights 

The surface roughness effects are analysed on mixed convective nanofluid flow.



Quasilinearization technique is used to linearize the governing equations.



The transport rates defined at the wall exhibit sinusoidal variations.



The addition of nanoparticles control the wall heat transfer rate, remarkably.



The surface roughness increases the skin-friction coefficient & heat transfer rate.

Abstract A numerical study is carried out to display the effects of surface roughness on mixed convective nanofluid flow along an exponentially stretching surface in presence of suction/injection. The dimensional coupled nonlinear partial differential equations are transformed into dimensionless form by using suitable non-similar transformations. The resulting equations are solved by utilizing the Quasilinearization technique as well as the implicit finite difference scheme. The influence of several non-dimensional parameters on various profiles and gradients is examined. The results are presented graphically, which are analyzed to depict the effects of various physical parameters, for example, Brownian diffusion parameter Nb, thermophoresis parameter Nt, suction/blowing parameter A and Lewis number Le.

In order to analyse the influence of surface roughness on mixed

convective nanofluid flow, the major part of this research paper is devoted to investigate the effects of the small parameter  and frequency parameter n over the gradients defined at the wall. The results reveal that an increase in the values of Nb and Nt, enhances the velocity and temperature of the fluid. The increasing value of suction parameter (A > 0) reduces the velocity of the fluid. Further, the increasing values of Nb and Le decrease the nanoparticle volume fraction profile. The sinusoidal variations are observed in the skin-friction 1

coefficient, Nusselt number as well as the nanoparticle Sherwood number. Moreover, with the addition of nanoparticles, the magnitude of the skin-friction coefficient increases, while the magnitude of heat transfer rate decreases, significantly. Keywords:

Exponentially

stretching

surface;

Mixed

convection;

Nanofluid;

Quasilinearization; Suction/injection; Surface roughness. * Corresponding author. E-mail addresses: [email protected] (P. M. Patil), [email protected] (Madhavarao Kulkarni) and [email protected] (P. S. Hiremath). 1 Introduction The surface roughness is a component part of surface texture. The roughness can be observed in terms of minute irregularities over the surface texture, which occur during the manufacturing processes [1]. The surface roughness has greater importance in research due to its vital role in heat and mass transfer rates between the fluid and the bounding surface. The surface roughness has numerous applications in the area of science and technology, such as, in the heat transfer devices, polymer fibre coating, electronic cooling techniques, nuclear reactor cooling, and heat transfer exchangers [2 – 4]. Many of the researchers have carried out the analytical and experimental works to examine the effect of surface roughness. An experimental data has been collected by Gupta, et al. [2] to examine the solar air heater of rectangular ducts with a transverse wire roughness operated transitionally through the rough region. Liu, et al. [3] have examined the impacts of surface roughness on fluid flow and heat transport characteristics in micro channels and their experimental results indicate that the friction factor and the heat transfer rate increase with the increase of the relative roughness over the bounding surface. Zhang [4] has carried out an analytical work to study the fluid mass flow rate through nanoslit pores in Poiseuille flow by using fractional approach model and in which, he studied the results of wall roughness for weak, medium-level and strong fluid-wall interaction in extensively varying channel heights. Smith and Epstein [5] have studied the influence of surface roughness on local skin-friction and Nusselt number, in which, friction measurements were made experimentally for air flow through a smooth copper pipe by using double pipe heat exchanger. The influence of artificial surface roughness on heat and mass transfer was experimentally examined by Savage and Myers [6], in which, the effects of drag forces and heat transfer were measured when water flows over a surface fitted with protuberances. An elementary analysis of momentum and heat transfer on an analytical model for the fluid flow

2

over hydraulic rough surface was studied by Lewis [7]. A numerical and experimental analysis was carried out to study the surface roughness on natural convection on a horizontal plate by Pretot, et al. [8] and in which, the Nusselt number was higher for protuberance and was minimum in the hollow profile. The influence of relative surface roughness, roughness distribution and gas refraction on the nitrogen flow in a microchannel was modeled by Sun and Faghri [9]. Dierich and Nikrityuk [10] have studied the impacts of surface roughness on a cylindrical particle and their study estimated the thickness of the surface roughness layer on the heat transfer rate as well as the friction coefficient for the cylindrical surface. The surface roughness effect on the flow behavior and heat transfer characteristics for circular microchannels was examined experimentally by Xing, et al. [11] and their results show that an increment in the friction factor is observed with an increasing surface roughness, and the Nusselt number increases with the increase in the values of Reynolds number. Ehsan, et al. [12] have investigated the effects of high roughness and their study reveal that the optimum volume fraction of nanofluid can be implemented for enhancing the heat transfer rate with the minimum pumping power required over the rough parallel plate. The relationship between the roughness parameter and the effective boundary slip has been studied with the simulation method based on COMSOL5.3 by Pan, et al. [13] and their results indicate that the surface roughness increases the drag of the liquid flow at the interface, significantly. The study of nanometer-sized particles in the fluid was carried out by Choi and Eastman [14]. The intention of using nanoparticles is to achieve maximum thermal conductivity and that too with smaller concentration. Nanofluids have numerous engineering and industrial applications in the areas of metal spinning, nuclear reactor cooling, cooling capability, controlling the heat transfer rate in electrical devices, etc. In the recent times, Ferdows, et al. [15] have studied the nanofluid mixed convection along an exponentially stretching surface and in their investigation, they have discussed the velocity, temperature, concentration distributions and also the skin-friction coefficient, wall heat and mass transfer rates with graphical representations. Eid [16] has carried out a numerical work on two-phase nanofluid flow over an exponentially stretching surface in presence of magnetic field and his study involves the analysis of various governing parameters, such as, the porous medium parameter, magnetic field parameter, heat generation or absorption parameter, chemical reaction parameter, etc. Besthapu, et al. [17] have analyzed the change in thermal properties of the nanofluid on an exponentially stretching surface and their investigation exhibit the combined effects of mixed convection, viscous dissipation and thermal conductivity over a stretching sheet. The doubly stratified MHD nanofluid was analyzed by Daniel, et al. [18] 3

and their results reveal that the rate of heat transfer reduces with an increase in the values of thermophoresis as well as Brownian diffusion parameters. Ahmad, et al. [19] have used Buongiorno model to analyze the nanoparticles influence on fluid flow and they have carried out computations for axial wall shear and local Nusselt number. Shen, et al. [20] have investigated the nanofluid flow with Cattaneo heat flux and their study reveals the influence of particle shape on heat and fluid flow characteristics. Rashad, et al. [21] have studied the entropy generation with the Cu-water based nanofluid flow in an inclined porous enclosure. Rostami, et al. [22] have investigated the heat transfer characteristics of mixed convection flow and their study exhibit the dual solutions of hybrid nanofluid flow for assisting as well as opposing buoyancy flows. Further, Hussain, et al. [23] have investigated the magnetic field effects on mixed convective nanofluid flow and their results show that the heat transfer rate increases for higher nanoparticle volume fraction concentration. In the present investigation, the wall suction/injection [24 - 26] is considered in order to control the flow separation at the boundary layer region. From the comprehensive literature survey, it is observed that no investigations have been carried out to study the combined influence of nanoparticles and surface roughness on mixed convection flow along an exponentially stretching surface in presence of suction/injection. This work is found to be an innovative study. The present mathematical modelling is pertinent to the processes observed in the manufacturing of polymer sheets. Initially, the polymer is in the molten phase state, which is drawn from a slit to achieve desired shapes. Then, in order to solidify such sheets, cooling is required which is done by allowing a fluid medium (water) around such material. This situation brings out the phenomenon known as mixed convection. A sudden cooling could cause breakage or cracks in the final manufactured product. Therefore, as a mechanism to control the heat and mass transfer rates, the nanoparticles are considered in the base fluid, which is used to cool the polymer product. Also, the heated fluid in the ambience needs be replaced continuously with the cool fluid, which is managed by considering phenomenon of suction/injection at the wall. Moreover, during the production, the polymer sheet emerging from the slit may not be smooth enough and it may have some surface roughness. Sine wave form is considered to model such surface roughness in terms of small parameter  that represents the amplitude and frequency parameter n. Thus, the results of the present analysis are of interest to the design engineers in the polymer industries and assist them by providing the design parameters to control the heat and mass transport rates that decide the ultimate quality of the manufactured product. In the present analysis, the governing equations along with the 4

boundary conditions are solved by using the non-similar transformations. Then, for further mathematical simplifications, the Quasilinearization technique is used along with the implicit finite difference method [27 - 30]. 2 Formulation of the problem In this analysis, the flow is considered along an exponentially stretching surface. The stretching of the sheet is considered along the x-direction and the boundary layer is measured along the y-direction. The velocity of the plate is taken as U w (x), while that of free stream is taken as U  (x). The surface roughness is assumed on the stretching surface and which is modeled by using the sine wave form. Deterministic approach is considered to model the surface roughness [31 - 32]. The flow geometry is shown in the Fig. 1.

Fig. 1. Graphical representation of the computational domain and coordinate system. The temperature of the stretching sheet is considered to be higher than that of ambient fluid ( Tw > T ). Here, Tw , w indicate, respectively, the temperature and nanoparticle volume fraction at the wall and T ,  indicate, respectively, the temperature and nanoparticle volume fraction, away from the surface. The suction/blowing is considered along the stretching surface. The density effects are considered in the modeling by taking a body force term in the momentum equation by utilizing the Boussinesq’s approximation [33] and other all physical quantities are assumed to be constant. The Buongiorno two phase model [34] is considered to study the impacts of nanoparticles on the fluid flow. The water is taken as a 5

base fluid and the nanoparticles such as Cu, Ag and Au are assumed to be added into it. The fluid flow is assumed to be caused due to the stretching of boundary sheet from a slit with enough force and the velocity of that sheet is taken in terms of an exponential order of the fluid flow in the x- axis direction. Thus, the suitable governing equations are as below [15 17]: The equation for mass conservation:

u v   0, (1) x y The equation for momentum conservation: u

    dU e u u  2u v  Ue   2  (1   ) g  (T  T )  g  p  (   ), x y dx y   

(2)

The equation for energy conservation: 2  DT  T   T T  2T  T u v   m 2  J  DB     ,  x y y  y  y T  y     

(3)

The equation for nanoparticle volume fraction conservation:

u

   2 D  2T v  DB 2  T . (4) x y T y 2 y

All the variables and non-dimensional parameters are defined in the nomenclature. The physical boundary conditions are given by,

  nx   y  0 : u  U w ( x)  U 0 e x / L 1   sin    ,  L  

v  vw ( x),

y   : u  Ue  U e x / L ,

T  Tw ,

T  T ,

  w ,    .

(5)

Here, the first term in the RHS of Eq. (2) indicates the pressure term due to mainstream velocity, the second term indicates the viscous diffusion effects, the third and fourth terms indicate the buoyancy effects due to heat and nanoparticles diffusions, respectively. Equation (3) states that heat can be transported in a nanofluid by convection (LHS terms), by conduction (first term on RHS), and also by virtue of nanoparticle diffusion (second and third terms on RHS). It is important to emphasize that  Cnf is the heat capacity of the nanofluid, and thus already accounts for sensible heat of the nanoparticles as they move homogeneously with the fluid. Therefore, the last two terms on the RHS truly account for the additional contribution associated with the nanoparticle motion relative to the fluid. Equation (4) states that the nanoparticles can 6

move homogeneously with the fluid (LHS terms), but they also possess a slip velocity relatively to the fluid (RHS terms), which is due to the Brownian diffusion and thermophoresis. In the present research paper, no slip condition is considered at the wall and nanoparticles behaviours are studied through Buongiorno two-phase model [34]. In the said model, the slip due to nanoparticles at the wall is studied through two major slip mechanisms, namely, thermophoresis and Brownian diffusion. Thus, in the present analysis, we have taken no slip condition and the ordinary boundary conditions are considered to study the nanofluid mixed convection flow past an exponentially stretching surface. Further, the slip due to nanoparticles is studied with the help of thermophoresis and Brownian diffusion parameters, which are obtained during the analysis of nanoparticles through Buongiorno two phase model. Applying following non-similar transformations: 1/2

x U    ,    0  e x /2 L y, L  x 

f  F ,

  x, y    U 0 x  e x /2 L f  ,  , 1/2

T  T  Tw  T  G  ,  ,

Tw  T   Tw

    w    S  ,  ,

w     w    e2 x / L ,

u

 , y

v

 . x

0



 T e2 x / L ,

0

(6)

on Eqs. (1) – (4), we find that the Eq. (1) is satisfied identically, and Eqs. (2) – (4) reduces as below:

f  F     1   f  F   F 2   F F   2   Ri  G  NrS   0, 2 

(7)

f  G  P r    1   f  G  P r Nb e2 S G  P r Nt e2 G 2  P r  F G  2P r  G  0, 2  (8)

Nt f  S  Le    1   f  S  2 Le S F   Le S F  G  0. (9) Nb 2  From Eq. (5), the non-dimensional boundary conditions can be written as,

F  1   sin(n  ), G  1, S  1 at   0   0 F   , G  0 , S  0 as   

(10) 7

The cross-diffusion terms, which are observed in the governing equations occur due to the Brownian diffusion and thermophoresis effects and these cross-diffusion terms are similar to the Soret and Dufour diffusion terms which exist in the binary fluid [35]. The nondimensional parameters such as, Grashof number Gr, Reynolds number Re, Richardson number Ri, nanoparticle buoyancy ratio parameter Nr, Prandtl number Pr, velocity ratio parameter  , Brownian diffusion parameter Nb, thermophoresis parameter Nt and Lewis number Le are defined as below:

g  (1   )(Tw0  T ) L3 Gr  ; 2

Re 

     (w0   ) 1 Nr   p ;      (Tw0  T ) (1   )

Pr 

Nb  JDB

(w0   )

where, J 



U0 L



 ; m

Nt  JDT

;

Ri 

;



(Tw0  T ) ;  T

Le 

Gr ; R e2

U ; U0

 DB

;

(11)

 p C pp is the ratio of specific heat of nanoparticle material to specific heat of the  Cnf 

fluid. Further,

f ( , )   Fd  f w ; where f

w

can be obtained from following

0

transformations as follows: From the continuity equation,

1   U 0      2  x 

1/ 2

w

  e x / 2 L  f   1  2 f    .  

Let us assume that the suction/injection at the wall be,

vw  v0e

x 2L

, which is non-uniform

and varies exponentially at the wall. 1/2

 L  1/2 1/2 Thus, f (1   )  2 f  2v0     A ,  U 0   1/ 2

 L  where, A   2v0   = constant, such that A > 0 indicates the suction, A < 0 indicates  U0 

the blowing, while A = 0 represent an impermeable surface. The gradients defined at the wall are as below:  u  2   1/2  y  y 0 2 Cf   2  e  R e  1   sin( )  F  , 0  , 2 Uw 8

i.e., R e1/2 C f  2( e )1/2 1   sin( )  F  ,0  . (12) 2

 T   y  1/2 1/2   y 0 Nu   x    e  R e  G  , 0  , i.e., R e 1/2 Nu    e  G  , 0  (13) Tw  T      y  1/2 1/2   y 0 NSh   x    e  R e  S  , 0  , i.e., R e 1/2 NSh    e  S  , 0  (14) w    3 Method for non-similar solutions The Quasilinearization technique has a second order convergence, which is used to linearize the set of non-dimensional governing equations (7) – (9) along with the boundary conditions (10). The resulting set of linear partial differential equations is given below:

Fi 1  A1i Fi 1  A2i F i 1  A3i Fi 1  A4i Gi 1  A5i S i 1  A6i ,

(15)

Gi 1  B1i Gi 1  B2i Gi 1  B3i Gi 1  B4i F i 1  B5i S i 1  B6i ,

(16)

Si 1  C1i Si 1  C2i S i 1  C3i Si 1  C4i F i 1  C5i G i 1  C6i ,

(17)

where, the coefficient function with iterative index i are known and that with (i+1) are unknown. The corresponding boundary constraints are given by,

F i 1  1   sin(n  ), Gi 1  1, S i 1  1, at,   0 ,   0 .

F i 1   , Gi 1  0, S i 1  0, at,    . (18) The coefficients in Eqs. (15) – (17) are given below: f  A1i     1   f  ; 2 

A4i   Ri;

A2i     2 F  F  ;

A5i   Ri Nr;

A3i   F ;

A6i    F 2  F F   2  ;

f  B1i  P r    1   f   2 P r Nt e2 G  P r Nb e2 S ; B2i   2 P r F ; 2  B3i    P r F ; B4i    Pr G  2 P r G  ; B5i  P r Nt e2 G ;

B6i  P r Nt e2 G2   P r  2G  G  F  P r Nb e 2 S G ; f  C1i  Le    1   f  ; C2i   2 Le F ; C3i    Le F ; 2 

C4i    Le {2S  S }; C5i 

Nt ; Nb

C6i  C4i F .

9

In finite difference scheme, the backward difference formula is used in the  direction, while the central difference formula is used in the  - direction. The obtained linear partial differential equations are discretized with the help of implicit finite difference scheme, which results into a block tri-diagonal matrix. Then, this matrix is solved using the Vargas’s algorithm [36]. The step sizes  and  are optimized and a convergence criterion is employed to guarantee the convergence of numerical solutions. Thus, when the difference between the present and previous iterations reaches value lesser than 10 -4, the iteration process is terminated, i.e., Max

F 

i 1

 w



  F w ,  G w   G w ,  S w   S w  10 4 . (19) i

i 1

i

i 1

i

4 Results and discussion In the present work, the water is considered as the working fluid and thus, the Prandtl number Pr value is taken as 7.0. The values   0 and   0 indicate the similarity and nonsimilarity solutions, respectively. The values of Richardson number, Ri> 0 and Ri< 0 indicate the aiding and opposing buoyancy flow, respectively. The value   1 presents the dominance of wall velocity,   1 represents the same strength of wall as well as free stream velocities, while   1 indicates the dominance of free stream velocity. The negative values of parameter A indicate the injection case, the positive values of parameter A indicate the suction case, while A = 0 indicates the impermeable surface. Since the surface roughness is modelled as a sine wave with low amplitude and high frequency, the surface roughness is modelled deterministically as a low-amplitude high-frequency sine wave. Therefore, the values of the small parameter  and frequency parameter n are taken in the ranges

10  n  100 ,

 0    0.5

and

respectively. The value   0 corresponds to smooth surface, while   0

corresponds to surface roughness of stretching sheet. In this problem, the ordinary fluid and nanofluid indicate the working fluid in absence and presence of nanoparticles, respectively. In view of this, the values of Brownian diffusion parameter Nb, thermophoresis parameter Nt, nanoparticle buoyancy ratio parameter Nr and Lewis number Le are taken as 0.1, 0.1, 0.1 and 10, respectively, for the presence of nanoparticles, whereas, for the case of absence of nanoparticles, these parameters are considered as zero. 4.1 Non-similar profiles in presence of nanoparticles The Figs. 2 and 3 present the effects of Brownian diffusion parameter Nb, thermophoresis parameter Nt and velocity ratio parameter



on velocity F ( , ) and

temperature G ( , ) profiles, respectively. The increasing values of Nb and Nt raises the 10

velocity and temperature of the fluid. The increasing values of Nb increase the random motion of nanoparticles and thermophoresis parameter Nt causes the movement of nanoparticles from hot wall to the cold ambient fluid. This induces the faster movement of fluid particles and in turn increases the velocity of the fluid. The random motion reduces the kinetic energy of nanoparticles and thus, results into deeper penetration of nanoparticles into the ambient fluid. This process raises the temperature of the fluid. Further, the velocity ratio parameter



increases the fluid’s velocity. For the case of   0 , the wall as well as free

stream velocities have significant impact on the fluid flow as compared to the case of   0 , where the wall velocity has zero impact on the fluid flow.

Fig.2. Effect of Nb, Nt and 10, Nr = 0.1,

 on velocity profile

 = 0.5, n = 50 and A = 0.5.

11

F ( , ) for  = 0.5, Ri = 10, Pr = 7.0, Le =

Fig.3. Effect of Nb and Nt on temperature profile G( , ) for  = 0.5, Ri = 10, Pr = 7.0, Le = 10, Nr = 0.1,

 = 0.5,  = 0.5, n = 50 and A = 0.5.

Fig.4. Effect of Nb and Le on-nanoparticle volume fraction profile S ( , ) for  = 0.5, Ri = 10, Pr = 7.0, Nr = 0.1, Nt = 0.1,

 = 0.5,  = 0.5, n = 50 and A = 0.5.

The Fig.4 depicts the variation of Nb and Le on a nanoparticle volume fraction profile

S ( , ) . For an increasing value of Nb and Le, the nanoparticle volume fraction profile decreases. This is due the fact that he increasing values of Nb and Le cause the increase in the

12

specific heat of nanoparticles, which in turn, decrease the nanoparticle volume fraction profile.

Fig.5. Effect of Nr and A on velocity profile F ( , ) for  = 0.5, Ri = 10, Pr = 7.0, Le = 10, Nt = 0.1, Nb = 0.1,

 = 0.5,  = 0.5 and n = 50.

The Fig.5 displays the influence of nanoparticle buoyancy ratio parameter Nr and wall suction/blowing parameter A on the velocity profile F ( , ) . The velocity of the fluid reduces with an increase in the values of nanoparticle buoyancy ratio parameter Nr and suction/injection parameter A. This behaviour is due to the reason that the increasing values of Nr act as a negative pressure gradient on the velocity profile. On the other hand, the wall suction causes the friction between the fluid particles near the boundary surface, which reduces the velocity of the fluid. 4.2 Surface roughness effect on gradients 4.2.1 Skin-friction coefficient

13

Fig.6. Effect of Nb and Nt on skin friction coefficient ( R e1/2C f ) for Ri = 10, Pr = 7.0, = 10, Nr = 0.1,

 = 0.5, 

Le

= 0.03, n = 20 and A = 0.5.

The Fig.6 presents the impact of Brownian diffusion parameter Nb and thermophoresis parameter Nt on the skin-friction coefficient ( R e1/2C f ) . The increasing values of Nb and Nt increase the friction between the wall and fluid particles. The reason being is that the increasing values of Nb increase the random motion nanoparticles and thus, increases the velocity of fluid particles near the wall. Further, the thermophoresis parameter Nt causes the movement of nanoparticles from the hot wall to the cold ambient fluid. This increases the friction between the wall and fluid particles, significantly.

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Fig.7. Effect of  and n on skin friction coefficient ( R e1/2C f ) for, Ri = 10, Pr = 7.0, Le = 0, Nr = 0, Nt = 0, Nb = 0,

 = 0.5, and A = 0.5.

Fig.8. Effect of  and n on skin friction coefficient ( R e1/2C f ) for Ri = 10, Pr = 7.0, Le = 10, Nr = 0.1, Nt = 0.1, Nb = 0.1,

 = 0.5, and A = 0.5. 15

Figs. 7(a) – 7(c) and 8(a) – 8(c) display the variations of skin-friction coefficient ( R e1/2C f ) along  for different values of small parameter  (  = 0, 0.03, 0.05 and 0.1) and

frequency parameter n (n = 10, 15, 20), for the cases of absence and presence of nanoparticles, respectively. The Fig. 8 (d) shows the comparison in between the ordinary fluid and nanofluid in presence of smooth as well as rough surfaces. In particular, in the Figs. 7(a) to 7(c), in absence of nanoparticles, the increasing values of small parameter  and frequency parameter n increase the friction between the wall and fluid particles, significantly. More number of cavities on the stretching wall are observed with more depth due to the increasing values of  and n. Thus, the skin-friction coefficient increases with the values of

 and n. On the other hand, same effects of  and n on skin friction coefficient are observed in the Figs. 8(a) - 8(c) for in presence of nanoparticles. However, in the Fig. 8(d), it is perceived that the magnitude of skin-friction coefficient increases with the addition of nanoparticles into the base fluid. Due to the addition of nanoparticles into the base fluid, the nanoparticles increase the momentum of fluid particles near the wall. Thus, the friction between the wall and fluid particles increases in the presence of both smooth as well as rough surfaces. Further, it is observed that the influence of surface roughness is more prominent near the orifice. 4.2.2 Wall heat transfer rate

16

Fig.9. Effect of Nb and Nt on wall heat transfer rate ( R e1/2 Nu ) for, Ri = 10, Pr = 7.0, Le= 10, Nr = 0.1,

 = 0.5,  = 0.5, n = 50 and A = 0.5.

The Fig.9 shows the effects of Brownian diffusion parameter Nb and thermophoresis parameter Nt on the wall heat transfer rate ( R e1/2 Nu ) . The rate of heat transfer from hot wall to the cold ambient fluid decreases for the increasing values of Nb and Nt, significantly. The physical reason being is that the increasing values of Nb increase the random motion of nanoparticles. The random motion results into the collision between the nanoparticles, which reduces their kinetic energy. This leads into the deeper penetration of nanoparticles into the base fluid, which increases the fluid’s temperature. Consequently, the transfer of heat from wall to the fluid reduces, significantly. On the other hand, the thermophoresis parameter Nt causes the movement of nanoparticles from hotter region of wall to the colder ambient fluid and which reduces the transfer of heat from the wall to the fluid. Figs.10 and 11 show the effects of small parameter  (  = 0, 0.01, 0.1, 0.2, 0.3, 0.4, 0.5) and frequency parameter n (n = 50, 75, 100) on the wall heat transfer rate ( R e1/2 Nu ) along with  for in absence and presence of nanoparticles, respectively.

17

Fig.10. Effect of  and n on wall heat transfer rate ( R e1/2 Nu ) for, Ri = 10, Pr = 7.0, Le = 0, Nr = 0, Nt = 0, Nb = 0,

 = 0.5, and A = 0.5.

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Fig.11. Effect of  and n on wall heat transfer rate ( R e1/2 Nu ) for, Ri = 10, Pr = 7.0, Le = 10, Nr = 0.1, Nt = 0.1, Nb = 0.1,

 = 0.5, and A = 0.5.

Particularly, in Fig. 10, the heat transfer rate exhibit sinusoidal variations in presence of surface roughness, which enhance with the increase in the values of  and n. The cavities with more depth ease the transfer of heat from the hot region to the cold one. In the Fig. 11, in presence of nanoparticles, for increasing values of  and n, the heat transfer rate increases near the origin and reaches a maximum value and then reduces away from the origin. With the addition of nanoparticles into the base fluid, the nanoparticles initially accumulate on a stretching wall and increases the heat transfer rate. After a certain time, they acquire enough energy and move away from the hot wall. This leads to the reduction in the heat transfer rate, significantly. Further, the surface roughness effects are more prominent for   0.5 , which can be seen in the Figs. 10 and 11. Further, the magnitude of the wall heat transfer rate is higher for the case of absence of nanoparticles (ordinary fluid) as compared to the presence of nanoparticles (nanofluid). The nanoparticles in the fluid increase the temperature of the fluid and thus, decrease the wall heat transfer rate. For example, at   1 , the addition of nanoparticles into the base fluid reduces the heat transfer rate about 88% in case of smooth as well as rough surfaces.

19

4.2.3 Nanoparticle surface wall mass transfer rate

Fig.12. Effect of Nb and Le on nano particle mass transfer rate ( R e1/2 NSh) for, Ri = 10, Pr = 7.0, Nr = 0.1, Nt = 0.1,  = 0.5,

 = 0.5, n = 50 and A = 0.5.

The Fig.12 shows the effects of Nb and Le on-nanoparticle mass transfer rate

( R e1/2 NSh) . For higher values of Nb and Le, the nanoparticle mass transfer rate increases, noticeably. The increasing values of Nb and Le increase the specific heat of nanoparticles near the wall, which in turn, increases the temperature of the fluid. These facts increase the random motion of nanoparticles, which results into the observed behavior in the nanoparticle mass transfer rate. The Fig.13 depicts the effects of small parameter  (  = 0, 0.01, 0.1, 0.2, 0.3, 0.4, 0.5) and frequency parameter n (n = 50, 75, 100) on nanoparticle mass transfer rate

( R e1/2 NSh) along  . The magnitude and number of sinusoidal variations in nanoparticle mass transfer rate increase with an increase in small parameter  and frequency parameter n, respectively. The higher values of  and n indicate the wall with more surface area and which causes such variations in the nanoparticle mass transfer rate.

20

Fig.13. Effect of  and n on nanoparticle mass transfer rate ( R e1/2 NSh) for, Ri = 10, Pr = 7.0, Le = 10, Nr = 0.1, Nt = 0.1, Nb = 0.1, 21

 = 0.5, and A = 0.5.

From all the figures, it is observed that due to the roughness of stretching surface, the transport rates, i.e. wall gradients, computed at the surface are found to be sinusoidally varying with mean values close to the corresponding values at the smooth surface. In the Table 1, the numerical values of wall heat transfer rate ( R e1/2 Nu ) are computed for various values of Prandtl number Pr and are validated with the previous results obtained by Ferdows, et al. [15], Eid [16] and Besthapu, et al. [17], when  = 0, Ri = 0, Pr = 0, Le = 0, Nr = 0, Nt = 0, Nb = 0,  = 0,  = 0, n = 0 and A = 0. It is found that the results are in good agreement with the previously established results. The Tables 2 and 3 present the effects of skin-friction coefficient as well as the heat transfer rate in absence and presence of nanoparticles, respectively. In both the tables, the gradient values are obtained for both cases of smooth and rough surfaces for various values of suction/injection parameter A. From the tables, it is observed that for suction, the skin-friction coefficient reduces and the heat transfer rate increases irrespective of the presence and absence of surface roughness as well as the nanoparticles. Furthermore, the skin-friction coefficient increases and the heat transfer rate decreases in presence of nanoparticles, irrespective of the presence and absence of surface roughness and suction/injection. Moreover, the skin-friction coefficient and also the heat transfer rate enhance in presence of the surface roughness irrespective of the presence and absence of nanoparticles and suction/injection. Table 1: Comparison of wall heat transfer rate ( R e1/2 Nu ) values obtained in the present analysis for various values of Pr with that of Ferdows, et al. [15], Eid [16] and Besthapu, et al. [17] when  = 0, Ri = 0, Pr = 0, Le = 0, Nr = 0, Nt = 0, Nb = 0,  = 0,  = 0, n = 0 and A = 0. Pr

Ferdows, et al. [15]

Eid [16]

Besthapu, et al. [17]

Present results

1

0.9547

0.9548

0.9548

0.9546

2

1.4714

1.4715

1.4715

1.4715

3

1.8691

1.8691

1.8691

1.8690

5

-

-

2.5001

2.5002

10

-

-

3.6605

3.6606

Table 2: The values of skin friction coefficient ( R e1/2C f ) and wall heat transfer rate

( R e1/2 Nu) for the case of absence of nanoparticles obtained for various values of suction or

22

injection parameter A in the both cases of smooth and rough surfaces, when  = 0.5, Ri = 10, Pr = 7.0, Le = 0, Nr = 0, Nt = 0, Nb = 0,  = 0.5 and n = 50. In absence of nanoparticles Gradients

Re Cf

A = - 0.5 2.31080

=0 A=0 2.11142

A = 0.5 1.88890

R e1/2 Nu

1.58732

1.80751

2.05497

1/2

 = 0.5 A = - 0.5 A=0 A = 0.5 143.45587 143.19028 142.89966 1.73848

1.96203

2.21131

Table 3: The values of skin friction coefficient ( R e1/2C f ) and wall heat transfer rate

( R e1/2 Nu) for the case of presence of nanoparticles obtained for various values of suction or injection parameter A in the both cases of smooth and rough surfaces, when  = 0.5, Ri = 10, Pr = 7.0, Le = 10, Nr = 0.1, Nt = 0.1, Nb = 0.1,  = 0.5 and n = 50. In presence of nanoparticles

R e1/2C f

A = - 0.5 2.48626

=0 A=0 2.33701

A = 0.5 2.15555

A = - 0.5 143.65889

R e1/2 Nu

0.93941

1.01211

1.09167

1.03054

Gradients

 = 0.5 A=0 A = 0.5 143.44174 143.18907 1.10398

1.18382

5 Conclusions The important conclusions drawn from the present detailed analysis of surface roughness effects on mixed convection nanofluid flow along an exponentially stretching surface in presence of suction/injection are given as following: 

The Brownian diffusion and thermophoresis parameters increase the velocity as well as temperature of the ambient fluid.



The nanoparticles in the ambient base fluid enhance the friction at the vicinity of the wall and reduce the rate of heat transfer from hot wall to the cold ambient fluid.



At   1 , the nanoparticles added into the base fluid reduce the wall heat transfer rate by about 88% in both cases of smooth (  0) and rough (  0.5) surfaces.



The random motion of nanoparticles reduces its volume fraction in the boundary layer region, while it increases the nanoparticle mass transfer rate, significantly.

23



Hence, the nanoparticles added into the base fluid have significant impact on the fluid flow.



The uniform suction at the wall decreases the velocity of the fluid flow.



The surface roughness, analyzed in terms of small parameter  and frequency parameter n, has significant impact on wall gradients. It enhances the skin friction between the wall and the fluid flowing over it.



The gradient values obtained for the case of rough surface are oscillatory with a mean value close to that value obtained in the corresponding case of smooth surface.

Acknowledgement: This work is supported by UGC-SAP-DRS-III with No. F. 510/3/DRS-III/2016 dated 29-022016. Nomenclature: A

suction/injection parameter

Cpp

specific heat of nanoparticles

D

mass diffusivity

DB

Brownian diffusion coefficient

Cf

skin friction coefficient

DT

thermophoretic diffusion coefficient

Nb

Brownian diffusion parameter

f

dimensionless stream function

Nt

thermophoresis parameter

Cnf

heat capacity of the nanofluid

F

dimensionless velocity

Re

Reynolds number

g

acceleration due to gravity

T

temperature

J

ratio of specific heat of nanoparticle to the specific heat of the fluid

G

dimensionless temperature

Nr

nanoparticle ratio of buoyancy parameter

Gr

Grashof number

Le

Lewis number

S

dimensionless volume fraction profile

n

frequency parameter 24

Nu

Nusselt number

NSh

nanoparticle Sherwood number

Pr

Prandtl number

Ri

Richardson number

Tw

temperature at the wall

Ue

free stream velocity

T

the temperature of fluid far away from the wall

U

composite reference velocity

U0

reference velocity

U

free stream velocity constant

Uw

wall velocity

u

x component of velocity

v

y component of velocity

x

distance along x coordinate

y

distance along y coordinate

Greek symbols



small parameter



kinematic viscosity

m

thermal diffusivity



the volumetric coefficient of thermal expansion of nanofluid



nanoparticle volume fraction

w

nanoparticle volume fraction at the wall



density of nanofluid



dimensionless stream function

p

nanoparticle mass density



the ratio of free stream velocity to reference velocity

∞

ambient nanoparticle volume fraction

 ,

transformed variables

Subscripts 0

value at the wall surface

e

free stream state

w

conditions at the surface wall 25

 ,

partial derivatives with respect to these variables

0, 

condition at the wall and free stream, respectively.

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